Invariant measures for Burgers equation with stochastic forcing

Weinan E, K. Khanin, A. Mazel and Ya. Sinai

Review from Zentralblatt MATH:

The authors study the following Burgers equation $${\partial u\over\partial t}+{\partial\over\partial x} \Biggl({u^2 \over 2}\Biggr)= \varepsilon {\partial^2u\over\partial x^2}+ f(x,t),$$ where $f(x,t)= {\partial F\over\partial x} (x, t)$ is a random forcing function, which is periodic in $x$ with period $1$, and with white noise in $t$. The general form for the potentials of such forces is given by $$F(x,t)= \sum^\infty_{k=1} F_k(x)\dot B_k(t),$$ where the $\{B_k(t), t\in(-\infty,\infty)\}$'s are independent standard Wiener processes defined on a probability space $(\Omega,{\cal F},{\cal P})$ and the $F_k$'s are periodic with period $1$. The authors assume for some $r\ge 3$ $$f_k(x)= F_k'(x)\in \bbfC^r(S^1),\quad \|f_k\|_{\bbfC^r}\le {C\over k^2},$$ where $S^1$ denotes the unit circle, and $C$ a generic constant. Without loss of generality, the authors assume that for all $k$: $\int^1_0 F_k(x) dx= 0$. They denote the elements in the proabability space $\Omega$ by $\omega= (\dot B_1(\cdot),\dot B_2(\cdot),\dots)$. Except in Section 8, where they study the convergence as $\varepsilon\to 0$, the authors restrict their attention to the case when $\varepsilon= 0$: $${\partial u\over\partial t}+{\partial\over\partial x} \Biggl({u^2\over 2}\Biggr)= {\partial F\over\partial x} (x,t).\tag 1$$ Besides establishing existence and uniqueness of an invariant measure for the Markov process corresponding to (1) the authors give a detailed description of the structure and regularity properties for the solutions that live on the support of this measure.

Reviewed by Stanislaw Wedrychowicz

Keywords: Burgers equation; random forcing function; Wiener processes; probability space; existence; uniqueness; invariant measure; Markov process

Classification (MSC2000): 35R60 35Q53 37A50 35B10 60J25

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