In this talk, we consider the Allen-Cahn functional with stationary and ergodic coefficients in the regime where the length scale of the heterogeneities is much smaller (microscopic) than the interface width (mesoscopic). Results by Ansini, Braides, Chiado Piat in 2003 suggest - and prove under certain restrictions on the ratio of these two scales - that the homogenization limit takes place before the sharp interface limit. In this talk, I discuss recent work that, in setting of the periodic homogenization, extends the above result beyond the mentioned restrictions, and, in the setting of stochastic homogenization, shows that a more complex behavior occurs.
I will introduce multiple zeta values, which are a generalization of the values of the Riemann zeta function. They satisfy two different product expressions, which imply numerous relations among them. On the other hand, it is conjectured by Zagier that there is a weight-grading on the algebra of multiple zeta values.
A kind of deformation of multiple zeta values are multiple q-zeta values, these are particular q-series degenerating to multiple zeta values under the limit q to 1. We will also study their algebraic relations, and how to obtain a conjectural weight-grading on the algebra of multiple q-zeta values.
In the end, I will present how multiple q-zeta values can be used to renormalize multiple zeta values at negative integers. In particular, I will explain the Connes-Kreimer renormalisation procedure.
Consider a nonlinear wave equation with a double-well potential and stochastic forcing. How often does the field jump from one potential well to the other? We answer this question for a cubic equation and random initial data sampled from its invariant measure, the phi^4 quantum field theory. This low-temperature asymptotic behaviour is known as an Eyring--Kramers law.
In the talk, we outline how this question of dynamics can be reduced to computing some partition functions. We also briefly discuss two different notions of metastable transitions. This presentation relates to joint work (still unpublished) with Nikolay Barashkov.
In this talk, we revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi, Musil, and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell’s inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers. This is a joint work with Ali Barki, Sergey Bobkov, and Cyril Roberto.
In this talk I will review some mathematical tools developed to characterize the behavior of vortices in the magnetic Ginzburg-Landau model of superconductivity. Vortices are quantized topological singularities that appear above a certain critical intensity of the applied magnetic field. Particular attention will be put to describe their location and number for energy-minimizers.
In this talk I will study the stochastic flow induced by planar stochastic differential equations in which the radial part of the drift term depends on the distance to the origin ("shear"). In particular I will derive tight conditions under which the flow is or is not strongly complete, does or does not admit a pull-back atractor and under which strong synchronization does or does not occur. Many of the methods used are new and could be applicable to larger classes of S(P)DE.
Most of the results are based on ongoing joined work with Maximilian Engel and Michael Scheutzow.
The Landau-Lifschitz-Navier-Stokes equations are stochastic partial differential equations, which introduce a stochastic forcing term to describe the macroscopic fluctuations away from the deterministic Navier-Stokes equations. We consider the large deviations of a suitable solution theory in a scaling regime where the noise intensity and correlation length go to zero simultaneously, with a coupled rate. We show that the large deviations reproduce those of the lattice gas studied by Quastel and Yau on sufficiently integrable fluctuations. The large deviation rate function is connected to weak-strong uniqueness and the validity of the energy equality for forced Navier-Stokes with forcing in the Leray class $L^2_tH^{-1}_x$, and there is no possibility of extending the theory into stronger regularity classes or integrability conditions.
We introduce and study nonlinear Markov processes in the sense of McKean, initiate a theory of these processes, and present a large class of new examples. More precisely, we construct nonlinear Markov processes with one-dimensional time marginals given as solution flows of nonlinear Fokker-Planck equations. These processes are given by path laws of weak solutions to the corresponding distribution-dependent stochastic differential equation. Our results apply to nonlinear Fokker-Planck equations with locally density-dependent coefficients, which includes many important nonlinear parabolic PDEs. Thus, we establish a one-to-one correspondence between solution flows of such PDEs and nonlinear Markov processes. Important examples are porous media and Burgers equation, as well as the 2D vorticity Navier-Stokes equation.
On a mesoscale, suspensions of locally heated colloidal particles can to some extent be described like ordinary isothermal systems with effective parameters that can be predicted for sufficiently symmetric setups. This still holds, to a lesser degree, for asymmetrically heated ("Janus”) particles that turn into active, self-propelling microswimmers, due to unbalanced thermal gradients. I will discuss informative signatures of the local nonequilibrium, accessible to mesoscale observers. Some increasingly coarse-grained paradigmatic situations that we have recently considered comprise active Brownian heat engines, polarization-density patterns emerging in activity gradients, ensuing active ratcheting, and time-reversal symmetry breaking fluctuations in non-reciprocal field theories.
It is well-known that, despite their aptness for complicated tasks like image classification, modern neural networks are prone to insusceptible input perturbations (a.k.a. adversarial attacks) which can lead to severe misclassifications. Adversarial training is a method to obtain classifiers which are robust against such attacks. In this talk I will show that in the binary clas- sification setting the method can be rephrased as geometric regularization problem, involving a nonlocal perimeter of Minkowski type. I will present existence and regularity theorems for minimizers, study local asymptotics of the nonlocal perimeter using Gamma-convergence, and discuss probabilistic relaxations which correspond to more classical notions of nonlocal perimeters.
Nonlinear potential theory and elliptic regularity theory are two classical topics in the modern analysis of partial differential equations. In this talk I show how these themes merge to solve the longstanding open problem, dating back to the seminal contributions of O. A. Ladyzhenskaya & N. N. Ural’tseva, N. S. Trudinger, and L. Simon (1967-1976), of deriving Schauder estimates for minima of functionals (resp. solutions to elliptic equations) featuring polynomial nonuniform ellipticity. The sharp rate of nonuniform ellipticity for the validity of Schauder theory is also disclosed. From recent, joint work with Giuseppe Mingione (Parma).
We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles the open problem of proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain, while also providing a toolbox for constructing such smooth universal mixers in all dimensions. This is based on joint work with Alex Blumenthal (Georgia Tech) and Michele Coti Zelati (ICL).
Following the approach of Otto, Sauer, Smith and Weber, we describe solutions to driven ODEs in terms of power series of derivatives of the nonlinearities. This gives a notion of rough path based on multi-indices: Its underlying Hopf algebra is dual to the Lie universal envelope of the free Novikov algebra. We also discuss how to introduce local counterterms (algebraic renormalization) and their associated (multi) pre-Lie algebra, leading to groups of translations of rough paths.
In my talk I will briefly present the multi-marginal optimal transport problem of finding the h-Wasserstein barycenter, where h is a nonnegative strictly convex function, as a generalization of the better known 2-Wasserstein barycenter. It appeared for the first time in a work by Argueh and Carlier and was further studied by Pass. The focus of the talk is the sparsity of the optimal plan for the MMOT formulation, which for the 2-case has been proved by Gangbo and Święch. Here we provide a proof of the absolute continuity of the h-Waserstein barycenter, which implies that the optimal plan is of Monge type.
This is a joint ongoing work together with G. Friesecke and T. Ried.
We prove local Lipschitz regularity for local minimisers of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq \R^N$, $N\ge 2$ and $F:\R^N\to \R$ is a quasiuniformly convex integrand in the sense of [Kovalev and Maldonado, 2005], i.e.,a convex $C^1$-function such that the ratio between the maximum and minimum eigenvalues of $D^2F$ is essentially bounded. This class of integrands includes the standard functions $F(z)=|z|^p$ for any $p>1$ and arises as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations.
The (fractional) Kolmogorov equation \begin{equation*} \begin{cases} \partial_t u +v \cdot \nabla_x u + (-\Delta_v)^{\frac{\beta}{2}} u = f, \quad t>0 u(0) = g, \end{cases} \end{equation*}where $u = u(t,x,v)$ and $\beta \in (0,2]$ has gained more and more interest in the past years. This is mainly due to the following three facts. First, it can be seen as the prototype for many kinetic equations, such as the famous Boltzmann or the Landau equation. Second, even though the Laplacian only acts in half of the variables, i.e. the equation is degenerate, and despite the first-order term being unbounded, solutions of the Kolmogorov equation admit excellent regularity properties. Last but not least, it serves as a good example to study the regularity transfer (from $v$ to $x$), a special feature of kinetic equations.In the recent work (joint work with Rico Zacher), we introduce the concept of kinetic maximal $L^p_\mu$-regularity and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterised in terms of an anisotropic Besov space. Kinetic maximal $L^p_\mu$-regularity can be used to obtain local (in time) existence of solutions to a class of quasilinear kinetic equations by an abstract principle. We apply this principle to a kinetic toy model.
The non-parametric estimation of a non-linear reaction term in a semi-linear parabolic stochastic partial differential equation (SPDE) is discussed. The estimation error can be bounded in terms of the diffusivity and the noise level. The estimator is easily computable and consistent under general assumptions due to the asymptotic spatial ergodicity of the SPDE as both the diffusivity and the noise level tend to zero. The analysis of the estimation error requires the control of spatial averages of non-linear transformations of the SPDE, and combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE.
We consider the four-waves spatial homogeneous kinetic equation arising in weak wave turbulence theory. In this talk I will present some new results on the existence and long-time behaviour of solutions around the Rayleigh-Jeans thermodynamic equilibrium solutions. In particular, introducing a cut-off on the frequencies, I will present an $L^2$ stability of mild solutions for initial data close to Rayleigh-Jeans, when the dispersion relation is weakly perturbed around the quadratic one. If time permits, I will discuss a more recent result (joint work with Miguel Escobedo) on stability of radial solutions without the cut-off on the frequencies.
We study the phase transition phenomenon of the singular Gibbs measure associated with the Schr¨odinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence. The singularity of the Gibbs measure makes an additional difficulty in proving the non-convergence in the strong coupling case.
In recent times, mathematicians have increasingly formulated problems related to strongly interacting systems as multi-marginal optimal transport (MMOT) problems. From a theoretical point of view, this formulation is helpful because a MMOT problem is convex, and one can use a well-developed and strong duality theory.
Unfortunately, the problem suffers extremely from the curse of dimension in that one seeks a solution in the product space of the marginal spaces, making direct application of algorithms from regular OT theory difficult.
Therefore, [FSV22] introduced a genetic modification of the column generation algorithm to solve an MMOT problem from DFT. This algorithm, respectively its extension to general MMOT problems, reduces the data complexity from $O(\ell^N)$ to $O(N*\ell)$ ($N$ the number and $\ell$ the cardinality of the marginals). It can also be shown that in the classical case of N=2 marginals, the algorithm always converges to the global optimum and reduces the data complexity, which may also make it interesting for large classical OT problems.
Friesecke, Schulz, Vögler (2022). Genetic column generation: Fast computation of high-dimensional multimarginal optimal transport problems. SIAM Journal on Scientific Computing, 44(3), A1632-A1654.
Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors such as the mean-field equation and the large deviation rate function can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme which approximates the limiting first order Hamiltonian-Jacobi equations(HJE). The discrete Hamiltonian is a m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well posedness of chemical master equation. The convergence from the monotone schemes to the viscosity solution of HJE is proved via constructing barriers to overcome the polynomial growth coefficient in Hamiltonian. This convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup yields the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered. Moreover, the LDP for invariant measures can be used to construct the global energy landscape for non-equilibrium reactions. It is also proved to be a selected unique weak KAM solution to the corresponding stationary HJE. Based on this stationary solution, a decomposition into Hamiltonian flow and gradient flow for general mean-field reaction rate equation will also be discussed.
We consider a gas of weakly interacting bosons in three dimensions subject to an external potential in the mean field regime. Assuming that the initial state of our system is a product state, we show that in the trace topology of one-body density matrices, the dynamics of the system can be described by the solution to the corresponding Hartree type equation. Using a dispersive estimate for the Hartree type equation, we obtain an error term that is uniform in time. Moreover, the dependence of the error term on the particle number is optimal. We also consider a class of intermediate regimes between the mean field regime and the Gross-Pitaevskii regime, where the error term is uniform in time but not optimal in the number of particles. (Joint work with Charlotte Dietze)
In this talk I will present some results obtained with B. Merlet in recent years on a family of energies penalizing oscillations in oblique directions. These functionals, which first appeared in the study of an isoperimetric problem with non-local interactions, can be seen as a natural extension of the Bourgain-Brezis-Mironescu energies. A central insight is that these energies actually control second order derivatives rather than first order ones. Indeed, functions of finite energy have mixed (or oblique) derivatives given by bounded measures. The main focus of the talk is the study of the rectifiability properties of these 'defect' measures. Time permitting we will draw connections with branched transportation, PDE constrained measures and Aviles-Giga type differential inclusions.
In this talk, I will present Wasserstein barycenters. The Wasserstein barycenter corresponds to the Fréchet mean (a generalization of the mean to metric spaces) of a random variable on the Wasserstein space of order 2, that is the space of probability measures of finite second moment equipped with a metric induced by optimal transport theory, which is commonly called Wasserstein distance (of order 2) in the literature. I will start by giving a summary of optimal transport and the tools which we will need. After defining the Wasserstein barycenter, I will give an overview of its analytic properties and try to explain some of the difficulties which arise when studying this object. Finally, I will study its probabilistic properties such as the law of large numbers and a heuristic idea to prove a central limit theorem, which can be made rigorous if one introduces a suitable regularization.
My talk will be devoted to the convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative SPDEs. We will discuss the optimal convergence rate and derive a quantitative central limit theorem for such SPDEs. The results can be applied, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized neural networks. We will see that including the noise in the limiting equation improves the convergence rate and retains information about the fluctuations of stochastic gradient descent in the continuum limit. – The talk is based on joint work with Benjamin Gess and Rishabh S. Gvalani.
The centered model is a crucial ingredient in the theory of regularity structures, used for a solution theory of singular stochastic PDEs. It provides a parametrization of the solution manifold, of which we seek to get robust control as an artificial smoothing parameter is removed.
In this talk, we will give a robust characterization of the model in regularity structures, which persists for rough noise. We will then show, how this characterization can be used to propagate symmetries from the noise to the model. Furthermore, we show that a convergent sequence of noise ensembles, satisfying uniformly a spectral gap assumption, implies convergence of the associated models. Combined with the characterization, this establishes a universality-type result.
We consider the Allen-Cahn equation with white noise initial datum under critical rescaling. The usual approach of performing a Picard iteration of the solution yields an infinite series of stochastic iterated integrals. In contrast to considering initial datum under sub-critical rescaling, each term in the infinite expansion has a positive contribution to the solution. In this talk we will give a gentle introduction into the methods used to control such an infinite series. We start by identifying stochastic integrals with rooted trees. We will then treat some explicit examples that motivate a systematic approach of determining the statistics of each term in the expansion. Time permitting, we identify the distribution of the solution with the help of a series expansion of an explicit ODE, exploiting the structure of the initial PDE.
The talk is based on joint work with Tommaso Rosati and Nikos Zygouras.
In this blackboard talk I’ll describe three models of waves / dynamics in heterogeneous media: acoustic waves in a random medium, quantum wave in a random potential, and the Lorentz mirror walk on the lattice. For each model we are interested in the long time dynamics and wish to answer the following question: what is the effect of randomness on the dynamics? These three models display three typical behaviors: ballistic transport, localization, and diffusive transport. I will give arguments in favor of each behavior, and draw links between one another.
This talk is based on joint works with Mitia Duerinckx (Brussels) and Matthias Ruf (EPFL), with Guy David (Orsay) and Svitlana Mayboroda (Minneapolis), and with Felipe Hernandez (Stanford).
The Griffith energy is a functional introduced by Francfort and Marigo to model the equilibrium state of a fracture in a brittle fracture in linear elasticity. The formula $\mathcal{G}$ given below is a bit simplified. Let $\Omega$ be a bounded open set of $\mathbf{R}^N$, which stands for the reference configuration (without crack) of a homogeneous, isotropic, brittle body. We apply a deformation (constant over time) at the boundary of the solid and we assume that the solid only undergoes infinitesimal transformations. Francfort and Marigo formulate the problem as the minimization of \begin{equation*} \mathcal{G}(u,K) := \int_{\Omega \setminus K} | {e(u)}^2 \, dx + \mathcal{H}^{N-1}(K), \end{equation*} among pairs $(u,K)$ such that $K$ is a $(N-1)$-dimensional subset of $\Omega$, $u \colon \Omega \setminus K \to \mathbf{R}^N$ is a smooth function which satisfies a Dirichlet condition at the boundary $\partial \Omega$, and the matrix $e(u) := (D u + D u^T)/2$ is the symmetric part of the gradient of $u$. We interpret $u$ as a displacement field (the deformation of the solid is $x \mapsto x + u(x)$), the matrix $e(u)$ as its linear strain tensor (it describes the local deformation of $\Omega$), $K$ as a crack. The energy $\mathcal{G}(u,K)$ puts in competition the elastic energy stored outside of the crack and the surface energy required to create the crack. This formulation does not say anything about the topology and the regularity of the crack a priori and as of yet, we know very few regularity results about minimizers. Although it looks like the Mumford-Shah energy, the Griffith energy provides in fact a lot of suprising new difficulties as one works with the symmetrized gradient instead of the full gradient. The goal of the talk is to present a new $\varepsilon$-regularity theorem for Griffith minimizers in all dimensions $N$. This is a joint work with Antoine Lemenant.
We investigate the effective diffusivity of a random drift-diffusion operator that is at the borderline of standard stochastic homogenization theory: In two space-dimensions, we consider the divergence-free drift with stream function given by the Gaussian free-field, with an ultra-violet cut-off at scale unity and an infra-red cut-off at a scale L. We establish the precise scaling of how the effective diffusivity diverges in terms of L, specifying recent results based on a Wiener chaos decomposition and a mathematical physics-type analysis in the corresponding Fock space. This amounts to the study of convection-enhanced diffusion at the borderline to anomalous diffusion. It provides a quantitative stochastic homogenization perspective, and therefore yields quenched rather than annealed results. Joint work with Georgiana Chatzigeorgiou, Peter Morfe and Felix Otto (MPI).
A common thread in modern approaches to providing a solution theory to singular stochastic PDEs is that the probabilistic aspects of the problem are encoded in the construction of a random element of some nonlinear space of distributions. Once a realisation of this data is fixed, the remaining theory is then deterministic.
In this talk, we will discuss this probabilistic aspect within the framework of regularity structures (where the nonlinear data is known as a “model”). In this setting, it is highly non-obvious that typical driving noises for singular SPDEs yield suitable models and it is typically the case that some renormalisation procedure is needed.
The primary purpose of this talk will be to discuss a new approach to obtaining the necessary stochastic estimates in the setting of tree-based regularity structures via the spectral gap inequality, inspired by recent work of Linares, Otto, Tempelmayr and Tsatsoulis in the multi-index setting.
(Based on a joint work with Martin Hairer)
We obtain partial $C^{0,\alpha}$-regularity for bounded solutions of a certain class of cross-diffusion systems, which are strongly coupled, degenerate quasilinear parabolic systems. Under slightly more restrictive assumptions, we obtain partial $C^{1,\alpha}$-regularity. The cross-diffusion systems that we consider have a formal gradient flow structure, in the sense that they are formally identical to the gradient flow of a convex entropy functional. The main novel tool that we use is a "glued entropy density", which allows us to emulate the classical theory of partial Hölder regularity for nonlinear parabolic systems. We are, in particular, able to obtain partial $C^{1,\alpha}$-regularity for solutions of the Maxwell-Stefan system, as well as partial $C^{1,\alpha}$-regularity for bounded solutions of the Shigesada-Kawasaki-Teramoto model.This talk is based on joint work with Marcel Braukhoff and Nicola Zamponi.
In this talk, we address the regularity problem for stable solutions to elliptic equations. Here, a solution is "stable" if the the principal eigenvalue of the linearized equation is nonnegative. In particular, for variational problems, stability amounts to the nonnegativity of the second variation and hence it includes the class of minimizers. The smoothness of stable solutions turns out to be a delicate question which depends on the dimension of the space. For instance, given a bounded domain $\Omega \subset \mathbb{R}^n$ and a function $f \in C^1(\mathbb{R})$, the semilinear problem \[\left\{\begin{array}{cl}- \Delta u = f(u) & \text{ in } \Omega\\u = 0 & \text{ on } \partial\Omega,\end{array}\right.\] may admit singular stable solutions when $n \geq 10$. For $n \leq 9$, it was recently shown that stable solutions are smooth for \emph{all} (nonnegative, nondecreasing, and convex) nonlinearities. The main goal of the talk will be to discuss some extensions of this optimal result to more general operators, including non-variational problems. We will emphasize the precise regularity assumptions needed on the coefficients and the domain.
The Mumford-Shah functional - being roughly speaking defined as the sum of the Dirichlet energy of a function plus the (d-1)-dimensional Hausdorff measure of its set of discontinuities - has been originally proposed by Mumford and Shah in the context of image segmentation; it is also an important prototypical model for the energy arising in mathematical models for fracture mechanics. Its minimization problem is one of the most important instances of a free discontinuity problem, a problem class in the calculus of variations for which the discontinuity set is obtained as a result of the energy minimization process. The Mumford-Shah functional is the subject of a number of intriguing conjectures, most famously the Mumford-Shah conjecture on the dimension and structure of the singular set of minimizers in the 2d case.
We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in 2d. Our monotonicity formula is formulated in terms of a truncation of an entropy introduced by David and Leger. It is in particular able to discriminate between singular points that are part of a C^1 interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.
This talk will advertise a course I will offer in March (pending interest) and contain some new results. In the 1920s Besicovitch asked the question: What can one say about the structure of sets $E$ in the plane, with the property that $\lim_{r \downarrow 0} \frac{ \mathcal{H}^{1}(B(x,r) \cap E)}{2r}= 1$ for almost every $x \in E$? Here, $\mathcal{H}^{1}$ denotes the Hausdorff $1$-measure, which should be thought of as measuring the "length" of a set. Besicovitch's work can be viewed as the founding idea behind the field of Geometric Measure Theory (GMT). After many results by Besicovitch, Federer, Marstrand, and Mattila, the groundbreaking work of Preiss in 1987 provided a very satisfying answer to the "density question" in the setting of measures in all dimensions. Preiss' work relies on the new notion of tangent measures. Due to a lack of flexibility in Preiss' work, much of it has been described as "searching for a needle in a haystack, and finding only needles". But, the tools related to tangent measures themselves are very flexible. In recent work with Tatiana Toro and Bobby Wilson, we show that if $\mu$ is a locally finite measure on $\mathbb{R}^{n}$ and there exists an $m$-so that for $\mu$ almost every $x$ there exists an ellipse $E_{x}$ so that $0
Introduced by Almgren in geometric measure theory, almost minimizers have seen applications in both parametric and non-parametric settings. There have also been many results on the regularity for almost minimizers in various contexts. In this talk, I will give a short introduction to this concept, which will be followed by a discussion about the regularity for BV almost minimizers.
So far there has not been a rigorous proof of perturbative renormalization of the massive scalar field theory with a quartic self-interaction on a half-space. In this talk, we will discuss the renormalisation problem of this model using the Polchinski flow equations in the case of the Robin boundary condition. We present the considered action and set up the system of perturbative flow equations satisfied by the connected amputated Schwinger distributions (CAS). To establish bounds on the CAS, being distributions, they have to be folded first with test functions. We introduce a suitable class of test functions together with the tree structures that will be used in the bounds to be derived on the CAS. We state the inductive bounds on the Schwinger functions which, being uniform in the cutoff, directly lead to renormalizability. Joint work with Christoph Kopper.
