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Frederic Rousset

Publications and source records attributed to Frederic Rousset.

12 recordsLinked to original sources
arXiv2017

Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit

In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for times which are polynomial in terms of the speed of light and of the inverse of size of initial perturbations. We build a kind of higher-order Vlasov-Darwin approximation which allows us to reach arbitrarily large powers of the speed of light.

math.AP↗
arXiv2017

The Nonlinear Schrodinger equation with a potential in dimension 1

We consider the cubic nonlinear Schrodinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, and does not have bound states, we obtain the long time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform - the so-called Weyl-Kodaira-Titchmarsh theory -, a precise understanding of the "nonlinear spectral measure" associated to the equation, and nonlinear stationary phase arguments as well as multilinear estimates in this distorted setting.

math.AP↗
arXiv2016

Long wave limit for Schrodinger maps

We study long wave limits for general Schrodinger maps systems into Kahler manifolds with a constraining potential vanishing on a Lagrangian submanifold. We obtain KdV type systems set on the tangent space of the submanifold. Our general theory is applied to study the long wave limit of the Gross-Pitaevskii equation, and of the Landau-Lifshitz systems for ferromagnetic and antiferromagnetic chains.

math.AP↗
arXiv2013

Global Regularity for some Oldroyd-B type Models

We investigate some critical models for visco-elastic flows of Oldroyd-B type in dimension two. We use a transformation which exploits the Oldroyd-B structure to prove an L^\infty bound on the vorticity which allows us to prove global regularity for our systems.

math.AP↗
arXiv2010

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 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 $L^\infty$. This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.

math.AP↗
arXiv2009

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 $\kappa \geq 0$ which may vanish.

math.AP↗
arXiv2008

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation "have the same sign". Our result applies to the generalized KP-I equation, the Nonlinear Schr\"odinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other contexts.

math.AP↗