Random data Cauchy problem for supercritical Schrödinger equations
arXiv:0901.4238
Abstract
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 is large enough and strongly ill-posed is is below some critical threshold . 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 . In the appendix we prove the equivalence between the smoothing effect for a Schrödinger operator with confining potential and the decay of the associate spectral projectors.
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