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

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 $.

Keywords

• hall-mhd system
• regularity
• energy equality