Energy equality for the multi-dimensional nonhomogeneous incompressible Hall-MHD equations in a bounded domain

Xun Wang; Qunyi Bie; Xun Wang; Qunyi Bie

doi:10.3934/era.2023002

Electronic Research Archive

2023, Volume 31, Issue 1: 17-36. doi: 10.3934/era.2023002

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Research article

Energy equality for the multi-dimensional nonhomogeneous incompressible Hall-MHD equations in a bounded domain

Xun Wang ¹,
Qunyi Bie ^{2
,
,}

1.
College of Science, China Three Gorges University, Yichang 443002, China
2.
College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China

Academic Editor: Alain.Miranville

Received: 16 August 2022 Revised: 30 September 2022 Accepted: 11 October 2022 Published: 19 October 2022

This paper focuses on the energy equality for weak solutions of the nonhomogeneous incompressible Hall-magnetohydrodynamics equations in a bounded domain $ \Omega \subset \mathbb{R}^n $ $ (n\geqslant2) $. By exploiting the special structure of the nonlinear terms and using the coarea formula, we obtain some sufficient conditions for the regularity of weak solutions to ensure that the energy equality is valid. For the special case $ n = 3 $, $ p = q = 2 $, our results are consistent with the corresponding results obtained by Kang-Deng-Zhou in [Results Appl. Math. 12:100178, 2021]. Additionally, we establish the sufficient conditions concerning $ \nabla u $ and $ \nabla b $, instead of $ u $ and $ b $.
- Hall-MHD system,
- regularity,
- energy equality
Citation: Xun Wang, Qunyi Bie. Energy equality for the multi-dimensional nonhomogeneous incompressible Hall-MHD equations in a bounded domain[J]. Electronic Research Archive, 2023, 31(1): 17-36. doi: 10.3934/era.2023002

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Abstract

This paper focuses on the energy equality for weak solutions of the nonhomogeneous incompressible Hall-magnetohydrodynamics equations in a bounded domain $ \Omega \subset \mathbb{R}^n $ $ (n\geqslant2) $. By exploiting the special structure of the nonlinear terms and using the coarea formula, we obtain some sufficient conditions for the regularity of weak solutions to ensure that the energy equality is valid. For the special case $ n = 3 $, $ p = q = 2 $, our results are consistent with the corresponding results obtained by Kang-Deng-Zhou in [Results Appl. Math. 12:100178, 2021]. Additionally, we establish the sufficient conditions concerning $ \nabla u $ and $ \nabla b $, instead of $ u $ and $ b $.

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