The dynamics of viscous waves is a very active subject in PDEs. When inertial forces are negligible compared to viscous forces, the fluids flow in the Stokes regime. In this talk, we will describe the dynamics of two incompressible, viscous and immiscible fluids filling a horizontally periodic strip. We consider the fluids to have different densities and to be subject to gravity forces. This induces the dynamics of the free boundary between the fluids. We will study the derivation of this model through a contour dynamics approach and we will address fundamental questions such as the well-posedness and stability of the free boundary.
Over the last twenty years there has been significant progress in the well-posedness study of singular stochastic PDEs in both parabolic and dispersive settings. In this talk, I will discuss some convergence problems for singular stochastic nonlinear PDEs. In a seminal work, Da Prato and Debussche (2003) established well-posedness of the stochastic quantization equation, also known as the parabolic Φ^k+1 _2 -model in the two-dimensional case. More recently, Gubinelli, Koch, Oh, and Tolomeo proved the corresponding well-posedness for the canonical stochastic quantization equation, also known as the hyperbolic Φ^k+1 _2 -model in the two-dimensional case. In the first part of this talk, I will describe convergence of the hyperbolic Φ^k+1 _2 -model to the parabolic Φ^k+1 _2 -model. In the dispersive setting, Bourgain (1996) established well-posedness for the dispersive Φ^4 _2 -model (=deterministic cubic nonlinear Schrödinger equation) on the two-dimensional torus with Gibbsian initial data. In the second part of the talk, I will discuss the convergence of the stochastic complex Ginzburg-Landau equation (= complex-valued version of the parabolic Φ^4 _2 -model) to the dispersive Φ^4 _2-model at statistical equilibrium.
During this talk, I will first recall a brittle damage model of a linearly elastic material, introduced by Francfort and Marigo, and its applications to the study of concentration and elastic degeneracy of weak material phenomena. I will explain the two main directions we intended to investigate during my thesis and the resulting research projects I wish to carry out.I will then explain in further detail the study of a static model which is based on the Gamma-convergence of brittle damage total energies to the Griffith functional. This first joint work with my director, Jean-François Babadjian, is the content of an article we submitted. The preprint is available on ArXiv (arXiv:2202.12152).If time allows it, I will finally present the convergence of a quasi-static brittle damage evolution, introduced by Francfort and Garroni, to a model of quasi-static perfect plasticity in the one dimensional setting.
We consider a particle process on the discrete torus, given by a rescaling of the particle sizes and time in the zero-range model with superlinear jump rate $g(k)=k^\alpha, \alpha\in [1,\infty)$, as well as the usual parabolic rescaling of space and time. Under a relationship between the two scalings, we derive a dynamical large deviation principle, with the same rate function as found for conservative SPDEs by Fehrman and Gess. In this context, we are able to avoid the usual superexponential estimate in the hydrodynamic limit, and the same role is played by stochastic estimates at the level of the particle system and an argument in the spirit of Aubin-Lions-Simons. We finally exhibit a gradient flow structure for the PME through the properties of the large deviation rate function.
In this talk, we will present a new construction of the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. It can be performed using either of the two combinatorial structures that have been proposed in the context of singular SPDEs: decorated trees and multi-indices. Our construction is inspired from multi-indices where the Hopf algebra was obtained as the universal envelope of a Lie algebra. We show that this Lie algebra comes from an underlying post-Lie structure.
We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller-Segel model, Hele-Shaw kinematic boundary condition, and the Young-Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. This formula greatly simplifies the computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability. It also leads to interesting mathematics due to non-selfadjointness of the linearized problem which is a signature of active matter out-of-equilibrium systems. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading. This is joint work with V. Rybalko and C. Safsten published in Transactions of AMS (to appear) and Phys. Rev.B, 2022.
We consider a family of processes obtained by randomly splitting the deterministic flows associated with some fluid models (e.g. Lorenz 96, 2d Galerkin-Navier-Stokes). These split dynamics conveniently separate the conservative and dissipative part of the underlying equation. We characterize some ergodic properties of these stochastic dynamical systems and prove their convergence to the original deterministic flow in the small noise regime, both in the conservative and in the dissipative setting. Finally, we show that the top Lyapunov exponent of these models is positive.
This is joint work with Jonathan Mattingly and Omar Melikechi.
We develop stability estimates for advection-diffusion equations in the DiPerna-Lions setting, i.e. with a velocity field that is Sobolev regular in the spatial variable. The estimates are formulated in terms of the distances from the optimal transport theory with logarithmic cost functions, and provide thus a bound for the rate of weak convergence. First, we obtain an estimate for distributional solutions given by different data. Second, we study the implicit finite volume discretization and we prove an error estimate for the rate of convergence of approximate solutions towards the unique exact solution. Both estimates are optimal invarious respects.
In this talk we investigate systems of particles immersed in a fluid and subject to gravitation. The particles interact with each other through the fluid in a very implicit way. I will show how, relying on previous results for mean-field limits of inertialess particles, a mesoscopic description of particles with inertia can be derived. In particular I will point out the connection to the Transport-Stokes and Vlasov-Brinkmann equations as mean-field limits. The results rely on the approximation of the system by a system with much more explicit interaction and a detailed understanding of the involved forces in comparison to the case of a single particle. This is based on joint work with Richard Hoefer (Paris).
Kac introduced a family of stochastic, many particle systems which model the behaviour of a spatially homogeneous, dilute gas, with evolution through binary elastic collisions. In the limit where the number of particles diverges, the empirical measures have the spatially homogeneous Boltzmann equation as a fluid limit. Although the Boltzmann equation itself is not explicitly probabilistic, we may use Kac’s process to study the Boltzmann Equation and vice versa, and in this talk I will discuss some recent works exploring this connection. This talk will also provide a preview of some of the topics I intend to discuss in more detail in the mini-course on the Boltzmann equation.
In this seminar we will study the asymptotic behaviour of phase-field functionals of Ambrosio and Tortorelli type allowing for small-scale oscillations both in the volume and in the diffuse surface term. The functionals under examination can be interpreted as an instance of a static gradient damage model for periodically heterogeneous materials. Depending on the mutual vanishing rate of the phase-field parameter and the oscillation parameter, the effective behaviour of the model will be fully characterised by means of Γ-convergence. This is joint work with T. Esposito, R. Marziani, and C. I. Zeppieri (Münster).
Considered one of the most important stochastic processes in the last decades, the KPZ equation has played an enormous role to motivate the study of singular SPDEs, becoming a foundational object in rough analysis. The presence of an explicit solution also made it possible to study its law and most of its properties in detail using standard tools of stochastic analysis. However, the connections between regularity structures and stochastic calculus are still not developed well enough to offer a joint perspective on the KPZ equation, even though both study the same object. In this talk, we will discuss how one can use regularity structures to derive a pathwise change of variable formula for the KPZ equation and some possible interpretations of it. Joint work with Tom Klose (TU Berlin).
Multiphase mean curvature is an important evolution equation in various geometrical or physical problems and its connection to the Allen-Cahn equations has a long history. In this talk, we will discuss a conditional convergence result for systems of Allen-Cahn equations to a De Giorgi type solution for multiphase mean curvature flow. Beforehand we will be looking at the gradient flow structure of both the mean curvature flow and the Allen-Cahn equation and discussing De Giorgi's optimal energy dissipation inequality for both equations. Lastly the question will be raised if the conditional convergence can be improved, or to be more specific, what happens if we drop the assumption of energy convergence.
In the two talks, we discuss the proof of invariance of the Gibbs measures for the three- dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ4 3- model. In the beginning of the first talk, we briefly review properties of Hamiltonian ODEs, which serve as a toy-model. Then, we state our main theorem and connect it with recent developments in constructive quantum field theory, dispersive PDEs, and stochastic PDEs. In the later parts of the first talk and all of the second talk, we discuss aspects of our proof. In particular, we discuss multilinear dispersive estimates, random operator bounds, and a hidden cancellation. During this discussion, we illustrate the main ideas through simple examples, which should make it accessible to participants with no background in dispersive equations or stochastic PDEs. This is joint work with Y. Deng, A. Nahmod, and H. Yue.
Let us consider a linear system on the Riemann sphere and assume that the coefficients of the system additionally depend on some parameter t (time). The following question is interesting: what can one say about the dependence of the coefficients on t, if monodromy is fixed? For a system with only simple poles, deformation equations have a very rich geometrical structure: it is a Hamiltonian system on a Lie algebra and the simplest non-trivial case corresponds to Painlev\'{e}~VI equation. I am going to talk about the simplest non-trivial case of a system with an irregular singular point. This system has the same structure and leads to Painlev\'{e}~V equation.
Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport and we explore algorithms like iterative scaling. This is joint work with Bernd Sturmfels, François-Xavier Vialard and Max von Renesse.
In the two talks, we discuss the proof of invariance of the Gibbs measures for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ4 3-model. In the beginning of the first talk, we briefly review properties of Hamiltonian ODEs, which serve as a toy-model. Then, we state our main theorem and connect it with recent developments in constructive quantum field theory, dispersive PDEs, and stochastic PDEs. In the later parts of the first talk and all of the second talk, we discuss aspects of our proof. In particular, we discuss multilinear dispersive estimates, random operator bounds, and a hidden cancellation. During this discussion, we illustrate the main ideas through simple examples, which should make it accessible to participants with no background in dispersive equations or stochastic PDEs. This is joint work with Y. Deng, A. Nahmod, and H. Yue.
The talk is devoted to the geometrical properties of the trajectories of the Wiener process. The main interest in the subject is inspired by the mathematical theory of polymers. It is known that the long molecules of linear polymers can be modeled as trajectories of non-smooth random processes. As a basic model Wiener trajectories are considered. The main problem arising there is the geometry of such random curves. We propose to study it considering the polygonal line inscribed into the trajectory. We will discuss the asymptotic behavior of a number of self-intersections in the line for dimension 2.
Integral currents are a weak generalization of smooth oriented manifolds with boundary and provide a natural setting in which to study the Plateau problem: ‘what are the surfaces of least m-dimensional area that span a given (m-1)-dimensional boundary?’ However, the weak nature of integral currents permit the formation of singularities. The problem of determining the size and structure of the interior singular set of an area-minimizer in this setting has been studied by many since the 1960s, with many ground breaking contributions. The codimension one case is significantly easier to handle, but in the higher codimension case, much less progress has been made since the celebrated (m-2)-Hausdorff dimension bound on the singular set due to Almgren, the proof of which has since been simplified by De Lellis and Spadaro. In this talk I will review the key features of the proof of Almgren/De Lellis-Spadaro and discuss how to strengthen this to an upper Minkowski dimension estimate. I will also discuss some work in progress with Camillo De Lellis (IAS) towards establishing rectifiability of the singular set in high codimension.
In this talk, I will discuss a new approach to the solvability of semilinear elliptic problems (in bounded and unbounded domains) supplemented by data assuming low regularity. The method exploits the nature of the nonlinearity as well as other intrinsic properties of the equation to select suitable functional frameworks where solutions are sought for. In the process, we borrow tools from modern harmonic analysis and function spaces theory. Two equations of interest will be mentioned: The weakly harmonic map problem and the stationary Navier-Stokes flow. I also plan to comment on the parabolic theory if time allows.
In this talk I will present recent works done in collaboration with B. Velichkov about the regularity of a free boundary problem that involves two positive harmonic functions glued along a free hypersurface with a so-called "transmission condition" in the jump of the derivative. I will explain in particular an epiperimetric inequality method that is an efficient way to obtain C^1 regularity.
We will first review the role of invariant measures in stochastic homogenization, and explain how the relative success in treating equations without drift, or with a drift that is either the gradient of a stationary field or mean-zero and divergence-free, is due in part to an explicit identification of the invariant measure or to uniform estimates that fail in the general case. We will then restrict our attention to the case of a mean-zero, divergence-free drift. We will prove that such environments homogenize weakly provided the drift admits a stationary, square-integrable stream matrix, thereby providing a simple PDE-based proof of recent optimal results in the discrete setting. Finally, under stronger integrability assumptions on the stream matrix, we will show that the environment satisfies a large-scale Hölder regularity estimate and first-order Liouville principle.
In this talk, we discuss the following question: Let us consider a function u in $L^1$ satisfying a differential constraint A u=0 (e.g. A =curl or A = div). Is it possible to modify u slightly, such that it still obeys the differential constraint and is in some better space (i.e. $L^{\infty}$)?This question can be seen as a generalization of Lipschitz extension/truncation results. I will briefly point out some applications of such a result in the Calculus of Variations. This talk is based on joint work with L. Behn (Bielefeld) and F. Gmeineder (Konstanz).
We study the Gibbs measure with log-correlated base Gaussian fields on the d-dimensional torus. When d = 2, the Gibbs measure corresponds to the well-studied Phi^k_2-measure. We first discuss the (non-)construction of the focusing Gibbs measure. As a Hamiltonian PDE system corresponding to the Gibbs measure, we consider the Zakharov-Yukawa system (Schr¨odinger-wave system with a Zakharov-type coupling) on the two-dimensional torus. We then present a phase transition and invariance of the Gibbs measure under the flow.
This is based on joint work with Tadahiro Oh (University of Edinburgh) and Leonardo Tolomeo (University of Bonn).
In the first part of the talk I shall discuss an approach to the localisation technique, for spaces satisfying the curvature-dimension condition, by means of L1-optimal transport. I shall present work on a generalisation of the technique to multiple constraints setting.
In the second part I shall discuss work in progress concerning general localisation scheme, which includes both the L1-localisation and the irreducible convex paving of the martingale transport theory.
In this talk I will present my recent works on free discontinuity and free boundary problems involving Robin or other similar boundary conditions. I will start with a scalar case that involves both free boundary and free discontinuity problems with obstacle constraints, whose motivation was originally to establish a quantitative version of the Saint-Venant inequality (dealing with the maximization of torsional rigidity among sets of fixed area), as well as a similar problem involved in thermal insulation. Then I will show some recent results on the regularity of a free boundary problem using an epiperimetric inequality method. Finally, I will discuss current works on vectorial free discontinuity problems in fluids mechanics, namely a theoretical approach to the drag minimization of an object immersed in a creeping flow with Navier conditions.
We study the approach to equilibrium in relative entropy of systems of gas particles modeled via the Kac master equation in arbitrary dimensions. First, we investigate the Kac system coupled to a thermostat, and secondly connected to a heat reservoir. As a result, we obtain exponential decay rates for the entropy and information that are essentially independent of the size of the systems. The use of the Fisher-information allows simple proofs with weak regularity assumptions.
Merriman, Bence and Osher's thresholding scheme is a time discretization of mean curvature flow. I restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoglu and Otto I show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional strong convergence results of Laux and Otto become unconditional in this case. The results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.
In this talk I plan to present new solvability techniques for elliptic problems subject to boundary data assuming a minimal regularity property. These techniques heavily rely on certain harmonic analysis tools and recent progress in the theory of function spaces. As an illustration, I will discuss the solvability theory for the weakly harmonic map system into a compact manifold (which is a "borderline" problem). Applications in the parabolic setting will also be mentioned with a particular focus on the chemotaxis Navier-Stokes equations.
Let u be the solution to the one-dimensional stochastic heat equation driven by a space-time white noise with constant initial condition. The purpose of this talk is to present a recent result on the uniform convergence of the density of the normalized spatial averages of the solution u on an interval [-R,R], as R tends to infinity, to the density of the standard normal distribution, assuming some non-degeneracy and regularity conditions on the diffusion coefficient. These results are based on the combination of Stein method for normal approximations and Malliavin calculus techniques.
This is a joint work with David Nualart.
In this talk, we will talk about our recent results on the traveling wave solution to the Burgers-Hilbert equation, specifically their long-term stability under small perturbations of the initial data, with a lifespan beyond the reach of local wellposedness. This is joint work with A. Castro and D. Cordoba.
We present some new and surprising examples of uniform integrability of sequences satisfying both a linear pde constraint and a (nonlinear) pointwise constraint. One example that we will explore would be higher integrability of hessian sequences that does not follow from the theory of quasiregular mappings. Earlier examples can be found in the works of D. Serre and G. De Philippis et al. We identify a new phenomenon that gives a counterexample to a conjecture in G. De Philippis et al. Some of our proofs rely on the theory of k-hessian equations, which we complement with new Sobolev regularity results. This is joint work with A. Guerra and M. Schrecker.
We consider the approach to regularity structures introduced by Otto, Sauer, Smith and Weber for a class of quasilinear SPDEs. This approach replaces Hairer's tree-based description by a greedier index set built from derivatives of the nonlinearity. We provide a Hopf-algebraic construction of the structure group within this framework. Considering the infinitesimal generators of certain actions in the space of nonlinearities, we build a pre-Lie algebra; its universal envelope, after a proper choice of basis, is dual to a Hopf algebra from which we can build the group. This more Lie-algebraic approach connects to already-existing constructions in regularity structures, which are rather combinatorial. Based on joint work with Felix Otto and Markus Tempelmayr.
We focus on the theory of fluctuations in stochastic homogenization for linear uniformly elliptic PDEs in divergence form. More precisely, we study the properties of the so-called standard homogenization commutator, a random field which plays a crucial role in this theory. Main tool in our approach is a covariance estimate -- a consequence of the Hellfer-Sjöstrand representation formula for covariances which holds in the Gaussian setting we adopt.
I will give an overview of some regularity results for minimisers of (p,q)-functionals, which are a class of functionals arising in nonlinear elasticity and other applications. A particular focus will be on the use of the difference quotient method to obtain global improved (Sobolev-)differentiability results. Further, I will discuss the occurrence of the Lavrentiev phenomenon in the case of non-autonomous integrands.
This talk is concerned with quantitative estimates in the context of stochastic homogenisation of random integral functionals defined on finite partitions. Assuming the integrand to be stationary, the techniques to determine the homogenised limiting functional in terms of Γ-convergence are by now quite well established. Here the homogenised limiting integrand is obtained via a blow-up formula by solving minmisation problems on larger and larger cubes. In this talk we show that the cubes in these minimisation problems can actually be replaced by almost flat hyperrectangles and we explain how to control quantitatively the fluctuation of the corresponding blow-up formulas.
This is joint work with Matthias Ruf (EPFL).
Many real-world networks exhibit the small-world phenomenon: their typical distances are much smaller than their sizes. A natural way to model this phenomenon is a long-range percolation graph on the lattice $Z^d$, in which edges are added between far-away vertices with probability falling off to the $s$-th power of the Euclidean distance. How does the resulting graph distance scale with the Euclidean distance? The question has been intensely studied in the past and the answer depends on the exponent $s$ in the connection probabilities, for which five regimes of behavior have been identified. In this talk I will give an overview on past results and then discuss our recent progress on the critical regime $s=d$. (Joint work with Marek Biskup.)
We consider the dynamical large deviations of an interacting particle system (Kac's process) whose hydrodynamic limit is the Boltzmann equation. In equilibrium, the particles are (at any given time) independent, and we recover the entropy from the large deviations at any one time. This gives us a new tool to investigate the entropy in the context of the Boltzmann equation, and we demonstrate some applications. One application will be a new derivation of Boltzmann's H-Theorem (decay of the entropy along the Boltzmann equation) from the reversibility of the microscopic dynamics.
We establish sharp upper and lower bounds for the Kantorovich optimal transport distance between the uniform measure and the occupation measure of a path of a fractional Brownian motion with Hurst index H taking values in a d-dimensional torus. Similar problems have been recently studied for diffusion processes taking values on a compact connected Riemmanian manifold. We give new insights in the case of fractional Brownian motion taking care of the absence of the Markovian structure by means of recently introduced PDE techniques and compare our result with the ones already known. In particular we show that a phase transition between rates occurs if d=1/H +2, in analogy with the random Euclidean bipartite matching problem, i.e. when the occupation measure is replaced by i.i.d. uniform points (formally given by infinite H).
Joint work with M. Huesmann (WWU Münster) and D. Trevisan (Università degli studi di Pisa)
When studying the regularity of surfaces which locally minimize the functional \int \| \nu_{E}\|_{p} for p > 2, one quickly runs into the pseudo p-Laplacian: a differential equation which, in this anisotropic setting, plays a role analogous to the role that the Laplacian plays for area minimizers. When the surface is a graph over the plane orthogonal to any standard basis vector, for instance e_{n}, the observation that D^{2}|_{(\cdot, 1)} \| \cdot \|_{p} \equiv 0 causes major problems for this regularity theory. Isolating the roles of homogeneity from the usual definition of ellipticity, we can consider any strictly convex norm \rho on \mathbb{R}^{n} and \gamma\in (1,n) , and recover De Giorgi-Nash-Moser theory whenever one considers weak solutions of \int \langle \rho(Du)^{\gamma-1} (D \rho)(Du), D \varphi \rangle = 0 . Time permitting, some preliminary results on 1st-order regularity of u stemming from a power-type convexity on \rho will also be discussed.
The study of periodic homogenization is a fundamental application of calculus of variations to material sciences. While there are many extensive and general results to cover energies for classical materials with periodic microstructure, it is only in the last 10 years that the periodic homogenization of modern materials have been considered. These materials are characterized by some non-classical behavior, and so the energies which model them do not fall under any classical framework. We will specifically present results on the homogenization limit for thin metamaterials (joint work with Elisa Davoli (TU Wien)) and a result on phase transitions of heterogenous temperature-responsive materials (joint work with Riccardo Cristoferi (Radboud University) and Irene Fonseca (Carnegie Mellon University).
I will present a new method of renormalizing singular stochastic PDEs based on the Wilson renormalization group theory and the Polchinski flow equation. The technique is applicable to a large class of semi-linear parabolic or elliptic SPDEs with fractional Laplacian, additive noise and polynomial non-linearity including equations arbitrarily close to criticality. A nice feature of the method is that it avoids the algebraic and combinatorial problems arising in different approaches.
I will discuss the periodic homogenization of diffuse interface energies and their L^2 gradient flows, or, in other words, the homogenization of Allen-Cahn-like equations with periodic coefficients. The main goal of the talk is to introduce the surface tension and the mobility, the two quantities that are expected to describe the homogenized behavior, and to explain what can be said about pulsating standing waves, which are so far the closest thing we have to correctors. Along the way, I will describe some particular examples that highlight pathologies that can occur.
We will first consider a perturbation theory for Markov processes, expanding expectations of the state at a given time in orders of some small perturbation parameter around equilibrium.
In many physical applications however, it is impossible to observe the full state space, and instead we can only distinguish a finite number of states. This results in a so called coarse grained process given as a function of the underlying Markov process.
We will show that the first order is analogous to the Markovian case and that we can extrapolate the second order from linear terms under certain conditions.
This talk is based on results of my master thesis and joint work with Urna Basu, Peter Sollich and Matthias Krüger https://arxiv.org/abs/2005.05169 .
A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. I will present an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.
We consider the long time behavior of solutions to the Burgers-FKPP equation $u_t +\beta u u_x = u_{xx} + u-u^2.$ The Burgers-FKPP equation solutions exhibit a phase transition phenomenon from being pulled to pushed as $\beta$ increases, and the analysis at the transition case $\beta=2$ is quite delicate. We show the convergence to a traveling wave for the whole spectrum of $\beta$. In particular, when $\beta\leq 2$, we introduce a weighted Hopf-Cole transform to construct upper and lower barriers in the self-similar variables for the linearized equation on the half line. This new transform differentiates the transition case $\beta=2$ from $\beta
In this talk, we survey some classical results on the speed of travelling waves in the FKPP equation, which is a stochastic reaction-diffusion equation arising in population genetics.
