Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i{\partial }_t+\Delta )u=\pm \partial ({\overline{u}}^m)$ on ${\mathbb{R}}^d$, $d\ge 1$, 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 $s>\max \left(\frac{d-1}{d}{s}_c,\frac{s_c}{2},s_c-\frac{d}{2(d+1)}\right)$ for $d+m\ge 5$, where $s$ is below the scaling critical regularity $s_c:=\frac{d}{2}-\frac{1}{m-1}$.

25 pages

References in corpus (5)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Random data Cauchy theory for supercritical wave equations II : A global existence result
• On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^d$, $d\ge 3$
• Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity
• A remark on global well-posedness of the derivative nonlinear Schrödinger equation on the circle

Cited by in corpus (9)

• Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation
• On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
• Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^3$
• Almost sure well-posedness for the cubic nonlinear Schrödinger equation in the super-critical regime on $\TTT^d$, $d\ge 3$
• Random data Cauchy problem for a generalized KdV equation in the supercritical case
• Probabilistic pointwise convergence problem of some dispersive equations
• Pointwise convergence problem of Ostrovsky equation with rough data and random data
• Random data theory for the cubic fourth-order nonlinear Schrödinger equation
• The Cauchy problem for the generalized KdV equation with rough data and random data