The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities

Shuai Zhang; Shaopeng Xu

doi:10.3934/cpaa.2020149

Article Contents

2020, Volume 19, Issue 6: 3367-3385. Doi: 10.3934/cpaa.2020149

This issue Previous Article On the Sobolev embedding properties for compact matrix quantum groups of Kac type Next Article Global solutions of shock reflection problem for the pressure gradient system

The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities

Shuai Zhang^1, , and
Shaopeng Xu^2,

1.
School of Mathematical Sciences, Peking University, Beijing 100871, China
2.
School of Science, Hainan University, Haikou 570228, China

^* Corresponding author
^* Corresponding author

Received: September 30, 2019

Revised: December 31, 2019

Published: March 2020

Shaopeng Xu is supported by Hainan Provincial Natural Science Foundation of China (No.2019RC168)

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Abstract

Chen and Zhang [7] consider the probabilistic Cauchy problem of the fourth order Schrödinger equation

$ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha u)_{|\alpha|\leq2},(\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq3, \end{align*} $

where $ P_m $ is a homogeneous polynomial of degree $ m $. The almost sure local well-posedness and small data global existence were obtained in $ H^s(\mathbb{R}^d) $ with the regularity threshold $ s_c-1/2 $ when $ d\geq3 $, where $ s_c: = d/2-2/(m-1) $ is the scaling critical regularity. For the lower regularity threshold $ (d-1)s_c/d $ with $ m = 2 $ and $ s_c-\min\{1,d/4\} $ with $ m\geq3 $, we get the corresponding well-posedness of the following fourth order nonlinear Schrödinger equation

$ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq2 \end{align*} $

on $ {\mathbb{R}}^d $ ($ d\geq2 $) with random initial data.

Keywords:

Mathematics Subject Classification: Primary: 35Q41, 35Q55; Secondary: 35R60, 60H15.

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