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

Zhong Tan; Leilei Tong

doi:10.3934/krm.2018010

Article Contents

2018, Volume 11, Issue 2: 1-23. Doi: 10.3934/krm.2018010

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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
* Corresponding author: Leilei Tong

Received: June 2015

Revised: February 2017

Published: April 2018

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

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Abstract

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.

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References

	F. Chen, Introduction to plasma physics and controlled fusion Plasma Physics, Vol. 1,1974.
	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.
	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.
	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.
	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.
	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.
	L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc. , Prentice Hall, 2004.
	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.
	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.
	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.
	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.
	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.
	J. Jerome , The Cauchy problem for compressible hydrodynamic-Maxwell systems: A local theory for smooth solutions, Differential Integral Equations, 16 (2003) , 1345-1368.
	T. Kato , The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975) , 181-205. doi: 10.1007/BF00280740.
	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.
	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.
	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.
	T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluids Dynamics, Publications Mathématiques d'Orsay, Université Paris-Sud, Orsay, 1978.
	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.
	R. Strain and Y. Guo , Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006) , 417-429. doi: 10.1080/03605300500361545.
	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.
	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.
	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.
	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.
	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.