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 socalled onesided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some nondegeneracy 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2000
 DOI:
 10.48550/arXiv.math/0005306
 arXiv:
 arXiv:math/0005306
 Bibcode:
 2000math......5306E
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 84 pages, published version, abstract added in migration