### Kunlong Shi and Tong Tang

College of Sciences, Nanjing Forestry University, 210037 Nanjing, P.R. China
e-mail:skl@njfu.edu.cn

School of Mathematical Science, Yangzhou University, 225002 Yangzhou, P.R. China
e-mail:tt0507010156@126.com

Abstract.   In this work, we prove the uniform regularity of smooth solutions to the full compressible MHD system in $$\mathbb{T}^3$$. Here our result is obtained by using the bilinear commutator and product estimates.

2020 Mathematics Subject Classification.   35Q30, 35Q35, 76D03

Key words and phrases.   MHD, compressible, uniform regularity.

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References:

1. T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), 1–73.
MathSciNet CrossRef

2. X. Blanc, B. Ducomet and Š. Nečasová, Global existence of a radiative Euler system coupled to an electromagnetic field, Adv. Nonlinear Anal. 8 (2019), 1158–1170.
MathSciNet CrossRef

3. B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266 (2006), 595–629.
MathSciNet CrossRef

4. B. Ducomet, M. Kobera and Š. Nečasová, Global existence of a weak solution for a model in radiation magnetohydrodynamics, Acta Appl. Math. 150 (2017), 43–65.
MathSciNet CrossRef

5. J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl. 10 (2009), 392–409.
MathSciNet CrossRef

6. J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal. 69 (2008), 3637–3660.
MathSciNet CrossRef

7. X. Blanc, B. Ducomet and Š. Nečasová, Global existence of a diffusion limit with damping for the compressible radiative Euler system coupled to an electromagnetic field, Topol. Methods Nonlinear Anal. 52 (2018), 285–309.
MathSciNet CrossRef

8. J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations 256 (2014), 2858–2875.
MathSciNet CrossRef

9. D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. 56 (2005), 791–804.
MathSciNet CrossRef

10. X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. 283 (2008), 255–284.
MathSciNet CrossRef

11. X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197 (2010), 203–238.
MathSciNet CrossRef

12. X. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows, Comm. Math. Phys. 324 (2013), 147–171.
MathSciNet CrossRef

13. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907.
MathSciNet CrossRef

14. D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam. 35 (2019), 23–100.
MathSciNet CrossRef

15. G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001), 61–90.
MathSciNet CrossRef

16. A. I. Vol'pert and S. I. Hudjaev, The Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.) 87(129) (1972), 504–528.
MathSciNet

17. Y. Wang, L. Du and S. Li, Blowup mechanism for viscous compressible heat-conductive magnetohydrodynamic flows in three dimensions, Sci. China Math. 58 (2015), 1677–1696.
MathSciNet CrossRef

18. Y. Wang, A Beale-Kato-Majda criterion for three dimensional compressible viscous non-isentropic magnetohydrodynamic flows without heat-conductivity, J. Differential Equations 280 (2021), 66–98.
MathSciNet CrossRef