Incompressible limit of Oldroyd-B fluids in the whole space

This paper is dedicated to the study of compressible Oldroyd-B model in the framework of critical spaces. We first prove the local well-posedness for the compressible Oldroyd-B model with initial data (ρ0−1,u0,τ0)∈B˙2,1d2×(B˙2,1d2−1)d×(B˙2,1d2)d×d. The global well-posedness is also established provided the initial data and coupling constant are sufficiently small. On the basis of this, we also consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to the compressible Oldroyd-B fluids converges to the solution to the corresponding incompressible model in some function spaces. In the meanwhile, the converge rates are obtained in some sense. To fulfill our proof, we give a new commutator estimate.