January  2016, 15(1): 161-183. doi: 10.3934/cpaa.2016.15.161

Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, China

Received  August 2015 Revised  September 2015 Published  December 2015

In this paper, we establish a Serrin-type blowup criterion for the Cauchy problem of the three dimensional compressible Navier-Stokes-Maxwell system, which states a classical solution exists globally, provided that the velocity satisfies Serrin's condition and that the $L_t^\infty L_x^\infty$ of density $\rho$ and the $L^2_tL_x^2$ of $\nabla^2 E$ are bounded. In particular, this criterion is analogous to the well-known Serrin's blowup criterion for the three-dimensional compressible Navier-Stokes equations. Moreover, it is independent of the temperature and magnetic field. It should be noted that it is the first result about the possibility of global existence of classical solution for the full Navier-Stokes-Maxwell system.
Citation: Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure & Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161
References:
[1]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids,, \emph{J. Math. Pures Appl.}, 83 (2004), 243. doi: 10.1016/j.matpur.2003.11.004. Google Scholar

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H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, \emph{J. Differential Equations}, 190 (2003), 504. doi: 10.1016/S0022-0396(03)00015-9. Google Scholar

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J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 337. doi: 10.1016/j.anihpc.2009.09.012. Google Scholar

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P. Germain, S. Ibrahim and N. Masmoudi, On the wellposedness of the Navier-Stokes-Maxwell equations,, preprint, (). Google Scholar

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B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations,, preprint, (). Google Scholar

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G. Y. Hong, X. F. Hou, H. Y. Peng and C. J. Zhu, Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum,, \emph{Sci. China Math.}, 57 (2014), 2463. doi: 10.1007/s11425-014-4896-x. Google Scholar

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X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows,, \emph{Comm. Math. Phys.}, 324 (2013), 147. doi: 10.1007/s00220-013-1791-1. Google Scholar

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X. D. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations,, preprint, (). Google Scholar

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X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 549. doi: 10.1002/cpa.21382. Google Scholar

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S. Ibrahim and S. Keraani, Global small solutions for the Navier-Stokes-Maxwell system,, \emph{SIAM J. Math. Anal.}, 43 (2011), 2275. doi: 10.1137/100819813. Google Scholar

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S. Ibrahim and T. Yoneda, Local solvability and loss of smoothness of the Navier-Stokes-Maxwell equations with large initial data,, \emph{J. Math. Anal. Appl.}, 396 (2012), 555. doi: 10.1016/j.jmaa.2012.06.038. Google Scholar

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H. X. Liu, T. Yang, H. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data,, \emph{SIAM J. Math. Anal.}, 46 (2014), 2185. doi: 10.1137/130920617. Google Scholar

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A. Matsumura and T. Nishida, The initial value problem for the equation of motion of viscous and heat-conductive gases,, \emph{J. Math. Kyoto. Univ.}, 20 (1980), 67. Google Scholar

[20]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids,, \emph{Comm. Math. Phys.}, 89 (1983), 445. Google Scholar

[21]

N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D,, \emph{J. Math. Pures Appl.}, 93 (2010), 559. doi: 10.1016/j.matpur.2009.08.007. Google Scholar

[22]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 187. Google Scholar

[23]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations,, \emph{J. Math. Pures Appl.}, 95 (2011), 36. doi: 10.1016/j.matpur.2010.08.001. Google Scholar

[24]

M. E. Taylor, Partial Differential Equations III: Nonlinear Equations,, Second edition. Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4419-7049-7. Google Scholar

[25]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method,, \emph{Ann. Sc. Norm. Super. Pisa CI.Sci.}, 10 (1983), 607. Google Scholar

[26]

H. Y. Wen and C. J. Zhu, Global symmetric classical solutions of the full compressible Navier-Stokes equations with vacuum and large initial data,, \emph{J. Math. Pures Appl.}, 102 (2014), 498. doi: 10.1016/j.matpur.2013.12.003. Google Scholar

[27]

H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum,, \emph{Adv. Math.}, 248 (2013), 534. doi: 10.1016/j.aim.2013.07.018. Google Scholar

[28]

H. Y. Wen and C. J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat-conducting fluids with vacuum,, \emph{SIAM J. Math. Anal.}, 45 (2013), 431. doi: 10.1137/120877829. Google Scholar

[29]

H. Y. Wen and C. J. Zhu, Global classical solution to 3D full compressible Navier-Stokes equations with vacuum at infinity,, preprint, (). Google Scholar

[30]

