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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  June 2018 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

P. -L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269.  doi: 10.1007/BF01759640.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

P. -L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998.

[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.

[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.

[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.

[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.

[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.

[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.

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