具临界非线性项的随机非线性Schrdinger方程的整体解

Global Solution for the Stochastic Nonlinear Schrdinger Equation with Critical Nonlinear Term

在二维空间中,讨论带调和势且具临界非线性项的一类随机非线性Schrdinger方程的Cauchy问题,在能量空间中研究其解整体存在的充分条件.借助于随机分析及偏微分方程的基本理论,利用It幃公式、鞅不等式和Gagliardo-Nirenberg不等式,通过估计能量泛函的期望得到在初值充分小时,Cauchy问题的解是整体存在的.

We discuss the Cauchy problem of the critical stochastic nonlinear Schrdinger equation with a harmonic potential in two dimensional space.By using Itó formula,Martingale inequality,Gagliardo-Nirenberg inequality,and estimating the expectation of energy functional,we derive global existence of the solution for the equation with sufficiently small initial data.