Invariant measures for Burgers equation with stochastic forcing
Abstract
In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.
Cite
@article{arxiv.math/0005306,
title = {Invariant measures for Burgers equation with stochastic forcing},
author = {Weinan E and K. M. Khanin and A. E. Mazel and Ya. G. Sinai},
journal= {arXiv preprint arXiv:math/0005306},
year = {2016}
}
Comments
84 pages, published version, abstract added in migration