The paper by Yuan Cai, Zhen Lei and their collaborators Fanghua Lin, Nader Masmoudi, entitled "Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions", has been published in the journal "Comm. Pure Appl. Math." in 2019.
The equations of the two-dimensional incompressible viscoelasticity is a system coupling the Navier-Stokes equations with a transport equation for the deformation tensor. The existence of global smooth solutions near the equilibrium with a fixed positive viscosity was known. The inviscid case was recently solved by Zhen Lei. While the latter was solely based on the techniques from the studies of hyperbolic equations, and hence the 2D problem is in general more challenging than that in higher dimensions, the former was relied crucially upon a dissipative mechanism. These two approaches are not compatible. In this work, they prove global existence of solutions, uniformly in both global time and viscosity. This allows 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, they introduce in the 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 cases 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.
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