Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

Abstract

In this paper, we study the local well-posedness of the cubic Schrödinger equation:with randomized initial data, and P being an operator of degree s ≥ 2. Using careful estimates in anisotropic spaces, we improve and extend known results for the standard Schrödinger equation (that is, P being Laplacian) to any dimension under natural assumptions on P , whose Fourier symbol might be sign changing. Quite interestingly, we also exhibit the existence of a new regime depending on s and d, which was not present for the Laplacian. 2000 Mathematics Subject Classification. 35Q41, 37L50. Key words and phrases. Schrödinger equation; almost-sure local well-posedness; random initial data. J.-B.C. supported by FCT -Fundação para a Ciência e a Tecnologia, under the project: UIDB/04561/2020, J. F. was partly supported by grant NSF-DMS-1816408.d+1 d−22 in the following sense: there exist c, C, γ > 0 such that for each 0 < T << 1, there exists a set Ω T ⊂ Ω with the following properties :