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