On a quasilinear Schrödinger equation: the small frequency limit
In this talk, I will present some recent results on a quasilinear Schrödinger equation with a power nonlinearity. After showing the uniqueness and the non-degeneracy of the positive radial solution uω for all ω>0, I will describe its asymptotic behavior in the limit ω-->0. This gives some important information about the orbital stability of uω and the uniqueness of normalized ground states. Joint work with François Genoud.
Luis Silvestre --- Lectures: Estimates for kinetic equations
We review some recent results on the analysis of kinetic equations in the homogeneous and inhomogeneous setting. We will discuss how the monotonicity of the Fisher information can be used to show that the Landau and Boltzmann equations have global-in-time smooth solutions in the space homogeneous setting. For space-inhomogeneous solutions, we will discuss scenarios of potential singularity formation and analyze what can be ruled out using conditional regularity results.
Nonlinear interpolation and the flow map of quasilinear equations
I will present an abstract result showing that for the flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data-to-solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms. Though I will not present the abstract general result, I will explain its proof and illustrate on some simple examples how we can easily check the assumptions. This is a joint work with T. Alazard, M. Ifrim, D. Tataru and C. Zuily.
The compressible Euler equations with damping: global well-posedness and relaxation limit in Lp spaces
In this talk, I will present recent results on the global well-posedness of the compressible Euler system with damping in Rd (d≥1) and its relaxation to the diffusive porous medium equation. Particular emphasis will be placed on the links between this singular relaxation problem, hypocoercive phenomena, and hyperbolic approximations of diffusive systems. In particular, we will see that the spectral structure of the linearized dynamics reveals hidden dissipative mechanisms governing both the large-time behavior of solutions and their convergence to the diffusive regime. I will then discuss nonlinear stability and convergence results obtained through a refined Littlewood–Paley frequency analysis.
Dynamics of the wave equation with conformally critical nonlinearity
This talk, based on joint works with Giuseppe Negro (Instituto Superior Técnico, Lisboa), concerns the focusing wave equation with a conformally critical nonlinearity (e.g. cubic in space dimension 3). We will show that global, nonscattering solutions of this equation are unstable, and construct a large family of solutions of this type with a self-similar asymptotic behaviour as t goes to infinity.
Determination of the long-time behaviour of solutions to the two-dimensional Keller-Segel system with critical mass
This presentation will describe the dynamics of solutions to the parabolic-elliptic Keller-Segel system in two dimensions. It will focus on the case of the critical mass of 8π, for solutions with finite second moment. Such solutions are global and concentrate in infinite time and an explicit example of a solution concentrating a stationary state at a scale log(t)-1/2 was constructed in the radial class and in the non-radial class. The main result that will be presented shows that this is the universal dynamics at critical mass, i.e. that all general solutions will converge to a stationary state which concentrates at scale log(t)-1/2 around the center of mass of the solution. The proof combines soliton resolution techniques, with the control of the evolution of the solution in various critical spaces, and with the perturbative analysis around an approximate solution involving multiple scales. This is joint work to appear with Federico Buseghin.
Luis Silvestre --- Lectures: Estimates for kinetic equations
We review some recent results on the analysis of kinetic equations in the homogeneous and inhomogeneous setting. We will discuss how the monotonicity of the Fisher information can be used to show that the Landau and Boltzmann equations have global-in-time smooth solutions in the space homogeneous setting. For space-inhomogeneous solutions, we will discuss scenarios of potential singularity formation and analyze what can be ruled out using conditional regularity results.
An explicit formula for the Benjamin–Ono hierarchy with applications to traveling waves and zero-dispersion limits
In this talk, we demonstrate how the Lax pair structure leads to an explicit formula for the Benjamin–Ono Hierarchy on the line. We then present two main applications of this formula. First, we obtain a complete classification of traveling wave solutions for all higher-order flows within the hierarchy. Second, we investigate the zero-dispersion limit of these flows and provide a precise characterization of the limit as an alternating sum of branches. This is a joint work with Patrick Gérard.
Luis Silvestre --- Lectures: Estimates for kinetic equations
We review some recent results on the analysis of kinetic equations in the homogeneous and inhomogeneous setting. We will discuss how the monotonicity of the Fisher information can be used to show that the Landau and Boltzmann equations have global-in-time smooth solutions in the space homogeneous setting. For space-inhomogeneous solutions, we will discuss scenarios of potential singularity formation and analyze what can be ruled out using conditional regularity results.
The Cauchy problem for quasi-linear parabolic systems revisited
In this talk, I'll present a joint work with I. Gallagher offering an alternative to Amann's theory for quasi-linear parabolic systems. As an example of use, I'll introduce the SKT system -- a cross-diffusion model introduced by Shigesada, Kawasaki and Teramoto -- and conclude with some related open questions.
Dissipative Solutions to a Compressible Non-Newtonian Navier-Stokes-Korteweg System with Density-Dependent Viscous Stress Tensor
The main objective of this paper is to prove that if capillarity effect is taken into account then there exist dissipative solutions to a system describing viscoplastic compressible flows with density dependent viscosities in a periodic domain Td with d=2, 3. We calculate the relative entropy inequality and in consequence show existence of dissipative solutions and the weak-strong uniqueness for this system.
We prove eigenvalue asymptotics and concentration of eigenfunctions of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of sound waves in gas planets. The talk is based on joint works with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat, and with Larry Read.
Diffusive Hamilton-Jacobi equations: gradient blow-up singularities, Liouville-type theorems and continuation after blow-up
We consider the diffusive Hamilton-Jacobi equation ut-Δu=|∇u|p with homogeneous Dirichlet boundary conditions, which plays an important role in stochastic optimal control theory and in certain models of surface growth (KPZ). Despite its simplicity, it displays a variety of interesting and surprising behaviors and significant progress has been made in the past ten years. We will discuss the following issues: - Gradient blow-up (GBU) on the boundary: time rate, single-point GBU, space and time-space profiles; - Liouville type theorems and their applications; - Continuation after GBU as a global viscosity solution with loss and recovery of boundary conditions.
Luis Silvestre --- Lectures: Estimates for kinetic equations
We review some recent results on the analysis of kinetic equations in the homogeneous and inhomogeneous setting. We will discuss how the monotonicity of the Fisher information can be used to show that the Landau and Boltzmann equations have global-in-time smooth solutions in the space homogeneous setting. For space-inhomogeneous solutions, we will discuss scenarios of potential singularity formation and analyze what can be ruled out using conditional regularity results.
The mass-critical cubic NLS with periodic initial data
I will present recent joint work with Beomjong Kwak on the Schrödinger equation with the cubic (focusing and defocusing) nonlinearity posed on the two-dimensional torus. First, the optimal L^4 Strichartz estimate will be discussed, leading to global well-posed for initial data which is small in L^2. This is based on a new counting argument which relies on incidence geometry, in particular the Szemeredi-Trotter Theorem. Second, the extension to large initial data will be discussed, which is based on an Inverse Strichartz Theorem. This is based on methods involving higher order Fourier analysis.
Chargement... 