J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations,, \emph{Sci. China Math.}, 57 (2014), 2153. doi: 10.1007/s11425-014-4792-4. Google Scholar

show all references

References:
[1]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids,, \emph{J. Math. Pures Appl.}, 83 (2004), 243. doi: 10.1016/j.matpur.2003.11.004. Google Scholar

[2]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities,, \emph{Manuscripta Math.}, 120 (2006), 91. doi: 10.1007/s00229-006-0637-y. Google Scholar

[3]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, \emph{J. Differential Equations}, 228 (2006), 337. doi: 10.1016/j.jde.2006.05.001. Google Scholar

[4]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, \emph{J. Differential Equations}, 190 (2003), 504. doi: 10.1016/S0022-0396(03)00015-9. Google Scholar

[5]

R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system,, \emph{Anal. Appl.}, 10 (2012), 133. doi: 10.1142/S0219530512500078. Google Scholar

[6]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces,, \emph{Math. Models Methods Appl. Sci.}, 17 (2007), 737. doi: 10.1142/S021820250700208X. Google Scholar

[7]

J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 337. doi: 10.1016/j.anihpc.2009.09.012. Google Scholar

[8]

E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004). Google Scholar

[9]

P. Germain, S. Ibrahim and N. Masmoudi, On the wellposedness of the Navier-Stokes-Maxwell equations,, preprint, (). Google Scholar

[10]

B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations,, preprint, (). Google Scholar

[11]

G. Y. Hong, X. F. Hou, H. Y. Peng and C. J. Zhu, Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum,, \emph{Sci. China Math.}, 57 (2014), 2463. doi: 10.1007/s11425-014-4896-x. Google Scholar

[12]

X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows,, \emph{Comm. Math. Phys.}, 324 (2013), 147. doi: 10.1007/s00220-013-1791-1. Google Scholar

[13]

X. D. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations,, preprint, (). Google Scholar

[14]

X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 549. doi: 10.1002/cpa.21382. Google Scholar

[15]

S. Ibrahim and S. Keraani, Global small solutions for the Navier-Stokes-Maxwell system,, \emph{SIAM J. Math. Anal.}, 43 (2011), 2275. doi: 10.1137/100819813. Google Scholar

[16]

S. Ibrahim and T. Yoneda, Local solvability and loss of smoothness of the Navier-Stokes-Maxwell equations with large initial data,, \emph{J. Math. Anal. Appl.}, 396 (2012), 555. doi: 10.1016/j.jmaa.2012.06.038. Google Scholar

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type,, Translated from the Russian by S. Smith., (1968). Google Scholar

[18]

H. X. Liu, T. Yang, H. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data,, \emph{SIAM J. Math. Anal.}, 46 (2014), 2185. doi: 10.1137/130920617. Google Scholar

[19]

A. Matsumura and T. Nishida, The initial value problem for the equation of motion of viscous and heat-conductive gases,, \emph{J. Math. Kyoto. Univ.}, 20 (1980), 67. Google Scholar

[20]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids,, \emph{Comm. Math. Phys.}, 89 (1983), 445. Google Scholar

[21]

N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D,, \emph{J. Math. Pures Appl.}, 93 (2010), 559. doi: 10.1016/j.matpur.2009.08.007. Google Scholar

[22]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 187. Google Scholar

[23]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations,, \emph{J. Math. Pures Appl.}, 95 (2011), 36. doi: 10.1016/j.matpur.2010.08.001. Google Scholar

[24]

M. E. Taylor, Partial Differential Equations III: Nonlinear Equations,, Second edition. Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4419-7049-7. Google Scholar

[25]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method,, \emph{Ann. Sc. Norm. Super. Pisa CI.Sci.}, 10 (1983), 607. Google Scholar

[26]

H. Y. Wen and C. J. Zhu, Global symmetric classical solutions of the full compressible Navier-Stokes equations with vacuum and large initial data,, \emph{J. Math. Pures Appl.}, 102 (2014), 498. doi: 10.1016/j.matpur.2013.12.003. Google Scholar

[27]

H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum,, \emph{Adv. Math.}, 248 (2013), 534. doi: 10.1016/j.aim.2013.07.018. Google Scholar

[28]

H. Y. Wen and C. J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat-conducting fluids with vacuum,, \emph{SIAM J. Math. Anal.}, 45 (2013), 431. doi: 10.1137/120877829. Google Scholar

[29]

H. Y. Wen and C. J. Zhu, Global classical solution to 3D full compressible Navier-Stokes equations with vacuum at infinity,, preprint, (). Google Scholar

[30]

J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations,, \emph{Sci. China Math.}, 57 (2014), 2153. doi: 10.1007/s11425-014-4792-4. Google Scholar

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