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September  2012, 5(3): 615-638. doi: 10.3934/krm.2012.5.615

Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$

Zhong Tan 1, , Yong Wang 1, and  Xu Zhang 1,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China, China, China

Received  January 2012 Revised  February 2012 Published  August 2012

We are concerned with the long-time behavior of global strong solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$, where the electric field is governed by the self-consistent Poisson equation. When the regular initial perturbations belong to $H^{4}(\mathbb{R}^{3})\cap \dot{B}_{1,\infty}^{-s}(\mathbb{R}^{3})$ with $s\in [0,1]$, we show that the density and momentum of the system converge to their equilibrium state at the optimal $L^2$-rates $(1+t)^{-\frac{3}{4}-\frac{s}{2}}$ and $(1+t)^{-\frac{1}{4}-\frac{s}{2}}$ respectively, and the decay rate is still $(1+t)^{-\frac{3}{4}}$ for temperature which is proved to be not optimal.
Keywords: optimal decay rates., Compressible Navier-Stokes-Poisson system.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: 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
References:
[1]

K. Deckelnick, $L^2$-decay for the compressible Navier-Stokes equations in unbounded domains, Comm. Partial Differential Equations, 18 (1993), 1445-1476. doi: 10.1080/03605309308820981.  Google Scholar

[2]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math, 61 (2003), 345-361.  Google Scholar

[3]

B. Ducomet and A. Zlotnik, Stabilization and stability for the spherically symmetric Navier-Stokes-Poisson system, Appl. Math. Lett., 18 (2005), 1190-1198. doi: 10.1016/j.aml.2004.12.002.  Google Scholar

[4]

B. Ducomet, A remark about global existence for the Navier-Stokes-Poisson system, Appl. Math. Lett., 12 (1999), 31-37. doi: 10.1016/S0893-9659(99)00098-1.  Google Scholar

[5]

B. Ducomet, E. Feireisl, H. Petzeltová and I. Straškraba, Global in time weak solution for compressible barotropic self-gravitating fluids, Discrete Contin. Dyn. Syst, 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113.  Google Scholar

[6]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, to appear in Commun. Part. Diff. Equ., 2012. Google Scholar

[7]

C. Hao and H.-L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equation, 246 (2009), 4791-4812. doi: 10.1016/j.jde.2008.11.019.  Google Scholar

[8]

L. Hsiao, H.-L. Li, T. Yang and C. Zou, Compressible non-isentropic bipolar Navier-Stokes-Poisson system in $\mathbbR^3$, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2169-2194.  Google Scholar

[9]

D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in $\mathbbR^3$, Comm. Math. Phy., 257 (2005), 579-619. doi: 10.1007/s00220-005-1351-4.  Google Scholar

[10]

H.-L. Li, A. Matsumura and G.-J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4.  Google Scholar

[11]

H.-L. Li, T. Yang and C. Zou, Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 1721-1736.  Google Scholar

[12]

H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbbR^3$, Math. Methods Appl. Sci., 34 (2011), 670-682. doi: 10.1002/mma.1391.  Google Scholar

[13]

H.-L. Li and T. Zhang, Large time behavior of solutions to $3D$ compressible Navier-Stokes-Poisson system, Sci. China Math., 55 (2012), 159-177. doi: 10.1007/s11425-011-4280-z.  Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schimeiser, "Semiconductor,'' Springer, 1990. Google Scholar

[15]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342. doi: 10.3792/pjaa.55.337.  Google Scholar

[16]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[17]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.  Google Scholar

[18]

V. A. Solonnikov, Evolution free boundary problem for equations of motion viscous compressible self gravitating fluid, Stability Appl. Anal. Contin. Media, 3 (1993), 257-275. Google Scholar

[19]

Z. Tan and G. C. Wu, Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World Appl., 13 (2012), 650-664. doi: 10.1016/j.nonrwa.2011.08.005.  Google Scholar

[20]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Diff. Equ., 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.  Google Scholar

[21]

G.-J. Zhang, H.-L. Li and C.-J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^3$, J. Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035.  Google Scholar

[22]

Y.-H. Zhang and Z. Tan, On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 30 (2007), 305-329. doi: 10.1002/mma.786.  Google Scholar

show all references

References:
[1]

K. Deckelnick, $L^2$-decay for the compressible Navier-Stokes equations in unbounded domains, Comm. Partial Differential Equations, 18 (1993), 1445-1476. doi: 10.1080/03605309308820981.  Google Scholar

[2]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math, 61 (2003), 345-361.  Google Scholar

[3]

B. Ducomet and A. Zlotnik, Stabilization and stability for the spherically symmetric Navier-Stokes-Poisson system, Appl. Math. Lett., 18 (2005), 1190-1198. doi: 10.1016/j.aml.2004.12.002.  Google Scholar

[4]

B. Ducomet, A remark about global existence for the Navier-Stokes-Poisson system, Appl. Math. Lett., 12 (1999), 31-37. doi: 10.1016/S0893-9659(99)00098-1.  Google Scholar

[5]

B. Ducomet, E. Feireisl, H. Petzeltová and I. Straškraba, Global in time weak solution for compressible barotropic self-gravitating fluids, Discrete Contin. Dyn. Syst, 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113.  Google Scholar

[6]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, to appear in Commun. Part. Diff. Equ., 2012. Google Scholar

[7]

C. Hao and H.-L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equation, 246 (2009), 4791-4812. doi: 10.1016/j.jde.2008.11.019.  Google Scholar

[8]

L. Hsiao, H.-L. Li, T. Yang and C. Zou, Compressible non-isentropic bipolar Navier-Stokes-Poisson system in $\mathbbR^3$, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2169-2194.  Google Scholar

[9]

D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in $\mathbbR^3$, Comm. Math. Phy., 257 (2005), 579-619. doi: 10.1007/s00220-005-1351-4.  Google Scholar

[10]

H.-L. Li, A. Matsumura and G.-J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4.  Google Scholar

[11]

H.-L. Li, T. Yang and C. Zou, Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 1721-1736.  Google Scholar

[12]

H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbbR^3$, Math. Methods Appl. Sci., 34 (2011), 670-682. doi: 10.1002/mma.1391.  Google Scholar

[13]

H.-L. Li and T. Zhang, Large time behavior of solutions to $3D$ compressible Navier-Stokes-Poisson system, Sci. China Math., 55 (2012), 159-177. doi: 10.1007/s11425-011-4280-z.  Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schimeiser, "Semiconductor,'' Springer, 1990. Google Scholar

[15]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342. doi: 10.3792/pjaa.55.337.  Google Scholar

[16]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[17]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.  Google Scholar

[18]

V. A. Solonnikov, Evolution free boundary problem for equations of motion viscous compressible self gravitating fluid, Stability Appl. Anal. Contin. Media, 3 (1993), 257-275. Google Scholar

[19]

Z. Tan and G. C. Wu, Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World Appl., 13 (2012), 650-664. doi: 10.1016/j.nonrwa.2011.08.005.  Google Scholar

[20]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Diff. Equ., 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.  Google Scholar

[21]

G.-J. Zhang, H.-L. Li and C.-J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^3$, J. Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035.  Google Scholar

[22]

Y.-H. Zhang and Z. Tan, On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 30 (2007), 305-329. doi: 10.1002/mma.786.  Google Scholar

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