On the Cauchy problem for two dimensional incompressible viscoelastic flows

We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a new $\textit{a priori}$ estimate on the space-time norm in $L^{\f32}_{loc}$ of the Cauchy-Green strain tensor $\tau=\F\F^\top$, or equivalently the $L^3_{loc}$ norm of the Jacobian of the flow map $\F$. It allows us to rule out possible concentrations of the energy due to deformations associated with the flow maps. Following the general compactness arguments due to DiPerna and Lions (\cite{DL}, \cite{FNP}, \cite{PL}), and using the so-called \textit{effective viscous flux}, $\mathcal{G}$, which was introduced in our previous work \cite{HL}, we are able to control the possible oscillations of deformation gradients as well.