Global regularity for the 3D compressible magnetohydrodynamics with general pressure

Anthony Suen

doi:10.3934/dcds.2022004

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2022, Volume 42, Issue 6: 2927-2943. Doi: 10.3934/dcds.2022004

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Global regularity for the 3D compressible magnetohydrodynamics with general pressure

Anthony Suen

Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong

Received: February 28, 2021

Revised: September 30, 2021

Published: June 2022

The work described in this paper was partially supported from the Dean's Research Fund of the Faculty of Liberal Arts and Social Science, The Education University of Hong Kong, HKSAR, China (Project No. FLASS/DRF 04634)

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Abstract

We address the compressible magnetohydrodynamics (MHD) equations in $ \mathbb R^3 $ and establish a blow-up criterion for the local-in-time smooth solutions in terms of the density only. Namely, if the density is away from vacuum ($ \rho = 0 $) and the concentration of mass ($ \rho = \infty $), then a local-in-time smooth solution can be continued globally in time. The results generalise and strengthen the previous ones in the sense that there is no magnetic field present in the criterion and the assumption on the pressure is significantly relaxed. The proof is based on some new a priori estimates for three-dimensional compressible MHD equations.

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Mathematics Subject Classification: Primary: 35Q35, 76D03.

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