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Mathematical Frontiers in the Analysis of Many-particle Systems
July 1 - 5, 2019 Cambridge, United Kingdom |
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Abstracts of the talks
Title: Dependence with respect to the data in incompressible optimal transport
Abstract: Incompressible optimal transport (or Brenier model) is a minimization problem introduced by Brenier in 89 in order to describe the behavior of an incompressible and inviscid fluid in a Lagrangian way. The data of the problem is the joint law of the initial and final positions of the particles, and the dynamics is guided by the Lagrange multiplier corresponding to the incompressibility constraint: the pressure field. In this talk, I will present a positive and a negative result concerning the continuous dependence of the pressure field with respect to the data. The negative part is related to the question of ill-posedness of the so-called kinetic Euler equation, a kinetic PDE known in plasma physics as the limit of the Vlasov-Poisson equation in a quasineutral regime.
Title: A baby version of the Landau damping with scalings. Application to the quasilinear approximation.
Abstract: Considering the solution of the Vlasov equation on a finite time interval and introducing some convenient and may be physical scalings one obtains an elementary version of the Landau Damping and a road map to the proof of the quasilinear approximation.
Title: Dynamics of a Coulomb gas in dimension two
Abstract: We study the dynamics of interacting planar Brownian particles subject to Coulomb pair repulsion. In contrat with the one dimensional log-gases associated with the Dyson Brownian motion, the system is well-posed for any temperature. We also discuss the invariant law and large time convergence to it in presence of an additional external force field. This is a joint wok with D. Chafaï and J. Fontbona.
Title: A two velocity hydrodynamic story
Abstract: In a series of papers over 9 years (2004-2012), Howard Brenner (1929--2013) [who was emeritus professor at MIT in chemical engineering] proposed a new theory in compressible fluid mechanics with high gradient of density based on the concept of two different velocities: the mass and the volume velocities. At the same time, D.B. with B. Desjardins discovered (with E. Zatorska later on) that a structure with two velocity hydrodynamics already exists in standard models (i.e. with one velocity field) if the shear and the bulk viscosities satisfy the BD algebraic relation. In this talk, I will try to give an historical overview of this mathematical story and explain at the end a recent mathematical result with A. Vasseur and C. Yu. Several comments and remarks will be related to kinetic equations and particle systems.
Title: Shock wave formation for the 2D isentropic compressible Euler equations
Abstract: In joint work with Steve Shkoller, and Vlad Vicol, we provide a new, elementary constructive proof of shock formation for the 2D isentropic compressible Euler equations in the presence of non-trivial vorticity.
Title: From particle methods to hybrid semi-Lagrangian schemes for transport equations.
Abstract: Particle methods for transport equations consist in pushing forward particles along the characteristic lines of the flow, and to describe then the transported density as a sum of weighted and smoothed particles. In this talk we present two classes of particle methods which aim at improving the accuracy of the numerical approximations with a minimal amount of smoothing. The idea of the Linearly Transformed Particle method is to transform the shape functions of particles in order to follow the local variation of the flow. This method has been adapted and analyzed for the Vlasov- Poisson system and for a compressible aggregation equation. In both cases the error estimate is improved compared to classical particle methods, with the gain of a strong convergence of the numerical solution. However, for long remapping periods, shapes of particles could become to much stretched out. The second method solve this problem of locality by combining a backward semi-Lagrangian approach and local linearizations of the flow. This is a joint work with Martin Campos-Pinto.
Title: Interplay between diffusion and coagulation processes
Abstract: Coagulation processes can be modeled thanks to the Smoluchowsky equations (in the spatially homogeneous case, they constitute an infinite system of ODEs). Blowup in those equations is related to the physical phenomenon of gelation. The interaction of coagulation with diffusion (once one considers the spatially inhomogeneous case) is complex because the diffusion coefficients of clusters depend on their size. We explain how refined duality lemmas enable to overcome this difficulty in some cases. This presentation is based on a work in collaboration with Maxime Breden.
Title: Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system
Abstract: We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between kinetic and reaction-diffusion models.
Title: Vortex models as limits of the evolution of solids in a perfect incompressible fluid.
