Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions

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 [35]. The inviscid case was solved recently by the second author [28]. While the latter was solely based on the techniques from the studies of hyperbolic equations, and hence the two-dimensional problem is in general more challenging 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 ∈ [0, +∞) and viscosity μ ≥ 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 that may apply to a wide class of physical systems of similar nature. Roughly speaking, the method works in the 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.