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Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
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 |
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.
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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.
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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. |
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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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