Our vision for this conference is to bring together experts in the mathematical theory of compressible and incompressible flows, with an emphasis on recent new ideas and directions. We are inviting speakers with diverse interests, including shock structure, kinetic theory and hydrodynamic limits, global well-posedness and long-time behavior, singular and inviscid limits, convex integration and non-uniqueness, kinetic formulations of conservation laws, collective behavior and biological systems, and numerical methods. These topics only partially reflect the diverse interests of our colleague, mentor, and friend, Alexis Vasseur, whose contributions, mathematical or otherwise, we recognize in his 50th birthday conference.
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Hyperbolic systems of conservation laws,
Guiding the flow of energy and matter,
Allowing for change and evolution,
But never letting things shatter.
From the smallest particles to the grandest galaxies,
The principle of least action has been used for a long while to find important equations of physics and mechanics through space-time optimization. Nevertheless, it is customary to say that recovering their solutions by space-time convex minimization doesn’t make much sense. However, this is indeed the case for systems of conservation laws with a convex entropy, at least in small time, in particular the Euler equations of isothermal gases. We will discuss how this idea can be (partly) extended to Einstein’s equations in vacuum (including with a cosmological constant). We obtain formulations that are very close to those of the Euler equations (provided that density and velocity fields with matrix values are used). However, convex optimization is lost, for lack of a convex entropy and we can recover only the equations but a priori not their solutions.
In this talk, we will discuss the intrinsic phenomena of cavitation/decavitation and concentration/deconcentration in the entropy solutions of the compressible Euler equations, the compressible Euler-Poisson equations, and related nonlinear PDEs, which are fundamental to the analysis of entropy solutions for nonlinear PDEs. We will start to discuss the formation process of cavitation and concentration in the entropy solutions of the isentropic Euler equations with respect to the initial data and the vanishing pressure limit. Then we will analyse a longstanding fundamental problem in fluid dynamics: Does the concentration occur generically so that the density develops into a Dirac measure at the origin in spherically symmetric entropy solutions of the multi-dimensional compressible Euler equations and related nonlinear PDEs? We will report our recent results and approaches developed for solving this problem for the Euler equations, the Euler-Poisson equations, and related nonlinear PDEs, and discuss its close connections with entropy methods and the theory of divergence-measure fields. Further related topics, perspectives, and open problems will also be addressed.
After a brief review on the state of the art regarding global well-posedness versus blow up in finite time for the homogeneous Landau equation, we present recent results on existence of solutions from the point of view of L^p theory. The talk will also touch upon open problems, future directions and possible approaches.
The homogeneous Landau-Coulomb equation is a nonlinear equation of parabolic type. We will see how to derive local estimates related to entropy dissipation and how to use it to prove that axi-symmetric solutions are smooth away from the axis. Joint work with François Golse and Alexis Vasseur.
Observability refers to the possibility of reconstructing the solution of an evolution PDE from knowing its restriction to some spatial subdomain and on some finite time interval. This is a property dual to controllability, and Bardos-Lebeau and Rauch have given a geometric condition for observability in the case of the wave equation posed on a bounded domain with smooth boundary. In this talk, I shall explain how a similar condition allows observing the solution of quantum dynamical equations, with explicit constants, and under minimal regularity conditions on the driving potential. The approach used here is based on a quantum analogue of the Wasserstein distance used to measure the proximity of a quantum density operator to its classical limit.
In recent times, we elaborated a tool, called Compensated Integrability, which provides Strichartz-like inequalities for some systems of conservation laws. We were able to derive space-time estimates for Gas dynamics, in terms of internal variables like pressure and density. We shall describe new developments, in particular a multilinear version of C.I., and derive additional estimates. These involve either the velocity field, or singular integrals.
The quantum Navier-Stokes model correspond to the classical Navier-Stokes model in which a quantum correction term called Bohm potential is added. In this talk I will first present some results about the existence of weak solutions which is obtained using renormalized solutions. The results allow us to obtain also the semi-classical limit i.e. the limit when the parameter behind the Bohm potential tends to zero. This limit lead to the classical Navier-Stokes model. In a second part I will speak about the viscous limit of the model for which we use the previous existence result, the notion of dissipative solutions and relative entropy estimates. In this limit we obtain a dissipative solution of the quantum Euler model.
Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon.
In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs.
In this talk, I will discuss the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy's law for incompressible porous media via buoyancy force. In contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions. The talk is based on work joint with Zhongtian Hu and Yao Yao.
The phenomenon of anomalous dissipation in turbulence predicts the existence of solutions to the incompressible Euler equations that enjoy regularity consistent with Kolmogorov’s 4/5 law and satisfy a local energy inequality. The "strong Onsager conjecture" asserts that such solutions do indeed exist. In this talk, we will discuss the background and motivation behind the strong Onsager conjecture. In addition, we outline a construction of solutions with regularity (nearly) consistent with the 4/5 law, thereby proving the conjecture in the natural L^3 scale of Besov spaces. This is based on joint work with Hyunju Kwon and Vikram Giri.
