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

## Decay of the compressible viscoelastic flows

 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

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.
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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