Frédéric Rousset

Global Well-Posedness for Euler–Boussinesq System with Critical Dissipation

In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global well-posedness results.

Uniform Regularity for the Navier–Stokes Equation with Navier Boundary Condition

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier–Stokes equation with the Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L∞. This allows us to obtain the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.

Generic types and transitions in hyperbolic initial–boundary-value problems

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition. Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system. Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data

In this paper we prove the global well-posedness for a three-dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier-Stokes equation with a transport-diffusion equation governing the temperature. Our result holds uniformly with respect to the heat conductivity coefficient

The KdV/KP-I Limit of the Nonlinear Schrödinger Equation

kappa geq 0

Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems

which may vanish.

Nonlinear stability of semidiscrete shock waves

We justify rigorously the convergence of the amplitude of solutions of nonlinear Schrodinger-type equations with nonzero limit at infinity to an asymptotic regime governed by the Korteweg–de Vries (KdV) equation in dimension 1 and the Kadomtsev–Petviashvili I (KP-I) equation in dimensions 2 and greater. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that there is no vortex formation for the global solution of the nonlinear Schrodinger equation in the energy space and deduce from this the convergence toward the unique solution in the energy space of the KdV equation. In arbitrary dimensions, we use a hydrodynamic reformulation of the nonlinear Schrodinger equation and recast the problem as a singular limit for a hyperbolic system. We thus prove that smooth

Landau Damping in Sobolev Spaces for the Vlasov-HMF Model

H^s

Geometric Optics and Boundary Layers for Nonlinear-Schrödinger Equations

solutions exist on a time interval independent of the small parameter. We then pass to the limit by a compactness argument and obtain the KdV/KP-I equation.

Multi-solitons and Related Solutions for the Water-waves System

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity e, u e t + F(u e ) x = e(B(u e )u e x) x . When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that u e converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for B invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlets boundary condition.