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On $ L^1 $ estimates of solutions of compressible viscoelastic system
Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Ookayama 2-12-1, Tokyo 152-8551, Japan |
We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to $ W^{2,1} $, and is sufficiently small in $ H^4\cap L^1 $, the solutions grow in time at the same rate as $ t^{\frac{1}{2}} $ in $ L^1 $ due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.
References:
[1] |
Q. Chen and G. Wu,
The 3D compressible viscoelastic fluid in a bounded domain, Commun. Math. Sci., 16 (2018), 1303-1323.
doi: 10.4310/CMS.2018.v16.n5.a6. |
[2] |
M. E. Gurtin, An Introduction to Continuum Mechanics, Math. Sci. Eng., vol. 158, Academic Press, New York-London, 1981 |
[3] |
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. |
[4] |
X. Hu,
Global existence of weak solutions to two dimensional compressible viscoelastic flows, J. Differential Equations, 265 (2018), 3130-3167.
doi: 10.1016/j.jde.2018.05.001. |
[5] |
X. Hu and D. Wang,
Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198.
doi: 10.1016/j.jde.2010.03.027. |
[6] |
X. Hu and D. Wang,
Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.
doi: 10.1016/j.jde.2010.10.017. |
[7] |
X. Hu and G. Wu,
Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833.
doi: 10.1137/120892350. |
[8] |
X. Hu and W. Zhao,
Global existence of compressible dissipative elastodynamics systems with zero shear viscosity in two dimensions, Arch. Ration. Mech. Anal., 235 (2020), 1177-1243.
doi: 10.1007/s00205-019-01443-z. |
[9] |
X. Hu and W. Zhao,
Global existence for the compressible viscoelastic system with zero shear viscosity in three dimensions, J. Differential Equations, 268 (2020), 1658-1685.
doi: 10.1016/j.jde.2019.09.034. |
[10] |
Y. Ishigaki,
Diffusion wave phenomena and $L^p$ decay estimates of solutions of compressible viscoelastic system, J. Differential Equations, 269 (2020), 11195-11230.
doi: 10.1016/j.jde.2020.07.020. |
[11] |
S. Kawashima, A. Matsumura and T. Nishida,
On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Commun. Math. Phys., 70 (1979), 97-124.
doi: 10.1007/BF01982349. |
[12] |
T. Kobayashi and Y. Shibata,
Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equation, Pacific J. Math., 207 (2002), 199-234.
doi: 10.2140/pjm.2002.207.199. |
[13] |
Y. Li, R. Wei and Z. Yao, Optimal decay rates for the compressible viscoelastic flows, J. Math. Phys., 57 (2016), 111506, 8 pp.
doi: 10.1063/1.4967975. |
[14] |
F.-H. Lin, C. Liu and P. Zhang,
On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.
doi: 10.1002/cpa.20074. |
[15] |
A. Matsumura, T. Nishida and P. Zhang,
The initial value problems for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.
|
[16] |
X. Pan, J. Xu and P. Zhang,
Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces, Discrete Contin. Dyn. Syst., 39 (2019), 2021-2057.
doi: 10.3934/dcds.2019085. |
[17] |
J. Qian,
Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.
doi: 10.1016/j.jde.2010.07.026. |
[18] |
J. Qian and Z. Zhang,
Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868.
doi: 10.1007/s00205-010-0351-5. |
[19] |
Y. Shibata,
On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226.
doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. |
[20] |
T. C. Sideris and B. Thomases,
Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788.
doi: 10.1002/cpa.20049. |
[21] |
R. Wei, Y. Li and Z. Yao,
Decay of the compressible viscoelastic flows, Commun. Pure Appl. Anal., 15 (2016), 1603-1624.
doi: 10.3934/cpaa.2016004. |
[22] |
G. Wu, Z. Gao and Z. Tan,
Time decay rates for the compressible viscoelastic flows, J. Math. Anal. Appl., 452 (2017), 990-1004.
doi: 10.1016/j.jmaa.2017.03.044. |
[23] |
F. Xu, X. Zhang, Y. Wu and L. Liu,
The optimal convergence rates for the multi-dimensional compressible viscoelastic flows, ZAMM Z. Angew. Math. Mech., 96 (2016), 1490-1504.
