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Volume 37, Issue 3
Existence and Approximation of Statistical Solutions of the 3D MHD Equations

Yuanyuan Zhang & Guanggan Chen

DOI: 10.4208/jpde.v37.n3.7

J. Part. Diff. Eq., 37 (2024), pp. 326-354.

Published online: 2024-08

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  • 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.

  • Keywords

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

  • AMS Subject Headings

35Q35, 60B05, 76D06, 28A33

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-326, author = {Zhang , Yuanyuan and Chen , Guanggan}, title = {Existence and Approximation of Statistical Solutions of the 3D MHD Equations}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {3}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n3.7}, url = {http://global-sci.org/intro/article_detail/jpde/23346.html} }
TY - JOUR T1 - Existence and Approximation of Statistical Solutions of the 3D MHD Equations AU - Zhang , Yuanyuan AU - Chen , Guanggan JO - Journal of Partial Differential Equations VL - 3 SP - 326 EP - 354 PY - 2024 DA - 2024/08 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n3.7 UR - https://global-sci.org/intro/article_detail/jpde/23346.html KW - 3D MHD equations, Vishik-Fursikov measures, stationary statistical solutions, convergence of measures. AB -

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.

Yuanyuan Zhang & Guanggan Chen. (2024). Existence and Approximation of Statistical Solutions of the 3D MHD Equations. Journal of Partial Differential Equations. 37 (3). 326-354. doi:10.4208/jpde.v37.n3.7
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