  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.
