  This paper studies the inviscid limit of the two-dimensional incompressible
viscoelasticity, which is a system coupling a Navier-Stokes equation with a
transport equation for the deformation tensor. The existence of global smooth
solutions near the equilibrium with a fixed positive viscosity was known since
the work of F. H. Lin, C. Liu, and P. Zhang in "On hydrodynamics of
viscoelastic fluids". The inviscid case was solved recently by the second
author Z. Lei. in "Global well-posedness of incompressible elastodynamics in
two dimensions". While the latter was solely based on the techniques from the
studies of hyperbolic equations, and hence the 2D problem is in general more
challenge than that in higher dimensions, the former was relied crucially upon
a dissipative mechanism. Indeed, after a symmetrization and a linearization
around the equilibrium, the system of the incompressible viscoelasticity
reduces to an incompressible system of damped wave equations for both the fluid
velocity and the deformation tensor. These two approaches are not compatible.
In this paper, we prove global existence of solutions, uniformly in both time
$t \in [0, \infty)$ and viscosity $\mu \geq 0$. This allows us to justify in
particular the vanishing viscosity limit for all time. In order to overcome
difficulties coming from the incompatibility between the purely hyperbolic
limiting system and the systems with additional parabolic viscous
perturbations, we introduce in this paper a rather robust method which may
apply to a wide class of physical systems of similar nature. Roughly speaking,
the method works in two dimensional case whenever the hyperbolic system
satisfies intrinsically a "Strong Null Condition". For dimensions not less than
three, the usual null condition is sufficient for this method to work.
