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Preprints, Working Papers, ... Year : 2022

Multilinear smoothing and local well-posedness of a stochastic quadratic nonlinear Schrödinger equation

Nicolas Schaeffer

Abstract

In this article, we study a d-dimensional stochastic quadratic nonlinear Schrödinger equation (SNLS), driven by a fractional derivative (of order −α < 0) of a space-time white noise: i∂ t u − ∆u = ρ 2 |u| 2 + ∇ −α Ẇ , t ∈ [0, T ] , x ∈ R d , u 0 = φ , where ρ : R d → R is a smooth compactly-supported function. When α < d 2 , the stochastic convolution is a function of time with values in a negative-order Sobolev space and the model has to be interpreted in the Wick sense by means of a time-dependent renormalization. When 1 ≤ d ≤ 3, combining both the classical Strichartz estimates and a deterministic local smoothing, we establish the local well-posedness of (SNLS) for a small range of α, in the spirit of [2]. Then, we revisit our arguments and establish multilinear smoothing on the second order stochastic term. This allows us to improve our local well-posedness result for some α. We point out that this is the first result concerning a Schrödinger equation on R d driven by such an irregular noise and whose local well-posedness results from both a stochastic multilinear smoothing and a deterministic local one combined with Strichartz inequalities.
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Dates and versions

hal-03632235 , version 1 (06-04-2022)

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Nicolas Schaeffer. Multilinear smoothing and local well-posedness of a stochastic quadratic nonlinear Schrödinger equation. 2022. ⟨hal-03632235⟩
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