Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3D

Abstract

In this paper we construct an invariant probability measure concentrated on $H^2(K)\times H^1(K)$ for a general cubic Klein–Gordon equation (including the case of the wave equation). Here K represents both the 3-dimensional torus or a bounded domain with smooth boundary in ${\mathbb {R}}^3$. That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic sense. We also establish qualitative properties of the constructed measure. This work extends the fluctuation–dissipation-limit approach to PDEs having only one (coercive) conservation law.

Global existence and uniqueness for the cubic KG equation in \({\mathscr {H}}^{1,0}(K)\)

Set the space $X:=X_T=C([0,T];{\mathscr {H}}^{1,0}(K))$ endowed with the norm $\Vert .\Vert _X=\sup _{t\in [0,T]}\Vert .\Vert _{1,0}$.

Proposition A.1

For any $y_0$ in ${\mathscr {H}}^{1,0}(K)$, there is a unique $y:=[y_1,y_2]$ belonging to $\cap _{T>0}X_T$ satisfying:

$$\begin{aligned} {\partial _t}y=Ay-[0,y_1^3], \end{aligned}$$

(A.1)

where $Ay=[y_2,\ \Delta _0y_1]$ and $y|_{t=0}=y_0$.

Sketch of the proof

The proof of the existence relies on the fixed point argument combined with the preservation of the energy

$$\begin{aligned} E(y)=\frac{1}{2}\Vert y\Vert _{1,0}^2+\frac{1}{4}\int y_1^4. \end{aligned}$$

We give it later. Now let us give the argument of the uniqueness, let x and y be two global in time solutions of (A.1), set $w=x-y$. Then

$$\begin{aligned} {\partial _t}w=Aw-[0,w_1^3-3x_1y_1^2+3x_1^2y_1]. \end{aligned}$$

We multiply this equation by ${\partial _t}w$ and integrate over K to find

$$\begin{aligned} {\partial _t}E(w)\le C_1\Vert {\partial _t}w_2\Vert (\Vert x_1\Vert _{L^6}^3+\Vert y_1\Vert _{L^6}^3)\le C_2\sqrt{E(w)}(\Vert x_1\Vert _{L^6}^3+\Vert y_1\Vert _{L^6}^3). \end{aligned}$$

Then for any $\epsilon >0$ and $t\in [0,T]$

$$\begin{aligned} {\partial _t}E(w(t))\le \frac{C_3}{\epsilon }E(w)\sup _{t\in [0,T]} (\Vert x_1\Vert _{L^6}^6+\Vert y_1\Vert _{L^6}^6)+\epsilon \end{aligned}$$

Using the fact that $H^{1}\subset L^6$ and the Gronwall inequality, we see that for any $T>0$ and the case $w_0=0$, we have

$$\begin{aligned} \sup _{t\in [0,T]}E(w){\ \lesssim \ }\epsilon \ \ \ for\ \ all\ \ \epsilon >0 \end{aligned}$$

which finishes the proof of the uniqueness.

In order to prove the existence part, for a given $y_0\in {\mathscr {H}}^{1,0}$ set the map

$$\begin{aligned} \phi _t y=e^{tA}y_0-\int _0^te^{-(t-s)A}[0,y_1^3]ds. \end{aligned}$$

It is not difficult to see that

$$\begin{aligned} \Vert \phi _t y\Vert _X&{\ \lesssim \ }\Vert y_0\Vert _{1,0}+T\Vert y\Vert _X^3;\\ \Vert \phi _t y-\phi _t x\Vert _X&{\ \lesssim \ }T\Vert y-x\Vert _X\sup _{t\in [0,T]}(\Vert x_1-y_1\Vert _1^6+\Vert x_1\Vert _1^6+\Vert y_1\Vert _1^6). \end{aligned}$$

Then for any $R>0$, there is $T_R>0$ such that $\phi _t$ is a contraction on an appropriately chosen ball of $X_{T_R}$. An iteration argument and the conservation of the energy finish the proof. $\square$