Well-posedness of a degenernate parabolic equation issuing from two-dimensional perfect fluid dynamics

Abstract

A degenerate parabolic equation of convection-diffusion type has been proposed by Robert and Sommeria in [12] to describe the relaxation towards statistical equilibrium states in 2D incompressible perfect fluid dynamics. The paper is concerned with the Cauchy problem for this equation. The local existence of a variational soluation is obtained in using the decrease of the (negative) mixing entropy and Schauder theorem. A smoothing effect is used when proving the uniqueness of the variational solutions by youdovitch's method. Finally, global existence of solutions and their asymptotic convergence towards Gibbs states are shown for a large class of initial data.

Keywords:

• Degenerate parabolic equation
• Statistical equilibrium
• Youdovitch's method

AMS:

• 35K65
• 35Q35
• 76C05
• 76F99