American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM
  • ERA Home
  • This Issue
  • Next Article
    Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case
doi: 10.3934/era.2020099

The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $

Yuhui Chen 1, , Ronghua Pan 2,, and  Leilei Tong 3,

1. 

School of Aeronautics and Astronautic, Sun Yat-Sen University, Guangzhou 510275, China
2. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
3. 

Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

* Corresponding author: Ronghua Pan

Received  November 2019 Revised  July 2020 Published  September 2020

Fund Project: The first author and third author are supported by China Scholarship Council. The second author is supported by National Science Foundation

We investigate the sharp time decay rates of the solution $ U $ for the compressible Navier-Stokes system (1.1) in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $ to the constant equilibrium $ (\bar\rho>0, 0) $ when the initial data is a small smooth perturbation of $ (\bar\rho,0) $. Let $ \widetilde U $ be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that $ \|U-\widetilde U\|_{L^2} $ decays at least at the rate of $ (1+t)^{-\frac54} $, which is faster than the rate $ (1+t)^{-\frac34} $ for the $ \widetilde U $ to its equilibrium $ (\bar\rho ,0) $. Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.

Keywords: Isentropic Navier-Stokes, sharp time decay rate, spectral analysis, energy method.
Mathematics Subject Classification: 35A01, 35B45, 35Q35, 76A05, 76D03.
Citation: Yuhui Chen, Ronghua Pan, Leilei Tong. The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $. Electronic Research Archive, doi: 10.3934/era.2020099
References:
[1]

K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains, Math. Z., 209 (1992), 115-130.  doi: 10.1007/BF02570825.  Google Scholar

[2]

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

[3]

R. Duan, H. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.  Google Scholar

[4]

R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758.  doi: 10.1142/S021820250700208X.  Google Scholar

[5]

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.  Google Scholar

[6]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003.  Google Scholar

[7]

D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614.  doi: 10.1007/s000330050049.  Google Scholar

[8]

Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Comm. Math. Phys., 266 (2006), 401-430.  doi: 10.1007/s00220-006-0017-1.  Google Scholar

[9]

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

[10]

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

[11]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensional exterior domain, J. Differential Equations, 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.  Google Scholar

[12]

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 $R^3$, Comm. Math. Phys., 200 (1999), 621-659.  doi: 10.1007/s002200050543.  Google Scholar

[13]

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

[14]

T.-P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418.  Google Scholar

[15]

T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), 120 pp. doi: 10.1090/memo/0599.  Google Scholar

[16]

A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids, MRC-Technical Summary Report, 2194 (1981), 1-16.   Google Scholar

[17]

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

[18]

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

[19]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.  doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[20]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.  Google Scholar

[21]

M. E. Schonbek, Lower bounds of rates of decay for solutions to the Navier-Stokes equations, J. Amer. Math. Soc., 4 (1991), 423-449.  doi: 10.1090/S0894-0347-1991-1103459-2.  Google Scholar

[22]

Y. Shibata and K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Japan, 55 (2003), 797-826.  doi: 10.2969/jmsj/1191419003.  Google Scholar

[23]

Y. Shibata and K. Tanaka, Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid, Comput. Math. Appl., 53 (2007), 605-623.  doi: 10.1016/j.camwa.2006.02.030.  Google Scholar

[24]

Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous $1$-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082.  doi: 10.1002/cpa.3160470804.  Google Scholar

show all references

References:
[1]

K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains, Math. Z., 209 (1992), 115-130.  doi: 10.1007/BF02570825.  Google Scholar

[2]

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

[3]

R. Duan, H. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.  Google Scholar

[4]

R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758.  doi: 10.1142/S021820250700208X.  Google Scholar

[5]

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.  Google Scholar

[6]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003.  Google Scholar

[7]

D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614.  doi: 10.1007/s000330050049.  Google Scholar

[8]

Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Comm. Math. Phys., 266 (2006), 401-430.  doi: 10.1007/s00220-006-0017-1.  Google Scholar

[9]

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

[10]

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

[11]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensional exterior domain, J. Differential Equations, 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.  Google Scholar

[12]

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 $R^3$, Comm. Math. Phys., 200 (1999), 621-659.  doi: 10.1007/s002200050543.  Google Scholar

[13]

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

[14]

T.-P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418.  Google Scholar

[15]

T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), 120 pp. doi: 10.1090/memo/0599.  Google Scholar

[16]

A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids, MRC-Technical Summary Report, 2194 (1981), 1-16.   Google Scholar

[17]

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

[18]

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

[19]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.  doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[20]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.  Google Scholar

[21]

M. E. Schonbek, Lower bounds of rates of decay for solutions to the Navier-Stokes equations, J. Amer. Math. Soc., 4 (1991), 423-449.  doi: 10.1090/S0894-0347-1991-1103459-2.  Google Scholar

[22]

Y. Shibata and K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Japan, 55 (2003), 797-826.  doi: 10.2969/jmsj/1191419003.  Google Scholar

[23]

Y. Shibata and K. Tanaka, Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid, Comput. Math. Appl., 53 (2007), 605-623.  doi: 10.1016/j.camwa.2006.02.030.  Google Scholar

[24]

Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous $1$-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082.  doi: 10.1002/cpa.3160470804.  Google Scholar

[1]

Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021005
[2]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[3]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[4]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[5]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001
[6]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[7]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[8]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[9]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408
[10]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128
[11]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[12]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180
[13]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[14]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061
[15]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
[16]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[17]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002
[18]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[19]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141
[20]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (36)
  • HTML views (163)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences