doi: 10.3934/era.2021067
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Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $

1. 

Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

2. 

School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China

* Corresponding author: yinghuizhang@mailbox.gxnu.edu.cn

Received  June 2021 Revised  July 2021 Early access September 2021

We are concerned with the Cauchy problem of the 3D compressible Navier–Stokes–Poisson system. Compared to the previous related works, the main purpose of this paper is two–fold: First, we prove the optimal decay rates of the higher spatial derivatives of the solution. Second, we investigate the influences of the electric field on the qualitative behaviors of solution. More precisely, we show that the density and high frequency part of the momentum of the compressible Navier–Stokes–Poisson system have the same $ L^2 $ decay rates as the compressible Navier–Stokes equation and heat equation, but the $ L^2 $ decay rate of the momentum is slower due to the effect of the electric field.

Citation: Guochun Wu, Han Wang, Yinghui Zhang. Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $. Electronic Research Archive, doi: 10.3934/era.2021067
References:
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P. Bella, Long time behavior of weak solutions to Navier–Stokes–Poisson system, J. Math. Fluid Mech., 14 (2012), 279-294.  doi: 10.1007/s00021-011-0051-4.  Google Scholar

[2]

Q. BieQ. Wang and Z. Yao, Optimal decay rate for the compressible Navier–Stokes–Poisson system in the critical $L^p$ framework, J. Differential Equations, 263 (2017), 8391-8417.  doi: 10.1016/j.jde.2017.08.041.  Google Scholar

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Q. Chen, G. Wu and Y. Zhang, Optimal large time behavior of the compressible bipolar Navier-Stokes-Poisson system with unequal viscosities, Preprint, arXiv: 2104.08565v1 [math.AP] 17 Apr 2021. Google Scholar

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R. Duan and X. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two–fluid Navier–Stokes–Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014.  doi: 10.3934/cpaa.2013.12.985.  Google Scholar

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[21]

W. Shi and J. Xu, A sharp time–weighted inequality for the compressible Navier–Stokes–Poisson system in the critical $L^p$ framework, J. Differential Equations, 266 (2019), 6426-6458.  doi: 10.1016/j.jde.2018.11.005.  Google Scholar

[22]

Z. Tan and G. 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

[23]

Z. Tan and X. Zhang, Decay of the non-isentropic Navier–Stokes–Poisson equations, J. Math. Anal. Appl., 400 (2013), 293-303.  doi: 10.1016/j.jmaa.2012.09.021.  Google Scholar

[24]

W. Wang and X. Xu, The decay rate of solution for the bipolar Navier–Stokes–Poisson system, J. Math. Phys., 55 (2014), 091502, 22 pp. doi: 10.1063/1.4894766.  Google Scholar

[25]

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

[26]

Z. Wu and W. Wang, Pointwise estimates for bipolar compressible Navier–Stokes–Poisson system in dimension three, Arch. Rational Mech. Anal., 226 (2017), 587-638.  doi: 10.1007/s00205-017-1140-1.  Google Scholar

[27]

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

[28]

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

[29]

F. Zhou and Y. Li, Convergence rate of solutions toward stationary solutions to the bipolar Navier–Stokes–Poisson equations in a half line, Bound. Value Probl., 2013 (2013), 22 pp. doi: 10.1186/1687-2770-2013-124.  Google Scholar

[30]

C. Zou, Large time behaviors of the isentropic bipolar compressible Navier–Stokes–Poisson system, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1725-1740.  doi: 10.1016/S0252-9602(11)60357-3.  Google Scholar

show all references

References:
[1]

P. Bella, Long time behavior of weak solutions to Navier–Stokes–Poisson system, J. Math. Fluid Mech., 14 (2012), 279-294.  doi: 10.1007/s00021-011-0051-4.  Google Scholar

[2]

Q. BieQ. Wang and Z. Yao, Optimal decay rate for the compressible Navier–Stokes–Poisson system in the critical $L^p$ framework, J. Differential Equations, 263 (2017), 8391-8417.  doi: 10.1016/j.jde.2017.08.041.  Google Scholar

