On conditional regularity criteria for the 3D generalized MHD system via partial components

Jinhuan Wang; Yongsheng Nie; Jinhuan Wang; Yongsheng Nie

doi:10.3934/era.2026149

Electronic Research Archive

2026, Volume 34, Issue 5: 3315-3333. doi: 10.3934/era.2026149

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Research article

On conditional regularity criteria for the 3D generalized MHD system via partial components

Jinhuan Wang ^1,2,
Yongsheng Nie ³

1.
School of Mathematics and Computational Sciences, Tangshan Normal University, Tangshan 063000, China
2.
College of Sciences, North China University of Science and Technology, Tangshan 063210, China
3.
Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China

Received: 04 February 2026 Revised: 25 March 2026 Accepted: 08 April 2026 Published: 16 April 2026

This paper focuses on the regularity criteria for the 3D generalized magnetohydrodynamic (MHD) equations, particularly those involving the partial components of the velocity $ u_3 $ and the magnetic field $ (b_1, b_2, b_3) $. For any $ l, i, j, k\in \{1, 2, 3\} $, we establish regularity criteria via $ \partial_l u_3 $ and $ (\partial_ib_1, \partial_jb_2, \partial_kb_3) $, $ u_3 $ and $ (\partial_ib_1, \partial_jb_2, \partial_kb_3) $, or $ \partial_l u_3 $ and $ (b_1, b_2, b_3) $. Furthermore, a comprehensive regularity condition is obtained via $ (\xi u_3, \eta b_1, \zeta b_2, \sigma b_3) $ and $ (\xi{'} \partial_{l}u_3, \eta{'} \partial_{i}b_1, \zeta{'} \partial_{j}b_2, \sigma{'} \partial_{k}b_3) $ with $ \xi+\xi{'} = \eta+\eta{'} = \zeta+\zeta{'} = \sigma+\sigma{'} = 1 $ and $ \xi, \eta, \zeta, \sigma, \xi{'}, \eta{'}, \zeta{'}, \sigma{'} \in \{0, 1\} $.
- generalized MHD equations,
- regularity criterion,
- partial components,
- weak solutions
Citation: Jinhuan Wang, Yongsheng Nie. On conditional regularity criteria for the 3D generalized MHD system via partial components[J]. Electronic Research Archive, 2026, 34(5): 3315-3333. doi: 10.3934/era.2026149

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Abstract

This paper focuses on the regularity criteria for the 3D generalized magnetohydrodynamic (MHD) equations, particularly those involving the partial components of the velocity $ u_3 $ and the magnetic field $ (b_1, b_2, b_3) $. For any $ l, i, j, k\in \{1, 2, 3\} $, we establish regularity criteria via $ \partial_l u_3 $ and $ (\partial_ib_1, \partial_jb_2, \partial_kb_3) $, $ u_3 $ and $ (\partial_ib_1, \partial_jb_2, \partial_kb_3) $, or $ \partial_l u_3 $ and $ (b_1, b_2, b_3) $. Furthermore, a comprehensive regularity condition is obtained via $ (\xi u_3, \eta b_1, \zeta b_2, \sigma b_3) $ and $ (\xi{'} \partial_{l}u_3, \eta{'} \partial_{i}b_1, \zeta{'} \partial_{j}b_2, \sigma{'} \partial_{k}b_3) $ with $ \xi+\xi{'} = \eta+\eta{'} = \zeta+\zeta{'} = \sigma+\sigma{'} = 1 $ and $ \xi, \eta, \zeta, \sigma, \xi{'}, \eta{'}, \zeta{'}, \sigma{'} \in \{0, 1\} $.

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