论文标题
关于阻尼/驱动的非线性Schrödinger方程（扩展版）的能量转移到高频
On The Energy Transfer To High Frequencies In The Damped/Driven Nonlinear Schrödinger Equation (Extended Version)
论文作者
论文摘要
我们考虑在$ n $ -cube $ k^{n} \ subset \ subset \ mathbb {r}^n $，$ n $中，在$ n $ -cube $ k^{n} \ subset $ k^{ k^{n}，\ quad u | _ {\ partial k^{n}} = 0，\quadν> 0，\]，其中$η（t，x）$是一种随机的力，是白色的，在太空中是白色的。众所周知，解决方案的Sobolev规范满足$ \ | u（t）\ | _m^2 \ lecν^{ - m}，$均匀地在$ t \ ge0 $和$ν> 0 $中。在这项工作中，我们证明，对于小$ν> 0 $和任何初始数据，概率很大，Sobolev规范$ \ | u（t，\ cdot）\ | _m $ a $ m> 2 $的解决方案的$ m> $至少与$ν^{ - κ_{n，m}} $的顺序至少相对于$ n，m} $ nime Inker $ \ MATHCAL {O}（\ frac {1}ν）$。
We consider a damped/driven nonlinear Schrödinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-νΔu+i|u|^2u=\sqrtνη(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad ν>0, \] where $η(t,x)$ 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 Cν^{-m}, $ uniformly in $t\ge0$ and $ν>0$. In this work we prove that for small $ν>0$ 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 $ν^{-κ_{n,m}}$ with $κ_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}ν)$.
