Optimal time decay rate for compressible viscoelastic equations in critical spaces

In this paper, we are concerned with the optimal time convergence rate of the global strong solution to some constant equilibrium states for the compressible viscoelastic fluids in the whole space. Green’s matrix method and energy estimate method are used to obtain the optimal time decay rate under the critical Besov space framework. Our result implies the optimal (Formula presented.) -time decay rate and only need the initial datum to be small in some critical Besov space which have very low regularity compared with the classical Sobolev space.