Probabilistic well-posedness and Gibbs measure evolution for the non linear Schrödinger equation on the 2D sphere.
It is known that the minimal Sobolev regularity needed for the semi-linear local well-posedness of the non linear Schrödinger equation, posed on a two dimensional spatial domain depends heavily on the geometry of this domain. In this talk we will observe a similar phenomenon in the study of the probabilistic well-posedness. We will show that in the study of solutions with low regularity gaussian random initial data the structure of the solutions on flat tori and on the standard sphere are totally different. As one may expect this phenomenon is related to the existence of stable closed geodesics but this time it manifests in a new way via the structure of the resonant manifold. Our methods are strong enough to deal with data distributed according to the Gibbs measure. This is a joint work with Nicolas Burq, Nicolas Camps and Chenmin Sun.
We will see that the Fisher information is monotone decreasing in time along solutions of the space-homogeneous Boltzmann equation for a large class of collision kernels covering all classical interactions derived from systems of particles. For general collision kernels, a sufficient condition for the monotonicity of the Fisher information along the flow is related to the best constant for an integro-differential inequality for functions on the sphere, which belongs in the family of the Log-Sobolev inequalities. As a consequence, we establish the existence of global smooth solutions to the space-homogeneous Boltzmann equation in the main situation of interest where this was not known, namely the regime of very soft potentials. Joint work with Luis Silvestre and Cédric Villani.
Stability of Rayleigh-Jeans equilibria in the kinetic FPUT equation.
In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from the microscopic Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the global existence and stability of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibrium solutions. This is a joint work with Pierre Germain (Imperial College London) and Joonhyun La (KIAS).
Deep- and shallow-water limits of statistical equilibria for the intermediate long wave equation.
The intermediate long wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, providing a natural connection between the deep-water regime (= the BO regime) and the shallow-water regime (= the KdV regime). Exploiting the complete integrability of ILW, I will discuss the statistical convergence of ILW to both BO and KdV, namely the convergence of the higher order conservation laws for ILW and their associated invariant measures. In particular, as KdV possesses only half as many conservation laws as ILW and BO, we observe a novel 2-to-1 collapse of ILW conservation laws to those of KdV, which yields alternative modes of convergence for the associated measures in the shallow-water regime. This talk is based on joint work with Guopeng Li and Tadahiro Oh.
Propagation of global analyticity and unique continuation for semilinear wave equations.
In this talk, I will first present the known results of unique continuation for wave-like equations. I will explain the difficulties of obtaining global results under natural geometrical assumptions. Then, I will present a recent result, in collaboration with Cristobal Loyola, where we prove unique continuation for semilinear wave equations under the geometric control assumption. A crucial step is the global propagation of analyticity in time from open sets verifying the geometric control condition. The proof uses control methods associated with Hale-Raugel ideas concerning attractor regularity.
Semiclassical defect measure of magnetic Laplacians on hyperbolic surfaces.
The purpose of this talk is to describe the quantum limits of eigenstates of magnetic Laplacians on hyperbolic surfaces. We will see that three distinct regimes appear, depending on the energy level: low, critical, and high. The main result is a quantitative version of Quantum Unique Ergodicity at the critical energy: at this energy level, eigenstates concentrate at a quantitative rate towards the Liouville measure. This is joint work with Laurent Charles.
Stability of discontinuous flow for incompressible inviscid fluid.
The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effects of the viscosity. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities (Annals of Math. 2005). However, achieving this limit with physical viscosities remained an open question up to our recent result together with Geng Chen and Moon-Jin Kang. In this presentation, we will provide a basic overview of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. We will describe the basic ideas and difficulties involved in the study of physical inviscid limits in the contex
Freidlin-Gartner formula and asymptotic profile in reaction-diffusion equations
We address the question of the large-time behavior of solutions of reaction-diffusion equations in periodic media. We will start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin-Gartner formula. We will outline a proof of the formula that holds true for general types of reaction terms. We will then present some recent results for the bistable equation, obtained in collaboration with H. Guo and F. Hamel, about a "regular" version of the Freidlin-Gartner formula and the convergence of the profile of the solution towards pulsating traveling fronts.
