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

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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 $\mathbb{R}^3$

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

References:

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[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.
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[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.
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[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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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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