We review techniques used to prove the existence of a deterministic asymptotic wave speed in the FKPP equation. We focus in particular on the arguments used by Mueller & Sowers (1995) and Conlon & Doering (2005).
Time permitting, we will discuss the difficulties associated with generalizing results obtained for this model to travelling wave solutions in other SPDE.
One of the main topics in optimal transport theory is the regularity of optimal transport maps. As is classically observed by Caffarelli, the regularity of maps is closely related to the geometry of densities; in particular, for general densities (with possibly-nonconvex support) the corresponding map is not necessarily fully smooth and one can expect only partial or conditional regularity. In this talk I will first briefly review previous regularity results and then present a recent joint work with Felix Otto on a variational boundary $\varepsilon$-regularity theory.
We provide an analytic approach to study the long term behavior of rough differential/evolution equations, with the driving noises of Hoelder continuity. Such systems can be solved either with Lyons’ theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths, or with fractional calculus. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of a global pullback attractor.
We are interested in computing the electrical field generated by a charge distribution localized on scale $\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $L\gg\ell$ around the support of the charge. We propose an artificial boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of $\ell$ and $L$, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and in the regime $1\ll\ell$). The boundary condition is motivated by stochastic homogenization that allows for a multipole expansion [Bella, Giunti, Otto 2020]. This work extends [Lu, Otto] from two to three dimensions, which requires to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [Gloria, Otto 2015].
In this talk, we discuss the construction and invariance of the Gibbs measure for a threedimensional wave equation with a Hartree-nonlinearity.
In the rst part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free eld. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free eld.
In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. At the moment, this is the only theorem proving the invariance of any singular Gibbs measure under a dispersive equation.
Various physical systems (fluids, nonlinear waves, etc.) exhibit turbulent, generally chaotic behavior when subject to forcing and weak damping. A natural problem when studying such systems is to establish unique ergodicity and quantify the convergence of generic time averages to the unique stationary statistics. In this talk, we discuss recent work on the long-time behavior of a class of hypoelliptic SDEs that covers some prototypical chaotic/turbulent systems such as Lorenz-96 and Galerkin truncations of the stochastic Navier-Stokes equations. Our main result is an optimal quantitative estimate on the exponential convergence to equilibrium in the limit of vanishing, balanced noise and dissipation. Exponential convergence for the model under consideration has been known for some time, but our quantitative estimates are new. Our proof uses a scheme that combines a weak Poincaré inequality argument with a quantitative hypoelliptic regularization estimate for the associated time-dependent Kolmogorov equation. As a necessary step in carrying out our approach, we obtain quantitative pointwise estimates (uniform in the small noise/dissipation parameter) on the stationary density. This is accomplished with hypoelliptic De Giorgi and Moser type iterations.
This is joint work with Jacob Bedrossian (University of Maryland).
Due to topology changes and geometric singularities, the existence theory for interface evolution problems typically relies on weak solution concepts. In the absence of a comparison principle, the question of their uniqueness properties however remained essentially unexplored for a long time. In this talk, I will present weak-strong uniqueness principles for two important interface evolution problems not admitting a comparison principle: i) the flow of two incompressible, viscous and immiscible fluids in the presence of surface tension, and ii) multiphase mean curvature flow. More precisely, for these models we establish uniqueness of a suitable class of weak solutions within the class of classical solutions prior to the first topology change. The key ingredient to these qualitative uniqueness results are quantitative stability estimates in terms of a novel notion of relative entropies for interface energies. I will explain this concept first in a simple two-phase setting. In a second step, I discuss how the corresponding multiphase analogue leads to a gradient flow generalization of a well-known concept from minimal surface theory: the notion of calibrations. I conclude with an outlook on possible future directions. The talk is based on joint works with Julian Fischer, Tim Laux and Theresa Simon.
We consider a stochastic many-particle system introduced by Kac as a mean field approximation to the spatially homogeneous Boltzmann equation. A first question one can ask is to prove a hydrodynamic limit for the many-particle system as the number of particles $N\to\infty$, which justifies Boltzmann’s assumption of molecular chaos, and to quantify the rate of convergence. We will address this problem in the cases where the dynamics are governed by a range of collision kernels, which represent either a) localised interactions (hard spheres), or b) a class of long-range repulsive potentials (hard potentials). The available results and required techniques depend strongly on the collision kernel in question. For the case a), many previous results are available and we use a `top-down’ approach in the spirit of Mischler and Mouhot, using the stability of the Boltzmann equation to obtain estimates with good time dependence. In the case b), we introduce a coupling of the kind first introduced by Tanaka, and show that the coupling is stable for a Wasserstein-type optimal transportation problem with a well-chosen cost function; this leads to a law of large numbers and stability for the Boltzmann equation.
We study the large time behavior of solutions for the Burgers-FKPP equation. When the coefficient $\beta$ of the Burgers nonlinearity increases, it leads to a phase transition from pulled fronts to pushed fronts. By introducing a weighted Hopf-Cole transform, we capture the criticality of phase transitions at $\beta=2$. With that, we can show the convergence of a solution to a single traveling wave in the Burgers-FKPP equation for all $\beta$. I will further show how our new approach can improve the spreading speed results for the Keller-Segel-FKPP equation.Close to the end of the talk, I will switch the topic and mention works in analyzing stochastic gradient algorithms from the continuous time limit perspective. I will discuss how such an approach can provide explanations of why two common bias-correcting methods in sampling, resampling and reweighting, can have different performance when stochastic gradient algorithms are applied.
I will present results on the Jacobian equation det Du = f where f\in L^p. In particular, I will present results on generic non-existence of solutions on bounded domains as well as results on symmetry properties of the solution answering questions by Ye, Hogan-McIntosh-Zhang and Hélein. An important tool will be a nonlinear open mapping principle that is of independent interest. In fact, as an application of this principle I will show that the set of initial data for which there are dissipative weak solutions in L^p_t L^2_x of the incompressible Euler equations is meagre in the space of solenoidal L^2 fields.
In the limit of small diffusivity, the large deviations of the space distribution of a large population of independent Brownian particles lead to quadratic optimal transport. This ten-year-old result by Léonard brought up to date the analysis of an entropic minimization problem introduced by Schrödinger in 1931, both for its relevance in applications and for its numerical properties. On the other hand, the large variety of natural phenomena exhibiting both transport and variations of mass - for instance in population dynamics - was a strong motivation for the development in the last few years of many so-called unbalanced optimal transport models, each of them proposing a quantitative compromise between these two behaviors. In this talk, based on a work in progress with Hugo Lavenant from Bocconi University, I will show how to derive a regularized version of an unbalanced optimal transport model from the large deviations of the branching Brownian motion. I will also present briefly how this study was motivated by a question in developmental biology.
We study a uniformly elliptic equation with high oscillatory (at scale S1. This operation makes the corrector equation easy to solve numerically and provide a natural approximation of the homogenized coefficient by the one coming from periodic homogenization. The variance of the difference suffers from two types of error: a random (that is fluctuation around its expectation) and a systematic one. We focus in this work on the systematic error and we characterize its asymptotic behaviour as L goes to infinity. We show, in the particular case where the law is generated by a stationary Gaussian field, that the asymptote is characterized by a deterministic matrix depending on the first and second-order correctors, the gradient of the covariance function and a fourth-order tensor involving the whole space Green function of the homogenized elliptic operator.
Anti-concentration inequalities provide limits on the extent to which random variables can be concentrated: for example, they commonly give uniform upper bounds on the probability that a random variable takes any particular value. In this talk I'll discuss some of the many connections between anti-concentration and combinatorics, initially focusing on applications to Ramsey graphs but also touching on a few other topics such as the polynomial Littlewood-Offord problem and permanents of random matrices.
We study the time evolution of the strongly coupled polaron which is a model for an electron moving in an ionic crystal. Its microscopic description is given by the Fr\"ohlich Hamiltonian. For initial data of Pekar product form with coherent phonon field and with sufficiently small energy, we provide an effective dynamics for the strongly coupled polaron. The effective dynamics is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. The proof is based on an adiabatic theorem for the Landau-Pekar equations and the persistence of the spectral gap. This is joint work with D. Feliciangeli, N. Leopold, D. Mitrouskas, B. Schlein and R. Seiringer.
In the first part of this talk, we focus on the proof of the existence of global-in-time weak solutions to the Shigesada-Kawasaki-Teramoto cross-diffusion system in population dynamics for an arbitrary number of population species. This model was first studied by Shigesada, Kawasaki and Teramoto in 1979 in the context of population dynamics describing competing species in a heterogeneous environment, and has since then attracted some attention in the context of the mathematical analysis of cross-diffusion phenomena, segregation effects and pattern formation. We proved that global existence for this model follows under a detailed balance or weak cross-diffusion condition, where the detailed balance condition is related to the symmetry of the mobility matrix in the formal gradient-flow structure, which mirrors Onsager's principle in thermodynamics. The second part of this talk deals with different derivation techniques of the Shigesada-Kawasaki-Teramoto cross-diffusion system and other related cross-diffusion models from the microscopic level. We present how to link at the formal level the entropy structure of this cross-diffusion system satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. Moreover, we present a rigorous proof of a many-particle limit from a moderately interacting stochastic many-particle system to the cross-diffusion model using techniques of K. Oelschläger. Finally, we describe how to generalize and extend these approaches and discuss some open questions.
We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions. After giving an overview of work by Bailleul, Mesnager and Norris on the small-time fluctuations for sub-Riemannian diffusion bridges, which assumes the initial and final positions to lie outside the sub-Riemannian cut locus, we extend their results to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. We show that, in a suitable chart and after a suitable rescaling, the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limit operator. We further consider small-time fluctuations for the bridge in a model class of diffusion processes satisfying a weak Hörmander condition, where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly.
Stochastic and deterministic dynamical systems are widely used to model complex real-world processes. From a statistical point of view, such a system can be described by the semigroup of linear Koopman operators, or its infinitesimal generator. Numerous different methods to model these linear operators based on simulation data have been developed in past years. However, many of those methods require a fair amount of prior knowledge about the system of interest. In this talk, I will present some of my recent work to overcome these limitations. I will present data-driven models for the Koopman semigroup using the tensor train format, and also on reproducing kernel Hilbert spaces. I will also show how these models can be used to calculate eigenvalues and eigenfunctions of the operators in question, and how the underlying dynamical system can be further analyzed and manipulated based on them. I will conclude by sketching future research directions along these lines.
The concept of self-organized criticality (SOC) postulates a power-law distribution for the size of intermittent events in a large class of stochastic particle processes. We present a specific discrete model which satisfies many typical properties of this class, and which can be formally identified as the finite difference approximation of a stochastic porous medium equation with a singular-degenerate nonlinearity. We will pursue this aspect by proving that a sequence of suitably rescaled discrete processes of this kind indeed converges in law (in a weak topology) to the solution of the SPDE. This can be viewed as a contribution to the search for a universal limit equation of models displaying SOC.
In this talk, I will present some recent results regarding isometric embedding. I first review some conclusions on $C^{1,\theta}$ isometric immersions and isometric extension and related problem. Then I will show our global extensions of the celebrated Nash-Kuiper theorem for $C^{1,\theta}$ isometric immersions of compact manifolds with optimal H\"older exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $\theta
It is known that the thin film equation is formally a gradient flow with respect to the Dirichlet energy and the Wasserstein metric. Then it is natural to consider the formal Fokker-Planck equation to make sure that the associated stochastic equation satisfies the detailed balance condition which means that the invariant measure is the Gibbs measure with respect to the energy. We make this argument rigorous by discretizing in space via a Galerkin scheme. This leads to a high dimensional stochastic differential equation in Ito form which corresponds to the discretized stochastic thin film equation. Contrary to the literature our ansatz yields an additional Ito correction term.
This is ongoing work with Benjamin Gess and Felix Otto.
We generalize the normalized combinatorial Laplace operator for graphs by defining two Laplace operators for hypergraphs that can be useful in the study of chemical reaction networks. We also investigate some properties of their spectra.
We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface without boundary and flows along that surface. Local-in-time well-posedness is established in the framework of Lp-Lq-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields and we show that each equilibrium is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.
The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical bound on the number of bound states for Schrödinger operators. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth.
In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.
Joint work with D. Hundertmark, P. Kunstmann, and S. Vugalter
This talk is concerned with modern methods for computing the effective material properties of heterogenenous materials.
In the first part, we shall discuss computational methods that became increasingly popular in the last decade and are specifically designed for complex microstructures, the so-called FFT-based computational homogenization methods, where FFT stands for the fast Fourier transform. We also discuss recent developments on numerical solution methods that are both memory-efficient and fast.
In the second part of the talk, we focus on fiber-reinforced composites which are frequently used in lightweight design. The corresponding microstructures feature (non-overlapping) cylindrical inclusions dispersed at a specific volume fraction and orientation state within a volume element, serving as a cut-out of a specific realization of a stochastic fiber-reinforced microstructure. We discuss how to generate such microstructures, and also shed light on the ensuing ramifications for the effective elastic moduli for such structures (using the computational methods discussed earlier).
One of the main questions in regularisation by noise literature is to understand whether an additive perturbation restores well-posedness of an ODE, i.e. under which conditions there exists a unique solution to $\dot{x}=b(x)+\dot{w}$ even if this is not the case for $w=0$. Davie first addressed the problem of identifying the analytical properties of a path $w$ which provide a regularising effect; Catellier and Gubinelli answered the problem by introducing the key concepts of averaging operators and nonlinear Young integrals. Remarkably, this allows to provide a consistent solution theory even when $b$ is merely distributional and to deduce that generic continuous functions have an arbitrarily high regularisation effect. In this talk I will first review their work and then present its more recent extensions. Based on a joint work with Massimiliano Gubinelli.
Small particles suspended in a fluid are ubiquitous in nature and technology. It is well-known that the particles change the effective viscosity of the fluid. The problem has been addressed by Einstein in his PhD dissertation in 1906. He obtained a quantitative result known as Einstein's law for the effective viscosity for spherical particles to first order in the volume fraction $\phi$ of the particles. Rigorous mathematical results have only been obtained in the last years. I will review these results and present recent improvements where we were able to relax the assumptions on the particle configurations considerably. This covers physically relevant random distributions of particles. A big challenge consists in the analysis of a dynamic version of Einstein's law. Indeed, the interaction between the particles accounting for Einstein's law is very singular ($1/|x|^3$ in three dimensions), and we presently do not know how to obtain the corresponding mean field-result for fixed volume fraction $\phi$ as we lose control over the interparticle distances. Nevertheless, I will present a perturbative result in the case $\phi \to 0$, that incorporates Einstein's law. This talk is based on joint works with David Gérard-Varet and Richard Schubert.
In this talk we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one - the classical configuration giving rise to the Rayleigh-Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.One day before the seminar, an announcement with the link will be sent to the mailing list of the AG seminar. If you are not on the mailing list, but still want to join the broadcast, please contact Pavlos Tsatsoulis.
We consider a variational two-dimensional Landau-de Gennes model in the theory of nematic liquid crystals on a disk. We investigate symmetry of minimizers under a symmetric boundary condition carrying a topological defect of degree k/2. One day before the seminar, an announcement with the link will be sent to the mailing list of the AG seminar. If you are not on the mailing list, but still want to join the broadcast, please contact Pavlos Tsatsoulis.
We will review aspects of the theory of Compensated Compactness, starting with the fundamental work of Murat and Tartar and concluding with recent results obtained jointly with A. Guerra, J. Kristensen, and M. Schrecker. Broadly speaking, the object of this study is to gain a better understanding of the interaction between weakly convergent sequences and nonlinear functionals. The general framework will be that of variational integrals defined on spaces of vector fields satisfying linear pde constraints that satisfy Murat's constant rank condition. We will focus on the weak continuity and lower semi-continuity of these integrals, as well as the Hardy space regularity of the integrands. One day before the seminar, an announcement with the link will be sent to the mailing list of the AG seminar. If you are not on the mailing list, but still want to join the broadcast, please contact Pavlos Tsatsoulis.
Inbetween elliptic PDEs, which do not depend on time (think of the Poisson equation), and honest parabolic PDEs, which do depend on time and are started at a given initial value (think of the heat equation), there are time-periodic parabolic PDEs: On the one hand, time-independent solutions to the elliptic PDE are also trivially time-periodic, which gives periodic problems an elliptic touch, on the other hand solutions to the initial value problem which are not constant in time might very well be periodic. I want to advocate for time-periodic problems not being the little sister of either elliptic or parabolic problems, but being a connector between the two and a class of its own right, by introducing a direct method for showing a priori $L^p$ estimates for time-periodic, linear, partial differential equations. The method is generic and can be applied to a wide range of problems. In the talk, I intend to demonstrate it on the heat equation and on boundary value problems of Agmon-Douglas-Nirenberg type. The talk is based on joint works with Yasunori Maekawa and Mads Kyed.
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilising the stochastic sewing lemma [K. Le, ’18] . This approach allows one to exploit regularisation by noise effects in obtaining convergence rates.In our first application we show convergence (to our knowledge for the first time)of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift.When the Hurst parameter is $H\in(0,1)$ and the drift is $C^\alpha$, $\alpha>2-1/H$, we show the strong $L_p$ and almost sure rates of convergence to be $(1/2+\alpha H) \wedge 1-$. As another application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $1/2-$ of the Euler-Maruyama scheme for $C^\alpha$ drift, for any $\alpha>0$. This is a joint work with Oleg Butkovsky and Máté Gerencsér.
For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the best choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382-386 (2018).
In general, inverse problems are notoriously ill-posed. As inverses of well-posed problems this, in particular, is reflected in their (in-)stability properties. In this talk I will discuss three robust, new mechanisms of proving (optimal) instability results for some classes of elliptic, parabolic and hyperbolic inverse problems. This is based on joint work with Herbert Koch and Mikko Salo.
Skyrmions are topologically nontrivial patterns in the magnetization of extremely thin ferromagnets. Typically thought of as stabilized by the so-called Dzyaloshinskii-Moriya interaction (DMI), or antisymmetric exchange interaction, arising in such materials, they are of great interest in the physics community due to possible applications in memory devices.
In this talk, I will characterize skyrmions as local minimizers of a two-dimensional limit of the full micromagnetic energy, augmented by DMI and retaining the nonlocal character of the stray field energy. In the regimeof dominating Dirichlet energy, I will provide rigorous predictions for their size and ``wall angles''. The main tool is a quantitative stability result for harmonic maps of degree ± 1 from the plane to the two-dimensional sphere, relating the energy excess of any competitor to the homogeneous H¹-distance to the closest harmonic map.
This is joint work with Anne Bernand-Mantel and Cyrill B. Muratov.
We consider a class of weakly interacting particles/diffusion processes living either on the torus or in Euclidean space. We first introduce the corresponding mean-field PDE which is an aggregation-diffusion equation. We then proceed to study the existence of nontrivial stationary solutions and of phase transitions for both the case with linear diffusion and the case with porous medium-type degenerate diffusion. We provide necessary and sufficient conditions on the interaction term for the existence of phase transitions for the associated free energy and characterise interactions for which the phase transitions are continuous or discontinuous. We also provide a sufficient set of conditions for the existence of saddle points of the free energy. Finally, we discuss the behaviour of the underlying particle system under the diffusive rescaling in the large particle limit and the effect the presence of phase transitions has on this limit.
In [Reinken et al., 2018], the authors derived a fourth-order continuum theory capable of reproducing mesoscale turbulence in a three-dimensional suspension of microswimmers. This is in some way a refinement of a phenomenological model proposed in [Wensink et al., 2012], that also describes an active fluid consisting of polar moving active particles like, e.g., self-propelling bacteria, suspendend in a fluid. The novelty of the former approach is, that it is capable of distinguishing between the orientation of the particles and the dynamics of the surrounding fluid.
In this talk, we study the mathematical properties of this system, in particular the global existence and weak-strong uniqueness of weak solutions. The latter is shown via the relative energy approach which is then applied to further examine the connection to the phenomenological model, for which strong solutions are available, as was shown in [Zanger et al., 2016].
Consider a system of N particles moving according to Brownian motions and branching at rate one. Each time a particle branches, the particle in the system furthest from the origin is killed. It turns out that we can use results about a related partial differential equation known as a free boundary problem to control the long term behaviour of this particle system for large N.
This is joint work with Julien Berestycki, Eric Brunet and James Nolen.
It is believed that in the 1860s Riemann proposed the function \[ R(x) = \sum_{n=1}^{\infty}\frac{\sin(n^2 x)}{n^2}, \qquad x \in \mathbb{R}, \] as an example of a continuous but nowhere differentiable function, but no proof was given at the time. Since then, many authors have worked on its analytic properties, first trying to solve Riemann's guess (proven a century later, in 1970) and studying further regularity properties afterwards.However, recent results suggest that the generalisation of Riemann's function \[ \phi(x) = \sum_{k \in \mathbb{Z}}\frac{e^{-4\pi^2ik^2x}-1}{-4\pi^2k^2}, \qquad x \in \mathbb{R} \] is a surprisingly precise approximation to a temporal trajectory in an experiment concerning vortex filaments. It can be argued then that it has also an intrinsic physical and geometric structure.In this talk, I will present my work concerning geometric properties of $\phi(\mathbb{R})$. We will see that its Hausdorff dimension is upper bounded by 4/3. %, while to prove lower bound strictly larger than 1 is an interesting but tougher challenge. Also, motivated by the fact that tangent vectors should be the representatives of the velocity of a particle following the physical trajectory, we will see that no such tangents exist in this case. Finally, adapting Frisch's definition of intermittency and flatness, we will see that Riemann's function is intermittent, albeit in a weak form.
After a brief explaination of the meaning and scope of inverse problems, in this talk I will introduce the most famous of them, the classical Calderón problem. I will give an idea of the main known results, the used techniques, some variants of the problem and a small collection of open questions. I will then consider the fractional Calderón problem, whose characteristics and available results are largely different than those of its classical counterpart. I will also show some real-life applications.
Much work has been dedicated to relating the phenomenological Landau-de Gennes model for liquid crystalline materials to its simpler large-domain limit, the Oseen-Frank model. As Landau-de Gennes is, however, phenomenological, it begs the question as to whether Oseen-Frank may be justi ed from models with a more sound physical justi cation. In this talk we outline some recent work on obtaining Oseen-Frank, for both nematic and cholesteric liquid crystals, as a large-domain limit of a more fundamental mean- eld model, where material constants appearing in the limiting gradient-type energy can be derived from more fundamental pairwise interactions of molecules.
For hyperbolic systems of conservation laws in one space dimension, the best theory of well-posedness is restricted to solutions with small total variation (Bressan et al. 2000). Looking to expand on this theory, we push in new directions. One key difficulty is that for many systems of conservation laws, only one nontrivial entropy exists. In 2017, in joint work with A. Vasseur, we proved uniqueness for the solutions to the scalar conservation laws which verify only a single entropy condition. Our result was the first result in this direction which worked directly on the conservation law. Further, our method was based on the theory of shifts and a-contraction developed by Vasseur and his team. These theories are general theories and apply also to the systems case, leading us to hope the framework we built for the scalar conservation laws will apply to systems. In this talk, I review the current progress on using the theory of shifts and a-contraction to push forward the theory of well-posedness for systems of conservation laws in one space dimension. This is joint work with A. Vasseur.