Emergence is a phenomenon of formation of collective outcomes in systems where communications between agents has local range. For a wide range of applications, such as swarming behavior of animals or exchange of opinions between individuals, such outcomes result in a globally aligned state or congregation of clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with non-integrable radial tails. Proving a similar statement for purely local interactions is a challenging mathematical problem. In this talk we will overview three programs of research directed on understanding the emergent phenomena: statistical approach to generic alignment for agent-based systems, kinetic approach based on relaxation and hypocoercivity, and hydrodynamic models incorporating a novel way of interaction based on topological communication.
The classical obstacle problem is to find the equilibrium position of an elastic membrane membrane whose boundary is fixed and forced to lie above a given obstacle.
According to Caffarelli's classical results, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set can generally be as large as the regular set. This talk will introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.
We study solutions of the non-cutoff Boltzmann equation whose hydrodynamic quantities stay bounded and away from vacuum. We discuss pointwise upper bounds for the equation in a bounded domain with different physical boundary conditions.
The incompressible and inhomogeneous Navier-Stokes system (INS) governs the evolution of fluids which, although incompressible, have non constant density. This is a coupling between a transport equation for the density, and an evolution equation similar to the "classical" Navier-Stokes equation for the velocity. It is known since Kazhikhov's seminal paper in 1974 that any initial data with finite energy velocity and strictly positive bounded density generates a global weak solution of finite energy for (INS). But, except in the constant density case and in dimension two, it is not known whether these solutions are unique.
We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neurons and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.
It was a well established fact that a naive treatment of zeroth oder source terms, like gravity forces, in the numerical simulation of conservation laws leads to instable or unphysical solutions.
In a remarkable paper with R. Botchorishvili and B. Perthame, A. Vasseur introduced an elegant way to fix this issue, by setting up a suitable definition of the numerical fluxes which is compatible with the equilibrium solutions of the problems.
We revisit this approach in the framework of staggered discretizations for the Euler equations, where densities and velocities are stored in dual discrete locations.
We derive first and second order versions of a well-balanced scheme, constructed by using the principles of kinetic schemes and hydrostatic reconstructions.
Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics.
We first propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.
We then consider other physically relevant scenarios in which the slender body PDE framework applies and shed light on the analysis of such problems.
I will discuss some aspects of the relationship between singularity formation and the multiscale nature of solutions of incompressible Navier-Stokes and Euler equations.
Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data.
In this talk, I will first discuss the L2 stability result for systems with two unknowns and non-isentropic Euler equations with three unknowns. The main idea in these joint works with Krupa and Vasseur are to use the method of weighted relative entropy norm and modified front tracking sheme with shifts. As an application, we proved all BV solutions must statisfy the Bounded Variation Condition, which is a condition added by Bressan and etc to show uniqueness of solution. Hence we showed the uniqueness of BV solution.
Then I will briefly introduce the recent result on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations. This is a famous open problem after Bressan-Bianchini’s seminal vanishing viscosity limit result for BV solutions of system of hyperbolic conservation laws. This is a joint work with Kang and Vasseur. And Kang will continue discussing it in his talk.
The convex integration shows that the compressible Euler system in multi-D is ill-posed in the class of entropy solutions, especially, non-uniqueness of entropy solutions containing a shock wave.
Recently, for the 1D isentropic case, we showed the uniqueness and stability of entropy solutions of small BV in the class of vanishing physical viscosity limits, that is, inviscid limits from the associated Navier-Stokes system.
These results use the so-called 'a-contraction with shifts' method for handling the uniform stability of a viscous shock.
In this talk, I will explain the key ideas of the method of a-contraction with shifts for a single viscous shock, and extension of the method to more general situations for two shocks and small BV solutions. Also, I will briefly mention its application to the study on long-time behavior of Navier-Stokes flows perturbed from Riemann data.
The results of this talk are based on the joint works with G. Chen, A. Vasseur, Y. Wang and other collaborators in Korea.
In this talk, we will consider the Euler equations of gas dynamics and applications in transonic flows. First the basic theory of Euler equations will be reviewed. Then we will present the results on the transonic flows past obstacles, transonic flows in the fluid dynamic formulation of isometric embeddings, and the transonic flows in nozzles. We will discuss global solutions and stability obtained through various techniques and approaches.
We consider the (spatially homogeneous) Landau equation modeling the effect of collisions of charged particles in a plasma. We present results obtained lately concerning the smoothness and the large time behavior of solutions to this equation in different contexts, in which entropy dissipation estimates play a significant role.
We consider a diffusion process with a random time-independent and spatially stationary drift. The two-dimensional case is scaling-wise critical; we focus on a divergence-free drift, which can be written as the curl of the Gaussian free field.In the presence of an unavoidable small-scale cut-off, we prove that the process is borderline super-diffusive: Its annealed second moments grow like $t\sqrt{\ln t}$.This refines a recent result of Cannizzaro, Haunschmid-Sibitz and Toninelli; the method however is completely different and appeals to quantitative stochastic homogenization of the generator that can be reformulated as a divergence-form second-order elliptic operator.In fact, it embeds homogenization techniques into a renormalization group argument reminiscent of the heuristics in the physics literature.This is joint work with Georgiana Chatzigeorgiou, Peter Morfe, and Lihan Wang.
