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On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation
Stochastics and Partial Differential Equations: Analysis and Computations ( IF 1.4 ) Pub Date : 2021-01-02 , DOI: 10.1007/s40072-020-00187-2
Guan Huang , Sergei Kuksin

We consider a damped/driven nonlinear Schrödinger equation in \(\mathbb {R}^n\), where n is arbitrary, \({\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,\) under odd periodic boundary conditions. Here \(\eta (t,x)\) is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy \( \Vert u(t)\Vert _m^2 \le C\nu ^{-m}, \) uniformly in \(t\ge 0\) and \(\nu >0\). In this work we prove that for small \(\nu >0\) and any initial data, with large probability the Sobolev norms \(\Vert u(t,\cdot )\Vert _m\) 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 \(\mathcal {O}(\frac{1}{\nu })\). It proves that solutions of the equation develop short space-scale of order \(\nu \) to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation.



中文翻译:

关于阻尼/驱动非线性Schrödinger方程中的能量到高频的传递

我们考虑\(\ mathbb {R} ^ n \)中的阻尼/驱动非线性Schrödinger方程,其中n是任意的,\({\ mathbb {E}} u_t- \ nu \ Delta u + i | u | ^ 2u = \ sqrt {\ nu} \ eta(t,x),\ quad \ nu> 0,\)在奇周期边界条件下。这里\(\ eta(t,x)\)是一个随机力,时间为白色,空间为平滑。已知解的Sobolev范数在\(t \ ge 0 \)\(\中均满足\(\ Vert u(t)\ Vert _m ^ 2 \ le C \ nu ^ {-m},\)nu> 0 \)。在这项工作中,我们证明了对于较小的\(\ nu> 0 \)和任何初始数据,Sobolev范数\(\ Vert u(t,\ cdot)\ Vert _m \)具有\(m> 2 \ )至少成为大到的顺序\(\ NU ^ { - \卡帕_ {N,M}} \)\(\卡帕_ {N,M}> 0 \) ,上的顺序的时间间隔\(\ mathcal {O}(\ frac {1} {\ nu})\)。证明了该方程的解在正数上发展了\(\ nu \)阶的短空间尺度,并严格建立了方程的(直接)能量级联。

更新日期:2021-01-02
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