Limit behaviors of weak solution and statistical solution for 3D Bénard-α model

Abstract

This paper is concerned with weak solutions and statistical solutions of the Bénard-α model in three-dimensional domain. Based on uniform estimates with respect to the regularization parameter α, we first prove that a sequence of weak solutions of the 3D Bénard-α model converges to a counterpart of the classical Bénard model in the given topology as the regularization parameter α vanishes. Applying the Topsoe lemma on the appropriate trajectory space with some special topological properties, we further show that a sequence of α-Vishik-Fursikov measures of the 3D Bénard-α model converges to a Vishik-Fursikov measure of the classical Bénard model as α decreases to zero.