Global Solutions of Two-Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem, and it is studied in this paper. We show global existence if the initial deformation gradient is close to the identity matrix in L-2 boolean AND L-infinity and the initial velocity is small in L-2 and bounded in L-p for some p > 2. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd-B model, the additional assumption on the initial velocity being bounded in L-p for some p > 2 may due to techniques we employed. The smallness assumption on the L-2 norm of the initial velocity is, however, natural for global well-posedness. One of the key observations in the paper is that the velocity and the effective viscous flux G are sufficiently regular for positive time. The regularity of G leads to a new approach for the pointwise estimate for the deformation gradient without using L-infinity bounds on the velocity gradients in spatial variables. (C) 2015 Wiley Periodicals, Inc.