Abstract: We consider the evolution of N solids in a perfect incompressible fluid in a two-dimensional domain. The fluid is driven by the incompressible Euler equation, while the solids evolve according to Newton's laws, under the influence of the the pressure on their boundary. This generates a well-posed system as long as no collision occurs. We investigate the asymptotic limit of this system as the solids shink to points, while their mass either goes to 0 or is fixed. We find in the limit vortex models in the spirit of Marchioro and Pulvirenti's wave-vortex system, in which the vortices are either classical point-vortices (following the flow of the regular part of the velocity) or massive point-vortices (fulfilling a second-order equation with a right-hand side reminiscent of the Kutta-Joukowski force of the irrotational theory). This is a joint work with Franck Sueur (Bordeaux).
Title: Partial Regularity in Time for the Landau Equation (with Coulomb Interaction) (joint work with M.P. Gualdani, Cyril Imbert and Alexis Vasseur)
Abstract: We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1/2.
Title:A Particle Approximation for a Kinetic Euler Equation.
Abstract: The kinetic incompressible Euler equation is a model for plasma. It is the formal limit of the classical Vlasov-Poisson system in the `quasi-neutral’ limit where the Debye length tends to zero. The Vlasov-Poisson system can itself be derived formally from a system of interacting particles, in the limit as the number of particles tends to infinity. The rigorous justification of this `mean field’ limit remains a major open problem. However, in recent years, researchers have derived the Vlasov-Poisson equation rigorously from various regularised microscopic systems. In this talk, I will present a joint work with Mikaela Iacobelli, in which we give a rigorous derivation of the kinetic incompressible Euler equation from a regularised particle system, using a combined mean field and quasi-neutral limit.
Title: On large time behaviour for the Vlasov-Navier-Stokes system
Abstract: We will review some recent results describing the large time behavior of solutions to the Vlasov-Navier-Stokes system, which is a fluid-kinetic coupled system modelling the dynamics of sprays. Joint works with O. Glass, A. Moussa and I. Moyano.
Title: Quantitative estimates of propagation of chaos for large systems of interacting particles
Abstract: We present a new method to derive quantitative estimates proving the propagation of chaos for large stochastic or deterministic systems of interacting particles. Our approach requires to prove large deviations estimates for non-continuous potentials modified by the limiting law. But it leads to explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. The method can be applied to very singular kernels that are only in negative Sobolev spaces and include the Biot-Savart law for 2d Navier-Stokes or repulsive gradient flows with less than Poisson singularity. This is a joint work with Z. Wang
Title: Study of Fokker-Planck operator with an external magnetic field.
Abstract: In this talk, I will present results about the exponential decay of the semi-group associated to Fokker-Planck operator with an external magnetic field, in Banach spaces with a polynomial weight. I will first introduce the hypocoercivity method with exponential weight, and then explain how enlargement theory’s help us achieve our goal.
Title: Rigorous Continuum Limit for the Discrete Network Formation Problem
Abstract: Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. We study the existence of solutions to this model, propose an adaptation so that a macroscopic system can be obtained as its formal continuum limit and show the global existence of weak solutions of the macroscopic gradient flow. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. This is joint work with J. Haskovec and P. Markowich.
Title: Semiclassical limit from Hartree to Vlasov-Poisson equation
Abstract: The Hartree equation is the mean field equation which describes the evolution of a system of particles in interaction in quantum mechanics. It can be proved that it converges in some weak sense to the Vlasov equation when the Planck constant \hbar becomes negligible. In this talk, I will present how this convergence can be quantitatively measured by introducing the Wigner transform, a semiclassical version of the Wasserstein-Monge-Kantorovitch distance introduced by F. Golse and T. Paul, and also a semiclassical analogue of the kinetic Lebesgue norms. One of the key step to reach this result is the propagation in time of semiclassical moments, in the spirit of the proof of existence for the Vlasov-Poisson equation by P.-L. Lions and B. Perthame, and weighted Schatten norms of the solution, which implies the boundedness of the spatial density of particles. This can be proved by using the formal analogies between the density operator formulation of quantum mechanics and kinetic theory.
Title: Unique continuation for waves and applications
Abstract: In this talk, we shall first present quantitative unique continuation estimates for wave equations. We shall then explain how such inequalities apply to the approximate and exact controllability of waves, and to hypoelliptic equations. This is based on joint works with Camille Laurent.
