Global solutions of two dimensional incompressible viscoelastic flows with discontinuous initial data

Abstract

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a long standing open problem, and it is studied in this paper. We show the global existence if the initial deformation gradient is close to the identity matrix in $L^2\cap L^{\infty }$, 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 the global well-posedness . One of the key observations in the paper is that the velocity and the \textquotedblleft effective viscous flux\textquotedblright $\mathcal{G}$ are sufficiently regular for positive time. The regularity of $\mathcal{G}$ leads to a new approach for the pointwise estimate for the deformation gradient without using $L^{\infty }$ bounds on the velocity gradients in spatial variables.