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April  2022, 42(4): 1835-1853. doi: 10.3934/dcds.2021174

On $ L^1 $ estimates of solutions of compressible viscoelastic system

Yusuke Ishigaki

Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Ookayama 2-12-1, Tokyo 152-8551, Japan

Received  July 2021 Published  April 2022 Early access  November 2021

Fund Project: This work was partially supported by JSPS KAKENHI Grant Number 19J10056

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.

Keywords: Compressible viscoelastic system, diffusion wave, large time behavior.
Mathematics Subject Classification: Primary: 76N10, Secondary: 35B40, 35Q35, 76A10.
Citation: Yusuke Ishigaki. On $ L^1 $ estimates of solutions of compressible viscoelastic system. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1835-1853. doi: 10.3934/dcds.2021174
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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