We discuss the dynamical large deviations of three models: Kac’s model of the Boltzmann Equation, a rescaled zero-range process converging to the porous medium equation, and the Landau-Lifschitz-Navier-Stokes equations. For these models, we discuss the challenges and successes of finding matching bounds, the links to well- or ill-posedness in low regularity classes, and the connection to the equality case of classical inequalities.
We review recent results on the weak and strong solution theory for the stochastic thin-film equation driven by Gaussian noise, which is white in time and colored in space. In particular, we discuss the effect of intermolecular van der Waals forces between the fluid and the substrate in more detail. We demonstrate that they result in a strictly positive and smooth film height which persists for all times. We also touch upon open questions and possible extensions of the presented results. The talk is based on joint work with Antonio Agresti, Konstaninos Dareiotis, Benjamin Gess and Manuel Gnann.
Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained by planetary rotation at large scales. Rapidly rotating Rayleigh-Bénard convection (RRRBC) provides a paradigm for direct numerical simulations (DNS) and laboratory studies of such flows, but the accessible parameter space remains restricted to moderately fast rotation (Ekman numbers $Ek \gtrsim 10^{-8}$), while realistic $Ek$ for astro-/geophysical applications are significantly smaller. Reduced equations of motion, the non-hydrostatic quasi-geostrophic equations (NHQGE) describing the leading-order behavior in the limit of rapid rotation ($Ek \to 0$) cannot capture finite rotation effects, leaving the physically most relevant part of parameter space with small but finite $Ek$ currently inaccessible. Here, we introduce the rescaled incompressible Navier-Stokes equations (RiNSE) – a reformulation of the Navier-Stokes-Boussinesq equations informed by the scaling laws valid for $Ek \to 0$. We provide the first full DNS of RRRBC at unprecedented rotation strengths down to $Ek=10^{-15}$ and below, and show numerically that the RiNSE converge to the NHQGE. This work opens the door to the exploration of a large region in the parameter space of rotating convection, and many aspects remain to be examined including low Prandtl numbers typical of stars and planets and different choices of boundary conditions.
Beyond the specific example of rotating convection, turbulence is a widely observed state of fluid flows, characterized by complex, nonlinear interactions between motions across a broad spectrum of length and time scales. While turbulence is ubiquitous, from teacups to planetary atmospheres, oceans and stars, its manifestations can vary considerably between different physical systems. For instance, three-dimensional (3D) turbulent flows display a forward energy cascade from large to small scales, while in two-dimensional (2D) turbulence, energy cascades from small to large scales. In various physical systems, a transition between such disparate regimes of turbulence can occur when a control parameter reaches a critical value. Specifically for turbulent flow confined in a thin fluid layer, there is a critical layer height where the fluid flow becomes two-dimensional under suitable conditions. Close to this threshold, 3D variations in the velocity are highly intermittent, and display jumps in the linear growth rate. Modeling of this problem motivates the study of stochastic ODE driven by Lévy noise, representing the dynamics of the amplitude of 3D modes. The probability density function (PDF) of the amplitude is then governed by a fractional Fokker-Planck equation containing a nonlocal, linear fractional diffusion operator. I describe certain results which I derived on this problem, including the asymptotic behavior of the PDF which lead to heuristic estimates of the moments that agree favorably with numerical solutions and outline some remaining questions.
References:[1] AvK et al., Bridging the Rossby number gap in rapidly rotating thermal convection. (in prep.)[2] AvK, Alexakis, A., Brachet, M.E., 2021. Lévy on-off intermittency. Phys. Rev. E, 103(5), p.052115.[3] AvK, Pétrélis, F., 2023. 1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency. J. Stat. Mech., 2023(1), p.013204.
The Sobolev regularity of solutions to the Monge-Ampère equation in the plane can be rephrased in terms of a unique continuation property of differential inclusions. After an overview of the known results concerning the Monge-Ampère equation, I will explain this connection to differential inclusions and sketch a proof of the unique continuation property. Based on joint work with G. De Philippis and A. Guerra.
