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June  2022, 42(6): 2927-2943. doi: 10.3934/dcds.2022004

## Global regularity for the 3D compressible magnetohydrodynamics with general pressure

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

Received  March 2021 Revised  October 2021 Published  June 2022 Early access  January 2022

Fund Project: 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)

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.

Citation: Anthony Suen. Global regularity for the 3D compressible magnetohydrodynamics with general pressure. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2927-2943. doi: 10.3934/dcds.2022004
##### References:
 [1] H. Bahouri and J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Rational Mech. Anal., 127 (1994), 159-181.  doi: 10.1007/BF00377659. [2] H. Cabannes, Theoretical magnetofluiddynamics, Applied Mathematics and Mechanics, Academic Press, New York; London, 13 (1970). [3] J. Fan, S. Jiang and Y. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 337-350.  doi: 10.1016/j.anihpc.2009.09.012. [4] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111. [5] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2. [6] 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.  doi: 10.1007/s00205-010-0295-9. [7] X. Huang, J. Li and Z. Xin, Serrin-type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886.  doi: 10.1137/100814639. [8] S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 9 (1984). [9] F. Li and H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126.  doi: 10.1017/S0308210509001632. [10] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. [11] M. Lu, Y. Du and Z.-A. Yao, Blow-up phenomena for the 3D compressible MHD equations, Discrete Contin. Dyn. Syst., 32 (2012), 1835-1855.  doi: 10.3934/dcds.2012.32.1835. [12] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762-1774.  doi: 10.1016/j.jde.2008.07.007. [13] R. Sart, Existence of finite energy weak solutions for the equations MHD of compressible fluids, Appl. Anal., 88 (2009), 357-379.  doi: 10.1080/00036810802713933. [14] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970. [15] A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805.  doi: 10.3934/dcds.2013.33.3791. [16] A. Suen, Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy, Discrete Contin. Dyn. Syst., 40 (2020), 1775-1798.  doi: 10.3934/dcds.2020093. [17] A. Suen, Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations, J. Differential Equations, 268 (2020), 2622-2671.  doi: 10.1016/j.jde.2019.09.037. [18] A. Suen, Existence, stability and long time behaviour of weak solutions of the three-dimensional compressible navier-stokes equations with potential force, J. Differential Equations, 299 (2021), 463-512.  doi: 10.1016/j.jde.2021.07.027. [19] A. Suen and D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 205 (2012), 27-58.  doi: 10.1007/s00205-012-0498-3. [20] Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001. [21] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. [22] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] H. Bahouri and J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Rational Mech. Anal., 127 (1994), 159-181.  doi: 10.1007/BF00377659. [2] H. Cabannes, Theoretical magnetofluiddynamics, Applied Mathematics and Mechanics, Academic Press, New York; London, 13 (1970). [3] J. Fan, S. Jiang and Y. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 337-350.  doi: 10.1016/j.anihpc.2009.09.012. [4] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111. [5] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2. [6] 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.  doi: 10.1007/s00205-010-0295-9. [7] X. Huang, J. Li and Z. Xin, Serrin-type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886.  doi: 10.1137/100814639. [8] S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 9 (1984). [9] F. Li and H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126.  doi: 10.1017/S0308210509001632. [10] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. [11] M. Lu, Y. Du and Z.-A. Yao, Blow-up phenomena for the 3D compressible MHD equations, Discrete Contin. Dyn. Syst., 32 (2012), 1835-1855.  doi: 10.3934/dcds.2012.32.1835. [12] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762-1774.  doi: 10.1016/j.jde.2008.07.007. [13] R. Sart, Existence of finite energy weak solutions for the equations MHD of compressible fluids, Appl. Anal., 88 (2009), 357-379.  doi: 10.1080/00036810802713933. [14] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970. [15] A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805.  doi: 10.3934/dcds.2013.33.3791. [16] A. Suen, Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy, Discrete Contin. Dyn. Syst., 40 (2020), 1775-1798.  doi: 10.3934/dcds.2020093. [17] A. Suen, Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations, J. Differential Equations, 268 (2020), 2622-2671.  doi: 10.1016/j.jde.2019.09.037. [18] A. Suen, Existence, stability and long time behaviour of weak solutions of the three-dimensional compressible navier-stokes equations with potential force, J. Differential Equations, 299 (2021), 463-512.  doi: 10.1016/j.jde.2021.07.027. [19] A. Suen and D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 205 (2012), 27-58.  doi: 10.1007/s00205-012-0498-3. [20] Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001. [21] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. [22] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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