Well-posedness and time-decay for compressible viscoelastic fluids in critical Besov space

In this paper, we construct local solution with highly oscillating initial velocity and then get the global strong solution in the L^p$L^p$ based Besov space which improves the result of J. Qian, Z. Zhang (2010) 25] and X. Hu, D. Wang (2011) 14]. The local existence and uniqueness lies on the Lagrange coordinate transform and the contraction mapping theorem. The global result lies on a decomposition of the system and some commutator estimates. In the last part, we prove a time-decay in the critical Besov space framework which seems to have little investigation. The proof is based on the properties of the Green's matrix and various interpolations between Besov type spaces.