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Asymptotic Limits for Strain-Gradient Viscoelasticity with Nonconvex Energy

Aseel AlNajjar, Stefano Spirito, Athanasios E. Tzavaras

Abstract

We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. We establish the asymptotic limits when the viscosity $ν\rightarrow 0$ or when the dispersion coefficient $δ\rightarrow 0$. For the latter problem, it is worth noting that, for the case of two space dimensions, we also establish a rate of convergence. This result bears analogies to a result of Chemin \cite{jean1996remark} on the rate of convergence of the zero-viscosity limit for the two-dimensional Navier-Stokes equations with bounded vorticity.

Asymptotic Limits for Strain-Gradient Viscoelasticity with Nonconvex Energy

Abstract

We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. We establish the asymptotic limits when the viscosity or when the dispersion coefficient . For the latter problem, it is worth noting that, for the case of two space dimensions, we also establish a rate of convergence. This result bears analogies to a result of Chemin \cite{jean1996remark} on the rate of convergence of the zero-viscosity limit for the two-dimensional Navier-Stokes equations with bounded vorticity.

Paper Structure

This paper contains 8 sections, 9 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Derivation of the dissipative structure estimate
  3. Existence theory for the system of viscoelasticity with strain-gradient dependence
  4. From Strain-Gradient Viscoelasticity to Viscoelasticity; a rate of Convergence in the 2d case
  5. Limit as $\delta \rightarrow 0$; a compactness argument
  6. Strain-Gradient Elasticity; Zero viscosity limit
  7. Existence via Faedo-Galerkin Approximation
  8. Auxiliary results

Key Result

Theorem 3.4

Assume the stored energy $W$ satisfies (H1)-(H3) for some $p \geq 2$. Let $(u_0^{\nu,\delta},F_0^{\nu,\delta})\in L^2(\mathbb{T}^d) \times L^p(\mathbb{T}^d)\cap H^1 (\mathbb{T}^d)$ . Then there exists a weak solution of eqn:Visco in the sense of Definition def. Moreover, if $W$ additionally satisfie

Theorems & Definitions (17)

  • Remark 3.1
  • Remark 3.2
  • Definition 3.3: Weak solution
  • Theorem 3.4: Global Existence of Weak Solutions
  • Theorem 4.1: Rate of convergence from Strain-Gradient Viscoelasticity to Viscoelasticity
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1: Zero-viscosity limit
  • proof
  • ...and 7 more