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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.

MSC:

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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