On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation

Abstract

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.

Appendices

Appendix A. Some estimates

For any integer $l\in {\mathbb {N}}$ and $F\in H^l$ we have that

$$\begin{aligned} \Vert \exp (iF(x))\Vert _{l} \le C_{l} (1+ |F|_\infty )^{l-1} \Vert F\Vert _{l}. \end{aligned}$$

(A.1)

Indeed, to verify (A.1) it suffices to check that for any non-zero multi-indices $\beta _1,\dots ,\beta _{l'}$, where $1\le l'\leqslant l$ and $|\beta _1| +\cdots + |\beta _{l'}| =l,$ we have

$$\begin{aligned} \Vert \partial _x^{\beta _1} F\cdots \partial _x^{\beta _{l'}} F \Vert _0 \le C |F|_\infty ^{l'-1} \Vert F\Vert _l. \end{aligned}$$

(A.2)

But this is the assertion of Lemma 3.10 in [17]. Similarly,

$$\begin{aligned} \Vert F G \Vert _r \leqslant C_r ( |F|_\infty \Vert G\Vert _r + |G|_\infty \Vert F\Vert _r), \quad F,G \in H^r,\;\; r\in {\mathbb {N}}, \end{aligned}$$

(A.3)

see [17, Proposition 3.7] (this relation is known as Moser’s estimate). Finally, since for $| \beta | \le m$ we have $| \partial _x^\beta v|_{2m/ \beta |} \le C |v|_\infty ^{1- |\beta | /m} \Vert v\Vert _m^{|\beta | /m}$ (see relation (3.17) in [17]), then

$$\begin{aligned} |\langle \!\langle |v|^2 v, v\rangle \!\rangle _m| \le C_m \Vert v\Vert _m^2 |v|_\infty ^2, \qquad |\langle \!\langle |v|^2 v, v\rangle \!\rangle _m| \le C'_m \Vert v\Vert _{m+1}^{\frac{2m}{m+1}} |v|_\infty ^{\frac{2m+4}{m+1}} . \end{aligned}$$

(A.4)

Appendix B. Proof of Theorem 8

Applying Ito’s formula to a solution $v(\tau )$ of Eq. (2.1) we get a slow time version of the relation (5.1):

$$\begin{aligned} \begin{aligned} \Vert v(\tau )\Vert _m^2=\Vert v_0\Vert _m^2+2 \int _{0}^{\tau } \big ( -\Vert v \Vert ^2_{m+1} - \nu ^{-1} \langle \!\langle i|v |^2v , v \rangle \!\rangle _m \big )ds +2 B_m \tau +2 M(\tau ), \end{aligned} \end{aligned}$$

(B.1)

where $M(\tau )= \int _0^{\tau }\sum _{d} b_d|d|^{2m} \langle v_d(s), d\beta _d(s)\rangle .$ Since in view of (A.4)

$$\begin{aligned} {\mathbb {E}}\big | \langle \!\langle |v|^2v, v\rangle \!\rangle _m \big | \le C_m \big ( {\mathbb {E}}\Vert v\Vert _{m+1}^{2}\big )^{\frac{m}{m+1}} {\mathbb {E}}\big ( |v|_\infty ^{2m+4}\big )^{\frac{1}{m+1}}, \end{aligned}$$

then denoting ${\mathbb {E}}\Vert v(\tau )\Vert _r^2 =: g_r(\tau ), \ r \in {\mathbb {N}}\cup \{0\},$ taking the expectation of (B.1), differentiating the result and using (2.3), we get that

$$\begin{aligned} \frac{d}{d\tau } g_m \le -2 g_{m+1} +C_m \nu ^{-1} g_{m+1}^{\frac{m}{m+1}} +2B_m \le -2 g_{m+1}\big ( 1- C'_m \nu ^{-1} g_{m}^{-\frac{1}{m}} +2B_m\big ), \end{aligned}$$

(B.2)

since $g_m \le g_0^{1/(m+1)} g_{m+1}^{m/(m+1)} \le C_m g_{m+1}^{m/(m+1)} .$ We see that if $g_m \ge (2\nu ^{-1} C'_m)^m$, then the r.h.s. of (B.2) is

$$\begin{aligned} \le -g_{m+1} +2B_m \le -C_m^{-1} g_m^{(m+1)/m} +2B_m \le - {\bar{C}}_m \nu ^{-m-1} + 2B_m, \end{aligned}$$

(B.3)

which is negative if $\nu \ll 1$. So if

$$\begin{aligned} g_m(\tau ) < (2\nu ^{-1} C'_m)^m \end{aligned}$$

(B.4)

at $\tau =0$, then (B.4) holds for all $\tau \ge 0$ and (2.4) follows. If $g_m(0)$ violates (B.4), then in view of (B.2) and (B.3), for $\tau \ge 0$, while (B.4) is false, we have that

$$\begin{aligned} \frac{d}{d\tau } g_m \le -C_m g_{m}^{(m+1)/m} +2B_m, \end{aligned}$$

which again implies (2.4). Besides, in view of (B.2),

$$\begin{aligned} \frac{d}{d\tau } g_m \le - g_{m} +C_m (\nu , |v_0|_\infty , B_{m_*},B_m). \end{aligned}$$

This relation immediately implies (2.5).

Now let us return to Eq. (B.1). Using Doob’s inequality and (2.4) we find that

$$\begin{aligned} {\mathbb {E}}(\sup _{0\le \tau \le T} |M(\tau )|^2 ) \le C<\infty . \end{aligned}$$

Next, applying (A.4) and Young’s inequality we get

$$\begin{aligned} \int _{0}^{\tau } \big ( -\Vert v \Vert ^2_{m+1} - \nu ^{-1} \langle \!\langle i|v |^2v , v \rangle \!\rangle _m \big )ds \le C_m \int _{0}^{\tau } | v(s)|_\infty ^{2m+3} ds, \quad \forall \ 0\le \tau \le T. \end{aligned}$$

Finally, using in (B.1) the last two displayed formulas jointly with (2.3) we obtain (2.6).