Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity

Abstract

We consider the Cauchy problem for the fourth order nonlinear Schrödinger equation with derivative nonlinearity $(i{\partial }_t+{\Delta }^2)u=\pm \partial (|u{|}^{2}u)$ on ${\mathbb{R}}^d$, $d\ge 3$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_d}$ or $|\nabla |={\mathcal{F}}^{-1}[|\xi |\mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s({\mathbb{R}}^d)$ with $\max (\frac{d-5}{2},\frac{d-5}{6})<s$, whose lower bound is below the scale critical regularity $s_c=\frac{d-3}{2}$.

References in corpus (1)

• On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^d$, $d\ge 3$

Cited by in corpus (1)

• Random data Cauchy problem for a generalized KdV equation in the supercritical case