Almost Sure Global Well-posedness for the Fourth-order nonlinear Schrodinger Equation with Large Initial Data

Abstract

We consider the following fourth-order nonlinear Schr{\”o}dinger equation(4NLS) \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u=c_1u^m+c_2(\partial u)u^{m-1}+c_3(\partial u)^2u^{m-2}, \end{align*} and establish the conditional almost sure global well-posedness for random initial data in $H^s(\mathbb{R}^d)$ for $s\in (s_c-1/2,\ s_c]$ when $d\geq3$ and $m\geq5$, where $s_c:=d/2-2/(m-1)$ is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new $M$-norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for $d=2,\ m\geq4$ and $ d\geq3,\ 3\leq m<5$.

Locations

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