Mean-field limits for systems of interacting biological neurons.
The dynamics of interacting biological neurons is commonly described through multi-agent models on graphs. We only assume an extended mean-field scaling but otherwise allow any arbitrary choice of synaptic weights. We are nevertheless able to identify an emerging structure through a new notion of extended graphons and novel regularity estimates on the system. This allows to derive limiting equations that are similar to the famous neural field models, but without having to assume some redefined spatial structure on the connection maps between neurons. Even more remarkably, the new trajectories of individual neurons and to quantify how similar two neurons are. This corresponds to a series of joint work with D. Poyato, V. Schmutz, J. Soler, and D. Zhou.
Mean-field limits for systems of interacting biological neurons.
The dynamics of interacting biological neurons is commonly described through multi-agent models on graphs. We only assume an extended mean-field scaling but otherwise allow any arbitrary choice of synaptic weights. We are nevertheless able to identify an emerging structure through a new notion of extended graphons and novel regularity estimates on the system. This allows to derive limiting equations that are similar to the famous neural field models, but without having to assume some redefined spatial structure on the connection maps between neurons. Even more remarkably, the new trajectories of individual neurons and to quantify how similar two neurons are. This corresponds to a series of joint work with D. Poyato, V. Schmutz, J. Soler, and D. Zhou.
Smoluchowski coagulation equation with a flux of dust particles.
We construct a time-dependent solution to the Smoluchowski coagulation equation with a constant flux of dust particles entering through the boundary at zero. The dust is instantaneously converted into particles and these solutions, that we call flux solutions, have linearly increasing mass. The construction is made for a general class of non-gelling coagulation kernels for which stationary solutions do exist. In the complementary regime, no flux solution is expected to exist. Flux solutions are expected to converge to a stationary solution in the large time limit. We show that this is indeed true in the particular case of the constant kernel with zero initial data. (Based on a joint work with Aleksis Vuoksenmaa - U. Helsinki).
Commutator estimates and mean-field limits for Coulomb/Riesz gases.
I will discuss the interplay between entropy, energy, and functional inequalities in the form of commutator estimates in establishing the mean-field convergence/propagation of chaos at the optimal rate for the first-order dynamics of repulsive Coulomb/Riesz gases in the full range of allowable potentials. Time permitting, I will also discuss applications to the derivation of the Lake equation as a supercritical mean-field limit of such systems under optimal scaling assumptions. This talk is based on joint work with Elias Hess-Childs and Sylvia Serfaty.
Mean-field limits for systems of interacting biological neurons.
The dynamics of interacting biological neurons is commonly described through multi-agent models on graphs. We only assume an extended mean-field scaling but otherwise allow any arbitrary choice of synaptic weights. We are nevertheless able to identify an emerging structure through a new notion of extended graphons and novel regularity estimates on the system. This allows to derive limiting equations that are similar to the famous neural field models, but without having to assume some redefined spatial structure on the connection maps between neurons. Even more remarkably, the new trajectories of individual neurons and to quantify how similar two neurons are. This corresponds to a series of joint work with D. Poyato, V. Schmutz, J. Soler, and D. Zhou.
Mean-field limits for systems of interacting biological neurons.
The dynamics of interacting biological neurons is commonly described through multi-agent models on graphs. We only assume an extended mean-field scaling but otherwise allow any arbitrary choice of synaptic weights. We are nevertheless able to identify an emerging structure through a new notion of extended graphons and novel regularity estimates on the system. This allows to derive limiting equations that are similar to the famous neural field models, but without having to assume some redefined spatial structure on the connection maps between neurons. Even more remarkably, the new trajectories of individual neurons and to quantify how similar two neurons are. This corresponds to a series of joint work with D. Poyato, V. Schmutz, J. Soler, and D. Zhou.
Chargement... 
