A KAM Theorem for Higher Dimensional Wave Equations Under Nonlocal Perturbation

Abstract

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equation under nonlocal perturbation

$$\begin{aligned} u_{tt}-\triangle u +M_\xi u +\left( \int _{\mathbb {T}^d} u^2 dx\right) u=0,\quad t\in \mathbb {R},\ x\in \mathbb {T}^d\ \end{aligned}$$

where $M_\xi$ is a real Fourier multiplier.

Appendix

Lemma 7.1

$$\begin{aligned} \Vert FG\Vert _{ D(r,s) }\le \Vert F\Vert _{ D(r,s) }\Vert G\Vert _{ D(r,s) }. \end{aligned}$$

Proof

Since $(FG)_{kl\alpha \beta }=\sum _{k',l',\alpha ',\beta '}F_{k-k',l-l',\alpha -\alpha ',\beta -\beta '} G_{k'l'\alpha '\beta '}$, we have

$$\begin{aligned} \Vert FG\Vert _{ D(r,s) }= & {} \mathop {\mathop {\sup }\limits _{\Vert z\Vert _\rho<s}}\limits _{\Vert \bar{z}\Vert _\rho<s}\sum _{k,l,\alpha ,\beta }|(FG)_{kl\alpha \beta }|s^{2l}|z^{\alpha }| |\bar{z}^{\beta }|e^{|k|r}\\\le & {} \mathop {\mathop {\sup }\limits _{\Vert z\Vert _\rho<s}}\limits _{\Vert \bar{z}\Vert _\rho <s} \sum _{k,l,\alpha ,\beta }\sum _{k',l',\alpha ',\beta '}|F_{k-k',l-l',\alpha -\alpha ',\beta -\beta ' }G_{k'l'\alpha '\beta '}|s^{2l}|z^{\alpha }| |\bar{z}^{\beta }|e^{|k|r}\\\le & {} \Vert F\Vert _{ D(r,s) }\Vert G\Vert _{ D(r,s) } \end{aligned}$$

and the proof is finished. $\square$

Lemma 7.2

(Cauchy inequalities)

$$\begin{aligned}&\Vert F_{\theta }\Vert _{D(r-\sigma ,s)}\le \frac{c}{\sigma }\Vert F\Vert _{ D(r,s)},\\&\Vert F_{I}\Vert _{D(r,\frac{1}{2} s)}\le \frac{c}{s^2}\Vert F\Vert _{ D(r,s) }, \end{aligned}$$

and

$$\begin{aligned}&\Vert F_{z_n}\Vert _{D(r,\frac{1}{2} s)}\le \frac{c}{s}\Vert F\Vert _{ D(r,s) }|n|^ae^{|n|\rho },\\&\Vert F_{\bar{z}_n}\Vert _{D(r,\frac{1}{2} s)}\le \frac{c}{s}\Vert F\Vert _{ D(r,s) }|n|^ae^{|n|\rho }. \end{aligned}$$

Let $\{\cdot ,\cdot \}$ denote the Poisson bracket of smooth functions, i.e.,

$$\begin{aligned} \{F,G\}=\left\langle \frac{\partial F}{\partial I}, \frac{\partial G}{\partial \theta }\right\rangle -\left\langle \frac{\partial F}{ \partial \theta },\frac{\partial G}{\partial I}\right\rangle +\mathrm{i}\sum _{n}\left( \frac{\partial F}{\partial z_n}\frac{\partial G}{\partial {\bar{z}_n}}- \frac{\partial F}{\partial {\bar{z}_n}}\frac{\partial G}{\partial { z_n}}\right) , \end{aligned}$$

then we have the following lemma:

Lemma 7.3

If

$$\begin{aligned} \Vert X_F\Vert _{ D(r,s) }< \varepsilon ',\ \Vert X_G\Vert _{ D(r,s) }< \varepsilon '', \end{aligned}$$

then

$$\begin{aligned} \Vert X_{\{F,G\}}\Vert _{D(r-\sigma ,\eta s)}<c\sigma ^{-1}\eta ^{-2}\varepsilon '\varepsilon '',\quad \ \eta \ll 1. \end{aligned}$$

In particular, if $\eta \sim \varepsilon ^{\frac{1}{3}}$, $\varepsilon ', \varepsilon ''\sim \varepsilon$, we have $\Vert X_{\{F,G\}}\Vert _{D(r-\sigma ,\eta s)}\sim \varepsilon ^{\frac{4}{3}}$.

For the proof, see [13]. $\square$