Probabilistic local well-posedness for the Schr{\''o}dinger equation posed for the Grushin Laplacian

Louise Gassot, Mickaël Latocca

Published 2021-03-05Version 1

We study the local well-posedness of the nonlinear Schr{\''o}dinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in the Sobolev spaces $H^k$ when $k \leq \frac{3}{2}$. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in $H^k$, $k \in (1,\frac{3}{2}]$, almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.