[3]

Q. Chen, G. Wu and Y. Zhang, Optimal large time behavior of the compressible bipolar Navier-Stokes-Poisson system with unequal viscosities, Preprint, arXiv: 2104.08565v1 [math.AP] 17 Apr 2021. Google Scholar

[4]

N. Chikami and R. Danchin, On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system, Math. Nachr., 290 (2017), 1939-1970.  doi: 10.1002/mana.201600238.  Google Scholar

[5]

D. Donatelli, Local and global existence for the coupled Navier–Stokes–Poisson problem, Quart. Appl. Math., 61 (2003), 345-361.  doi: 10.1090/qam/1976375.  Google Scholar

[6]

R. Duan, et al. Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-33. doi: 10.1016/j.jde.2007.03.008.  Google Scholar

[7]

R. Duan and X. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two–fluid Navier–Stokes–Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014.  doi: 10.3934/cpaa.2013.12.985.  Google Scholar

[8]

B. DucometE. FeireislH. Petzeltova 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

[9]

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

[10]

Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\mathbb{R}^3$, Arch. Rational Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x.  Google Scholar

[11]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330.  doi: 10.1007/s00205-005-0365-6.  Google Scholar

[12]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbb{R}^3$, J. Differ. Equ., 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.  Google Scholar

[13]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659.  doi: 10.1007/s002200050543.  Google Scholar

[14]

H.-L. LiA. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Arch. Ration. Meth. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4.  Google Scholar

[15]

H.-L. LiT. Yang and C. Zou, Time asymptotic behavior of the bipolar Navier–Stokes–Poisson system, Acta Math. Sci. B, 29 (2009), 1721-1736.  doi: 10.1016/S0252-9602(10)60013-6.  Google Scholar

[16]

Y. Li and J. Liao, Existence and zero-electron-mass limit of strong solutions to the stationary compressible Navier-Stokes-Poisson equation with large external force, Math. Methods Appl. Sci., 41 (2018), 646–663. doi: 10.1002/mma.4634.  Google Scholar

[17]

Y. Li and N. Zhang, Decay rate of strong solutions to compressible Navier-Stokes-Poisson equations with external force, Electron. J. Differential Equations, (2019), Paper No. 61, 18 pp.  Google Scholar

[18]

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.   Google Scholar

[19]

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.  doi: 10.1215/kjm/1250522322.  Google Scholar

[20]

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

[21]

W. Shi and J. Xu, A sharp time–weighted inequality for the compressible Navier–Stokes–Poisson system in the critical $L^p$ framework, J. Differential Equations, 266 (2019), 6426-6458.  doi: 10.1016/j.jde.2018.11.005.  Google Scholar

[22]

Z. Tan and G. 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

[23]

Z. Tan and X. Zhang, Decay of the non-isentropic Navier–Stokes–Poisson equations, J. Math. Anal. Appl., 400 (2013), 293-303.  doi: 10.1016/j.jmaa.2012.09.021.  Google Scholar

[24]

W. Wang and X. Xu, The decay rate of solution for the bipolar Navier–Stokes–Poisson system, J. Math. Phys., 55 (2014), 091502, 22 pp. doi: 10.1063/1.4894766.  Google Scholar

[25]

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

[26]

Z. Wu and W. Wang, Pointwise estimates for bipolar compressible Navier–Stokes–Poisson system in dimension three, Arch. Rational Mech. Anal., 226 (2017), 587-638.  doi: 10.1007/s00205-017-1140-1.  Google Scholar

[27]

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

[28]

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

[29]

F. Zhou and Y. Li, Convergence rate of solutions toward stationary solutions to the bipolar Navier–Stokes–Poisson equations in a half line, Bound. Value Probl., 2013 (2013), 22 pp. doi: 10.1186/1687-2770-2013-124.  Google Scholar

[30]

C. Zou, Large time behaviors of the isentropic bipolar compressible Navier–Stokes–Poisson system, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1725-1740.  doi: 10.1016/S0252-9602(11)60357-3.  Google Scholar

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