Multilinear smoothing and local well-posedness of a stochastic quadratic nonlinear Schr{ö}dinger equation

Abstract

In this article, we study a $d$-dimensional stochastic quadratic nonlinear Schrödinger equation (SNLS), driven by a fractional derivative (of order $-\alpha <0$) of a space-time white noise: $$\{\begin{array}{l}i{\partial }_{t}u-\Delta u={\rho }^2|u{|}^2+⟨\nabla {⟩}^{-\alpha }\dot{W},t\in [0,T],x\in {\mathbb{R}}^d,\\ u_0=\varphi ,\end{array}$$ where $\rho :{\mathbb{R}}^d\to \mathbb{R}$ is a smooth compactly-supported function. When $\alpha <\frac{d}{2}$, 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 $1\le d\le 3$, combining both the classical Strichartz estimates and a deterministic local smoothing, we establish the local well-posedness of (SNLS) for a small range of $\alpha$, in the spirit of \cite{Schaeffer1}. 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 $\alpha$. We point out that this is the first result concerning a Schrödinger equation on ${\mathbb{R}}^d$ 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.