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Optimal Time Decay Rate for the Compressible Viscoelastic Equations in Critical Spaces

In this paper, we are concerned with the convergence rates of the global strong solution to constant equilibrium state for the compressible viscoelastic fluids in the whole space. We combine both analysis about Green's matrix method and energy estimate method to get optimal time decay rate in critical Besov space framework. Our result imply the optimal $L^{2}$-time decay rate and only need the initial data to be small in critical Besov space which have very low regularity compared with traditional Sobolev space.