A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity

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June 2018, 23(4): 1757-1766. doi: 10.3934/dcdsb.2018079

A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity

Jishan Fan ^1, , Fucai Li ^2,, and Gen Nakamura ^3,

1.	Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2.	Department of Mathematics, Nanjing University, Nanjing 210093, China
3.	Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

F. Li is the corresponding author

Received August 2016 Revised October 2017 Published March 2018

We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain.

Keywords: Compressible magnetohydrodynamic equations, zero heat conductivity, regularity criterion.

Mathematics Subject Classification: Primary: 76W05; Secondary: 35Q60, 35B44.

Citation: Jishan Fan, Fucai Li, Gen Nakamura. A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1757-1766. doi: 10.3934/dcdsb.2018079

References:

[1]	P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269. doi: 10.1007/BF01759640. Google Scholar
[2]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. Google Scholar
[3]	H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differ. Equ., 5 (1980), 773-789. doi: 10.1080/03605308008820154. Google Scholar
[4]	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. doi: 10.1007/s00220-006-0052-y. Google Scholar
[5]	J. Fan, S. Jiang and Y. Ou, A blow-up criterion for the compressible viscous heat-conductive flows, Ann. Inst. H. Poincaré Anal. Non linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012. Google Scholar
[6]	J. Fan, F. Li and G. Nakamura, A blow-up criterion to the 2D full compressible magnetohydrodynamic equations, Math. Meth. Appl. Sci., 38 (2015), 2073-2080. doi: 10.1002/mma.3205. Google Scholar
[7]	J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875. doi: 10.1016/j.jde.2014.01.021. Google Scholar
[8]	J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001. Google Scholar
[9]	J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. doi: 10.1016/j.na.2007.10.005. Google Scholar
[10]	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. Google Scholar
[11]	X. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows, Commun. Math. Phys., 324 (2013), 147-171. doi: 10.1007/s00220-013-1791-1. Google Scholar
[12]	X. Huang and Y. Wang, L^∞ continuation principle to the non-baratropic non-resistive magnetohydrodynamic equations without heat conductivity, Math. Methods Appl. Sci., 39 (2016), 4234-4245. doi: 10.1002/mma.3860. Google Scholar
[13]	T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differential Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036. Google Scholar
[14]	S. Jiang and F. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asympt. Anal., 95 (2015), 161-185. doi: 10.3233/ASY-151321. Google Scholar
[15]	S. Jiang, Q. Ju and F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365. doi: 10.1088/0951-7715/25/5/1351. Google Scholar
[16]	S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420. doi: 10.1016/j.aim.2014.03.022. Google Scholar
[17]	P. -L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. Google Scholar
[18]	L. Lu, Y. Chen and B. Huang, Blow-up criterion for two-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows, Nonlinear Anal., 139 (2016), 55-74. doi: 10.1016/j.na.2016.02.021. Google Scholar
[19]	X. Pu and B. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538. doi: 10.1007/s00033-012-0245-5. Google Scholar
[20]	Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Rational Mech. Anal., 201 (2011), 727-742. doi: 10.1007/s00205-011-0407-1. Google Scholar
[21]	Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pure Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001. Google Scholar
[22]	W. Von Wahl, Estimating ∇u by div u and curl u, Math. Meth. Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206. Google Scholar
[23]	H. Wen and C. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534-572. doi: 10.1016/j.aim.2013.07.018. Google Scholar

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References:

[1]	P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269. doi: 10.1007/BF01759640. Google Scholar
[2]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. Google Scholar
[3]	H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differ. Equ., 5 (1980), 773-789. doi: 10.1080/03605308008820154. Google Scholar
[4]	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. doi: 10.1007/s00220-006-0052-y. Google Scholar
[5]	J. Fan, S. Jiang and Y. Ou, A blow-up criterion for the compressible viscous heat-conductive flows, Ann. Inst. H. Poincaré Anal. Non linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012. Google Scholar
[6]	J. Fan, F. Li and G. Nakamura, A blow-up criterion to the 2D full compressible magnetohydrodynamic equations, Math. Meth. Appl. Sci., 38 (2015), 2073-2080. doi: 10.1002/mma.3205. Google Scholar
[7]	J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875. doi: 10.1016/j.jde.2014.01.021. Google Scholar
[8]	J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001. Google Scholar
[9]	J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. doi: 10.1016/j.na.2007.10.005. Google Scholar
[10]	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. Google Scholar
[11]	X. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows, Commun. Math. Phys., 324 (2013), 147-171. doi: 10.1007/s00220-013-1791-1. Google Scholar
[12]	X. Huang and Y. Wang, L^∞ continuation principle to the non-baratropic non-resistive magnetohydrodynamic equations without heat conductivity, Math. Methods Appl. Sci., 39 (2016), 4234-4245. doi: 10.1002/mma.3860. Google Scholar
[13]	T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differential Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036. Google Scholar
[14]	S. Jiang and F. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asympt. Anal., 95 (2015), 161-185. doi: 10.3233/ASY-151321. Google Scholar
[15]	S. Jiang, Q. Ju and F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365. doi: 10.1088/0951-7715/25/5/1351. Google Scholar
[16]	S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420. doi: 10.1016/j.aim.2014.03.022. Google Scholar
[17]	P. -L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998. Google Scholar
[18]	L. Lu, Y. Chen and B. Huang, Blow-up criterion for two-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows, Nonlinear Anal., 139 (2016), 55-74. doi: 10.1016/j.na.2016.02.021. Google Scholar
[19]	X. Pu and B. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538. doi: 10.1007/s00033-012-0245-5. Google Scholar
[20]	Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Rational Mech. Anal., 201 (2011), 727-742. doi: 10.1007/s00205-011-0407-1. Google Scholar
[21]	Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pure Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001. Google Scholar
[22]	W. Von Wahl, Estimating ∇u by div u and curl u, Math. Meth. Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206. Google Scholar
[23]	H. Wen and C. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534-572. doi: 10.1016/j.aim.2013.07.018. Google Scholar

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