Probabilistic local well-posedness for the Schr\"odinger equation posed for the Grushin Laplacian

TL;DR

This paper establishes almost-sure local well-posedness for the nonlinear Schrödinger equation with the Grushin operator for low regularity initial data in Sobolev spaces, using probabilistic methods.

Contribution

It proves the first known local well-posedness results in Sobolev spaces with regularity below 1.5 for the Schrödinger equation associated with the Grushin operator, via probabilistic techniques.

Findings

Almost-sure local well-posedness for $k \, \in\, (1, \frac{3}{2}]$

Existence of a large family of initial data with probabilistic well-posedness

Use of bilinear and trilinear estimates in the proof

Abstract

We study the local well-posedness of the nonlinear Schr\"odinger 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.