Random data Cauchy problem for supercritical Schrödinger equations

Laurent Thomann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

In this paper we consider the Schrödinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $H^{s}$ if $s$ is large enough and strongly ill-posed is $s$ is below some critical threshold $s_{c}$. Here we use the randomisation method of the inital conditions, introduced by N. Burq-N. Tzvetkov and we are able to show that the equation admits strong solutions for data in $H^{s}$ for some $s < s_{c}$. In the appendix we prove equivalence between smoothing effect for a schrödinger operator with confining potential and decay of the associate spectral projectors.

Mots clés: Nonlinear Schrödinger equation; potential; smoothing effect; random data; supercritical equation

Domaines: Equations aux dérivées partielles [math.AP]

Laurent Thomann. Random data Cauchy problem for supercritical Schrödinger equations. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2009, 26 (6), pp.2385--2402. ⟨hal-00357319v2⟩