In this talk, I will give an overview of the Schramm-Loewner Evolutions (SLE) theory and present new results on this theory based on the analysis of a Singular Differential Equation that appears naturally in this context. This equation appears when extending the conformal maps to the boundary and can be thought of as a singular Rough Differential Equation (RDE), as in Rough Path Theory. In the study of RDEs, questions such as continuity of the solutions, the uniqueness/non-uniqueness of solutions depending on the behavior of parameters of the equation, appear naturally. We adapt these type of questions to the study of the backward Loewner differential equation in the upper half-plane, and the conformal welding homeomorphism. This view will allow us to obtain some new structural and geometric information about the SLE traces in the regime where they have double points.
This first part is a joint work with Dmitry Belyaev and Terry Lyons.
Also, I aim to present an analysis of the Stochastic Taylor approximation for this singular RDE that is part of a project with James Foster and Terry Lyons, and, if time allows, to cover the main ideas of an independent project that uses ideas from Quasi-Sure Stochastic Analysis through Aggregation in order to study SLE theory quasi-surely. This quasi-sure study will allow us to overcome some of the difficulties with the previous analysis that I will emphasize throughout the talk.
The KPZ equation, a model for the random growth of rough interfaces, has been the subject of great physical and mathematical interest since its introduction in 1986. By simple changes of variables, it is closely related to the stochastic heat equation, which models the partition function of a random walk in a random environment, and the stochastic Burgers equation, a simple model for turbulence. I will explain several recent results about the existence, classification, and properties of spacetime-stationary solutions to these equations on $R^d$ for various values of d. These solutions thus represent the behavior of the models in large domains on long time scales. Most of the results are joint work with various combinations of C. Graham, Y. Gu, L. Ryzhik, and O. Zeitouni.
In shape-memory alloys, it is common to analyze pattern formation induced by the existence of different zero free energy phases in the material. These microstructure provokes the shape-memory effect. The classical model used to study the problem is \begin{equation}\label{1} \inf_{\substack{y\in W^{1,\infty}(\Omega)\\ y|_{\partial \Omega}= Fx}} \ \int_\Omega \phi\left(\nabla y\right) \,dx, \quad %\begin{array}{l} \ker\, \phi = \{u_1,\, u_2,\, \dots,\, u_n\}\, \oplus\, \mathbb{R}^{2\times 2}_{skew},%\\ \{u_1,\, u_2,\, u_3\} \subset \mathbb{R}^{2\times 2}_{sym} \end{array} \end{equation} where $\{u_1,\, u_2,\, \dots,\, u_n\} \subset \mathbb{R}^{2\times 2}_{sym}$ and the function $\phi:\mathbb{R}^{2\times 2}\rightarrow \mathbb{R}$ satisfies mild growth conditions and it is invariant under addition of skew-symmetric matrices to its argument. In this talk, I will present some recent results about the relaxed problem via quasiconvexification. More precisely, it will be proved that the quasiconvex hull of $\ker \phi$ equals its convex hull if the $n$ wells are pairsewise symmetrized-rank-one connected. Particularly, in the three-well problem if one of this connection is lost, then the contention of the quasiconvex hull of $\ker \phi$ in its convex hull is strict.
We will discuss Keller-Segel-type models of chemotaxis, whose prominent feature is a competition between aggregating and diffusive mechanisms. Roughly, in case the aggregation prevails, short-time smooth solutions blow-up in finite time, whereas if the diffusion wins, short-time smooth solutions can be continued indefinitely. We will recall both the seminal results on the classical Keller-Segel system, and certain new ones concerning fractional and semilinear diffusions. Our focus will be the most-interesting `fair-competition regime', where the diffusion and aggregation appear to be in a balance.
In this talk, I will be discussing the results obtained for the stochastic evolution equation, which describes the system governing the nematic liquid crystals perturbed by pure jump noise in the Marcus canonical form.
A briefing on the existence of a martingale solution in two and three dimensions will be presented. In addition, the pathwise uniqueness of the martingale solution in two dimensions will be presented, from which the existence of a strong solution will be deduced.
The final part of the talk concerns the large deviation theory for the above-said model. I start with the stochastic two-dimensional nematic liquid crystal model influenced by multiplicative Gaussian noise. The Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions will be analyzed using the weak convergence method. Then using a similar technique, I will establish a large deviation principle for stochastic nematic liquid crystals driven by pure jump noise in the Marcus canonical form in two dimensions.
Noise sensitivity is a concept that measures if the outcome of a Boolean function can be predicted when one is given its value for a perturbation of the input. A sequence of functions is noise sensitive when this is asymptotically not possible. A non-trivial example of a sequence that is noise sensitive is the crossing functions in critical two-dimensional Bernoulli percolation. In this setting, noise sensitivity can be understood via the study of randomized algorithms. Together with a discretization argument, these techniques can be extended to the continuum setting. In this talk, we prove noise sensitivity for critical Voronoi percolation in dimension two, and derive some consequences of it. Based on a joint work with D. Ahlberg.
In the optimal transport problem, it is well-known that the geometry of the target domain plays a crucial role in the regularity of the optimal transport. In the quadratic cost case, for instance, Caffarelli showed that having a convex target domain is essential in guaranteeing the optimal transport’s continuity. In this talk, we shall explore how, quantitatively, important convexity is in producing continuous optimal transports.
This talk is concerned with homogenization problems from the perspective of numerical analysis and approximation theory. By means of a linear elliptic model diffusion problem, we will introduce a numerical homogenization method that is based in the computation of operator-dependent subspaces with a quasi-local basis and uniform approximation properties for arbitrary rough diffusion coefficients. The key result in the corresponding error analysis is the exponential decay of the Green's function associated with numerical corrector problems. A non-standard application of the numerical homogenization method and its analysis is the sparse approximability of the expected solution operator in prototypical random diffusion problems.
Atmospheric gravity waves play a significant role in numerical weather and climate prediction. Especially, waves with rather short wavelengths, which are omnipresent and influence predictions, remain spatially unresolved by the numerical simulations, such that they need to be parametrized, i.e. represented somehow by resolved quantities.
In order to improve weather and climate forecasting, my work aims to enhance gravity wave parametrizations by focusing on nonlinear effects. Nonlinear wave dynamics may lead to counterintuitive properties that are not available in linear theory. For instance, the group velocity, as defined usually by the derivative of the dispersion relation, may not coincide with the wave’s actual envelope velocity. Also, stability properties may change notably. Nonlinear traveling wave packets, e.g., turn out to be unconditionally prone to modulational instabilities.
For my investigations, I use numerical methods such as finite volume schemes with operator splitting and analytical techniques like linear stability analysis in combination with Fredholm operator theory.
From disordered to ordered spatial structures, complex geometries can define new states of matter, rigorously formulated in mathematics and experimentally observed in physics. Surprising probabilistic and geometric phenomena can lead to both fundamental insights and innovative material designs. By presenting recent examples of intriguing geometries, this talk spans a bridge from an anomalous suppression of large-scale density fluctuations to stable matchings and from a hidden order in tessellations to photonic networks, which shape the flow of light.
This work is motivated by the fact that classical homogenization theory poorly takes into account interfaces or boundaries. It is particularly unfortunate when one is interested in phenomena arising at the interfaces or the boundaries of the periodic media (the propagation of plasmonic waves at the surface of metamaterials for instance). To overcome this limitation, we have constructed an effective model which is enriched near the interfaces and the boundaries. For now, we have treated and analysed the case of simple geometries : for instance a plane interface between a homogeneous and a periodic half spaces. We have derived a high order approximate model which consists in replacing the periodic media by an effective one but the transmission conditions are not classical. The obtained conditions involve Laplace- Beltrami operators at the interface and requires to solve cell problems in periodicity cell (as in classical homogenization) and in infinite strips (to take into account the phenomena near the interface). We establish well posedness for the approximate model and error estimates which justify that this new model is more accurate near the interface and in the bulk. This will be illustrated by numerical results.
In this talk I would like to present some connections between the isoperimetric inequality, the stability of Möbius transformations of the sphere and the well known geometric rigidity estimates of Friesecke, James and Müller regarding the stability of the special orthogonal group. The main result is of local nature and asserts that for a Lipschitz map that is apriori close to a Möbius transformation of the sphere, an average conformal-isoperimetric type of deficit controls the deviation (in an average sense) of the map in question from a particular Möbius transformation. Its link to the geometric rigidity of SO(n) is subsequently discussed. This is joint work with Prof. Stephan Luckhaus.
Stochastic delay differential equations (SDDE) are prominent examples of stochastic differential equations on infinite-dimensional spaces which, in general, do not generate a stochastic flow. Consequently, a dynamical theory to study these equations seemed to be impossible for a long time. In this talk, we solve this problem by showing that SDDE indeed induce random dynamical systems on an infinite dimensional fiber bundle. On this structure, we can prove a Multiplicative Ergodic Theorem which yields a spectral theory for linear SDDE. As an application, we can prove a stable manifold theorem for SDDE. Joint work with Mazyar Ghani Varzaneh and Michael Scheutzow (both TU Berlin).
We introduce a new family of interpolation inequalities, with two radial power-law weights and exponents in the subcritical range. They are related with the so-called entropy- entropy production inequalities in the problem of intermediate asymptotics for nonlinear diffusions, and play a role for the porous medium equation similar to some standard Caffarelli-Kohn-Nirenberg inequalities for the fast diffusion equation.
We address the question of symmetry breaking: are the extremal functions radially symmetric or not? By extremal functions we mean functions that realize the equality case in the inequality, written with optimal constants. Although the Euler-Lagrange equations are invariant under rotation, we prove that the extremal functions are not radially symmetric, provided the power laws of the weights are chosen appropriately. Our proof of the symmetry breaking is variational and relies on the stability analysis of optimal solutions in the class of radially symmetric functions. The core of the proof consists of finding the optimal constant in a weighted Hardy-Poincaré inequality.
This work is a collaboration with Jean Dolbeault (CEREMADE, Université Paris Dauphine) and Matteo Muratori (Politecnico di Milano).
Small particles moving in a fluid are encountered in various situations in nature and technology. In many cases, gravitation is the driving force for the dynamics of the particles. If the particles are not too small, the system can be microscopically modeled by the Navier-Stokes equations coupled with a system of ODEs for the particles according to Newton's laws. Although the force acting on each particle due to the gravity is directly proportional to its mass, and we do not include direct (e.g. electromagnetic) interaction between the particles themselves, the motion of the particles will be quite complex in many situations. The complexity arises from the interaction of the particles through the fluid. Indeed, the presence of each particle induces a disturbance in the fluid flow which again influences all the other particles.
Assuming that the fluid inertia is negligible, we focus on three closely related microscopic models for spherical and non-spherical particles with and without inertia. We will then discuss corresponding macroscopic models which consist of systems that couple a Vlasov equation to Stokes equations. In the case of inertialess spherical particles, the macroscopic system can be rigorously derived from the microscopic dynamics in the limit of many small particles.
This talk is based on joint work with Juan Velázquez and Arianna Giunti.
We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calderón problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.
The Mullins-Sekerka problem for closed interfaces is widely studied since it appears naturally as a gradient flow of the area functional, as a sharp interface limit of the Cahn-Hilliard equation, and in physical models of phase changes. In this talk I will address the Mullins-Sekerka problem for interfaces with a ninety degree contact angle. In particular, I will show existence and uniqueness of strong solutions and discuss stability properties. This is joint work with Helmut Abels, Harald Garcke, and Mathias Wilke.
Point-like sources are commonly used in physics. They usually represent compact bodies whose inner structure is not physically significant. This idealization simplifies the description of the motion of the bodies. However, because of the distributional character of the sources the equations of motion governing the evolution of the system typically contain ill-defined products of distributions. In the talk, I will present a method which allows to consistently renormalize PDEs describing classical fields coupled to pointlike bodies in the approximation in which the non-linear terms are treated as a perturbation of the linear equation. The motivation behind the ongoing project is the description of the motion of two compact bodies interacting gravitationally. The problem is of great importance especially because of the need to construct accurate template waveforms which are necessary to detect and interpret gravitational waves emitted by coalescing binary systems.
Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes it according to the space-time scale in such a way that the system rescales to the SPDE in the macroscopic limit. We justify this belief for a class of continuous microscopic growth models (for KPZ) and phase co-existence models (for \Phi^4_3), allowing general microscopic nonlinear mechanisms beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussian random fields. This essentially allows one to reduce the problem of a general function to that of a polynomial.
We study the continuous-time random walk in a random degenerate environment: the infinite supercritical percolation cluster. This subject has been a topic of interest over the past decades and a number of results pertaining to the random walker have been established: Gaussian heat kernel bounds for the transition kernel, quenched invariance principle for the walk, local limit theorem etc. An important ingredient of the proofs is to implement a renormalization structure for the infinite cluster. In this talk, we will present some of the results aforementioned, will introduce a renormalization structure for the infinite cluster which was used to adapt the recently developed theory of quantitative stochastic homogenization to this degenerate setting, and will present some new results which can be derived from this construction. This is joint work with S.Armstrong and C.Gu.
Non-Euclidean, or incompatible elasticity, is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes plastic deformations or inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional.
Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them. In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results and open problems on the relations between their elastic behavior and their curvature.
Based on joint work with Asaf Shachar.
We begin with a review of well-posedness results for the inhomogeneous wave equation with variable coefficients on a bounded domain in $\mathbb{R}^3$. After the warm up, we consider coupled systems of semi-linear wave equations and discuss the notion of short time well-posedness for the system. Under the assumption of small Cauchy data, we show the source to solution map for the nonlinear problem determines the source to solution map for the linear problem. We discuss why the assumption of small data and short times allows us to recover multiple sound speeds uniquely for the nonlinear problem in some natural settings. A variety of open problems are mentioned during the talk.
We study an instance of diffusion and mixing in the incompressible 2d Navier-Stokes equations. Broadly put, mixing causes a transfer of energy to high frequencies. This adds to the dissipative forces, giving rise to what we refer to as "enhanced dissipation". While such effects are well known in the physics literature and experimentally studied, mathematically rigorous results are much more scarce.
This talk is based on recent work for the case of the Poiseuille flow. We first demonstrate enhanced dissipation for the Navier-Stokes equations linearized around this flow by identifying a time scale that is faster than the purely diffusive one. Subsequently, we give a transition stability threshold below which this behavior persists in the full, nonlinear setting.
In the first part we study the existence and uniqueness of solutions to the higher order parabolic Cauchy problems on the upper half space, given by $\partial_t u = (-1)^{m+1} \mbox{div}_m A(t,x)\nabla^m u$ and $L^p$ initial data space. The (complex) coefficients are only assumed to be elliptic and bounded measurable. Our approach follows the recent developments in the field for the case $m=1$. In the second part we consider the $BMO$ space of initial data. We will see that the Carleson measure condition $$\sup_{x\in \mathbb{R}^n} \sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}\int_0^{r}|t^m\nabla^m u(t^{2m},x)|^2\frac{dxdt}{t}
In this talk, we consider random heterogeneous problems. Inspired by the recents works of F. Otto, M. Duerinckx and A. Gloria, our aim is to understand how much the response fluctuates around its homogenized limit. In other words, we wish to estimate the distribution of the response, on the basis of the knowledge of the distribution of the material coefficients, and without resorting to a costly Monte Carlo approach. In the first part of the talk, we show that, in a weakly random setting, the fluctuations of the response are governed by a tensor that we identify. In the second part of the talk, we consider strongly random settings, and we present numerical simulations that confirm our theoretical findings.
Joint work with F. Legoll (ENPC).
In this talk I will discuss an epiperimetric inequality associated to the parabolic Signornini problem, and show how it can be used to study the asymptotic behavior of the solution around certain free boundary points, as well as the regularity of the free boundary.
We considered the asymptotic behavior of one-phase free boundary graph in one exterior domain. We obtained that the solution is Lipschitz continuous, non-degenerate, and has one specific version of Weiss's monotonicity formula by adding extra terms. In planar case, we proved the asymptotic flatness of the solution in some sense, and we also discussed higher dimensional cases under additional conditions.
Martensitic transformations are phase transitions between different crystalline states occurring in certain alloys. These are often characterised by complex microstructures, which have been widely studied in the context of the calculus of variations. However, it may occur that infinitely many minimisers exist, and it is an open problem how to select the physically relevant ones. In this talk I present a moving mask hypothesis that can be used as a selection mechanism for physically relevant microstructures in thermally induced martensitic phase transitions. The moving mask hypotheses allows to prove a rigidity result for the two-well problem, and to better understand the importance of the cofactor conditions, particular conditions of supercompatibility between phases, which are believed to influence reversibility.
Invariance principles à la Donsker are well understood for many random walks in random environments. But if we encounter the random walk as noise acting on a differential equation, then we have to study the invariance principle in a fine topology in order to understand the convergence properties of the solutions: The so called rough path topology. In my talk I will briefly present this topology and then discuss a general invariance principle that describes the fluctuations in the ergodic theorem for stationary Markov processes in rough path topology. As an application, we will see a rough path invariance principle for the random conductance model (a particular class of random walks in random environments). The talk is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.
In 1970s Burkholder, Davis, and Gundy proved the following inequalities which connect the $L^p$-norm of a martingale with its quadratic variation: \[ \mathbb E \sup_{t\geq 0}|M_t|^p \eqsim_p \mathbb E [M]_{\infty}^{p/2},\;\;\; 1\leq p
We give an a priori bound for solutions of the dynamic $\Phi^4_3$ equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.We will show the different techniques used for the small and large scale bounds. For small scales we use techniques derived from Hairer’s theory of regularity structures.. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.
A lot of attention has been devoted to the entropic regularization of optimal transport in recent years for its links with the theory of large deviations and for its formidable efficiency in terms of numerical computations. The purpose of this talk will be to introduce and to study the zero noise limit of the incompressible version of this regularized problem. With this aim in view, I will present an elementary way to modify an absolutely continuous curve with values in the Wasserstein space to get an admissible curve for the dynamical entropic optimal transport. This construction allows to recover in a variational way two famous theorems:the displacement convexity of the entropy along the optimal transport (due to McCann),the Gamma-convergence of the dynamical entropic optimal transport towards the classical optimal transport (due to Léonard).Besides, our modification preserves incompressibility, so these proofs adapt transparently to the incompressible case, and we get as new results these theorems in that setting. Doing so, we extend a result by Lavenant for the convexity of the entropy, and a result by Benamou-Carlier-Nenna for the Gamma-convergence.
This is a joint work with L. Monsaingeon (Lisbon and Nancy University).
The aim of this talk is to review recent advances building on the fundamental work of Bourgain, Brezis, Mironescu, and Van Schaftingen in the study of linear (elliptic) systems $B u=f$, where the source term is a Radon measure. The results discussed will cover aspects of critical Sobolev regularity, boundary estimates, and fine properties of the solutions. Special attention will be given to the case of operators of order $n$ on $R^n$, in which case the operators $B$ for which the solution $u$ is bounded or continuous are characterized.
We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander's bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris's lemma to work directly on the Malliavin matrix, instead of the "reduced Malliavin matrix" which is not available in this context.
On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Itô formula for rough PDEs.
This is a joint work with Martin Hairer.
Diffusion processes on the L2-Wasserstein space were introduced by Sturm and von Renesse in 2009 and have been subject of much interest in recent years. In particular, Konarovskyi proposed in 2017 a construction of a diffusion based on a system of massive coalescing particles. The aim of my talk is to present a Girsanov Theorem and a Bismut-Elworthy formula for a diffusion process inspired by Konarovskyi's model.
We will first introduce a diffusion on the Wasserstein space which is a regularized version of Konarovskyi's model. It can be viewed as a continuum of particles evolving on the real line according to a Gaussian interaction kernel weighted by the mass associated to each particle.
Second, we will prove a Girsanov-type Theorem on this process, using an appropriate Fourier inversion of the Gaussian kernel. As a Corollary, we obtain weak existence and weak uniqueness of the solution to a Fokker-Planck equation on the L2-Wasserstein space with very irregular drift.
Third, we will present a regularization result on the semi-group associated to this process with non-smooth drift and prove a Bismut-Elworthy formula which provides an upper bound for the gradient of the semi-group in smooth directions of differentiation.
Bessel processes are a classical family of stochastic diffusions obeying singular dynamics which, in a certain regime, involve a remarkable renormalization procedure. More recently a family of stochastic PDEs related to Bessel processes has been introduced, the dynamics of which involve similar but more acute renormalizations. In my talk I shall introduce these processes and explain the remarkable underlying structure. Applications to scaling limits of dynamical critical wetting models will be mentionned. This is based on joint work with Lorenzo Zambotti.
In this talk, we prove global well-posedness of a system describing behavior of dilute flexible polymeric fluids. This model is based on kinetic theory, and a main difficulty for this system is its multi-scale nature. A new function space, based on moments, is introduced to address this issue, and this function space allows us to deal with larger initial data.
The Dean-Kawasaki (DK) model is an important nonlinear stochastic equation in the fluctuating hydrodynamics field. It describes the evolution of the density function for a system of finitely many particles undergoing Langevin dynamics.
This equation is formally obtained, in a Schwartz distribution setting, on the hydrodynamic scale: due to its complicated noise structure, well-posedness for the equation is open except for the simplest, purely diffusive, case, corresponding to overdamped Langevin dynamics. In this case it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist.
We derive and analyse a suitably regularised DK model under two relevant sets of assumptions concerning the Langevin particle system. We thus address formal mathematical issues associated with the distributional setting of the original DK model.
We further prove a high-probability result for the existence and uniqueness of mild solutions to this regularised DK model, and also discuss relevant open questions.
This is joint work with Tony Shardlow and Johannes Zimmer.
I will present results about scaling limits of Hamilton-Jacobi equations with highly oscillatory, mixing spatio-temporal dependence. When the Hamiltonian is centered, then the long time, large space behavior of the solutions is governed by a spatially homogenous, stochastic Hamilton-Jacobi equation. The result relies on explicit convergence rates for the stochastic homogenization of Hamilton-Jacobi equations and new stability estimates for stochastic viscosity solutions. I will also discuss a class of problems for which the hyperbolic scaling limit leads to spatio-temporal stochastic homogenization.
Germanium antimony tellurides play an important role both as phase-change materials for data storage and as thermoelectric materials for the interconversion of heat and electrical energy. They are characterized by disordered cubic high-temperature phases. Upon cooling, they become rhombohedral. This can be a simple lattice distortion or a diffusion-controlled process that is associated with vacancy ordering. This leads to nanostructures with twin domains that significantly influence physical properties. These nanostructures can be influences by different cooling rates or annealing processes as wenn as by the introduction of additional chemical elements. Oriented (endotaxial) intergrowth of different crystal structure types occurs if indium is added. This adds additional possibilities for phase transitions. Synchrotron raditiation enables the in situ observation of phase transitions in a space- and time resolved manner. However, transport properties depend on a complex interplay of electronic and phononic effects.
There are a number of classical results on the approximation of Sobolev functions by Lipschitz continuous mappings on Euclidean spaces in the sense of Lusin, as well as Lusin-type approximations of measurable vector fields by gradients of Lipschitz-continuous functions. However, the classical proofs rely on the doubling property of the Lebesgue measure and some other techniques which are specific to the finite-dimensional setting.
Nevertheless, it turns out that for a infinite-dimensional space equipped with a Gaussian measure there are natural counterparts of these results, although some questions remain open. We will discuss some new observations in this area based on the estimates for heat semigroups and some dimension-independent bounds.
General relativity, our modern theory of gravity, connects the matter content of the universe to spacetime geometry through the Einstein equation. It endows geometry with rich dynamics: solutions describe cosmology, black holes, neutron stars and gravitational waves, all of which came together in the binary merger events detected by LIGO and Virgo. The nonlinear nature of the Einstein equation gives rise to these dynamics, but at the same time makes finding solutions a challenge.
