Global strong solutions for the multi-dimensional compressible viscoelastic flows with general pressure law

In this paper, we mainly focus on the compressible viscoelastic flows of Oldroyd type with the general pressure law, with one of the non-Newtonian fluids exhibiting the elastic behavior. For the viscoelastic flows of Oldroyd type with the general pressure law, P′(ρ̄)+α>0, with α > 0 being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data u⃗0 and the low frequency part of ρ0, τ0 are small enough compared to the viscosity coefficients. In particular, when the viscosity is large, the part of the initial data can be large. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces.