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 & 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, (1984). Google Scholar

[2]

G. Q. Chen, J. W. Jerome and D. H. Wang, Compressible Euler-Maxwell equations,, \emph{Transport Theory Statist. Phys.}, 29 (2000), 311. doi: 10.1080/00411450008205877. Google Scholar

[3]

P. A. Davidson, An Introduction to Magnetohydrodynamics,, Cambridge Texts in Applied Mathematics, (2001). doi: 10.1017/CBO9780511626333. Google Scholar

[4]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: relaxation case,, \emph{J. Hyperbolic Differ. Equ.}, 8 (2011), 375. doi: 10.1142/S0219891611002421. Google Scholar

[5]

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

[6]

Y. H. Feng, Y. J. Peng and S. Wang, Asymptotic behavior of global smooth solutions for full compressible Navier-Stokes-Maxwell equations,, \emph{Nonlinear Anal. Real World Appl.}, 19 (2014), 105. doi: 10.1016/j.nonrwa.2014.03.004. Google Scholar

[7]

Y. H. Feng, S. Wang and S. Kawashima, Global existence and asymptotic decay of solutions to the non-isentropic Euler-Maxwell system,, \emph{Math. Models Methods Appl. Sci.}, 14 (2014), 2851. doi: 10.1142/S0218202514500390. Google Scholar

[8]

P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system,, \emph{Ann. Sci. $\acuteE$c. Norm. Sup$\acutee$r.}, 3 (2014), 469. Google Scholar

[9]

Y. Guo, Smooth irrotational fluids in the large to the Euler-Poisson system in $\mathbbR^{3+1}$,, \emph{Comm. Math. Phys.}, 195 (1998), 249. doi: 10.1007/s002200050388. Google Scholar

[10]

M. L. Hajjej and Y. J. Peng, Initial layers and zero-relaxation limits of Euler-Maxwell equations,, \emph{J. Differential Equations}, 252 (2012), 1441. doi: 10.1016/j.jde.2011.09.029. Google Scholar

[11]

F.-M. Huang, M. Mei, Y. Wang and H.-M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors,, \emph{SIAM J. Math. Anal.}, 43 (2011), 411. doi: 10.1137/100793025. Google Scholar

[12]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, \emph{Arch. Ration. Mech. Anal.}, 58 (1975), 181. Google Scholar

[13]

S. Kawashima, System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics,, Ph.D thesis, (1983). Google Scholar

[14]

H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$,, \emph{Arch. Rational Mech. Anal.}, 196 (2010), 681. doi: 10.1007/s00205-009-0255-4. Google Scholar

[15]

Q. Q. Liu and C. J. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations,, \emph{Indiana Univ. Math. J.}, 4 (2013), 1203. doi: 10.1512/iumj.2013.62.5047. Google Scholar

[16]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd-multi dimensions,, \emph{Comm. Math. Phys.}, 196 (1998), 145. doi: 10.1007/s002200050418. Google Scholar

[17]

T. Luo, R. Natalini and Z. P. Xin, Large-time behavior of the solutions to a hydrodynamic model for semiconductors,, \emph{SIAM J. Appl. Math.}, 59 (1998), 810. doi: 10.1137/S0036139996312168. Google Scholar

[18]

P. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[19]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations,, \emph{Chin. Ann. Math. Ser. B}, 28 (2007), 583. doi: 10.1007/s11401-005-0556-3. Google Scholar

[20]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations,, \emph{Comm. Partial Differential Equations}, 33 (2008), 349. doi: 10.1080/03605300701318989. Google Scholar

[21]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations,, \emph{SIAM J. Math. Anal.}, 40 (2008), 540. doi: 10.1137/070686056. Google Scholar

[22]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system,, \emph{Methods Appl. Anal.}, 18 (2011), 245. doi: 10.4310/MAA.2011.v18.n3.a1. Google Scholar

[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,, \emph{SIAM J. Math. Anal.}, 44 (2012), 2002. doi: 10.1137/100806515. Google Scholar

[24]

W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimension,, \emph{J. Differential Equations}, 173 (2001), 410. doi: 10.1006/jdeq.2000.3937. Google Scholar

[25]

J. W. Yang, R. X. Lian and S. Wang, Incompressible type Euler as scaling limit of compressible Euler-Maxwell equations,, \emph{Commun. Pure Appl. Anal.}, 1 (2013), 503. doi: 10.3934/cpaa.2013.12.503. Google Scholar

[26]

J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations,, \emph{Sci. China Math.}, 10 (2014), 2153. doi: 10.1007/s11425-014-4792-4. Google Scholar

show all references

References:
[1]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Vol. 1, (1984). Google Scholar

[2]

G. Q. Chen, J. W. Jerome and D. H. Wang, Compressible Euler-Maxwell equations,, \emph{Transport Theory Statist. Phys.}, 29 (2000), 311. doi: 10.1080/00411450008205877. Google Scholar

[3]

