Existence and Approximation of Statistical Solutions of the 3D MHD Equations (Journal of Partial Differential Equations, 2024)
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 3DMHD equations by taking advantage of the Krein-Milman theorem along with somefunctional and measure theories. Then by applying the Topsoe lemma on the constructed trajectory space possessing some special topological properties, we show thatthe Vishik-Fursikov measure and the stationary Vishik-Fursikov statistical solution ofthe 3D MHD system are approximated by the counterparts of the 3D MHD-$α$ system,respectively, as the parameter $α$ decreases to zero.
Keywords: 3D MHD equations; Vishik-Fursikov measures; stationary statistical solutions; convergence of measures.