We study a class of dissipative nonlinear PDE's forced by a random force ??(t, x), with the space variable x varying in a bounded domain. The class contains the 2D Navier-Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in x and stationary, short-correlated in time t. In this paper, we confine ourselves to "kick forces" of the form ??(t, x) = S+8k=-8 d(t - kT)?k(x), where the ?k's are smooth bounded identically distributed random fields. The equation in question defines a Markov chain in an appropriately chosen phase space (a subset of a function space) that contains the zero function and is invariant for the (random) flow of the equation. Concerning this Markov chain, we prove the following main result (see Theorem 2.2): The Markov chain has a unique invariant measure. To prove this theorem, we present a construction assigning, to any invariant measure, a Gibbs measure for a 1D system with compact phase space and apply a version of Ruelle-Perron-Frobenius uniqueness theorem to the corresponding Gibbs system. We also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced PDE.
|Number of pages||40|
|Journal||Communications in Mathematical Physics|
|Publication status||Published - Sep 2000|