Nicolas Burq

Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local well-posedness results in any dimension, as well as global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of cubic defocusing nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential

Abstract We prove spacetime weighted- L 2 estimates for the Schrodinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.

Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes

Resume On demontre que la condition de controle geometrique de C. Bardos, G. Lebeau et J. Rauch est une condition necessaire et suffisante pour la eontrolabilite exacte des ondes avec conditions de Dirichlet sur le bord.

Random data Cauchy theory for supercritical wave equations II: a global existence result

We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in

Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

\bigcap_{s<1/2} (H^s(\Theta)\times H^{s-1}(\Theta))

Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces

. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2, 3] on the non linear Schrödinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.

On the water-wave equations with surface tension

We give estimates for the

Random data Cauchy theory for supercritical wave equations I: local theory

L^p

LOWER BOUNDS FOR SHAPE RESONANCES WIDTHS OF LONG RANGE SCHRÖDINGER OPERATORS

norm (

Probabilistic well-posedness for the cubic wave equation

2\leq p \leq +\infty