NoDEA 1 (1994) 389-402 1021-9722/94/040389-14 $1.50-[-0.20 @1994 Birkh/iuser Verlag, Basel Stochastic Burgers' equation Giuseppe DA PRATO Scnola Normale Superiore di Pisa Piazza dei Cavalieri 6, 56126 Pisa, Italy Arnaud DEBUSSCHE CNRS et Universit6 Paris Sud, 91405 Orsay, France Roger TEMAM Universit6 Paris Sud, 91405 Orsay, France Abstract We consider a Burgers' equation perturbed by white noise. We prove the existence and uniqueness of the global solution as well as the existence of an invariant measure for the corresponding transition semigroup. 1 Introduction It is well known that the Burgers' equation is not a good model for turbulence. It does not display any chaos; even when a force is added to the right hand side all solutions converge to a unique stationary solution as time goes to infinity. However the situation is totally different when the force is a random one. Several authors have indeed suggested to use the stochastic Burgers' equation as a simple model for turbulence, [1],[2],[4],[6]. The equation has also been proposed in [7] to study the dynamics of interfaces. Here we consider the Burgers' equation with a random force which is a space- time white noise (or Brownian sheet ) N 02W Oucot -- cO2u(t'X)ox ~ + 21 ozcO (u2(t,x)) + __Otcox. (1.1) N We recall that W(t,x), t _> 0, x E R is a zero mean Gaussian process whose covariance function is given by E[W(~,x)W(~,y)] = (~A s)(x A y), ~,s _>O, x,y e R.