In this talk I will describe two nonlinear frontiers of general relativity research. At the "dynamical frontier" weakly dissipative systems give rise to new nonlinear dynamics, including instabilities and even turbulence in the spacetime geometry. A major finding is the emergence of new approximate conservation laws that control turbulent cascades. These systems are often relevant to conjectured holographic dualities, and may manifest astrophysically. At the "deep learning frontier" I will describe data challenges facing gravitational-wave astronomy, and approaches to try to solve them with deep neural networks.
In this talk we will introduce two models for the movement of a small droplet over a substrate: the thin film equation and the quasi static approximation. By tracking the motion of the support of solutions to the thin film equation, we connect this two models. This connection was expected from experimental data which suggested Tanner's law: the edge velocity of a spreading thin film on a pre-wetted solid is approximately proportional to the cube of the slope at the inflection. This is joint work with Prof. Antoine Mellet.
In this talk, we will consider a tamed version of the 3-dimensional (stochastic) magnetohydrodynamics (MHD) equations, following the ideas of M. Röckner and X.C. Zhang. We will look at both the deterministic and the stochastic version of the equation and discuss questions of well-posedness for these equations.
In a population where individuals reproduce at different rates (i.e. have different “fitness”), the fraction of high-fitness types naturally increases over time—this is what Darwin coined "natural selection". This process can be represented abstractly as a non-linear yet exactly soluble integro-differential equation. I will show that this equation possesses self-similar solutions and describe their basins of attractions. The presentation will be guided by analogies with extreme value statistics on the one hand and and mean-field coarsening dynamics on the other.
The model consists of a system of two diffusion equations one of which is called a signal and another is an observation process. The task is for any moment of time to compute the best in square mean estimator of the signal given observations to this time, which is called filter and for which linearized SPDE equations have been derived (long ago). In the talk it will be shown how to derive these equations "by hands" without any big theory. If time allows, approximation issues will be also discussed.
About joint work with Martin Tassy and/or Andrew Krieger. Previous works have shown that arctic circle phenomenons and limiting behaviors of some integrable discrete systems can be explained by a variational principle.
In this talk we present a method to deduce variational principles for non-integrable discrete systems. We illustrate the method by considering two different models. In the first model, we consider graph homomorphisms form Z^d to a regular tree. In the second model, we derive a quenched variational principle for height functions exposed to a random field.
We use the finite volume method to discretely approximate the Kantorovich distance W_2 on the space of probability measures in Euclidean space. This method gives the discrete space a Riemannian structure. However, the question of Gromov-Hausdorff convergence was unanswered except for cubic finite volumes on the torus (Gigli-Maas 2013). We show that the limit distance is in general lower than the Kantorovich distance due to cost-decreasing oscillations. However, under a simple geometric condition on the finite volumes, we show Gromov-Hausdorff convergence.
In order to quantify weak convergence in passive or active scalar problems one commonly uses analytic or geometric mixing scales. While not equivalent, we show that after some modifications both notions are comparable. Here, we further introduce a dyadic model problem. In a second part, we consider decay rates of these scales for Sobolev regular initial data when evolving under transport type dynamics.
In this talk, I consider periodic homogenization of non-convex integral functionals that are motivated by non-linear elasticity. In this situation long wavelength buckling can occur which mathematically implies that the homogenized integrand is given by an asymptotic multi-cell formula. From this formula it is difficult to deduce qualitative or quantitative properties of the effective energy. Under suitable assumptions, in particular that the integrand has a single, non-degenerate, energy well at the set of rotations, we show that the multi-cell formula reduces to a much simpler single-cell formula in a neighbourhood of the rotations. This allows for a more refined, corrector based, analysis. In particular, for small data, we obtain a quantitative two-scale expansion and uniform Lipschitz estimates for energy minimizer. This is joint work with Stefan Neukamm (Dresden).
We prove well-posedness for singular semilinear SPDEs on a smooth bounded domain $D$ in $\mathbb{R}^n$ of the form \[ dX(t) + AX(t)\,dt + \beta(X(t))\,dt \ni B(t,X(t))\,dW(t)\,, \qquad X(0)=X_0\,. \] The linear part is associated to a linear coercive maximal monotone operator $A$ on $L^2(D)$, while $\beta$ is a (multivalued) maximal monotone graph everywhere defined on $\mathbb{R}$ on which no growth nor smoothness conditions are required. Moreover, the noise is given by a cylindrical Wiener process on a Hilbert space $U$, with a stochastic integrand $B$ taking values in the Hilbert-Schmidt operators from $U$ to $L^2(D)$: classical Lipschitz-continuity hypotheses for the diffusion coefficient are assumed. A comparison with the corresponding deterministic equation and possible generalizations are discussed.This study is based on a joint work with Carlo Marinelli (University College London).
The Weertman equation is a nonlinear integrodifferential equation which models steadily-moving dislocations in materials science. Its solution can be interpreted as the (unique) traveling wave of an "artificial" dynamical system (a nonlocal reaction-diffusion
equation). Under reasonable hypotheses, we prove that, for any initial condition, the dynamical system actually converges to the solution of the Weertman equation. That convergence provides a way of approximating numerically this solution.
Joint work with: Y.-P. Pellegrini (CEA-DAM), F. Legoll (ENPC), C. Le Bris (ENPC).
In this talk, I plan to study convergence rates in $L^2$ norm for elliptic homogenization problems in Lipschitz domains. It involves some new weighted-type inequalities for the smoothing operator at scale $\varepsilon$, as well as, layer and co-layer type estimates, and the related details will be touched. In order to obtain a sharp result, a duality argument will be imposed. Here we do not require any smoothness assumption on the coefficients, and the main ideas may be extended to other models, such as Stokes systems and parabolic systems, arising in the periodic homogenization theory.
This talk addresses a higher-order variational problem with an obstacle constraint, which models thin elastic bodies adhering to non-flat solid substrates. Focusing on a one-dimensional periodic setting, we consider how physical parameters in the model affect the shapes of least energy solutions. Our main concern is the regime of small bending rigidity. To this end we first establish a Gamma-convergence result with an appropriate compactness property and then obtain a stronger convergence of least energy solutions beyond fundamental consequences of the Gamma-convergence.
When there is a big applause after a good concert, sometimes the audience starts to clap in a synchronised manner even though there is no one synchronising the different people. Similarly, in biology, the actions of individual neural cells in the heart are synchronised to create a joint periodic impulse.
These examples can be modelled as a system of coupled oscillators. This leads to the Kuramoto model, where the evolution of each oscillator is determined by its natural frequency and the coupling. On the one hand, the coupling works towards a synchronisation. On the other hand, the natural frequencies differ, which desynchronises the system. Using the mean-field limit, a large number of oscillators can be described by a density over the phase space. Its evolution is then described by a PDE, which is a non-linear transport equation without any dissipation.
Despite the lack of dissipation, stable stationary states are observed in numerical simulations. Here, the stability is due to the frequency differences and can be understood as phase mixing. This is the same stability mechanism as in Landau damping for the Vlasov-Poisson equation or inviscid damping for the Euler equation. It is a delicate stability mechanism because there is no stability in strong topology but only in weak topology.
In this talk, I will introduce the topic and describe this fascinating stability mechanism. Then I will present my results on the stability of inhomogeneous stationary states and explain the main ideas of the proof.
We study the asymptotic behavior of the top eigenvectors and eigenvalues of the random conductance Laplacian in a large domain of $\mathbb{Z}^d$ ($d\geq 2$) with zero Dirichlet conditions. Let the conductances $w$ be positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. Then we show that the spectrum of the Laplacian displays a sharp transition between a completely localized and a completely homogenized phase. A simple moment condition distinguishes between the two phases.In the homogenized phase we can even generalize our results to stationary and ergodic conductances with additional jumps of arbitrary length. Here, our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincar\'e inequalities, Moser iteration and two-scale convergence. The investigation of the homogenized phase is joint work with M. Slowik and M. Heida.
The numerical approximation of solutions to stochastic partial differential equations (SPDEs) with additive spatial white noise on bounded domains is considered. The differential operator is given by the fractional power of an integer order elliptic differential operator and is therefore non-local. Its inverse operator is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator is approximated by a weighted sum of non-fractional resolvents at certain quadrature nodes. These resolvents are then discretized in space by a standard finite element method. By combining this approach with approximate realizations of the white noise, which are based only on the mass matrix of the finite element discretization, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, explicit rates of strong and weak convergence are derived. A key property of the presented scheme is that it does not require the knowledge of the eigenfunctions of the differential operator, which is necessary, e.g., for approximations based on truncated spectral Karhunen-Loève expansions of the noise term. For this reason, the method is particularly interesting for real-world applications in spatial statistics, such as to employ solutions to fractional order SPDEs as approximations of Gaussian Matérn fields. This application is taken up in numerical experiments to illustrate and attest the theoretical results.
Topological properties of random attractors are essential to understand the asymptotic behavior of random dynamical systems. In the deterministic case, set attractors of continuous-time systems are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space.
We prove that random attractors of continuous-time systems, which attract compact sets almost surely, are connected. Moreover, we present an example where compact sets converge to the attractor in probability and the attractor is not connected.
This is joint work with M. Scheutzow.
We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers. (Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)
I will give an overview of recent results about behavior of random walks on infinite percolation clusters, such as quenched invariance principle and Gaussian heat kernel bounds, particularly in the presence of strong spatial correlations.
It is well known from the literature that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is "very bad" and has no solutions (or has multiple solutions), then the addition of a random noise leads almost surely to a "nice" ODE with a unique solution. The first part of the talk will be devoted to SDEs with distributional drift driven by alpha-stable noise. These equations are not well-posed in the classical sense. We define a natural notion of a solution to this equation and show its existence and uniqueness whenever the drift belongs to a certain negative Besov space. This generalizes results of E. Priola (2012) and extends to the context of stable processes the classical results of A. Zvonkin (1974) as well as the more recent results of R. Bass and Z.-Q. Chen (2001).
In the second part of the talk we investigate the same phenomenon for a 1D heat equation with an irregular drift. We prove existence and uniqueness of the flow of solutions and, as a byproduct of our proof, we also establish path-by-path uniqueness. This extends recent results of A. Davie (2007) to the context of stochastic partial differential equations.
[1] O. Butkovsky, L. Mytnik (2016). Regularization by noise and flows of solutions for a stochastic heat equation. arXiv 1610.02553. To appear in Annal. Probab.[2] S. Athreya, O. Butkovsky, L. Mytnik (2018). Strong existence and uniqueness for stable stochastic differential equations with distributional drift. arXiv 1801.03473.
(Joint work with Siva Athreya & Leonid Mytnik).
A global approximation theorem for a differential operator P is a result ensuring that a solution v of P[v] =0 in a closed set S satisfying some topological assumptions can be approximated by a global solution u of the equation P[u]=0. The theory for elliptic equations has been widely developed. In this talk I will show global approximation theorems for parabolic equations. In addition, I will apply these results to prove the existence of solutions of the heat equation with local hot spots with prescribed behavior as well as minimal graphs with micro-oscillations. This is a joint work with A. Enciso and D. Peralta-Salas.
We will review some results --obtained in collaboration with Sven Jarohs (Frankfurt) and Alberto Saldaña (Karlsruhe)-- concerning higher-order fractional Laplacians, which are nonlocal operators of non-integer order larger than 2. These pertain the validity (or rather the failure) of maximum principles, the extension of Boggio's formula for the Green function, the structure and the boundary behaviour of corresponding harmonics, the formulation of suitable natural boundary conditions, and the pointwise evaluation of the operators.
We consider the dynamic $\Phi^4_2$ model (or stochastic Allen--Cahn equation) formally given by the SPDE \begin{equation*} (\partial_t - \Delta) \phi_\varepsilon = - \phi_\varepsilon^3 + \phi_\varepsilon +\sqrt{2\varepsilon} \xi \end{equation*} where $\xi$ is a space-time white noise. When $\varepsilon$ is small the solutions of the equation spend long time intervals in metastable states before reaching equilibrium. This phenomenon is known as metastability. We discuss a coupling argument for solutions started from suitable initial conditions in space dimension 2 as a consequence of metastability. Such a result is already known in space dimension 1. The basic obstacle in our case is that classical solution theory for SPDEs is not applicable here since the non-linear term is ill-defined due to the irregularity of $\xi$. Hence a renormalization is required to compensate the divergences of the non-linear term. This destroys the "nice" structure of the non-linear term and a deeper analysis on the level of the so-called "remainder" term is required. An interesting application of such a coupling argument appears in the proof of the Eyring--Kramers law in the theory of metastability.
The cubic nonlinear Schrodinger equation (NLS) is energy-critical (s_c = 1) with respect to the scaling symmetry, where s_c is the scaling critical regularity. The initial value problem (IVP) of cubic NLS is scaling invariant in the Sobolev norm H^1 of scaling critical regularity. First this talk introduce the deterministic global well-posedness result of cubic NLS on 4d-torus (T^4) in the critical regime (with H^1 initial data). Second we consider the cubic NLS in the super-critical regime (with H^s data, s=3).
In the talk we will present an interacting particle model on the real line which has a connection with the geometry of Wasserstein space. The model is a natural generalization of the coalescing Brownian motions but now particles can sticky-reflected from each other. The fundamental new feature is that particles carry mass which is aggregated as more and more particles occupy the same position and which determines the diffusivity of the individual particle in inverse proportional way. We are going to present the infinite dimensional SDE with discontinuous coefficients which describes the particle model. We will discuss the existence of a weak solution such an equation, using a finite particle approximation. Also the stationary case will be considered and an invariant measure will be constructed. Joint work with Max von Renesse.
We present a general framework for the optimal design of surface energies on networks. We give sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal microgeometries, and we show that there holds a localization principle which allows to reduce to the periodic setting in the general nonperiodic case. Furthermore we discuss the issue of crystallinity of the homogenized energy densities of spin systems in the periodic setting. This is joint work, in progress, with Andrea Braides and Antonin Chambolle.
We discuss the behavior of singular stochastic PDEs in the presence of boundaries and boundary conditions (Dirichlet, Neumann, Robin). With the theory of regularity structures we derive solution theories for a number of equations, some of which behave exactly as one would naively expect, while some do not, and certain `boundary renormalisations' take place. Joint work with Martin Hairer.
This talk aims to give a brief introduction to certain inverse problems for PDE. We will more specifically discuss three types of inverse problems for elliptic PDE, which are inverse boundary problems and some inverse problems in scattering and spectral theory. This will include a discussion of some more specific research problems, which are the use of monotonicity based methods for Helmholtz type equations, magnetic potentials in inverse scattering and singular potentials in the multi dimensional Borg-Levinson problem.
Specialized functions of bounded deformation are established as an setting for elasticity problems in which deformations are allowed to jump on some $(n-1)$-dimensional set. We are interested in a specific subspace of these functions (in 2d), in which we fix the possible direction of the jump set for each component. In this setting we are able to provide different Korn-Poincaré-type estimates, independent of the size of the jump set. We apply these estimates to approximate such functions with functions that are smooth up to a jump set of finitely many segments and that satisfy the same directionally constraint.
In the talk, I will give an introduction to my research based on a work by Bella, Fehrman, and Otto on stochastic homogenization. Consider the random differential operator $\nabla\cdot a\nabla $ where the random matrix (coefficient field) $a$ is assumed to be stationary and ergodic. By making use of the extended correctors $(\phi,\sigma)$ and choosing a reasonable homogenization error, they can obtain a regularity estimate, namely the excess decay, which implies a Liouville principle for $a$-harmonic functions, i.e. functions satisfying $\nabla \cdot a \nabla u=0$. It is interesting to know whether their ideas work in the discrete case (the random conductance model on the lattice). The answer is positive: By using several analytic and numerical methods, it is possible to implement their ideas in the discrete case.
The iterated-integral signature from rough path theory is a powerful representation of the shape of a path in space. Recurrent neural networks are among the machine learning algorithms of choice when you want to learn a function whose inputs are sequences. Enabling them to notice patterns in a long sequence of input requires tweaks, the most popular of which is Long Short Term Memory,(Schmidhuber & Hochreiter 1997), which can be slow to learn but is very effective. I'll introduce these two tools, and explain why it might be a good idea to use the signature as an extended way to form a memory in a network, showing some results on how well this works.
During this talk we will consider the waves equation in highly inhomogeneous media. More precisely we will be interested in the homogeneization theory that is derive a space independent equation which is a good approximation of the initial waves equation when the scale of the inhomogeneity, namely $\epsilon $, goes to infinity.It is a classical and well-known result that homogeneization process holds on time sales of order T. Moreover, up to the appearance of some dispersive phenomenon this result can be extended to time scales $\epsilon^{-2}$T. This analysis is based on Blocs waves that diagonalize the spatial part in the d'Alembertien.Unfortunatly, Blocs waves theory is restricted to the periodic framework. However, independently of the periodicity assomption, a truncaded version of these waves can be defined, the so-called Taylor-Bloch waves. These waves "almost" diagonalize the spatial operator. This construction is based upon the introduction of a new set of extended correctors.Thanks to Taylor-Bloch waves we will extend the homogeneization result to polynomial time scales.
I will talk about the existence of martingale solutions to the stochastic heat equation in a Riemannian manifold by using suitable Dirichlet form on the Riemannian path and loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.
I will give a gentle introduction to Martin Hairer's theory of regularity structures which a focus on how "renormalization" appears there. Then I will describe the mathematical challenges involved in developing making renormalization systematic in this theory, here I will focus on how one obtains stochastic estimates which are fundamental inputs to the local existence theorems proved by the theory.
This talk will introduce a pathwise well-posedness theory for a stochastic porous medium equation with nonlinear, conservative noise. Since the noise will be described by a rough path, taking for instance a generic sample path of fractional Brownian motion, such equations are not classically well-posed. Furthermore, owing to the nonlinear structure, it is not possible to make sense of the equation through a simple transformation. Therefore, we will instead pass to the kinetic formulation which, for smooth driving paths, yields an equivalent linear equation. On this level, the noise can be understood in terms of the underlying characteristics, which are described by a well-posed and stable system of rough differential equations. A meaningful interpretation of the original equation can then be obtained by propagating test functions along the corresponding inverse characteristics. Time permitting, applications of these methods to fast diffusion and porous medium equations with multiplicative noise will also be discussed.
In this talk I give an overview of new existence and regularity results for functionals of linear growth which either depend on the full gradient (leading to the space BV) or specific combinations of derivatives (leading to the spaces BVA). Since in general there are no Korn--type inequalities available for p=1 - which is usually referred to as Ornstein-type Non-inequalities - the latter spaces are genuinely strictly larger than BV and encompass, e.g., the space BD of functions of bounded deformation which plays an important role in perfect plasticity.
I will present a general introduction to some variational approaches for the analysis of line defects in crystals, the dislocations. In its generality the problem can be very complex. These defects may be relevant at different scales (in space and time) and interact creating complicated structures and patterns. The aim of the talk is to provide an overview of some of the models that can be mathematically treated, pointing some of the most interesting features of them (as for instance the formation of microstructure), as well as some of the main difficulties and open problems.
We consider homogenization problems for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. The main result is for a single path and with a Hamiltonian that is smooth, convex, and positively homogenous. For such equations, we discuss some recent methods for obtaining well-posedness and stability estimates. We also address equations involving multiple driving signals, and show that homogenization or blow-up can occur.
In this talk we consider the Muskat problem for the incompressible porous media equation. By this we mean a special initial value problem, where the density is given by two different constants separated by a curve. Recent results of Castro, Cordoba and Faraco showed the existence of infinitely many weak solutions, whenever the initial curve is in H^5(R). In particular, these weak solutions show a mixing behavior in a zone around the curve. I will show an alternative proof of this result. The main difference is to use a piecewise constant density. Therefore, instead of solving a nonlinear equation for the curve, it will be sufficient to make a power series ansatz up to order two. Moreover I will show that the necessary regularity of the initial curve is the same as before. If time allows, I give some remarks about an analogous ansatz for the Euler equations.
The numerical simulation of many-body quantum nuclear dynamics has been a long standing scientific challenge. Semiclassical methods, like ab initio molecular dynamics based on Born-Oppenheimer approximation, provide practical strategies to overcome the curse of dimensionality. The Born-Oppenheimer approximation breaks down however in the non-adiabatic regime. In this talk, we discuss some recent mathematical understanding and development of stochastic semiclassical methods for non-adiabatic quantum dynamics. In particular, we will present a rigorous surface hopping algorithm in the spirit of fewest switches surface hopping, based on a novel path integral representation of the semiclassical Schroedinger equations.
We study the nonlocal energy \begin{align*} \mathcal{E}(\Omega) = \mathcal{P}(\Omega) + \int_{\mathbb{R}^n} \left|\nabla \Phi_\Omega\right|^2 \text{\rm d} x \end{align*} where $\mathcal{P}(\Omega)$ denotes the perimeter of a set $\Omega \subset \mathbb{R}^n$ with prescribed volume $|\Omega|=V >0$ and $\Phi_\Omega \in \dot H^1(\mathbb{R}^n)$ is the weak solution of $\Delta \Phi_\Omega = \partial_1 \chi_\Omega$. This energy is related to the study of long and slender "needle" domains in uniaxial ferromagnetic materials where the second term models dipole interactions. We prove existence of minimizers and show that they are ($\mathcal{L}^n$ equivalent to) bounded, connected sets with smooth boundary. In particular we show that $\nabla \Phi_\Omega \in L^\infty$ for local minimizers. For the physically important case $n=3$, we furthermore establish a scaling law for the minimal energy in terms of the prescribed mass $V$.
For a geometrically linear variational model for a shape memory alloy undergoing cubic-to-tetragonal transformations involving a small non-dimensional parameter we prove a rigidity theorem for generic sequences whose energy is on the order of well-known branching constructions of habit planes. Without assuming any further regularity of their limits, we recover rank-one connectedness of the average strains at macroscopic interfaces between mixtures of martensite variants. The proof proceeds via a non-convex differential inclusion for the limit and exploits a balance of the inclusion and “discontinuity” of the strain: If the strain is less regular, the differential inclusion provides more information.
This talk will be concerned with the notion of heat kernel and Brownian motion on compact Lie groups and their relation with the Yang-Mills measure in two dimensions. We shall review some open problems and recent advances in their study, as the dimension of the compact Lie groups goes to infinity. Some of the latter are part of a joint work with J. Norris.
(Joint work with Harsha Hutridurga (Imperial College London) and Jeffery Rauch (University of Michigan). We consider the problem of homogenising convection dominated parabolic equations with rapidly oscillating locally periodic coefficients. Here the drift appears as (1/eps)b(x,x/eps). In the case that the drift vector field b(x,y) is purely periodic, i.e. b(x,y)=b(y), and of zero mean in y, homogenisation of such problems is classical and well understood, for example through the methods of asymptotic expansions and two-scale convergence. More recently, this has been extended to the case where b is purely periodic but not of zero mean. The general case, where b(x,y) is locally periodic, has remained open. We give a partial answer to this question, that under assumptions on the coefficients, the equation homogenises in the Lagrangian frame given by the flow associated to the mean of the drift vector field b. To achieve this we introduce a new notion of two-scale convergence along flows. Time permitting, I will discuss work in preparation on how this new notion of convergence can be applied to the Friedlin-Wentzell averaging principle in 2d Hamiltonian flows.
