Optimal temporal decay rates for the compressible viscoelastic flows

For the Cauchy problem of the three-dimensional compressible viscoelastic flows, we establish the optimal temporal decay rates of the all-order spatial derivatives of the global strong solution in the weaker initial condition. The main novelty of this paper is that the optimal decay estimates of the highest-order derivatives of the solution is obtained by using spectral analysis and energy method, which can be considered as the further investigation to [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] with only the lower-order derivative estimates.