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

Yusuke Ishigaki

doi:10.3934/dcds.2021174

Article Contents

2022, Volume 42, Issue 4: 1835-1853. Doi: 10.3934/dcds.2021174

This issue Previous Article Fujita type results for quasilinear parabolic inequalities with nonlocal terms Next Article Sublacunary sets and interpolation sets for nilsequences

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: June 30, 2021

Early access: November 2021

Published: April 2022

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 76N10, Secondary: 35B40, 35Q35, 76A10.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.