P. A. Davidson, An Introduction to Magnetohydrodynamics,, Cambridge Texts in Applied Mathematics, (2001). doi: 10.1017/CBO9780511626333. Google Scholar

[4]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: relaxation case,, \emph{J. Hyperbolic Differ. Equ.}, 8 (2011), 375. doi: 10.1142/S0219891611002421. Google Scholar

[5]

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

[6]

Y. H. Feng, Y. J. Peng and S. Wang, Asymptotic behavior of global smooth solutions for full compressible Navier-Stokes-Maxwell equations,, \emph{Nonlinear Anal. Real World Appl.}, 19 (2014), 105. doi: 10.1016/j.nonrwa.2014.03.004. Google Scholar

[7]

Y. H. Feng, S. Wang and S. Kawashima, Global existence and asymptotic decay of solutions to the non-isentropic Euler-Maxwell system,, \emph{Math. Models Methods Appl. Sci.}, 14 (2014), 2851. doi: 10.1142/S0218202514500390. Google Scholar

[8]

P. Germain and N. Masmoudi, Global existence for the Euler-Maxwell system,, \emph{Ann. Sci. $\acuteE$c. Norm. Sup$\acutee$r.}, 3 (2014), 469. Google Scholar

[9]

Y. Guo, Smooth irrotational fluids in the large to the Euler-Poisson system in $\mathbbR^{3+1}$,, \emph{Comm. Math. Phys.}, 195 (1998), 249. doi: 10.1007/s002200050388. Google Scholar

[10]

M. L. Hajjej and Y. J. Peng, Initial layers and zero-relaxation limits of Euler-Maxwell equations,, \emph{J. Differential Equations}, 252 (2012), 1441. doi: 10.1016/j.jde.2011.09.029. Google Scholar

[11]

F.-M. Huang, M. Mei, Y. Wang and H.-M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors,, \emph{SIAM J. Math. Anal.}, 43 (2011), 411. doi: 10.1137/100793025. Google Scholar

[12]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, \emph{Arch. Ration. Mech. Anal.}, 58 (1975), 181. Google Scholar

[13]

S. Kawashima, System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics,, Ph.D thesis, (1983). Google Scholar

[14]

H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$,, \emph{Arch. Rational Mech. Anal.}, 196 (2010), 681. doi: 10.1007/s00205-009-0255-4. Google Scholar

[15]

Q. Q. Liu and C. J. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations,, \emph{Indiana Univ. Math. J.}, 4 (2013), 1203. doi: 10.1512/iumj.2013.62.5047. Google Scholar

[16]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd-multi dimensions,, \emph{Comm. Math. Phys.}, 196 (1998), 145. doi: 10.1007/s002200050418. Google Scholar

[17]

T. Luo, R. Natalini and Z. P. Xin, Large-time behavior of the solutions to a hydrodynamic model for semiconductors,, \emph{SIAM J. Appl. Math.}, 59 (1998), 810. doi: 10.1137/S0036139996312168. Google Scholar

[18]

P. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[19]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations,, \emph{Chin. Ann. Math. Ser. B}, 28 (2007), 583. doi: 10.1007/s11401-005-0556-3. Google Scholar

[20]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations,, \emph{Comm. Partial Differential Equations}, 33 (2008), 349. doi: 10.1080/03605300701318989. Google Scholar

[21]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations,, \emph{SIAM J. Math. Anal.}, 40 (2008), 540. doi: 10.1137/070686056. Google Scholar

[22]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system,, \emph{Methods Appl. Anal.}, 18 (2011), 245. doi: 10.4310/MAA.2011.v18.n3.a1. Google Scholar

[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,, \emph{SIAM J. Math. Anal.}, 44 (2012), 2002. doi: 10.1137/100806515. Google Scholar

[24]

W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimension,, \emph{J. Differential Equations}, 173 (2001), 410. doi: 10.1006/jdeq.2000.3937. Google Scholar

[25]

J. W. Yang, R. X. Lian and S. Wang, Incompressible type Euler as scaling limit of compressible Euler-Maxwell equations,, \emph{Commun. Pure Appl. Anal.}, 1 (2013), 503. doi: 10.3934/cpaa.2013.12.503. Google Scholar

[26]

J. W. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations,, \emph{Sci. China Math.}, 10 (2014), 2153. doi: 10.1007/s11425-014-4792-4. Google Scholar

[1]

Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic & Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002

[2]

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

[3]

Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005

[4]

Xueke Pu, Min Li. Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5149-5181. doi: 10.3934/dcdsb.2019055

[5]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010

[6]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[7]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056

[8]

John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333

[9]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[10]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

[11]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[12]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129

[13]

Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032

[14]

Yinxia Wang, Hengjun Zhao. Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating. Communications on Pure & Applied Analysis, 2018, 17 (2) : 347-374. doi: 10.3934/cpaa.2018020

[15]

Wenjun Wang, Weike Wang. Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513

[16]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615

[17]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

[18]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229

[19]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[20]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]