The regularity structures introduced by Martin Hairer allow us to describe the solution of a singular SPDEs by a Taylor expansion with new monomials. We present the two Hopf Algebras used in this theory for defining the structure group and the renormalisation group. We will point out the importance of recursive formulae with twisted antipodes. This is a joint work with Martin Hairer and Lorenzo Zambotti.
We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities in measure solutions to linear PDE’s and of the corresponding generalized convexity classes. (with G. De Philippis, F. Rindler)
We will present L∞-estimates for solutions of Stochastic PDEs (SPDEs) of parabolic type obtained by techniques motivated by the works of De Giorgi and Moser in the deterministic setting. The global estimates will then be applied in order to prove solvability for a class of semilinear SPDEs, while the local estimates will be used in order to obtain a weak Harnack-type inequality for solutions of linear equations, which is in turn used to deduce information about the oscillation of the solutions. The results are from joint work with Mate Gerencser (IST, Austria).
In this talk we will be interested in the homogenization of the oscillating Dirichlet problem for linear elliptic systems with periodic coefficients. Curiosity about this problem finds roots within the study of the homogenization of the Dirichlet problem with non-oscillating boundary data; Here, there is a boundary layer phenomenon as the oscillating term in the two-scale expansion induces strong gradients at the boundary. As is described by Allaire and Amar, in order to refine the two-scale expansion to yield the expected asymptotic estimates up to the boundary, it is necessary to study the homogenization of the oscillating Dirichlet problem. To treat the oscillating Dirichlet problem one tries to approximate the solution by functions solving Dirichlet problems on half- planes which approximate the boundary. This procedure is very sensitive to the geometry of the domain as the approximating hyperplanes, depending on their normal, may ruin the periodicity of the coefficients. Our main goal will be to discuss some recent results of Armstrong, Kuusi, Mourrat, and Prange that are contained within their paper, ``Quantitative Analysis of Boundary Layers in Periodic Homogenization.'' In the case of smooth, bounded, uniformly convex domains and with a smoothness assumption on the coefficients and the boundary data, for $p \in \left[2, \infty \right)$ they obtain $L^p$-convergence rates for the homogenization of the oscillating Dirichlet problem for $d \geq 2$ that are optimal for $d\geq3$ and also obtain results on the regularity of the homogenized boundary data.
Motivated from a biological model, we consider the problem of minimising Willmore's curvature energy in the class of connected surfaces with prescribed area embedded into a bounded domain. From a computational point of view, it may be favourable to approximate the curvature energy by a phase field model. Diffuse Interfaces, however, can easily separate into multiple components along a gradient flow evolution. This is overcome using a topological penalty term in the functional involving a geodesic distance function. I will present a proof of Gamma-convergence to the sharp interface limit in two and three dimensions and numerical evidence of the effectiveness of our method in two dimensions. Time permitting, I will also demonstrate that it is not possible to get a sharper control of the topology of a surface (i.e. genus) even in the sharp interface limit.
Ever since the seminal work of Bethuel-Brezis-Helein(BBH) on a simplified 2d Ginzburg-Landau system, a very active field in nonlinear analysis and mathematical physics that connects the theory of harmonic maps, energy renormalization and hard-analysis concentration estimates has been developed. However, up to this date, the beautiful description of vortex configurations achieved in the 2d setting is yet to find an exact analog in higher dimensions; this is due to a weaker characterization of vortex defects and the delicate interplay between the geometry of these codimension 2. In this talk I will present a framework that allows to capture an "effective interaction energy" of nearly parallel vortex filaments in certain 3d settings: this provides a next order or "renormalization" of the energy of clustering-filaments configurations together with a more accurate description of the vortex region in the spirit of the BBH study in 2d. This is joint work with Robert Jerrard.
We consider the convex lift of Mumford-Shah type functionals in the space of rectifiable currents and we prove a generalized coarea formula in dimension one, for finite linear combinations of SBV graphs. Then we use this result to prove the equivalence between the minimum problems for the Mumford-Shah functional and the lifted one and, as a consequence, we obtain a weak existence result for calibrations in one space dimension.
We consider rough differential equations with a possibly unbounded drift term. It turns out that the usual one-sided growth conditions are not sufficient to guarantee non-explosion of the solution in finite time. We then provide a further condition under which non-explosion can be assured. Some applications in stochastic analysis are discussed.
Joint work with M. Scheutzow (Berlin).
The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flows, there are no rigorous results confirming or infirming those predictions. Here, we consider a well-known simplified model for 3D turbulence: Burgulence, or turbulence for the 1D or multi-dimensional potential Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum.
We show an optimal upper bound for the transition probability of a tagged particle in the symmetric exclusion process. This provides a first step towards the proof of a quenched invariance principle for the walk of the tagged particle. This talk is based on a joint work with Yu Gu and Jean-Christophe Mourrat.
We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show a link between Kantorovich duality and the permutation polytope by characterizing the transfer plan in two different ways. We prove that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a characterization for convex modified log sobolev inequality and a class of weak transport-entropy inequalities.
This talk contains results of a join work with N. Gozlan, C. Roberto, P-M. Samson and P. Tetali, and a join work of M. Strzelecki.
We consider a mathematical model for the Rayleigh-Benard convection, the stochastic Swift-Hohenberg equation: \[ \partial_t u = -(1+\partial_x^2)^2u+\varepsilon^2\nu u-u^3+\varepsilon^{\frac{3}{2}}\xi(t,x). \] Near its change of stability, the fluid's motion can be described in a multiscale setting as the product of a slowly varying amplitude equation and a faster periodic wave. After an introduction to the problem in its deterministic setting, we'll review some known stochastic results and see some recent developments in the unbounded space domain setting.
The membrane or bilaplacian model was first introduced in the physics literature to model random interfaces with constant curvature, and studied mathematically for the first time by Sakagawa and Kurt. It is a centered multivariate Gaussian whose covariance is given by the discrete bilaplacian operator on the lattice. It is tempting to think of it as a kin of the discrete Gaussian free field (GFF), and indeed many results can be deduced with the same methods for both, as for example the fluctuations of the maximum in higher dimensions. Moreover, as the GFF, it also can be seen as a generalised Gaussian variable arising as scaling limit of discrete models, for example in the odometer of the divisible sandpile or in height fluctuations of spanning forests. We will discuss some of the interesting features of the model and some open conjectures.
The talk is an overview of the area and should be accessible to everybody.
Based on joint works with Alberto Chiarini, Rajat Subhra Hazra and Wioletta Ruszel.
The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth isometric immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based on the novel observations on the decay properties of the Riemann invariants of the Gauss-Codazzi system. The weighted Riemann invariants are introduced and a comparison principle is applied with properly chosen control functions.
We investigate the mean-field limit for the gradient flow evolution of particle systems with pairwise singular interactions, as the number of particles tends to infinity. Based on a modulated energy method, we prove a new weak-strong stability principle for the limiting equation. This yields a quantitative proof of the mean-field result in dimensions 1 and 2 for a family of interactions for which this problem was still open.
For equilibrium systems pressure has been demonstrated to have a microscopic origin related to momentum of the particles. Through kinetic theory and statistical mechanics a link among macroscopic and microscopic world has been derived. When we leave the particular case of equilibrium a general theory does not exist yet and different nonequilibrium issue can lead to different results. In this talk I will focus on active particles, i.e. particles capable of converting (internal or environmental) energy into directed motion. Through this supply of energy due to internal degrees of freedom entails a non-conservation of mechanical energy and non-elasticity of interaction among particles, assumed condition on kinetic theory for derivation of the pressure on a wall. Therefore, we expect that pressure exerted by active particles could present new and stimulating features, due to the nonequilibrium nature of the microscopic model.
In this talk we aim at quantifying the maximal distance between a given time series and a minimal model, which is given in terms of a scalar SDE driven by additive Lévy noise. For this purpose we introduce a metric on the formal space of Lévy measures, which is essentially based on a family of weighted Wasserstein distances of suitably renormalized tails of the Levy measures. Due to its optimal coupling property, we call this distance a coupling distance. In the main theorem of this work we prove an estimate between the laws in path space of two Lévy diffusions driven by two Lévy processes in terms of the coefficients of the equations and the coupling distance of their Lévy triplets. These results are applied in the sequel in order to show that the law of certain paleoclimate data is close in path space to the law of an asymmetric heavy-tailed Lévy diffusion with polynomial tails for a certain index, which is of interest in the climate sciences.
In this talk we consider a model for a nucleon interacting with the $\sigma$ and $\omega$ mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics nonrelativistic limit where is described by a nonlinear Schrödinger-type equation with a mass which depends on the solution itself.After discussing some previous results on the existence of positive solutions, I will prove the uniqueness and non-degeneracy of these ones. As an application, I will construct solutions to the relativistic $\sigma$ and $\omega$ model, which consists of one Dirac equation coupled to two Klein-Gordon equations.The talk is based on joint works with M.J. Esteban, L. Le Treust and Mathieu Lewin.
In this talk, we will discuss the continuous version of a variational problem proposed by Giuliani, Lebowitz and Lieb. The model has a perimeter term and a non-local term, which are in competition. We compute the $\Gamma$-limit as the parameter $J$ converges to its critical value $J_c$. The $\Gamma$-limit will turn out to be finite only on stripes. By using the convergence of the minimizers, we show that minimizers converges to periodic stripes. This work is in collaboration with Michael Goldman.
We present a convergence result for solutions of the vector-valued Allen-Cahn equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit.
This is joint work with Thilo Simon (MPI MIS).
I will consider the motion by curvature of a network of curves in the Euclidean plane and I will discuss existence and asymptotic behavior of the evolution presenting some old, new and possibly "incoming" results. The project is in collaboration with Carlo Mantegazza, Matteo Novaga and Felix Schulze.
A key challenge in the theory of inverse problems is the derivation of classical methods of regularization from Bayesian formulations. We consider this issue in the context of recovering binary one dimensional functions; think, for example, of the problem of barcode reconstruction from the blurred and noisy signal which comes from the scanner.
This is joint work with Andrew Stuart (University of Warwick) and Viet Ha Hoang (NTU Singapore).
The main ideas of Inverse Problems, Bayesian Method and other concepts would be explained along with the current state of research.
This is an easy introductory talk to the mathematics of active biosystems. Before jumping into mathematics we present experimental videos of striking biological phenomena that motivate our study. The talk consists of three parts. First we give a brief introduction to homogenization theory which is a powerful mathematical tool for studying multiscale problems. Next we describe a PDE model of collective behavior of swimming bacteria and present mathematical results on effective viscosity of bacterial suspensions. Finally we present a phase field PDE model of a crawling cell on a substrate and describe its surprising features such as self-sustained motion (traveling wave solutions) and spontaneous breaking of symmetry. In conclusion, we briefly describe the concept of interdisciplinary mathematics.
The classical Boltzmann equation provides a mesoscopic description of a large number of identical, interacting particles. Namely, it is an evolution equation for the particle density which accounts for both free transport and binary collisions. In this talk, we discuss a variant of the classical model in which particles undergo a stochastic transport mechanism in between collisions. This may be interpreted as accounting for an external random field, or background noise acting on the particles. We focus in particular on the well posedness of the mesocopic, stochastic PDE description of the system. Our main result is the global existence of so-called renormalized, weak martingale solutions. This is a joint work with Sam Punshon-Smith (UMD-College Park).
In this talk we introduce briefly the problem of regularity for harmonic maps and present the new regularity results proved for the singular sets of minimizing and stationary harmonic maps in collaboration with Aaron Naber (see arXiv:1504.02043).
We prove that the singular set of a minimizing harmonic map is rectifiable with effective n-3 volume estimates. The results are based on an improved quantitative stratification technique, which consists in a detailed analysis of the symmetries and almost symmetries of the map u and its blow-ups at different scales, and rely on a new $W^{1,p}$ version of Reifenberg's topological disk theorem. The application of this theorem in the situation of harmonic maps hinges on the monotonicity formula for the normalized energy.
Motivated by the study of a micromagnetic enery, Jabin, Otto and Perthame proved in a 2002's paper that in dimension 2, we can select curl-free unit-lenght vector field through a kinetic formulation. Roughly speaking, this kinetic formulation excludes curl-free unit-lenght vector field which are singular along $\mathcal{H}^1$-set to keep only solutions with point-like singularities. We will give a natural generalization to this kinetic formulation in any dimension and show that if the dimension is strictly greater than 2, it becomes much more rigid as it characterizes constant vector field or vortices, that is $\pm\frac{x-P}{|x-P|}$; if we have enough time, we will show how this is related with totally umbilical surfaces.
I will recall what the Plateau's problem is and summarize the state of the art. I will present a compactness principle valid under fairly general assumptions on the class of competitors, which can be applied to solve different formulations of the Plateau's problem. The approach is the minimization of the d-Hausdorff measure among a family of d-rectifiable closed subsets of $R^n$. Such class is then specified to give meaning to boundary conditions. The obtained minimizers are regular up to a set of dimension less than $d-1$. I will also present a variant of this compactness principle for the minimization of more general energies, generated by elliptic integrands (in the sense of Almgren).
We study the existence of sets that minimize their measure amongst a family stable under homotopy in a compact, boundaryless Riemannian manifold. We consider the problem in the category of sets without resorting to the weakened notions of Federer (currents) or Almgren (varifolds). This setting allows to free oneself of additional regularity assumptions such as orientability, or even rectifiability of the competitors. In this talk, I will try to explain how a polyhedral approximation process inspired by Federer can be generalized to this setup and make up for the lack of compactness of the set-based approach.
The investigation of stability and instability properties for nonlinear elliptic PDEs has been the target of intense investigation in the last decades, the most celebrated results being probably the compactness and noncompactness results for the set of solutions of the Yamabe equation. Instability results are usually obtained through a constructive method, called a finite-dimensional reduction method. In this talk, to investigate non-compactness issues for a highly coupled system of elliptic PDEs where the usual methods are ineffective, I will describe a new finite-dimensional reduction method which merges the traditional energy approach with asymptotic analysis techniques.
In this talk I discuss a robust, new approach of proving (almost) optimal regularity for the (variable coefficient) thin obstacle problem. The central tool here consists of a Carleman inequality. This allows to control the vanishing order of solutions to the problem and to deduce compactness of blow-up solutions also in the presence of metrics with low regularity. This is joint work with Herbert Koch and Wenhui Shi.
We are interested in the large-scale regularity of random linear elliptic operators on the half-plane with homogeneous Dirichlet boundary conditions. This is expressed in a first order Liouville theorem, which is obtained through means of a corresponding excess-decay. The main difficulty is the construction of a sublinear solution to an appropriate corrector-problem.
I will give an overview of my works on the pathwise investigation of SPDEs. On the one hand they include rough path considerations for (semi)linear equations with finite dimensional driving noise. On the other hand they cover singular equations on manifold using techniques from the theories of paracontrolled distributions and regularity structures. In the end I will give an overview of current and future research projects.
Due to sensitivity with respect to initial data and parameters, individual solutions of the basic equations of fluid mechanics are unpredictable and seemingly chaotic. However, some of their statistical properties are robust. As early as the 19th century it was conjectured that turbulent flow cannot be solely described by deterministic methods, and indicated that a stochastic framework should be used. In this framework, invariant measures of the stochastic equations of fluid dynamics presumably contain the statistics posited by the basic theories of turbulence.
In this talk we investigate properties of invariant measures for the Boussinesq equations in the presence of a degenerate random forcing acting only in the temperature component. The main goal is to prove convergence of invariant measures in singular limits when Prandtl numbers approach infinity. As an application we recover estimates for the Nusselt number in stochastic setting.
More precisely, we show a general framework for converting the problem of convergence of measures to the question of finite time convergence of solutions. Then we analyze singular limit problems in a stochastic setting.
This is a joint work with S. Friedlander (U. of Southern California), N. Glatt-Holtz (Virginia Tech), G. Richards (U. of Rochester), and E. Thomann (Oregon State).
We study a symmetric diffusion in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched invariance principle and a quenched local central limit theorem for such a diffusion, under some moment conditions on the environment; the key tools are a local maximal inequality and a local parabolic Harnack inequality obtained with Moser’s iteration technique. As a further application of Moser’s iteration scheme, we study a continuous-time random walk on the lattice in an environment of time-dependent random conductances. We assume that the law of the conductances is ergodic with respect to space-time shifts. Also in this case, we prove a quenched invariance principle under some moment conditions on the environment.
My talk will be divided in two parts. In the first part I will present a short (and far complete) overview of the general existence-uniqueness-stability theory of hyperbolic systems of conservation laws in one space dimension, with particular emphasis on the role of the so-called "interaction estimates" and "interaction potentials" in the development of such theory. In the second part of the talk I will focus on one of these interaction estimates, recently proved in a joint work with S. Bianchini, and I will provide a sketch of the proof of such estimate in a simplified setting, in order to make it accessible also to people who are not experts in this field.
This talk consists of two parts. In the first part, I will give a very brief introduction to the model of random interlacements. In the second part, I will investigate the asymptotic behaviour of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk on Z^d (d bigger or equal to 3), and explain how this problem is linked with the model of random interlacements.
A classical way to detect "holes" in a compact manifold M is to check whether every continuous function u from a k-dimensional sphere Sk to M extends to a continuous function from the (k+1)-dimensional ball to M. The obstruction to this is the k-th fundamental group of M. The question I consider in this talk is the natural analogue of this qualitative question when we replace the qualitative property of being a continuous function by the quantitative property of having Lp-integrable gradient. After formulating it, I will describe the following features of this Sobolev extension problem: the p corresponding the conformally invariant energy ||Du||_{Lk(Sk)} is the critical one, providing the most intriguing behaviors extensions will tend to create singularities, a big difference to the C0 casethe setting is complementary to that of VMO functions and to so-called epsilon-regularity results points 2, 3 are symptoms of the fact that our function spaces are "not rectifiable"* uniformization using hyperbolic geometry provides short and elegant extension constructionsThe talk is based on a recent result in collaboration with Jean Van Schaftingen.(*: if time constraints will allow it)
The theory of regularity structure and the notion of paracontrolled distribution introduced respectively (and independently) by M.Hairer and M.Gubinelli, P.Imkeller, N.Perkowski allow to study the well-posedness of a large class of singular parabolic stochastic partial differential equations. The aim of the talk is to discuss the context for which theses new technique can be used and what kind of existence result is obtained by using this two approach and namely we will focus on the spectral theory of Schrödinger operator with white noise potential.
We will consider lattice and continuous-space birth-and-death Markov dynamics. The underlying Markov process is obtained as a unique solution to a certain stochastic integral equation. The existence and uniqueness theorem as well as martingale characterization, the existence of an invariant distribution, recurrence properties and maximal irreducible measure will be discussed. We will also talk about shape results in continuous-space settings.
Motivated by the Schrödinger problem, we study reciprocal classes. Given a continuous time Markov chain, the associated reciprocal class is the convex hull of its bridges. Its elements are in general non Markovian, but rather Markov fields indexed over time.
We propose two different characterisations of it: the first one base on short time asymptotic and the second one based on Integration by parts(duality formulae). A key notion behind both approaches is that of reciprocal characteristics which, in some sense, replaces that of the generator for the original chain.
If time allows, the relation between global bounds on the characteristics and the fluctuations of a bridge are discussed.
For some differential equations the addition of a carefully chosen, random noise term can produce a regularizing effect (e.g. solutions are more regular, or restored uniqueness).
I will first mention a few easy examples (ODEs) to introduce some of these regularizing mechanisms, then detail two cases where we have regularization for a PDE: the linear transport equation and a kinetic equation with force term. I will present some classical results for these two equations, related to well-posedness and regularity of solutions, that in the stochastic setting can be obtained under weaker hypothesis. These results are based on a careful analysis of the stochastic characteristics and the regularising properties of some associated parabolic/elliptic PDE.
If time allows, I will conclude by introducing a new different strategy of proof based on stochastic exponentials and an associated parabolic PDE, which allows to obtain wellposedness for stochastic PDEs in a class of solutions which are only regular in mean. This will illustrated by the application to the transport equation.
This work is supported by LABEX MILYON / ANR-10-LABX-0070.
We shall first discuss the Hörmander theorem for general $It^o$ processes and related (Kolmogrov) forward/backward stochastic PDEs, which may be beyond the scope of Markovian framework. Furthermore, we would also present the maximum principle for forward stochastic PDEs under a Hörmander type condition, which states the $L^p$ ($p\geq 2$) estimates for the time-space uniform norm of weak solutions.
The dynamics of slightly undercooled crystal-melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment-detachment limited kinetics finds expression through a certain singularly perturbed, hyperbolic-parabolic, geometric partial differential equation.Among its solutions, we discover a remarkable family of 1D convex- and concave- translating fronts whose fixed asymptotic angles deviate from the thermodynamically expected Wulff angles in direct proportion to the degree of undercooling: a non equilibrium (thermokinetic) effect.We also present a novel geometric matched-asymptotic analysis that demonstrates that the slow evolution of the large-scale features of 1D solutions $\mathcal{I}$ are captured by a Wulff-faceted interface $\mathcal{A}$ evolving under an intrinsic facet dynamics. This emergent dynamics possesses a Peclet length $L_\text{p}$ below which a spatio-temporal symmetry of parabolic type appears. We thereby theoretically predict, and numerically verify, that within the sub-Peclet regime the universal scaling law $\mathcal{L} \sim t^{1/2} $ governs the time $t$ evolution of the characteristic length $\mathcal{L}$ of the interface $\mathcal{I}$.Related Article: Stephen J. Watson, "Emergent Parabolic Scaling of Nano-Faceting Crystal Growth", Proceedings of the Royal Society A, Vol. 471 (Issue 2174) , DOI: 10.1098/rspa.2014.0560
What is the best way to enclose and separate N regions of volume 1 in the plane with the minimum amount of perimeter?
The solutions to this particular problem are called "minimizers planar N-clusters" and, so far, their structure is still mostly unknown except for the case N=1,2,3. In 2001 Thomas Hales proved the so-called '' hexagonal honeycomb conjecture '': the regular hexagonal tiling provides the partition of the plane in equal-area chambers having the minimum amount of localized perimeter. His result somehow provides an answer for the case N=infinity and it turns out to be a powerful instrument for the study of the asymptotic behavior (in N) of minimizing planar N-clusters. After a brief overview of the problem and of the main open questions I will discuss the consequence of Hales's result in the study of such objects. In the second part of the seminar I will give a sketch proof of a result obtained in collaboration with prof. Alberti about the asymptotic behavior of some classes of minimizing planar N-cluster from an energetic point of view.
The tangential touch problem in elliptic theory consists of exposing the dynamics of the free boundary near the fixed boundary in obstacle problems. The solution of this problem is discussed for fully nonlinear elliptic operators in two dimensions -- based on joint work with Andreas Minne.
We study a random conductance problem on a d-dimensional discrete torus of size L>0. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance AL of the network is a random variable, depending on L, and the main result is a quantitative central limit theorem for this quantity as L→∞. In terms of scalings we prove that this nonlinear nonlocal function AL essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of AL.
Joint work with J. Nolen
An important example of an interacting particle system is the exclusion process. It consists of particles performing simple random walks on Z^d, with the rule that any attempted jump of one particle onto another is suppressed.
The talk will start with a more precise definition and construction of this process, and a derivation of some of its basic properties. Next, we will paint one of the particles red and follow its trajectory. It is known that the trajectory of this red particle "homogenizes", in the sense that it looks like a Brownian motion over large scales. We will discuss quantitative versions of this statement, which are for the most part open questions.
