Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions
Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions
复制标题
二维不可压缩粘弹性的消失粘度极限
DOI：
10.1002/cpa.21853
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发表时间：
2017-03
影响因子：
3
通讯作者：
Yuan Cai;Zhen Lei;F. Lin;N. Masmoudi
中科院分区：
文献类型：
--
作者：
Yuan Cai;Zhen Lei;F. Lin;N. Masmoudi
本文研究了二维不可压粘弹性的无粘极限问题，这是一个耦合Navier-Stokes方程和变形张量的迁移方程的系统。自[35]的工作以来，已知具有固定正粘性的平衡点附近整体光滑解的存在性。第二作者最近解决了无粘性情况[28]。虽然后者完全基于双曲方程研究的技术，因此二维问题通常比高维问题更具挑战性，但前者主要依赖于耗散机制。事实上，在对称化和围绕平衡的线性化之后，不可压缩粘弹性的系统减少到流体速度和变形张量的阻尼波方程的不可压缩系统。这两种方法是不相容的。本文证明了该方程在时间t ∈ [0，+∞）和粘性μ ≥ 0上整体一致存在解.这使我们能够证明，特别是消失的粘度极限的所有时间。为了克服纯双曲型极限系统和附加抛物粘性扰动系统之间的不相容性带来的困难，本文介绍了一种相当鲁棒的方法，它可以适用于广泛的一类类似性质的物理系统。粗略地说，该方法在二维情况下只要双曲系统内在地满足“强零条件”就有效。对于不小于3的维数，通常的零条件足以使该方法起作用。© 2019 Wiley Periodicals，Inc.
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.