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doi: 10.3934/era.2020099

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

 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.

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
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