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Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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