Title: Effective Hamiltonians and long-time semiclassical dynamics
Abstract: Effective Hamiltonians are an important tool in solid state physics; here, effective Hamiltonians are derived for the the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. By means of a Floquet-Bloch decomposition we reduce the dynamics to those of an infinite family of pseudo-differential dispersive equations, for which the scaling limit under consideration amounts to performing simultaneously the semiclassical and the long-time limit. Our approach to this problem is based on the quantification of dispersive effects via a two-microlocal analysis on critical points on the Bloch energies. In particular, we are able to deal with degenerate critical points and codimension-one crossings. This is joint work with V. Chabu and C. Fermanian-Kammerer.
Title: A free boundary problem for cell motility
Abstract: We derive a new free boundary problem to model the crawling motion of cells on a substrate. Our starting point is a system of equations coupling a fourth order diffuse interface approximation of Cahn-Hilliard type to a second order parabolic equation modeling some chemo-repulsive effects (which account for the formation on protrusions along the membrane of the cell). The resulting free boundary problem is of Hele-Shaw type. It combines the (regularizing) effects of surface tension with the (destabilizing) effects of the chemo-repulsive potential. It exhibits interesting properties (e.g. symmetry breaking and hysteresis phenomena) which are in good agreement with observations.
Title: The regularity theory for degenerate Kolmogorov equations
Abstract: The aim of my talk is to present a survey of results concerning a family of degenerate linear second order Partial Differential Equations, that includes some non homogeneous kynetic equations. In the first part of my talk, I describe the main points of the regularity theory for classical solutions. In particular, the existence of a fundamental soltion, the interior Schauder estimates, the boundary value problem and the Harnack inequality will be discussed. I will conclude with some recent achievements on the theory of weak solutions.
Title: Backward clusters and Wild sum in a low-density regime for Hard-sphere systems.
Abstract: When studying the dynamics of a tagged particle in a $N$-particle system it is natural and useful the introduction of the backward cluster of that particle. We want to estimate the size of a backward cluster in a low-density regime. In doing this we establish a rigorous connection with the Wild's sum for the Boltzmann equation and this justifies heuristic assumptions in previous papers. This is a research in collaboration with S. Simonella.
Title: Asymptotic stability of homogeneous equilibria for screened Vlasov-Poisson systems
Abstract: We shall describe a recent alternative proof of the result obtained by Bedrossian, Masmoudi and Mouhot on the Landau damping in the whole space. This approach is based on the characteristic method and the derivation of pointwise in time dispersive estimates for the linearized equation. Joint work with D. Han-Kwan (Polytechnique) and T. Nguyen (Penn State)
Title: Regularity estimates for the Boltzmann equation without cutoff.
Abstract: We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of the hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exists after certain time T only if at least one of these hydrodynamic quantities blows up. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate. We apply them to gain regularity iteratively and we combine them with a pointwise decay estimate for large velocities. This talk is based on a series of papers. Part of them are joint work with Cyril Imbert, and part are joint with Clement Mouhot as well.
Title: Global nonlinear stability of Minkowski space for the massive and massless Einstein–Vlasov systems
Abstract: The Einstein—Vlasov system describes an ensemble of collisionless particles interacting via gravity, as modelled by general relativity. Under the assumption that all particles have equal mass the system is qualitatively different depending on whether this mass is zero or nonzero. I will present two theorems concerning the global nonlinear stability of the trivial solution, Minkowski space, in the two respective cases. The massive case is joint work with Hans Lindblad.
Title: Anomalous diffusion for kinetic Fokker-Planck equation
Abstract: We consider a one dimensional particle having Langevin dynamics where the velocity $(v_t)_{t\geq 0}$ is a reversible diffusion process with invariant probability measure being of Cauchy type : i.e is asymptoticly equivalent to $l v l^{-\beta}$, $\beta >0$. We prove that, under a suitable rescaling, the position process resembles a Brownian motion if $\beta>5$, a stable process if $\beta \in [1,5)$. This recovers a work by G. Lebeau and M. Puel with an alternative probabilist approach.
Title: Blow-up solutions to 3D Euler are hydrodynamically unstable
Abstract: We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. We explain why the prediction of such a blow-up, via direct numerical experiments, is so difficult. We show how, in such a scenario, the solution becomes unstable as time approaches the blow-up time. This is a joint work with Misha Vishik.
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