ON THE EQUATIONS DESCRIBING A RELAXATION TOWARD A STATISTICAL EQUILIBRIUM STATE IN THE 2-DIMENSIONAL PERFECT FLUID-DYNAMICS

The large scale evolution of a two-dimensional (2D) incompressible ide al fluid can be modeled by introducing eddy-viscosity terms. This proc edure introduces a new convection-diffusion equation for vorticity. Su ch relaxation equations have a structural similarity with the 2D Navie r-Stokes equations in the ''stream function-vorticity'' formulation bu t also contain an additional degenerate transport term being essential for conserving the kinetic energy. Using the negative entropy as the Lyapunov functional and after performing the precise estimates for the degenerate transport, we prove existence and uniqueness of solutions to the relaxation equation for a large class of initial data. Furtherm ore, we study the long time dynamics of the solution, making a link wi th the statistical equilibrium theory.