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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 $\mathbb{R}^{3}$

 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.
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

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##### 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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