An interacting particle system on the real line which is a generalization of the Arratia flow will be considered. We will suppose that particles start from all points of a finite interval, move independently up to the moment of the collision then coalesce and move together. Every particle has a mass influencing on the diffusion and obeying the mass conservation. We will discuss construction of the stochastic flow describing the evolution of such a system as a thermodynamic limit. Also, uniqueness of distribution of such a flow as a solution of a martingale problem, properties of filtration generated by the flow, the large deviation principle and connection with the Wasserstein diffusion will be presented.
We show the existence of the growth exponent for loop-erased random walk (LERW) in three dimensions. We also present exponential tail bounds for the number of steps of the LERW in three dimensions. In the talk, we present the sketch of the proof via non-intersecting random walks.
In the first half of this talk a new approach to construct solutions to the two dimensional stochastic quantisation equation in the whole plane will be discussed. The quantisation equation is a non-linear stochastic PDE that can only be interpreted in a ``renormalised'' sense. Following an idea by Da Prato and Debussche we reduce the problem to solving a PDE with random coefficients. We then show existence and uniqueness for this equation in a suitable Besov space with weights. In the second half I will discuss how some ``infinite constants'' that appear in the renormalisation procedure disappear on the level of Freidlin-Wentzell type large deviations.
We consider an inviscid, incompressible fluid whose vorticity is concentrated along a smooth curve, the "vortex filament". Assuming that the fluid motion is described by the Euler equation, one may ask if there is a geometric law that describes the motion of the vortex filament. This question was answered in 1906 by da Rios, who formally showed that to leading order and modulo a rescaling in time, the curve evolves by an equation known as the binormal curvature flow.
In joint work with Bob Jerrard, we rediscover da Rios' result by means of rigorous arguments and under quite weak regularity assumtions on the vorticity.
First-passage percolation is a random growth model on the cubic lattice Z^d. It models, for example, the spread of fluid in a random porous medium. Quantitatively describing the ``average time'' required for the fluid to percolate through the medium ---known as the time-constant of first-passage percolation--- is a classical, but unsolved problem in probability. We view first-passage percolation as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman equation, and derive an exact variational formula for the time-constant. The random fluctuations of the model are (conjecturally) universal; we will present some new results in this direction.
We prove a quenched large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster on Zd, d ≥ 2. We take the point of view of the moving particle and first prove a quenched LDP for the distribution of the pair empirical measures of the environment Markov chain. The classical LDP for the distribution of the mean velocity of the random walk drops out easily via the contraction principle and both rate functions admit variational formulas.
Our results are based on invoking ergodicity arguments in this non-elliptic set up to control the growth of gradient functions (correctors) which come up naturally via convex variational analysis in the context of homogenization of random Hamilton-Jacobi Bellman equations.
This is joint work with Noam Berger (Munich).
Various discretizations of complex analysis have been actively studied since 1920s because of applications to numerical analysis, statistical physics, and inegrable systems. This talk concerns complex analysis on quadrilateral lattices tracing back to the works of J. Ferrand. We solve a problem of S.K. Smirnov on convergence of discrete harmonic functions on planar nonrhombic lattices to their continuous counterparts under lattice refinement. This generalizes the results of R.Courant-K.Friedrichs-H.Lewy, L.Lusternik,D.S.Chelkak-S.K.Smirnov, P.G.Ciarlet-P.-A.Raviart. We also prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts (this is a joint work with A.I. Bobenko). The proofs are based on energy estimates inspired by electrical network theory.
The classical Cauchy-Lipschitz theorem shows existence and uniqueness of the flow of any sufficiently smooth vector field in R^d. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption on the vector field which prevents the trajectories from blowing up in finite time; in particular, it does not apply to fast-growing, smooth vector fields.
In this seminar we give an overview of the topic and we introduce a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show existence and uniqueness under only local assumptions on the vector field and we apply the result to a kinetic equation, the Vlasov-Poisson system, where we describe the solutions as transported by a suitable flow in the phase space. This allows, in turn, to prove existence of weak solutions for general initial data.
In nematic liquid crystals, rod-like molecules tend to align. Locally this alignment can exhibit uniaxial symmetry, or be biaxial. In certain regimes, a "biaxial escape" phenomenon is predicted. We show that the uniaxial symmetry constraint is very rigid independently of the regime: biaxial escape has to be more than the mere breaking of this symmetry. Then we present a rigorous interpretation of biaxial escape at low temperature (joint with Andres Contreras).
We investigate the percolation model obtained by considering level sets of the Gaussian free field on the d-dimensional lattice above a given height h. It has recently been proved that, as h varies, this model exhibits a non-trivial percolation phase transition in all dimensions d greater or equal to 3. We show that the associated critical density behaves like $1/d^{1 + o(1)}$ as d goes to infinity. The proof gives the (principal) asymptotics of the corresponding critical height $h_*(d)$. Moreover, it shows that a related parameter $h_{**}(d)$, which characterizes a strongly subcritical regime, is in fact asymptotically equivalent to $h_*(d)$.
We discuss the dynamical sine-Gordon equation in two space dimension with parameter $\beta$. Our main result is that when $\beta^2 < 16\pi /3$, the Wick renormalised equation is locally well-posed. We aim to use this particular equation as an example to illustrate the general ideas in the theory of regularity structures recently developed by Martin Hairer. A main feature of this model is that when $\beta^2 < 4\pi$, previous solution theory applies, while for $\beta^2 \in [4\pi, 16\pi /3)$, one has to apply the theory of regularity structures. This is joint work with Martin Hairer.
Curvature driven flows arise in a number of models from materials science, physics, and biology. PDE's can be used to model these flows, but they tend to be non-linear and somewhat complicated. In the early 90's, Bence-Merriman-Osher (BMO) developed a very simple "thresholding algorithm" for producing mean curvature motion for a hypersurface. Their ideas have been extended over the years to mean curvature motion for filaments in space, and fourth order flows such as Willmore flow and Surface Diffusion. In this talk I will discuss the thresholding algorithms for mean curvature and Willmore flows. In particular I will detail a new proof of the convergence of the original BMO algorithm to mean curvature flow, which does not use the maximum principle. This new technique seems promising for proving the convergence of the algorithm for filament motion, which involves vector valued functions.
We investigate perturbations of traveling wave solutions to a thin-film equation with quadratic mobility and a zero contact angle condition at the triple junction between the three phases liquid, gas, and solid. This equation can be derived from the Navier Stokes system of a liquid droplet with a Navier-slip condition at the liquid-solid interface. Existence and uniqueness have been established joint with Giacomelli, Knüpfer, and Otto. As solutions are generically non-smooth, the approach relied on suitably subtracting the leading- order singular expansion at the free boundary. In this talk, we present a substantial improvement of this result by showing the regularizing effect of the degenerate-parabolic equation to arbitrary orders of the singular expansion. This result turns out to be natural in view of the properties of the source-type self-similar profile.
We study the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of a priori estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives explicit formulas for each derivative. In particular, the first term yields the celebrated (electric and elastic) Clausius-Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first rigorous proof of these formulas in the case of random inclusions.
In this seminar I would like to review and discuss Kirszbraun's classical theorem on the extension of Lipschitz functions and mention some recent, related results in this context. In particular, in the final part of this talk, I will present an example showing the non-existence of a Kirszbraun-type theorem in the case of Almgren's multiple valued functions.
In this seminar I will discuss the notion of isometric maps, i.e. maps preserving distances between points of metric spaces. Depending on the metric and on the regularity required, isometric maps behave very differently. This talk is devoted to the introduction to the cases of spaces with the same dimension and to the exposition of some recent results on the generic behavior of isometric maps.
Shadowing theory study properties of approximate trajectories: sequences of points, where next one is close to image of the previous one. Such sequnces naturally appears, for example, in the presence of noise or as a result of numerical simulation.
We introduce notions of shadowing, hyperbolicity and structural stablity and briefly discuss relation between them.
In the talk we concentrate on random finite length pseudotrajectories (and trajectories). where main questions will have probabilistic nature. We formulate several questions and conjectires. We consider model ``skew-product'' example which can be reduced to a 1-dimensional random walk. This simple example, however allows us to disprove one the long-posed conjecture
A variety of semiconductor oxide systems and their morphology will be discussed. Epitaxial layers deposited at low pressure show vicinal surfaces and layer-by-layer growth. At higher pressures, nanowire forest from with complicated dependence on growth conditions. Applications of such structures reach from high mobility transistors, magneto-tunneling junctions to coherent nano light sources and fingerprint sensors.
Shadowing theory study properties of approximate trajectories: sequences of points, where next one is close to image of the previous one. Such sequnces naturally appears, for example, in the presence of noise or as a result of numerical simulation.
We introduce notions of shadowing, hyperbolicity and structural stablity and briefly discuss relation between them.
In the talk we concentrate on random finite length pseudotrajectories (and trajectories) where main questions will have probabilistic nature. We formulate several questions and conjectires. We consider model "skew-product" example which can be reduced to a 1-dimensional random walk. This simple example, however allows us to disprove one the long-posed conjecture.
In this tallk I will present the notion of calibration, with some examples, and show its link with Area Minimizing Currents. Time permitting, I will address some regularity issues for these types of geometries. The talk will be introductory and kept as elementary as possible.
The question of finding a right formulation for conservation laws which has a good physics meaning and a convenient shape for doing mathematics remains an open one, linked with the 6th problem of Hilbert. We will first discuss an entropy definition and then a kinetic one that appear to be equivalent, focusing on the Burgers equation and on a recent application.
In Materials Science, dislocations are 1D defects in the periodic structure of crystals whose presence (and motion) is believed to be the main mechanism of plastic deformations in metals. we consider a 2D model in which dislocations are introduced as point topological defects of the strain field. We draw a parallel between them and the vortices in superconductors, studied within the Ginzburg-Landau framework, pointing out similarities and differences between these two kinds of singularities. Finally, we present some recent results relative to the statics and dynamics of screw dislocations using a purely variational approach based on Gamma-convergence and on the minimizing movements method. The results are obtained in collaboration with Roberto Alicandro, Adriana Garroni and Marcello Ponsiglione.
We will briefly consider the mean-field limit of systems of N bosons. Afterwards, we will introduce the Fock space representation and speak about the case when, instead of bosons, we have systems of N fermions. In particular, we will observe that, given certain initial data and regularity of the interaction potential, a system of many fermions will display Hartree-Fock dynamics in the mean-field regime.
I will present high-resolution numerical simulations of statistically steady high Rayleigh number convection in both two-dimensional and three-dimensional porous media, and will discuss in detail the relationship between the strength of convection, as described by the Rayleigh number, and the transfer of buoyancy, as described by the Nusselt number. In addition, I will examine the remarkable dynamical structure of high Rayleigh number porous convection, which takes the form of an ordered columnar exchange flow with a regular (Rayleigh-number dependent) horizontal wavelength. I will investigate the physical control of this structure. I will extend these results to consider the more physically relevant context of convection from below one boundary, and apply the results to the convective dissolution of CO_2 that has been injected into underground brine-saturated porous rock.
Let X be a compact metric space. We shall be concerned with the structure of attractors of continuous self-maps of X, and with questions involving continuity and stability in the setting of attractors.
Hot Brownian motion is the diffusive motion of a colloidal particle which is maintained at higher temperature than its solvent, e.g. by laser light absorption. After a brief review of the stochastic theory of diffusion, I will present its extension, based on fluctuating hydrodynamics, to the case of hot Brownian motion. In particular I will show that the random dynamics of the hot particle is governed by a frequency dependent effective temperature which results from the coupling of the hydrodynamic modes with the temperature inhomogeneities in the fluid. Conceptually, this lays the basis for out-of equilibrium statistical mechanics of Brownian motion under general non-isothermal conditions.
A set in a metric space is said to be porous if each of its points sees nearby holes in the set of radius proportional to their distance away, at arbitrarily small scales. In this talk I will review some properties of the notion of porosity and its applications in Geometric Measure Theory; in particular in the study of the differentiability of Lipschitz functions. No previous knowledge by the audience in Geometric Measure Theory will be assumed.
In this talk we characterize necessary and sufficient conditions on the stored energy density in order to assure weak* lower semicontinuity on the set of bi-Lipschitz functions in the plane. This problem is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of the admissible deformations are key requirements. Generally speaking, the main difficulty in finding such conditions is that the set of bi-Lipschitz functions is non-convex. Thus, standard cut-off techniques that modify the generating sequence to have the same boundary conditions as the limit generally fail; however, the standard proofs of in calculus of variations rely on such methods. We obtain this cut-off by following a strategy inspired by Daneri&Pratelli, i.e. we modify the generating sequence (on a set of gradually vanishing measure near the boundary) first on a one dimensional grid and then rely on bi-Lipschitz extension theorems. We also present method of modifying the sequence on the grid that could be extended to more general classes of mappings. This is joint work with Martin Kruzik (Prague) and Malte Kampschulte (Aachen).
In this talk I will present a recent result of P.L. Lions on the fine qualitative properties of correctors in homogenization of linear elliptic PDEs in quasi-periodic media. The behavior of these correctors is governed by some fine algebraic relations between the periods. This can be seen as one of the simplest instance of the so-called small divisors problem.
We review some aspects of stochastic calculus on graphs: Walsh Brownian motion, Ito's formula, Tsirelson theorem... We then study some examples of SDEs on graphs and their associated stochastic flows.
We consider a nonlinear parabolic equation with spatially distributed (discontinuous) hysteresis. Due to discontinuous nature of hysteresis well-posedness of the problem is not trivial.
Depending on the initial data we distinguish two cases: "transverse" and "non-transverse".
For the transverse case we show a connection of the above problem with free boundary problems. This allows us to find sufficient conditions that guarantee the well-posedness (existence, uniqueness and continuous dependence on initial data), which is generally not typical for systems with discontinuous hysteresis.
For the non-transverse case we consider spatial discretization of the problem, which leads us to a lattice differential equation. In that case we observe a strange "rattling" phenomenon, which leads to non-trivial pattern formation. The main technique for this case is careful estimate of the discrete parabolic Green function.
To a transient symmetric Markov jump process on a network is naturally associated an infinite measure on loops and the Poisson point process of loops of intensity proportional to this measure are sometimes called “loop soups”. We focus on the “loop soup” of parameter 1/2 and construct a coupling between the Poisson ensemble of loops and the Gaussian free field on the network satisfying two constraints. First of all half the square of the free field must be the occupation field of the loops. Besides that the sign of the free field must be constant on clusters of loops. This is an improvement over the relation between the Poisson ensemble of loops and the Gaussian free field obtained by Le Jan, which did not take in account the sign of of the free field. As a consequence of our coupling we deduce that loop clusters at parameter 1/2 do not percolate on periodic lattices.
We study the d-dimensional Cahn-Hilliard equation on the flat torus in the parameter regime in which the system size is large and the mean value is close---but not too close---to -1. In this setting, the uniform state is a local but not global energy minimizer, and we are interested in a quantitative description of the surrounding energy landscape. We derive a sharp leading order estimate of the size of the energy barrier surrounding the uniform state. A sharp interface version of the proof leads to a gamma limit that connects the Cahn Hilliard energy with a heuristic description proposed by Biskup, Chayes, and Kotecky. This work is joint with Michael Gelantalis (RWTH).
Rank-based interacting particles are systems of diffusion processes evolving on the real line with drift and variance coefficients depending only on their rank. For a choice of coefficients describing mean-field interactions, these systems provide a probabilistic representation of the solution to scalar nonlinear parabolic evolution equations through propagation of chaos results. After describing this phenomenon, we shall use this representation to derive convergence to equilibrium of the solution to the evolution equation. Then, we shall address the small noise limit of rank-based interacting particles, in order to obtain an approximation of scalar hyperbolic conservation laws, and finally extend this procedure to hyperbolic systems of conservation laws.
Many basic equations in physiscs can be written in the form of conservation law. However, as it is common in the field of PDEs and SPDEs, classical or strong solutions do not exist in general and, on the other hand, weak solutions are not unique. The notion of kinetic formulation and kinetic solution turns out to be a very convenient tool to overcome these difficulties.
In this talk, I will explain the main ideas on a deterministic model problem and show how this approach can be adapted to stochastic setting. In the remaining time I will present several well-posedness results for the case of hyperbolic, semilinear degenerate parabolic and quasilinear degenerate parabolic stochastic conservation laws.
The talk is based on joint works with Arnaud Debussche and Julien Vovelle.
We consider the Blackstock--Crighton rotational model equation which is a fourth-order in space and third-order in time nonlinear partial differential equation and arises in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups to investigate the linearization of the equation and show that the underlying semigroup is analytic which, together with a negative spectral bound of its generator, leads to exponential decay results for the linear homogeneous equation. Moreover, invoking the Banach fixed-point theorem, we prove local well-posedness of the Blackstock--Crighton model for sufficiently small initial data. Finally, we show how barrier's method is used to obtain global in time well-posedness and provide exponential decay results also for the nonlinear equation.
I will discuss some results on viscosity solutions of fully nonlinear parabolic equations associated to a uniformly elliptic operator. I derive an estimate which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. I will present two applications: I get a partial regularity theorem, and in a joint work with Scott Armstrong, we also show how this can serve to study the rate of convergence for a broad class of schemes.
In this talk, I will present a characterization result, in the spirit of Kinderlehrer & Pedregal, for Young measures generated by gradients of Sobolev maps which additionally satisfy various determinant constraints. Young measures are an important tool in nonlinear analysis as they allow one to characterise limits of nonlinear quantities depending on the sequence generating the Young measure. However, important determinant constraints in elasticity, e.g. the orientation-preserving condition, are non-convex and cannot be accounted for by the proof of Kinderlehrer & Pedregal. In joint work with Filip Rindler and Emil Wiedemann, we overcome this difficulty by employing a variant of convex integration to "correct" the Jacobian and provide generating sequences in Sobolev spaces of exponent less than the dimension where the determinant constraints prove to be soft. Several interesting applications are also presented.
We consider the Blackstock--Crighton rotational model equation which is a fourth-order in space and third-order in time nonlinear partial differential equation and arises in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups to investigate the linearization of the equation and show that the underlying semigroup is analytic which, together with a negative spectral bound of its generator, leads to exponential decay results for the linear homogeneous equation. Moreover, invoking the Banach fixed-point theorem, we prove local well-posedness of the Blackstock--Crighton model for sufficiently small initial data. Finally, we show how barrier's method is used to obtain global in time well-posedness and provide exponential decay results also for the nonlinear equation.
Abstract: Gibbs measures are fundamental quantities to be studied in Statistical Mechanics. We will consider very general local interaction and show that some large deviation principles hold and the corrisponding rate functions have an integral representation. Part of what I will be saying is inspired by a recent result of Roman Kotecký and Stephan Luckhaus.
I investigate a Kac-type many particle model that allows a reference-free description of plastic deformation.
In the framework of the model a solid body is described by a set of particle positions. A lattice is fitted to the particle configuration around each point on a mesoscopic scale. The lattice parameters are used as an argument of a non-linear elasticity energy functional.
There are three main results. First, there is an estimate for the upper bound of the energy barrier of plastic relaxation.
Secondly, there is an estimate for the difference between the fitted lattice parameters of points of low energy density that are sufficiently close to each other. Sequences of these points can be used for homotopy type arguments. In particular, it is possible to identify dislocations as topological defects in this framework.
Furthermore, I use the fitted lattice parameters as local Lagrangian coordinates and bound the energy from below with a functional of these coordinates.
The dynamics of nanomagnetic particles is described by the stochastic Landau-Lifshitz-Gilbert (SLLG) equation.
In the first part of the talk we will discuss the long time behaviour of the finite-dimensional SLLG equation. Firstly, we explain how statistical mechanics argument defines the form of the noise of the equation. Then we will consider different approximations of the equation such as structure preserving discretisation and penalisation approximation. We discuss the convergence of approximations and their consistency with the long time behaviour of the system.
In the second part of the talk we will look at the infinite dimensional case. Firstly, we present a numerical scheme convergent to the solution of SLLG equation. Then we show some numerical results and discuss open problems, such as existence of invariant measure, existence of solution in the case of space-time white noise, etc. In particular we will explain why Krylov-Bogoliubov Theorem is not directly applicable to the proof of existence of invariant measure even in the case of coloured noise. In the end we will present certain transformation of SLLG equation which allows to represent the noise as the sum of additive noise and energy conservative noise. Computational examples will be reported to illustrate the theory.
The talk is based on the recently published book (jointly with L. Banas, Z. Brzezniak, A. Prohl) and on the work in progress of the author.
There will be an introduction to classical (2 marginal) optimal trasport problem in 1-dimension, with the solution in the strictly convex case. Then the multimarginal problem in one dimension for a symmetric, repulsive convex cost function is treated, showing the existence of an optimal map also in this case.
Cattaneo’s law can be regarded as a formal first order Taylor expansion of a delay equation consisting of Fourier’s law of heat conduction in which a delay parameter (phase lag) is added within the time dependence of the heat flux. Similarly, by including two or three delay parameters, the dual- and three-phase-lag models (developed by Tzou and Choudhuri) allow combinations of Taylor approximations of different orders.
After presenting the phase-lag models, the talk will give an overview of well-posedness and stability results for dual- and three-phase-lag equations of heat conduction and thermoelasticity. We will see that the order of Taylor approximation can decide on well-posedness and that conditions on the phase-lags are needed to guarantee stability. Exemplarily, well-posedness and (in)stability will be shown for a particular three-phase-lag heat equation.
We study transition layers between two domains of constant magnetization vectors -- "domain walls" -- in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types.
In the first part of the talk, we present a reduced model for a single domain wall in an infinitely extended film that captures the optimal splitting of the wall into an asymmetric, stray-field free core and logarithmically decaying, symmetric wall tails (obtained via Gamma convergence, joint work with R. Ignat and F. Otto).
In the second part, we consider periodic domain patterns for which the wall width is not only confined by material anisotropy but also neighboring wall tails. Using methods similar to the single-wall case, we derive a reduced model that describes the influence of the tail interaction on the splitting into core and tails. Moreover, it yields a prediction for the increased rotation of the magnetization towards magnetic fields applied along the wall normal direction.
In this talk, I will present some results on convergence of sequences of reversible diffusion processes to a deterministic limit, with a particular emphasis on the large deviations. The method is based on the gradient flow formulation of the PDE satisfied by the flow of marginals of the diffusions. As an application, I will discuss the large deviation principle for the Ginzburg-Landau model with random conductances.
We present a method for the derivation of lower bounds on free boundary propagation for the thin-film equation, one of the most prominent examples of a higher-order degenerate parabolic equation. In particular, we obtain sufficient conditions for instantaneous forward motion of the free boundary, upper bounds on waiting times, as well as lower bounds on asymptotic propagation rates. Our estimates coincide (up to a constant factor) with the previously known reverse bounds and are therefore optimal. To the best of our knowledge, these results constitute the first lower bounds on free boundary propagation for any higher-order degenerate parabolic equation. Our technique is based on new monotonicity formulas for solutions to the thin-film equation which involve weighted entropies with singular weight functions. It turns out that our method is not restricted to the thin-film equation, but also applicable to other higher-order parabolic equations like quantum drift-diffusion equations.
In this talk, we outline a proof of the identification of the gradient flow of the Renyi entropy in the p-Wasserstein space with the flow of the q-heat flow, i.e. the parabolic q-Laplace equation. We work in the purely non-smooth setting of metric measure spaces and only require some growth bounds of the reference measure (for q2) and lower semicontinuity of the upper gradient of the Renyi entropy which holds under a curvature assumption shown in a previous talk. Those spaces include smooth Riemannian/Finsler manifolds with lower bound on the Ricci curvature.
