# MathSciDoc: An Archive for Mathematician ∫

#### Dynamical Systemsmathscidoc:2205.11003

Stoch PDE: Anal Comp, 2021.1
We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n\in\mathbb{N}$, under Dirichlet boundary conditions $u_t-\nu\Delta u+i|u|^2u=\sqrt{\nu}\eta(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad \nu>0,$ where $\eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $\| u(t)\|_m^2 \le C\nu^{-m},$ uniformy in $t\ge0$ and $\nu>0$. In this work we prove that for small $\nu>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ become large at least to the order of $\nu^{-\kappa_{n,m}}$ with $\kappa_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu})$.
No keywords uploaded!
[ Download ] [ 2022-05-17 13:34:26 uploaded by huangguan ] [ 37 downloads ] [ 0 comments ]
@inproceedings{guan2021on,
title={ON THE ENERGY TRANSFER TO HIGH FREQUENCIES IN THE DAMPED/DRIVEN NONLINEAR SCHRO ̈DINGER EQUATION},
author={Guan HUANG, and Sergei Kuksin},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517133426144744199},
booktitle={Stoch PDE: Anal Comp},
year={2021},
}

Guan HUANG, and Sergei Kuksin. ON THE ENERGY TRANSFER TO HIGH FREQUENCIES IN THE DAMPED/DRIVEN NONLINEAR SCHRO ̈DINGER EQUATION. 2021. In Stoch PDE: Anal Comp. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517133426144744199.
Please log in for comment!