Invariant measures for Burgers equation with stochastic forcing. (English) Zbl 0972.35196

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 \((\Omega,{\mathcal F},{\mathcal P})\) and the \(F_k\)’s are periodic with period \(1\). The authors assume for some \(r\geq 3\) \[ f_k(x)= F_k'(x)\in \mathbb{C}^r(S^1),\quad \|f_k\|_{\mathbb{C}^r}\leq {C\over k^2}, \] 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 probability 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.


35R60 PDEs with randomness, stochastic partial differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
35B10 Periodic solutions to PDEs
60J25 Continuous-time Markov processes on general state spaces
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