Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian
Résumé
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 wellposedness result is known in the Sobolev spaces H k when k 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 ∈ (1, 3 2 ], almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.
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