Global existence of weak solutions to two dimensional compressible viscoelastic flows

The global existence of weak solutions of the compressible viscoelastic flows in two spatial dimensions is studied in this paper. We show the global existence if the initial velocity u0 is small in Hη with an arbitrary η∈(0,12) and the perturbation of (ρ0,F0) around the constant state (1,I) are small in L2∩B˙p,12p with p∈(−1+1+16η2η,4). One of the main ingredients is that the velocity and the “effective viscous flux” Gi are sufficiently regular for positive time. The regularity of Gi helps to obtain the L∞ estimate of density and deformation gradient, and hence eliminates the possible concentration and oscillation issues.