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 L²∩L^∞ and the initial velocity is small in L² 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 L² 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^∞ bounds on the velocity gradients in spatial variables.