• Previous Article
    Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation
  • KRM Home
  • This Issue
  • Next Article
    Optimal time decay of the non cut-off Boltzmann equation in the whole space
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.  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.   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.  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.  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.  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.  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.   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.  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.  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.   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.  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.  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.  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.   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.  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.   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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.   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.  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.  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.   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.  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.  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.  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.   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.  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.   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.  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.  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.  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.  doi: 10.1002/mma.786.  Google Scholar

[1]

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

[2]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013

[3]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009

[4]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985

[5]

Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077

[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]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[8]

Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751

[9]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[10]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078

[11]

Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic & Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043

[12]

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

[13]

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

[14]

Debanjana Mitra, Mythily Ramaswamy, Jean-Pierre Raymond. Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension. Mathematical Control & Related Fields, 2015, 5 (2) : 259-290. doi: 10.3934/mcrf.2015.5.259

[15]

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

[16]

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

[17]

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

[18]

Tong Tang, Hongjun Gao. On the compressible Navier-Stokes-Korteweg equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2745-2766. doi: 10.3934/dcdsb.2016071

[19]

Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040

[20]

Baowei Feng. On the decay rates for a one-dimensional porous elasticity system with past history. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2905-2921. doi: 10.3934/cpaa.2019130

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]