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 u(0) is small in H-eta with an arbitrary eta is an element of (0, 1/2) and the perturbation of (rho(0), F-0) around the constant state (1, I) are small in L-2 boolean AND B (over dot)(p, 1)(2/P) with p is an element of (-1+root 1+16 eta/2 eta, 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-infinity estimate of density and deformation gradient, and hence eliminates the possible concentration and oscillation issues. (C) 2018 Elsevier Inc. All rights reserved.