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

2022/03/07 by Casteras, Jean-Baptiste, Foldes, Juraj, Uraltsev, Gennady
#35A01 #35G20 #35Q55 #35R60 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2203.03500

Abstract

In this paper, we study the local well-posedness of the cubic Schrödinger equation: (i ∂_t - \mathscrL) u = ± |u|² u on I × ℝ^d, with randomized initial data, and \mathscrL being an operator of degree σ≥ 2. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation (i.e. \mathscrL = Δ) to any dimension and obtain results under natural assumptions for general \mathscrL.

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