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 Lp 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 Lp 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.
ASJC Scopus subject areas
- Applied Mathematics