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

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

Year:    2024

Author:    Yuanyuan Zhang, Guanggan Chen

Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 3 : pp. 326–354

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jpde.v37.n3.7

Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 3 : pp. 326–354

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

Keywords:    3D MHD equations Vishik-Fursikov measures stationary statistical solutions convergence of measures.

Author Details

Yuanyuan Zhang

Guanggan Chen