doi: 10.1002/zamm.201500095. |
show all references
References:
[1] |
Q. Chen and G. Wu,
The 3D compressible viscoelastic fluid in a bounded domain, Commun. Math. Sci., 16 (2018), 1303-1323.
doi: 10.4310/CMS.2018.v16.n5.a6. |
[2] |
M. E. Gurtin, An Introduction to Continuum Mechanics, Math. Sci. Eng., vol. 158, Academic Press, New York-London, 1981 |
[3] |
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. |
[4] |
X. Hu,
Global existence of weak solutions to two dimensional compressible viscoelastic flows, J. Differential Equations, 265 (2018), 3130-3167.
doi: 10.1016/j.jde.2018.05.001. |
[5] |
X. Hu and D. Wang,
Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198.
doi: 10.1016/j.jde.2010.03.027. |
[6] |
X. Hu and D. Wang,
Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.
doi: 10.1016/j.jde.2010.10.017. |
[7] |
X. Hu and G. Wu,
Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833.
doi: 10.1137/120892350. |
[8] |
X. Hu and W. Zhao,
Global existence of compressible dissipative elastodynamics systems with zero shear viscosity in two dimensions, Arch. Ration. Mech. Anal., 235 (2020), 1177-1243.
doi: 10.1007/s00205-019-01443-z. |
[9] |
X. Hu and W. Zhao,
Global existence for the compressible viscoelastic system with zero shear viscosity in three dimensions, J. Differential Equations, 268 (2020), 1658-1685.
doi: 10.1016/j.jde.2019.09.034. |
[10] |
Y. Ishigaki,
Diffusion wave phenomena and $L^p$ decay estimates of solutions of compressible viscoelastic system, J. Differential Equations, 269 (2020), 11195-11230.
doi: 10.1016/j.jde.2020.07.020. |
[11] |
S. Kawashima, A. Matsumura and T. Nishida,
On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Commun. Math. Phys., 70 (1979), 97-124.
doi: 10.1007/BF01982349. |
[12] |
T. Kobayashi and Y. Shibata,
Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equation, Pacific J. Math., 207 (2002), 199-234.
doi: 10.2140/pjm.2002.207.199. |
[13] |
Y. Li, R. Wei and Z. Yao, Optimal decay rates for the compressible viscoelastic flows, J. Math. Phys., 57 (2016), 111506, 8 pp.
doi: 10.1063/1.4967975. |
[14] |
F.-H. Lin, C. Liu and P. Zhang,
On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.
doi: 10.1002/cpa.20074. |
[15] |
A. Matsumura, T. Nishida and P. Zhang,
The initial value problems for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.
|
[16] |
X. Pan, J. Xu and P. Zhang,
Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces, Discrete Contin. Dyn. Syst., 39 (2019), 2021-2057.
doi: 10.3934/dcds.2019085. |
[17] |
J. Qian,
Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.
doi: 10.1016/j.jde.2010.07.026. |
[18] |
J. Qian and Z. Zhang,
Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868.
doi: 10.1007/s00205-010-0351-5. |
[19] |
Y. Shibata,
On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226.
doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. |
[20] |
T. C. Sideris and B. Thomases,
Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788.
doi: 10.1002/cpa.20049. |
[21] |
R. Wei, Y. Li and Z. Yao,
Decay of the compressible viscoelastic flows, Commun. Pure Appl. Anal., 15 (2016), 1603-1624.
doi: 10.3934/cpaa.2016004. |
[22] |
G. Wu, Z. Gao and Z. Tan,
Time decay rates for the compressible viscoelastic flows, J. Math. Anal. Appl., 452 (2017), 990-1004.
doi: 10.1016/j.jmaa.2017.03.044. |
[23] |
F. Xu, X. Zhang, Y. Wu and L. Liu,
The optimal convergence rates for the multi-dimensional compressible viscoelastic flows, ZAMM Z. Angew. Math. Mech., 96 (2016), 1490-1504.
doi: 10.1002/zamm.201500095. |
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