Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian

Abstract

We study the local well-posedness of the nonlinear Schrödinger 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 Hk when k⩽32. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in Hk, k∈(1,32], almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.