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A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative
A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
We are concerned with the breakdown of strong solutions to the three-dimensional compressible magnetohydrodynamic equations with density-dependent viscosity. It is shown that for the initial density away from vacuum, the strong solution exists globally if the gradient of the velocity satisfies $ \|\nabla{\bf{u}}\|_{L^{2}(0,T;L^\infty)}<\infty $. Our method relies upon the delicate energy estimates and elliptic estimates.
References:
[1] |
J. T. Beale, T. Kato and A. Majda,
Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.
doi: 10.1007/BF01212349. |
[2] |
X. Cai and Y. Sun,
Blowup criteria for strong solutions to the compressible Navier-Stokes equations with variable viscosity, Nonlinear Anal. Real World Appl., 29 (2016), 1-18.
doi: 10.1016/j.nonrwa.2015.10.007. |
[3] |
Y. Chen, X. Hou and L. Zhu,
A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain, Math. Meth. Appl. Sci., 40 (2017), 5526-5538.
doi: 10.1002/mma.4407. |
[4] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[5] |
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.
![]() ![]() |
[6] |
E. Feireisl, A. Novotný and H. Petzeltová,
On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[7] |
E. Feireisl, A. Novotný and Y. Sun,
A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 301 (2014), 219-239.
doi: 10.1007/s00205-013-0697-6. |
[8] |
C. He and Z. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[9] |
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. |
[10] |
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. |
[11] |
X. D. 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.
doi: 10.1007/s00220-013-1791-1. |
[12] |
X. D. Huang, J. Li and Y. Wang,
Serrin-type blowup criterion for full compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 207 (2013), 303-316.
doi: 10.1007/s00205-012-0577-5. |
[13] |
X. D. Huang, J. Li and Z. Xin,
Blowup criterion for viscous baratropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35.
doi: 10.1007/s00220-010-1148-y. |
[14] |
X. D. 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. |
[15] |
X. D. Huang, J. Li and Z. Xin,
Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[16] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics (Ph. D. thesis), Kyoto University, 1983. |
[17] |
O. A. Lady$\check{z}$enskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
![]() ![]() |
[18] |
H. Li, X. Xu and J. Zhang,
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.
doi: 10.1137/120893355. |
[19] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅱ: compressible models, Oxford University Press, Oxford, 1998.
![]() |
[20] |
B. Lü, X. Shi and X. Xu,
Global well-posedness and large time asymptotic behavior of strong solutions to the compressible magnetohydrodynamic equations with vacuum, Indiana Univ. Math. J., 65 (2016), 925-975.
doi: 10.1512/iumj.2016.65.5813. |
[21] |
A. Novotný and I. Stra$\check{s}$kraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.
![]() ![]() |
[22] |
A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of
density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805; Corrigendum, Discrete Contin. Dyn. Syst., 35 (2015), 1387-1390.
doi: 10.3934/dcds.2013.33.3791. |
[23] |
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. |
[24] |
Y. Sun, C. Wang and Z. Zhang,
A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., 201 (2011), 727-742.
doi: 10.1007/s00205-011-0407-1. |
[25] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, American Mathematical Society, Providence, R. I., 2006.
doi: 10.1090/cbms/106. |
[26] |
A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130 (1982), 197-213; Correction, Ann. Mat. Pura Appl., 132 (1983), 399-400.
doi: 10.1007/BF01760990. |
[27] |
A. I. Vol'pert and S. I. Khudiaev,
On the Cauchy problem for composite systems nonlinear equations, Mat. Sb, 87 (1972), 504-528.
|
[28] |
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. |
[29] |
Z. Xin,
Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.
|
[30] |
Z. Xin and W. Yan,
On blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys., 321 (2013), 529-541.
doi: 10.1007/s00220-012-1610-0. |
[31] |
X. Xu and J. Zhang,
A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci., 22 (2012), 1150010, 23pp.
doi: 10.1142/S0218202511500102. |
[32] |
X. Zhong, On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction, to appear in Indiana Univ. Math. J., (2019). |
show all references
References:
[1] |
J. T. Beale, T. Kato and A. Majda,
Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.
doi: 10.1007/BF01212349. |
[2] |
X. Cai and Y. Sun,
Blowup criteria for strong solutions to the compressible Navier-Stokes equations with variable viscosity, Nonlinear Anal. Real World Appl., 29 (2016), 1-18.
doi: 10.1016/j.nonrwa.2015.10.007. |
[3] |
Y. Chen, X. Hou and L. Zhu,
A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain, Math. Meth. Appl. Sci., 40 (2017), 5526-5538.
doi: 10.1002/mma.4407. |
[4] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[5] |
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.
![]() ![]() |
[6] |
E. Feireisl, A. Novotný and H. Petzeltová,
On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[7] |
E. Feireisl, A. Novotný and Y. Sun,
A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 301 (2014), 219-239.
doi: 10.1007/s00205-013-0697-6. |
[8] |
C. He and Z. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[9] |
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. |
[10] |
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. |
[11] |
X. D. 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.
doi: 10.1007/s00220-013-1791-1. |
[12] |
X. D. Huang, J. Li and Y. Wang,
Serrin-type blowup criterion for full compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 207 (2013), 303-316.
doi: 10.1007/s00205-012-0577-5. |
[13] |
X. D. Huang, J. Li and Z. Xin,
Blowup criterion for viscous baratropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35.
doi: 10.1007/s00220-010-1148-y. |
[14] |
X. D. 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. |
[15] |
X. D. Huang, J. Li and Z. Xin,
Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[16] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics (Ph. D. thesis), Kyoto University, 1983. |
[17] |
O. A. Lady$\check{z}$enskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
![]() ![]() |
[18] |
H. Li, X. Xu and J. Zhang,
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.
doi: 10.1137/120893355. |
[19] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅱ: compressible models, Oxford University Press, Oxford, 1998.
![]() |
[20] |
B. Lü, X. Shi and X. Xu,
Global well-posedness and large time asymptotic behavior of strong solutions to the compressible magnetohydrodynamic equations with vacuum, Indiana Univ. Math. J., 65 (2016), 925-975.
doi: 10.1512/iumj.2016.65.5813. |
[21] |
A. Novotný and I. Stra$\check{s}$kraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.
![]() ![]() |
[22] |
A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of
density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805; Corrigendum, Discrete Contin. Dyn. Syst., 35 (2015), 1387-1390.
doi: 10.3934/dcds.2013.33.3791. |
[23] |
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. |
[24] |
Y. Sun, C. Wang and Z. Zhang,
A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., 201 (2011), 727-742.
doi: 10.1007/s00205-011-0407-1. |
[25] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, American Mathematical Society, Providence, R. I., 2006.
doi: 10.1090/cbms/106. |
[26] |
A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130 (1982), 197-213; Correction, Ann. Mat. Pura Appl., 132 (1983), 399-400.
doi: 10.1007/BF01760990. |
[27] |
A. I. Vol'pert and S. I. Khudiaev,
On the Cauchy problem for composite systems nonlinear equations, Mat. Sb, 87 (1972), 504-528.
|
[28] |
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. |
[29] |
Z. Xin,
Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.
|
[30] |
Z. Xin and W. Yan,
On blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys., 321 (2013), 529-541.
doi: 10.1007/s00220-012-1610-0. |
[31] |
X. Xu and J. Zhang,
A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci., 22 (2012), 1150010, 23pp.
doi: 10.1142/S0218202511500102. |
[32] |
X. Zhong, On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction, to appear in Indiana Univ. Math. J., (2019). |
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