The vanishing viscosity limit of statistical solutions of the Navier-Stokes equations. I. 2-D periodic case

Abstract

In this paper we present a result on the vanishing viscosity limit of the statistical solutions of the Navier-Stokes equations in the 2-D periodic domain. In this case the enstrophy type estimate combined with the strengthened energy estimate lead to the strong compactness for the family of statistical solutions. Thus by choosing a subsequence we can have a statistical solution of the Euler equation as the vanishing viscosity limit of the sequence. This provides us the global existence in time of statistical solution of the 2-D Euler equation in the periodic domain with the initial data of finite mean enstrophy. The uniqueness remains open.