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February  2018, 11(1): 191-213. doi: 10.3934/krm.2018010

Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$

Zhong Tan and  Leilei Tong ,

School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen 361005, China

*Corresponding author: Leilei Tong

Received  June 2015 Revised  February 2017 Published  August 2017

Fund Project: This work is Supported by the National Natural Science Foundation of China (Grant Nos. 11271305,11531010).

The compressible non-isentropic Navier-Stokes-Maxwell system is investigated in $\mathbb{R}^3$ and the global existence and large time behavior of solutions are established by pure energy method provided the initial perturbation around a constant state is small enough. We first construct the global unique solution under the assumption that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If further the initial data belongs to $\dot{H}^{-s}$ ($0≤ s<3/2$) or $\dot{B}_{2, ∞}^{-s}$ ($0< s≤3/2$), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the $L^p$-$L^2$ $(1≤ p≤ 2)$ type of the decay rates follows without requiring that the $L^p$ norm of initial data is small.

Keywords: Compressible Navier-Stokes-Maxwell system, global solution, time decay rate, energy method, interpolation.
Mathematics Subject Classification: Primary: 35Q35, 35Q30, 35Q61, 82D37, 76N10, 35B40.
Citation: Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic and Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010
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R. 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.

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G. Hong, X. Hou, H. Peng and C. 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.

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S. Ibrahim and S. Keraani, Global small solutions of the Navier-Stokes-Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295.  doi: 10.1137/100819813.

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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, J. Math. Anal. Appl., 396 (2012), 555-561.  doi: 10.1016/j.jmaa.2012.06.038.

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F. Li and Y. Mu, Low Mach number limit of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334-344.  doi: 10.1016/j.jmaa.2013.10.064.

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V. Sohinger and R. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbb{R}_{x}^{n}$, Advances in Mathematics, 261 (2014), 274-332.  doi: 10.1016/j.aim.2014.04.012.

[20]

R. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.  doi: 10.1080/03605300500361545.

[21]

Z. Tan and Y. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.  doi: 10.1016/j.na.2009.05.012.

[22]

Z. Tan and Y. Wang, Global solution and large-time behavior of the $3D$ compressible Euler equations with damping, J. Differential Equations, 254 (2013), 1686-1704.  doi: 10.1016/j.jde.2012.10.026.

[23]

Z. Tan, Y. Wang and Y. Wang, Decay estimates of solutions to the compressible Euler-Maxwell system in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 2846-2873.  doi: 10.1016/j.jde.2014.05.056.

[24]

Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, SIAM J. Math. Anal., 44 (2012), 3281-3323.  doi: 10.1137/120879129.

[25]

J. 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.

show all references

References:
[1]

F. Chen, Introduction to plasma physics and controlled fusion Plasma Physics, Vol. 1,1974.

[2]

R. 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.

[3]

R. Duan, Global smooth flows for the compressible Euler-Maxwell system. The relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413.  doi: 10.1142/S0219891611002421.

[4]

J. Fan and F. Li, Uniform local well-posedness to the density-dependent Navier-Stokes-Maxwell system, Acta Appl Math, 133 (2014), 19-32.  doi: 10.1007/s10440-013-9857-9.

[5]

Y. Feng, Y. 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.

[6]

P. Germain, S. Ibrahim and N. Masmoudi, Well-posedness of the Navier-Stokes-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 71-86.  doi: 10.1017/S0308210512001242.

[7]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc. , Prentice Hall, 2004.

[8]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.  doi: 10.1090/S0894-0347-2011-00722-4.

[9]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[10]

G. Hong, X. Hou, H. Peng and C. 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.

[11]

S. Ibrahim and S. Keraani, Global small solutions of the Navier-Stokes-Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295.  doi: 10.1137/100819813.

[12]

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.

[13]

J. Jerome, The Cauchy problem for compressible hydrodynamic-Maxwell systems: A local theory for smooth solutions, Differential Integral Equations, 16 (2003), 1345-1368. 

[14]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.

[15]

F. Li and Y. Mu, Low Mach number limit of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334-344.  doi: 10.1016/j.jmaa.2013.10.064.

[16]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.

[17]

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.

[18]

T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluids Dynamics, Publications Mathématiques d'Orsay, Université Paris-Sud, Orsay, 1978.

[19]

V. Sohinger and R. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbb{R}_{x}^{n}$, Advances in Mathematics, 261 (2014), 274-332.  doi: 10.1016/j.aim.2014.04.012.

[20]

R. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.  doi: 10.1080/03605300500361545.

[21]

Z. Tan and Y. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.  doi: 10.1016/j.na.2009.05.012.

[22]

Z. Tan and Y. Wang, Global solution and large-time behavior of the $3D$ compressible Euler equations with damping, J. Differential Equations, 254 (2013), 1686-1704.  doi: 10.1016/j.jde.2012.10.026.

[23]

Z. Tan, Y. Wang and Y. Wang, Decay estimates of solutions to the compressible Euler-Maxwell system in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 2846-2873.  doi: 10.1016/j.jde.2014.05.056.

[24]

Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, SIAM J. Math. Anal., 44 (2012), 3281-3323.  doi: 10.1137/120879129.

[25]

J. 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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