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

Hiroyuki  Hirayama; Mamoru  Okamoto

doi:10.3934/dcds.2016102

Article Contents

2016, Volume 36, Issue 12: 6943-6974. Doi: 10.3934/dcds.2016102

This issue Previous Article Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation Next Article Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function

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

Hiroyuki Hirayama^1, and
Mamoru Okamoto^2,

1.
Organization for Promotion of Tenure Track, University of Miyazaki, 1-1, Gakuenkibanadai-nishi, Miyazaki, 889-2192
2.
Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City, 380-8553

Received: January 31, 2016

Revised: February 29, 2016

Published: May 2017

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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 $s_c := \frac{d}{2}-\frac{1}{m-1}$.

Keywords:

Random data Cauchy problem,
Schrödinger equation.

Mathematics Subject Classification: Primary: 35Q55.

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