Decay of the compressible viscoelastic flows

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September 2016, 15(5): 1603-1624. doi: 10.3934/cpaa.2016004

Decay of the compressible viscoelastic flows

W. Wei ^1, , Yin Li ^2, and Zheng-An Yao ^3,

1.	Department of Mathematics, Guizou University, Guiyang, Guizhou Province
2.	School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China
3.	Department of Mathematics, Sun Yat-sen University, Guangzhou 510275

Received December 2014 Revised May 2016 Published July 2016

Abstract
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In this paper we study the time decay rates of the solution to the Cauchy problem for the compressible viscoelastic flows via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, and the $\dot{H}^{-s}(0\leq s<\frac{3}{2})$ negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.

Keywords: Viscoelastic flows, energy method, negative Sobolev space..

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76N15, 76P05, 82C4.

Citation: W. Wei, Yin Li, Zheng-An Yao. Decay of the compressible viscoelastic flows. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1603-1624. doi: 10.3934/cpaa.2016004

References:

[1]	J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.
[2]	K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains, Math. Z., 209 (1992), 115-130. doi: 10.1007/BF02570825.
[3]	K. Deckelnick, $L^{2}$-decay for the compressible Navier-Stokes equations in unbounded domains, Commun. Partial Differ. Equ., 18 (1993), 1445-1476. doi: 10.1080/03605309308820981.
[4]	R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal $L^p-L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-223.
[5]	R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for compressible Navier-Stokes equations with potential force, Math. Models. Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X.
[6]	Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.
[7]	X. P. Hu and D. H. 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.
[8]	X. P. Hu and D. H. 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.
[9]	X. P. Hu and G. C. 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.
[10]	X. P. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 3437-3461.
[11]	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 $R^{3}$, Commun. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.
[12]	T. Kobayashi and Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations, Pacific J. Math., 207 (2002), 199-234. doi: 10.2140/pjm.2002.207.199.
[13]	F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.
[14]	Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Comm. Math. Sci., 5 (2007), 595-616.
[15]	P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146. doi: 10.1142/S0252959900000170.
[16]	T. P. Liu and W. K. Wang, The pointwise estiamtes of diffusion waves for Navier-Stokes equations in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418.
[17]	L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
[18]	J. Z. Qian and Z. F. Zhang, Global well-posedness for the compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868. doi: 10.1007/s00205-010-0351-5.
[19]	M. E. Schonbek, $L^{2}$ decay of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. doi: 10.1007/BF00752111.
[20]	M. E. Schonbek, Large time behaviour of solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763. doi: 10.1080/03605308608820443.
[21]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[22]	T. Zhang and D. Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^p$ framework, SIAM J. Math. Anal., 44 (2012), 2266-2288. doi: 10.1137/110851742.
[23]	Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.
[24]	Y. J. Wang and Z. Tan, Global existence and optimal decay rate for the strong solutions in $H^{2}$ to the compressible Navier-Stokes equations, Appl. Math. Lett., 24 (2011), 1778-1784. doi: 10.1016/j.aml.2011.04.028.

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References:

[1]	J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.
[2]	K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains, Math. Z., 209 (1992), 115-130. doi: 10.1007/BF02570825.
[3]	K. Deckelnick, $L^{2}$-decay for the compressible Navier-Stokes equations in unbounded domains, Commun. Partial Differ. Equ., 18 (1993), 1445-1476. doi: 10.1080/03605309308820981.
[4]	R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal $L^p-L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-223.
[5]	R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for compressible Navier-Stokes equations with potential force, Math. Models. Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X.
[6]	Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.
[7]	X. P. Hu and D. H. 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.
[8]	X. P. Hu and D. H. 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.
[9]	X. P. Hu and G. C. 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.
[10]	X. P. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 3437-3461.
[11]	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 $R^{3}$, Commun. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.
[12]	T. Kobayashi and Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations, Pacific J. Math., 207 (2002), 199-234. doi: 10.2140/pjm.2002.207.199.
[13]	F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.
[14]	Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Comm. Math. Sci., 5 (2007), 595-616.
[15]	P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146. doi: 10.1142/S0252959900000170.
[16]	T. P. Liu and W. K. Wang, The pointwise estiamtes of diffusion waves for Navier-Stokes equations in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418.
[17]	L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
[18]	J. Z. Qian and Z. F. Zhang, Global well-posedness for the compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868. doi: 10.1007/s00205-010-0351-5.
[19]	M. E. Schonbek, $L^{2}$ decay of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. doi: 10.1007/BF00752111.
[20]	M. E. Schonbek, Large time behaviour of solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763. doi: 10.1080/03605308608820443.
[21]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[22]	T. Zhang and D. Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^p$ framework, SIAM J. Math. Anal., 44 (2012), 2266-2288. doi: 10.1137/110851742.
[23]	Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.
[24]	Y. J. Wang and Z. Tan, Global existence and optimal decay rate for the strong solutions in $H^{2}$ to the compressible Navier-Stokes equations, Appl. Math. Lett., 24 (2011), 1778-1784. doi: 10.1016/j.aml.2011.04.028.

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