On the equations describing a relaxation toward a statistical equilibrium state in the two-dimensional perfect fluid dynamics

The large scale evolution of a two-dimensional (2D) incompressible ideal fluid can be modeled by introducing eddy-viscosity terms. This procedure introduces a new convection-diffusion equation for vorticity. Such relaxation equations have a structural similarity with the 2D Navier--Stokes equations in the "stream function-vorticity" formulation but 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. Furthermore, we study the long time dynamics of the solution, making a link with the statistical equilibrium theory.