Existence and Approximation of Statistical Solutions of the 3D MHD Equations

Abstract

This paper focuses on the statistical characteristics of the 3D MHD equations. We firstly establish an existence theorem of a Vishik-Fursikov measure of the 3D MHD equations by taking advantage of the Krein-Milman theorem along with some functional and measure theories. Then by applying the Topsoe lemma on the constructed trajectory space possessing some special topological properties, we show that the Vishik-Fursikov measure and the stationary Vishik-Fursikov statistical solution of the 3D MHD system are approximated by the counterparts of the 3D MHD-$α$ system, respectively, as the parameter $α$ decreases to zero.