Almost sure well-posedness and orbital stability for Schr\"odinger equation with potential

Abstract

In this paper, we study the almost sure well-posedness theory and orbital stability for the nonlinear Schr\"odinger equation with potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u+|u|^{2}u=0,\ (x, t) \in \mathbb{R}^4 \times \mathbb{R}, \\ \left.u\right|_{t=0}=f \in H ^s(\mathbb{R}^4), \end{array}\right. \end{equation*} where $\frac{1}{3}<s<1$ and $V(x):\mathbb{R}^4\rightarrow \mathbb{R}$ satisfies appropriate conditions. The main idea in the proofs is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing and maximal function spaces. To our best knowledge, this is the first orbital stability result for this model.