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 H^η with an arbitrary η∈(0,[Formula presented]) and the perturbation of (ρ₀,F₀) around the constant state (1,I) are small in L²∩B˙_p,1 ^{[Formula presented]} with p∈([Formula presented],4). One of the main ingredients is that the velocity and the “effective viscous flux” G_i are sufficiently regular for positive time. The regularity of G_i helps to obtain the L^∞ estimate of density and deformation gradient, and hence eliminates the possible concentration and oscillation issues.