The Linear Sigma Model is the N-component and O(N)-invariant generalization of the well-known $\Phi^{4}_{2}$ model. In the present work, we show that on $T^{2}$ at large N, each marginal distribution is close to a massive Gaussian free field, quantified in the 2-Wasserstein distance. The proof is a simple application of classical tools in Euclidean field theory combined with the Feyel/Üstünel variant of Talagrand's inequality. In contrast to prior work using stochastic quantization, our proof avoids perturbative assumptions on the mass or the coupling constant. Based on joint work with Matías Delgadino.
The gravity-Stokes system serves as a fundamental model for understanding the dynamics of incompressible fluids in certain regimes. We focus on the scenario where two fluids of different densities evolve in a two-dimensional region without mixing. The density difference together with the gravity influence induce the dynamics of the two fluids and hence the evolution of the free interface arising between them. Through a contour dynamics approach, we address questions such as the existence and uniqueness of global solutions for this system and their asymptotic behavior, making emphasis on the properties of the free boundary. This talk is based on joint work with F. Gancedo and R. Granero-Belinchón.
Since astrophysical black holes are expected to be rotating, one would like to study quantum fields on such spacetimes. For linear theories, one of the biggest difficulties lies in identifying and constructing physically reasonable and well-motivated states for the quantum theory. In this talk, we discuss one example of such a state for rotating black holes, the Unruh state. We will detail its construction, as well as the proof for it being physically reasonable. If time permits, we will mention some applications of the Unruh state for the exploration of quantum effects on black hole spacetime.
Rayleigh-Benard convection - a fluid motion in a layer heated from below and cooled from above - is one of the paradigmatic systems in fluid dynamics. The key question here is how does the heat transport (Nusselt number Nu) depend on the thermal driving strength (Rayleigh number Ra) for extremely large Ra (in the ultimate regime, relevant in geophysical and astrophysical applications)?
We will discuss the existing physical models for the ultimate regime and their relation to the mathematically rigorous upper bounds for the heat transport.
The talk is based on papers by D. Lohse and O. Shishkina, ``Ultimate Rayleigh-Benard turbulence'' (to appear in Reviews of Modern Physics) and ``Ultimate turbulent thermal convection'' Physics Today 76 (2023).
We study the focusing Φ4-measure on the real line, namely, focusing Gibbs measure associated with the Schrödinger equation, initiated by Lebowitz, Rose, and Speer (1988). Through the process of taking the infinite volume limit, we analyze that the focusing Φ4-measure with appropriate scaling exhibits Gaussian fluctuation around scaled solitary waves, which are minimizers of the focusing Hamiltonian.
This talk is based on ongoing work with Philippe Sosoe (Cornell)
We explore the weak coupling limit for stochastic Burgers type equation in critical dimension, and show that it is given by a Gaussian stochastic heat equation, with renormalised coefficient depending only on the second order Hermite polynomial of the nonlinearity. We use the approach of Cannizzaro, Gubinelli and Toninelli (2024), who treat the case of quadratic nonlinearities, and we extend it to polynomial nonlinearities. In that sense, we extend the weak universality of the KPZ equation shown by Hairer and Quastel (2018) to the two dimensional generalized stochastic Burgers equation. A key new ingredient is the graph notation for the generator. This enables us to obtain uniform estimates for the generator. This is joint work with Nicolas Perkowski.
We extend the paradifferential appraoch to quasilinear wave eqauations of Bahouri and Chemin to the case of a quasilinear wave equation with a (random) distributional forcing. I order to do this, we combine their methods with recently developed tools from singular SPDEs, more precisely the Paracontrolled Calculus of Gubinelli, Imkeller and Perkowski. In two dimensions, we obtain a well-posedness result for noises with Hölder-regularity better than -1/8 for generic initial data and for noises with regularity better than -1 with prepared initial data.
In this talk, we consider the two-phase Mullins-Sekerka evolution of nearly circular simple closed curves in the plane. Transferring the method of O. Chugreeva, F. Otto and M. G. Westdickenberg from 2019 to graphs over equilibrium configurations with non-trivial background curvature, we capture and quantify the respective relaxation rates. Our result establishes the existence of two qualitatively different relaxation regimes, an initial regime of (lengthscale-independent) algebraic decay and a latter regime in which (lengthscale-dependent) exponential decay takes over. This talk is based on joint work with M. G. Westdickenberg and U. Hryniewicz.
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.
