American Institue of Mathematical Sciences

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.

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:
 [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, 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, 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, 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, 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, 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, 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, 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, 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, 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, Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545. [21] Z. Tan, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545. [21] Z. Tan, 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, 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, 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, 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.
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [9] Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75 [10] Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087 [11] Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 [12] 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 [13] 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 [14] Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433 [15] Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045 [16] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 [17] Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 [18] Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471 [19] Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851 [20] Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721

2016 Impact Factor: 1.261

Article outline