Instead of following Otto’s idea of using the p-Wasserstein space to construct the q-heat flow, we follow Ambrosio-Gigli-Savaré (Inv. Math. 2013) and show existence, mass preservation and other properties of the q-heat flow even in the non-compact setting and use this to show that it “solves" the gradient flow of the (3-p)-Renyi entropy in the p-Wasserstein space. Uniqueness will follow from some convexity assumptions of the upper gradient of the Renyi entropy.
The so-called damped Navier-Stokes equations in the whole 2D space will be considered. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces will be discussed. In addition, the applications of the developed theory to the classical Navier-Stokes problem in R2 will be also presented.
When the parameter describing the interaction between two trapped Bose-Einstein condensates is very large, the total system undergoes a phase transition. The condensates then segregate, break the symmetry of the trap and their supports became half-ball shaped. In this talk, we study this phenomenon using two phase transition models : optimal partition problems and a Cahn-Hilliard energy functional. We will first study the Gross-Pitaevskii functional describing the behaviour of a condensate and the main properties of its minimizers. Then, we will explain the analogous functional when two condensates are presents. We finally study the associated phase transition problems in two main regimes and explain the results obtained.
At first we will consider the Boussinesq equations for Rayleigh-Bénard convection in the limit of infinite Prandtl number and recall some well known upper bounds on the vertical heat transport, i.e the Nusselt number Nu, in terms of the Rayleigh number Ra. We will then introduce the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number and we will prove a new upper bound for the Nusselt number of the form $Ra^{1/3}$ (modulo logarithmic correction) under the assumption that the the ratio of Prandtl number and Rayleigh number is greater than c_0, where c_0 is a non-dimensional constant depending on the aspect ratio of the domain only. The method applied to derive the result is to view the Boussinesq system at large Prandtl number as a small perturbation of the infinite Prandtl number model for convection. The seminar is based on the following papers:"Infinite Prandtl number convection" by P. Constantin and C.Doering (1999),"Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh-Bénard convection" by C. Doering, F. Otto and M.G. Reznikoff (2006),"Bound on vertical heat transport at large Prandtl number" by X. Wang (2007).
Heating up a ferromagnetic body through its Curie temperature changes the material's permanent magnetism to induced magnetism and the material becomes paramagnetic. At the Curie temperature the material might consist of two seperate phases, a ferromagnetic and a paramagnetic phase. To understand the evolution of the phase boundary at this temperature, a quasistatic sharp interface model is introduced. The movement of the phase boundary is described by the mean curvature flow equation including a nonlocal outer force, which is induced by the magnetization structure in the ferromagnetic phase. Luckhaus and Sturzenhecker developed an implicit time discretization for curvature driven flows with a local outer force. This method has been used to construct short-time solutions to several problems. For this model, a phase field approximation of the micromagnetic energy provides functionals to which the discretization can be applied and so compactness of the sets can be derived.
I will present probabilistic ideas that arise in relation to homogenization, without assuming familiarity of the audience with probability. We will first see how to define a random walk on a graph, how to relate it to a PDE, and what the homogenization of this PDE means for the random walk. I will then define the concept of a martingale, and give a sketch of proof of homogenization based on this tool.
In a seminal work, Jordan, Kinderlehrer and Otto proved that the Fokker-Planck equation can be described as a gradient flow of the free energy functional in the Wasserstein space, bringing this way the underlying particle interpretation of the diffusion phenomenon in the forefront. The aim of the talk is to convince the audience that it is possible to retrieve this natural (from a statistical mechanics point of view) coupling of functional and metric, by studying the large deviations of particle models. The talk will contain an introduction to the Wasserstein space and its geometric properties, a brief discussion of large deviations and the presentation of my work on the subject.
In the early 90s Kohn and Müller established a model for the formation of domain pattern in martensitic shape-memory alloys. For a vectorial generalization of their model for solid-solid phase transitions with SO(2) invariance, we discuss the cases with one and two rank-1 connections and derive upper and lower bounds to the infimum energy with domain branching.
Optimal Transport and the Monge-Kantorovich problem recently revived both from a theoretical and a practical standpoint. Optimal mass transfer problems have many connections with PDE systems arising in meteorology, electrodynamics, crowd motion, optimal design.
After a brief introduction on the Monge-Kantorovich problem, a lagrangian class of numerical methods is presented to approximate its solution. It is compared in terms of formulation and computational costs with Benamou-Brenier and AHT methods.
Then, a numerical study on how the Wasserstein distance could be used to study low order representations of systems of PDEs is detailed and some numerical experiments discussed. The method proposed is effective when the solutions are featured by the transport of concentrated structures.
In the last part, a still ongoing work is presented on unbalanced optimal transport problems. Several mass sources models are described. In the end, an equivalence between mass sources models and metrics of the space is presented, aiming at modeling obstacles and constraints.
Shape memory alloys (SMAs) are smart materials able to undergo diffusionless martensitic transitions upon thermal or mechanical loadings. At the level of a single crystal, these transitions are provided by formation and propagation of mobile interfaces or interfacial microstructures (object composed of more mobile interfaces). Experimental observations of these objects and their consequent analyses by theoretical tools and numerical models enable a deeper insight into the macroscopic effective behaviour of SMAs, and improve the understanding of the underlying physical phenomena.
The talk will be focused on mobile microstructures in two particular SMAs: Cu-Al-Ni undergoing the cubic-to-orthorhombic transition and Ni-Mn-Ga undergoing the cubic-to-tetragonal transition with adaptive monoclinic modulations. For both alloys, the constructions of theoretical and numerical models of experimentally observed microstructures will be described, and the resulting models will be discusses with the emphasis on their contribution to understanding the mechanisms of transitions in SMAs.
Experimentally, one observes that an important parameter controlling the width of the thermal hysteresis accompanying certain martensitic transformations is the compatibility of the highly symmetric austenite phase and the martensitic variants. This observation forms the basis of a theory of hysteresis that assigns an important role to the energy of the transition layer (Zhang, James, Mueller, Acta mat. 57(15):4332-4352, 2009). Following this ansatz, we study the energy barriers leading to hysteresis and analyze the optimal microstructures in the transition layers for low-hysteresis alloys.
In this talk, we will begin with definitions of hyperbolic space and De Sitter space. After giving some known results for constant curvature hypersurfaces in hyperbolic space and constant curvature spacelike hypersurfaces in De Sitter space, we will show a strong duality theorem of hyperbolic space and De Sitter space. Finally we will show some applications for this duality theorem. This is a joint work with Bo Guan and Joel Spruck.
Steiner tree problem consists in finding a connected set of minimal 1-dimensional measure containing a (prescribed) finite set of points in R^n. I will show how to understand this as a mass minimization problem in a family of currents with coefficients in a suitable group. This allows to introduce calibrations in this problem and therefore to prove the absolute minimality of some concrete configurations. No previous knowledge by the audience in geometric measure theory will be assumed.
This is joint work with Annalisa Massaccesi.
In a set of remarkable experiments, Yves Couder and collaborators have shown that drops of silicone oil bouncing on a vibrating liquid bath can be made to move horizontally and exhibit many features analogous to quantum mechanics, such as tunnelling, single-particle diffraction or quantized orbits. Our aim was to provide the simplest theoretical model capable of quantitative prediction of the drop's behaviour, which would guide us in search of new phenomena. This was achieved using a logarithmic spring model for the drop-bath interaction. We show the rich regime diagrams of the drop's vertical and horizontal behaviour, compare our model with experiments, and explain why the drops walk only in a small region of system parameters.
Starting from the minimization formulation of 3d nonlinear elasticity we derive the homogenized plate model in the bending regime, by means of Gamma convergence. We obtain different models, depending on the relation between the oscillations of the material and the thickness of the body.
I will introduce a class of models of self-interacting random polymers and mention an approach to study it as a perturbation of a generalization of a free field. I will then show a simple new method to decompose free fields into sums of independent Gaussian fields with spatially localized correlations. The method uses the finite propagation speed of the wave equation and related properties of Chebyshev polynomials.
Kohn-Sham Density Functional Theory (KSDFT) is quantum mechanical theory for studying macroscopic and electronic structure properties of molecular systems. In the first half of the talk, I will sketch the origins of KSDFT from quantum mechanics, and motivate its importance as a practical computational tool for probing molecular systems at the subatomic level. In the second half, I will describe a linear scaling implementation of KSDFT developed at Caltech, and provide an elementary error analysis for the implementation.
A new algorithm for simulating the well-known Mullins model for polycrystalline grain growth is presented. This algorithm extends the distance function-based diffusion-generated motion algorithm to the case where the surface energy between each pair of grains may be specified. Well-resolved simulations beginning with over 500,000 grains in two dimensions and 64,000 grains in three dimensions are performed. The effect of the initial texture (e.g. fiber texture or random texture) and the form of the surface energy (as a function of misorientation) on the evolution of the misorientation distribution is studied. A new statistic, the surface energy distribution, is proposed and demonstrated to help predict the evolution of the misorientation distribution.
In this talk I will present some results obtained with B. Zwicknagl on a variational model for the epitaxial deposition of a film on a rigid substrate in the presence of a crystallographic misfit. I will show the scaling behavior of the minimal energy in terms of the volume of the film and the size of the misfit, and derive the reduced models in the various regimes by $\Gamma$-convergence methods. Depending on the relation between the thickness of the film and the size of the misfit, the surface or the elastic energy contribution dominate, and in the critical case, the two contributions balance. In particular, the formation of islands is proven if the misfit is large compared to the volume of the film.
In this seminar I will talk about bounds on convective heat transport in a porous layer heated from below. The results I will present are established by C.R. Doering and P. Constantin and are derived using the background field variational method. The marginal stability concept will be introduced in order to derive rigorous limits on the time-averaged convective heat transport, i.e the Nusselt Number Nu, as a function of the Rayleigh number Ra.
In this talk, I will give an introduction to the existence theory of nonlinear Schrödinger equations. Nonlinear Schrödinger equations (NLS) arise in physics in the study of Bose-Einstein condensation and nonlinear optics. They provide an important example for semilinear dispersive evolution equations and have received much attention over the past decades for their interesting mathematical effects such as blow-up of solutions, existence of solitons, and scattering. I will talk about local and global existence of solutions to NLS and present my own work in this direction with Paolo Antonelli (SNS, Pisa) and Christof Sparber (UIC, Chicago).
I will talk about two (and, if time permits, three) problems I worked on recently - all of them were a joint work with Prof. Robert V. Kohn.
In the first one we study the wrinkling of a thin elastic sheet caused by a prescribed non-Euclidean metric. This is a model problem for the patterns seen, for example, in torn plastic sheets and the leaves of plants. Following the lead of other authors we adopt a variational viewpoint, according to which the wrinkling is driven by minimization of an elastic energy subject to appropriate constraints and boundary conditions. We prove an upper bound and a matching lower bound with the conjectured scaling for the minimum of the energy. We also discuss the minimization of the same functional over a more restricted class, again proving matching upper and lower bounds.
The second problem involves an elastic sheet loaded in tension which wrinkles and the length scale of the wrinkles tends to zero with vanishing thickness of the sheet. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on a planar deformation) by introducing a cascade of wrinkles. The lower bound is more subtle, since it must be ansatz-free.
In this talk I will present new backward uniqueness results for the heat equation in conical domains improving previous results of Sverak and Li. Moreover, under additional assumptions on the initial data the optimal angular dependence is proved. After reviewing the different behavior of the heat equation in bounded and unbounded domains the central Carleman estimate will be motivated.
The most common shape memory alloys are monoclinic-I martensite. We study their zero energy states and have two surprising results:
First, there is a five-dimensional continuum in which the energy minimising microstructures are T3s, i.e. infinite-rank laminates. To our knowledge, this is the first real material in which T3s occur. We discuss some of the consequences of this discovery.
Second, there are in fact two types of monoclinic-I martensite, which differ by their convex polytope structure but not by their symmetry properties. It happens that all known materials belong to one of the two types. We explore whether materials belonging to the other type would have superior properties since they have different zero-energy states.
Our analysis uses algebraic methods, in particular the theory of convex polytopes.
We discuss a hierarchy of approximation methods developed for accelerating sampling microscopic dynamics of stochastic lattice systems. The approach is based on efficient coupling of different resolution levels, taking advantage of the low sampling cost in a coarse space and the local reconstructions.
We provide error estimates for (a) long-time stationary dynamics in terms of relative entropy, and (b) finite-time weak error estimates that control mesoscale observables.
From the computational point of view the multilevel nature of the method allows for speeding up sampling algorithms such as kinetic Monte Carlo applied to systems with complex lattice geometries and particle interactions.
We also briefly discuss related mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed kinetic Monte Carlo simulations.
We analyze the time harmonic Maxwell's equations in a complex geometry. The homogenization process is performed in the case that many small, thin conductors are distributed in a subdomain of $\mathbb{R}^3$. Each single conductor is, topologically, a split ring resonator, but we allow arbitrary flat shapes. In the limit of large conductivities in the rings and small ring diameters we obtain an effective Maxwell system. Depending on the frequency, the effective system can exhibit a negative effective permeability.
Let us consider a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof consists of a divide-and-conquer strategy, which mimics the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the Gibbs measure restricted to a basin of attraction has a ”good” Poincaré constant providing the fast convergence of the diffusion to metastable states. The second ingredient is a refinement of the mean-difference estimate introduced by Chafai and Malrieu. There, we propose a new weighted transportation distance, which contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also results in a sharp estimate of the constant of the logarithmic Sobolev inequality.
(joint work with Georg Menz)
The talk concerns the problem of hydrodynamic limits, where one tries to obtain a macroscopic PDE description from a microscopic discrete stochastic model. We review some results that have been obtained previously and present a new method that allows us to obtain uniform in time estimates on the rate of convergence. This approach is based on comparing the semigroups of the micro- and macroscopic processes via a Duhamel-type formula. Furthermore we present some results on the convergence of the microscopic entropy towards the macroscopic one.
This is joint work with Clement Mouhot.
I will speak about a problem of homogenisation of a linear PDE with periodic coefficients. Coefficients of the operator are not supposed to be uniformly elliptic, they degenerate on a periodic subset of the domain as the period tends to zero (hence the term high-contrast homogenisation). The limit homogenised operator in this case is two-scale and possesses a band-gap spectrum. One of the key elements in the analysis of such type of problems is two-scale convergence (Nguetseng).
I start by making a 15-minute introduction to the classical pronlem of the mathematical theory of homogenisation. This will be followed by the description of a series of experiments that suggest that the (suitably normalised) strength of a plastically deformed solid may increase with diminishing sample size. I will show how such effects can be accounted for by considering higher-order terms in the classical problem of periodic homogenisation and propose a related variational framework that yields the desired higher-order (or "strain-gradient") constitutive laws.
In the talk we shall discuss about plate models for different regimes obtained by the process of simultaneous homogenization and dimensional reduction with the special emphasis on the von Kármán case. Since we have two small parameters, namely the thickness of the plate $h$ and the oscillations of material $\varepsilon(h)$, the obtained models depend on the relation between these two parameters. We shall discuss the case when they are on the same scale i.e. $\lim_{h \to 0} \frac{h}{\varepsilon(h)}=\gamma \in \langle 0, \infty \rangle$. During the talk we shall also present three new models obtained by this procedure: homogenized von Kármán plate model, periodically wrinkled von Kármán plate and homogenized von Kármán shell model.
For every euclidean Radon measure $\mu$ we state an adapted version of Rademacher's theorem, which is, in a certain sense, optimal. We define a sort of fibre bundle (actually just a map $S$ that at each point $x\in\mathbb{R}^n$ associates a vector subspace $S(x)$ of $T_x\mathbb{R}^n$, possibly with non-costant dimension $k(x)$) such that every Lipschitz function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is differentiable at $x$, along the k(x)-plane $S(x)$, for $\mu$-a.e. $x$. We prove that $S$ is maximal in the following sense: if $T$ is a vector space such that $T(x)\not\in S(x)$ whenever $k(x)\neq n$, then there exists a Lipschitz function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ which doesn't admit any derivative in the direction $V(x)$ for $\mu$-a.e. $x$ satisfying $k(x)\neq n$. Joint work with Giovanni Alberti.
It is well known from the theory of classical homogenization that the large scale behavior of linear elliptic equations with periodically oscillating coefficients can be described by a limiting equation (of the same type) with homogenized coefficients that are constant. Moreover, the homogenized coefficients are determined by a certain homogenization formula, the evaluation of which requires to solve a linear cell-problem. The same statement is true in the case of stochastic homogenization, e.g. for linear elliptic equations with random coefficients that are ergodic and statistically homogeneous in space.
Part 1 of the lecture is introductory. We recall basic notions, such as stationarity and ergodicity, present two (classical) proofs of the discrete, stochastic homogenization result and motivate the notion of correctors.
The content of part 2 is work in progress: In contrast to the deterministic case, in the stochastic case the cell problem has to be solved on an infinite domain. Hence, in practice, one has to make approximations. We are interested in the convergence rate to zero of the associated error for the approximation and study the case of discrete elliptic equations on the d-dimensional lattice. In particular, in the special case of random coefficients that are identically distributed and independent, we prove the optimal decay rate of the approximation-error. We explain that this observation can be used to setup and analyze approximation schemes that combine periodization, regularization and empirical averaging.
In this talk I will present some interpolation inequalities that involves the BV-norm and some negative norms of a function u. These inequalities are the strong version of some already known estimates in weak form, which play a crucial role in the study of pattern formation. The main ingredient in the proof of these estimates is given by a geometric construction, that was first used by Choksi, Conti, Kohn and Otto in the context of branched patterns in superconductors, and which main idea goes back to De Giorgi.
This is a joint work with Felix Otto.
This is joint work with Constanza Rojas-Molina. For many elliptic partial differential equations it is known that a solution cannot vanish of arbitrary order unless it is identically zero. This can be proven using a Carleman estimate. The latter is also useful to prove a quantitative version of the unique continuation principle. We are interested in a particular quantitative and scale free version of this result: Consider an eigenfunction of a Schroedinger equation on cube of size L with periodic boundary conditions. Assume that the $L^2$-norm of the eigenfunction is one. The cube of size L can be decomposed into unit cubes. Place in each unit cube arbitrarily a ball with fixed but small radius. We wnat to derive a lower bound on the $L^2$-norm of the eigenfunction when integrated over the union of the small balls, which is independent of the size $L$, and depends in an explicit way on the other parameters of enterig the problem.
In a recent article, Grunewald, Otto, Villani and Westdickenberg have devised a new method to study the convergence of interacting continuous spin systems to a deterministic scaling limit. In this talk, I shall show how to use their results and technique to study the behavior of the microscopic entropy in this scaling limit, and obtain convergence in a stronger sense than the one obtained in [GOVW], extending results of Kosygina to a more general setting.
The aim of this talk is to provide some basic understanding of liquid-liquid systems with negative spreading coefficient. In particular stationary states, so called liquid lenses, are studied by introducing appropriate PDE problems. It is indicated how dimensionally reduced lubrication models can be derived from these flow problems and numerical solutions to them, both time-dependent and stationary are discussed.
Stationary solutions for the lubrication equations with suitable boundary conditions are studied using formal asymptotics and Gamma-convergence methods. In the latter case proof of existence and uniqueness of a global energy minimizer is given for the resulting sharp-interface model.
Kirchhoff's nonlinear plate theory arises as a thin film asymptotic limit from three dimensional elasticity. It lives on intrinsically flat surfaces, i.e., on deformations of two dimensional films into three dimensional euclidean space which do not stretch the film. In the first part of the talk I will explain the local and global structure enjoyed by such 'non-stretching' deformations.
The energy functional corresponding to Kirchhoff's plate theory agrees with a constrained version of the Willmore functional from differential geometry. In the second part of the talk I will recall some facts about the Willmore functional and I will discuss minimizers of Kirchhoff's plate functional.
A three-dimensional elastic rod is a body that occupies a cylindrical domain with small "thickness h". We consider rods made of composite materials with microstructure on a small "material fine-scale e". We are interested in the effective behavior for small thickness and small material fine-scale. Here, the zero-thickness limit corresponds to dimension reduction, where we expect to obtain a one-dimensional elasticity model. On the other side, in the homogenization limit "e -> 0" we expect that a material law emerges that is homogeneous-in space. In the talk we discuss the simultaneous limit. In a first part, we explore the coupling between relaxation patterns associated to dimension reduction and homogenization by several "mathematical experiments". In a second part, we briefly present a rigorous derivation of a homogenized, nonlinear bending-torsion theory for inextensible rods as Gamma-limit from 3d elasticity.
We are interested in the long-term behavior of deterministic and non-deterministic continuous-time dynamical systems. On the one hand, the statistical long-term behavior is well characterized by the stationary distribution of the system and by almost-invariant sets, i.e. sets from which the rate of outward mass transport is small. The computation of such objects supports the analysis of systems arising for example in oceanography, astrodynamics and molecular dynamics. On the other hand, engineering applications often require knowledge about the domain of attraction of some asymptotically stable fixed point.
It turns out that all this information can be computed from eigenpairs of the transfer operator associated with the underlying system. Transfer operator methods are often more reliable and efficient than techniques based on direct simulation. Sophisticated approaches have been developed for the discretization of transfer operators, however they still all require the simulation of (short) trajectories. In order to design computationally more efficient methods, we propose to discretize the infinitesimal generator of the transfer operator semigroup.
In my talk I will show the development of a discretization which has a physical meaning, and uses no trajectory simulation at all. Further, I will discuss its relation to the original system, and other properties as well.
In a Hilbert space setting homogenization of evolutionary equations is discussed. A suitable topology on material laws is introduced and several properties of that topology can be shown. With those properties homogenization theorems of a large class of linear evolutionary problems of classical mathematical physics can be obtained.
The vorticity of a two-dimensional perfect (inviscid, incompressible) fluid is transported by its flow. Thus, the space of vorticities is naturally "foliated". I will present a rigorous proof (under certain non-degeneracy conditions) that steady-states persist from one "orbit" of this "foliation" to nearby ones. (This type of stability result can be expected in view of the geometric interpretation of the incompressible Euler equations: the Euler flows are geodesics on an infinite-dimensional manifold equipped with a right-invariant metric.) In particular, I will focus on some analytical difficulties of the proof: the Nash-Moser inverse function theorem is used to overcome "loss of derivatives" of certain linear operators.
A new class of phase field models is derived and studied where the elastic energy, being quasiconvex, incorporates microstructure formation on smaller length scales. After a review of the used elasticity theory, global existence of weak solutions in certain microstructural regimes is proved with two different methods.
First numerical computations illustrate the characteristic properties of the solutions and allow a comparison to the existing models with linear elasticity.
In this talk the dynamics of coarsening liquid nanoscopic droplets will be described in the framework of the lubrication approximation for different slip regimes considered at the solid substrate. We start with the introduction of underlying lubrication equations and shortly summarize existence results for them. Next, a sharp interface model for the evolution of droplets is derived in two and three dimensional cases. By that one arrives at the closed set of reduced ODE models, analytical and numerical investigation of which allows for derivation of the coarsening rates dependence on the slip-length. Finally, we discuss a possibility for the rigorous justification of the observed coarsening rates.
In this talk I will present some classical heuristics and related recent rigorous results for the Kolmogorov-Obukhov spectral scaling for fluid flows at high Reynolds number.
