# American Institute of Mathematical Sciences

November  2015, 14(6): 2283-2313. doi: 10.3934/cpaa.2015.14.2283

## Large time behavior of solution for the full compressible navier-stokes-maxwell system

 1 Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R.China

Received  January 2015 Revised  July 2015 Published  September 2015

In this paper, the Cauchy problem for the compressible Navier-Stokes-Maxwell equation is studied in $R^3$, the $L^p$ time decay rate for the global smooth solution is established. Our method is mainly based on a detailed analysis to the Green's function of the linearized system and some elaborate energy estimates. To give the explicit representation of the Green's function, we use the Helmholtz decomposition by which we can decompose the solution into two parts and give the expression to each part. Our results show a sharp difference between the decay of solution for Navier-Stokes-Maxwell system and that for the Navier-Stokes equation.
Citation: Weike Wang, Xin Xu. Large time behavior of solution for the full compressible navier-stokes-maxwell system. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2283-2313. doi: 10.3934/cpaa.2015.14.2283
##### References:
 [1] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. [2] G. Q. Chen, J. W. Jerome and D. H. Wang, Compressible Euler-Maxwell equations, Transport Theory Statist. Phys., 29 (2000), 311-331. doi: 10.1080/00411450008205877. [3] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333. [4] R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413. doi: 10.1142/S0219891611002421. [5] R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl. (Singap.), 10 (2012), 133-197. doi: 10.1142/S0219530512500078. [6] Y. H. Feng, Y. J. Peng and S. Wang, Asymptotic behavior of global smooth solutions for full compressible Navier-Stokes-Maxwell equations, Nonlinear Anal. Real World Appl., 19 (2014), 105-116. doi: 10.1016/j.nonrwa.2014.03.004. [7] Y. H. Feng, S. Wang and S. Kawashima, Global existence and asymptotic decay of solutions to the non-isentropic Euler-Maxwell system, Math. Models Methods Appl. Sci., 14 (2014), 2851-2884. doi: 10.1142/S0218202514500390. [8] P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system, Ann. Sci. Éc. Norm. Supér., 3 (2014), 469-503. [9] Y. Guo, Smooth irrotational fluids in the large to the Euler-Poisson system in $\mathbbR^{3+1}$, Comm. Math. Phys., 195 (1998), 249-265. doi: 10.1007/s002200050388. [10] M. L. Hajjej and Y. J. Peng, Initial layers and zero-relaxation limits of Euler-Maxwell equations, J. Differential Equations, 252 (2012), 1441-1465. doi: 10.1016/j.jde.2011.09.029. [11] F.-M. Huang, M. Mei, Y. Wang and H.-M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025. [12] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. [13] S. Kawashima, System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics, Ph.D thesis, Kyoto University, Kyoto, 1983. [14] H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$, Arch. Rational Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4. [15] Q. Q. Liu and C. J. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations, Indiana Univ. Math. J., 4 (2013), 1203-1235. doi: 10.1512/iumj.2013.62.5047. [16] T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd-multi dimensions, Comm. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418. [17] T. Luo, R. Natalini and Z. P. Xin, Large-time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830. doi: 10.1137/S0036139996312168. [18] P. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, 1990. doi: 10.1007/978-3-7091-6961-2. [19] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chin. Ann. Math. Ser. B, 28 (2007), 583-602. doi: 10.1007/s11401-005-0556-3. [20] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations, 33 (2008), 349-376. doi: 10.1080/03605300701318989. [21] Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 540-565. doi: 10.1137/070686056. [22] Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal., 18 (2011), 245-268. doi: 10.4310/MAA.2011.v18.n3.a1. [23] Y. Ueda, S. Wang and S. Kawashima, Dissipative structure of the regularity type and time asymptotic decay of solutions for the Euler-Maxwell system, SIAM J. Math. Anal., 44 (2012), 2002-2017. doi: 10.1137/100806515. [24] W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimension, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937. [25] J. W. Yang, R. X. Lian and S. Wang, Incompressible type Euler as scaling limit of compressible Euler-Maxwell equations, Commun. Pure Appl. Anal., 1 (2013), 503-518. doi: 10.3934/cpaa.2013.12.503. [26] J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math., 10 (2014), 2153-2162. doi: 10.1007/s11425-014-4792-4.

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##### References:
 [1] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. [2] G. Q. Chen, J. W. Jerome and D. H. Wang, Compressible Euler-Maxwell equations, Transport Theory Statist. Phys., 29 (2000), 311-331. doi: 10.1080/00411450008205877. [3] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333. [4] R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413. doi: 10.1142/S0219891611002421. [5] R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl. (Singap.), 10 (2012), 133-197. doi: 10.1142/S0219530512500078. [6] Y. H. Feng, Y. J. Peng and S. Wang, Asymptotic behavior of global smooth solutions for full compressible Navier-Stokes-Maxwell equations, Nonlinear Anal. Real World Appl., 19 (2014), 105-116. doi: 10.1016/j.nonrwa.2014.03.004. [7] Y. H. Feng, S. Wang and S. Kawashima, Global existence and asymptotic decay of solutions to the non-isentropic Euler-Maxwell system, Math. Models Methods Appl. Sci., 14 (2014), 2851-2884. doi: 10.1142/S0218202514500390. [8] P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system, Ann. Sci. Éc. Norm. Supér., 3 (2014), 469-503. [9] Y. Guo, Smooth irrotational fluids in the large to the Euler-Poisson system in $\mathbbR^{3+1}$, Comm. Math. Phys., 195 (1998), 249-265. doi: 10.1007/s002200050388. [10] M. L. Hajjej and Y. J. Peng, Initial layers and zero-relaxation limits of Euler-Maxwell equations, J. Differential Equations, 252 (2012), 1441-1465. doi: 10.1016/j.jde.2011.09.029. [11] F.-M. Huang, M. Mei, Y. Wang and H.-M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025. [12] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. [13] S. Kawashima, System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics, Ph.D thesis, Kyoto University, Kyoto, 1983. [14] H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$, Arch. Rational Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4. [15] Q. Q. Liu and C. J. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations, Indiana Univ. Math. J., 4 (2013), 1203-1235. doi: 10.1512/iumj.2013.62.5047. [16] T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd-multi dimensions, Comm. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418. [17] T. Luo, R. Natalini and Z. P. Xin, Large-time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830. doi: 10.1137/S0036139996312168. [18] P. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, 1990. doi: 10.1007/978-3-7091-6961-2. [19] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chin. Ann. Math. Ser. B, 28 (2007), 583-602. doi: 10.1007/s11401-005-0556-3. [20] Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations, 33 (2008), 349-376. doi: 10.1080/03605300701318989. [21] Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 540-565. doi: 10.1137/070686056. [22] Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal., 18 (2011), 245-268. doi: 10.4310/MAA.2011.v18.n3.a1. [23] Y. Ueda, S. Wang and S. Kawashima, Dissipative structure of the regularity type and time asymptotic decay of solutions for the Euler-Maxwell system, SIAM J. Math. Anal., 44 (2012), 2002-2017. doi: 10.1137/100806515. [24] W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimension, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937. [25] J. W. Yang, R. X. Lian and S. Wang, Incompressible type Euler as scaling limit of compressible Euler-Maxwell equations, Commun. Pure Appl. Anal., 1 (2013), 503-518. doi: 10.3934/cpaa.2013.12.503. [26] J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math., 10 (2014), 2153-2162. doi: 10.1007/s11425-014-4792-4.
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