# American Institute of Mathematical Sciences

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 and 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, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004. [2] Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y. [3] Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 337-411. doi: 10.1016/j.jde.2006.05.001. [4] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9. [5] R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197. doi: 10.1142/S0219530512500078. [6] R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X. [7] J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012. [8] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004. [9] P. Germain, S. Ibrahim and N. Masmoudi, On the wellposedness of the Navier-Stokes-Maxwell equations,, preprint, (). [10] B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations,, preprint, (). [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, Sci. China Math., 57 (2014), 2463-2484. doi: 10.1007/s11425-014-4896-x. [12] 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. [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, (). [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, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382. [15] S. Ibrahim and S. Keraani, Global small solutions for the Navier-Stokes-Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295. doi: 10.1137/100819813. [16] S. Ibrahim and T. Yoneda, Local solvability and loss of smoothness of the Navier-Stokes-Maxwell equations with large initial data, J. Math. Anal. Appl., 396 (2012), 555-561. doi: 10.1016/j.jmaa.2012.06.038. [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., Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I., (1968). [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, SIAM J. Math. Anal., 46 (2014), 2185-2228. doi: 10.1137/130920617. [19] A. Matsumura and T. Nishida, The initial value problem for the equation of motion of viscous and heat-conductive gases, J. Math. Kyoto. Univ., 20 (1980), 67-104. [20] A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. [21] N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571. doi: 10.1016/j.matpur.2009.08.007. [22] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195 [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, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001. [24] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7. [25] A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Sc. Norm. Super. Pisa CI.Sci., 10 (1983) 607-647. [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, J. Math. Pures Appl., 102 (2014), 498-545. doi: 10.1016/j.matpur.2013.12.003. [27] H. Y. Wen and C. J. 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. [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, SIAM J. Math. Anal., 45 (2013), 431-468. doi: 10.1137/120877829. [29] H. Y. Wen and C. J. Zhu, Global classical solution to 3D full compressible Navier-Stokes equations with vacuum at infinity,, preprint, (). [30] J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math., 57 (2014), 2153-2162. doi: 10.1007/s11425-014-4792-4.

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##### References:
 [1] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004. [2] Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y. [3] Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 337-411. doi: 10.1016/j.jde.2006.05.001. [4] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9. [5] R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197. doi: 10.1142/S0219530512500078. [6] R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X. [7] J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012. [8] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004. [9] P. Germain, S. Ibrahim and N. Masmoudi, On the wellposedness of the Navier-Stokes-Maxwell equations,, preprint, (). [10] B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations,, preprint, (). [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, Sci. China Math., 57 (2014), 2463-2484. doi: 10.1007/s11425-014-4896-x. [12] 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. [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, (). [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, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382. [15] S. Ibrahim and S. Keraani, Global small solutions for the Navier-Stokes-Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295. doi: 10.1137/100819813. [16] S. Ibrahim and T. Yoneda, Local solvability and loss of smoothness of the Navier-Stokes-Maxwell equations with large initial data, J. Math. Anal. Appl., 396 (2012), 555-561. doi: 10.1016/j.jmaa.2012.06.038. [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., Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I., (1968). [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, SIAM J. Math. Anal., 46 (2014), 2185-2228. doi: 10.1137/130920617. [19] A. Matsumura and T. Nishida, The initial value problem for the equation of motion of viscous and heat-conductive gases, J. Math. Kyoto. Univ., 20 (1980), 67-104. [20] A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. [21] N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571. doi: 10.1016/j.matpur.2009.08.007. [22] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195 [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, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001. [24] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7. [25] A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Sc. Norm. Super. Pisa CI.Sci., 10 (1983) 607-647. [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, J. Math. Pures Appl., 102 (2014), 498-545. doi: 10.1016/j.matpur.2013.12.003. [27] H. Y. Wen and C. J. 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. [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, SIAM J. Math. Anal., 45 (2013), 431-468. doi: 10.1137/120877829. [29] H. Y. Wen and C. J. Zhu, Global classical solution to 3D full compressible Navier-Stokes equations with vacuum at infinity,, preprint, (). [30] J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math., 57 (2014), 2153-2162. doi: 10.1007/s11425-014-4792-4.
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