Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions

Abstract

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. © 2019 Wiley Periodicals, Inc.

Global well-posedness for the three dimensional simplified inertial Ericksen-Leslie systems near equilibrium

“…Unfortunately, the system (2.1) doesn't have any scaling invariance (nor Lorentz invariance). Inspired by [3,11], we are still able to use the scaling operator under this circumstance. Applying S + 1 onto (2.1) 1 , (2.1) 3 , and applying S + 2 onto (2.1), thanks to the commutation:…”

“…The main trouble in the estimate of E v κ is due to commutators between the scaling operator and the viscosity terms. Fortunately, we can take the approach borrowed from [3,11].…”

“…Next, we estimate the diffusion terms with indefinite sign. To this end, we need a technical lemma followed from [3].…”

“…Remark 1.1. Following the recent work [3,18,20], we can apply the ghost weight energy method [1] to even prove a uniform bound for the highest-order energy. However, as the current estimate is sufficient to obtain the global existences of solutions, we will not pursue this direction in this paper.…”

Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D

“…There is an extensive literature on the existence of solutions to the equations of viscoelastic fluids, see for example [1], [2], [3], [4], [5], [11], [12], [13], [14]. However, none of these works examine the question of the uniformity in the Reynolds number.…”