Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global solvability and stabilization in multi-dimensional small-strain nonlinear thermoviscoelasticity

Michael Winkler

TL;DR

The paper addresses the challenge of global solvability for multi-dimensional, small-strain nonlinear thermoviscoelastic evolution in Kelvin–Voigt materials with inertia, governed by a coupled system with a temperature-dependent heat capacity $\kappa$ and potential defect measures. It develops a robust variational framework based on generalized and integrable solutions, employing logarithmically enhanced entropy functionals to derive uniform bounds that are strong enough to pass to the limit in regularized approximate problems. The main results prove global existence of solutions without small-data restrictions, positivity of the temperature, and, under mild decay assumptions on external forcing and appropriate growth of $\kappa$, stabilization of the temperature to a spatially homogeneous state $\Theta_\infty>0$ along with decay of the mechanical parts $u_t$ and $u$ at large times. The analysis introduces novel entropy refinements and a careful limit-passage strategy that accommodates defect measures, providing a new pathway to handle high-dimensional thermoelastic models with nonlinear heat production and temperature-dependent heat capacity, with implications for the mathematical understanding of large-data global solvability and long-time behavior in complex continua.

Abstract

Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic evolution in Kelvin-Voigt materials with inertia, as given by \[ u_{tt} = μΔu_t + (λ+μ)\nabla\nabla\cdot u_t + \hatμ Δu + (\hatλ+\hatμ) \nabla\nabla\cdot u - B\nablaΘ, \qquad \qquad κΘ_t = DΔΘ+ μ|\nabla u_t|^2 + (λ+μ) |{\rm div} \, u_t|^2 - BΘ{\rm div} \, u_t, \qquad \qquad \qquad (\star) \] is globally solvable in multi-dimensional settings and for initial data of arbitrary size. The present manuscript addresses this in the context of an initial value problem in smoothly bounded $n$-dimensional domains with $n\ge 2$, posed under homogeneous boundary conditions of Dirichlet type for the displacement variable $u$, and of Neumann type for the temperature $Θ$. Within suitably generalized concepts of solvability, global existence of solutions is shown without any size restrictions on the data, and for a system actually more general than ($\star$) by, inter alia, allowing the heat capacity $κ$ to depend on $Θ$. Apart from that, results on large time behavior are derived which particularly assert stabilization of $Θ$ toward a spatially homogeneous limit. Besides on standard features related to energy conservation and entropy production, in its core parts the analysis relies on an evolution property of certain logarithmic refinements of classical entropy functionals, to the best of our knowledge undiscovered in precedent literature and possibly of independent interest.

Global solvability and stabilization in multi-dimensional small-strain nonlinear thermoviscoelasticity

TL;DR

The paper addresses the challenge of global solvability for multi-dimensional, small-strain nonlinear thermoviscoelastic evolution in Kelvin–Voigt materials with inertia, governed by a coupled system with a temperature-dependent heat capacity and potential defect measures. It develops a robust variational framework based on generalized and integrable solutions, employing logarithmically enhanced entropy functionals to derive uniform bounds that are strong enough to pass to the limit in regularized approximate problems. The main results prove global existence of solutions without small-data restrictions, positivity of the temperature, and, under mild decay assumptions on external forcing and appropriate growth of , stabilization of the temperature to a spatially homogeneous state along with decay of the mechanical parts and at large times. The analysis introduces novel entropy refinements and a careful limit-passage strategy that accommodates defect measures, providing a new pathway to handle high-dimensional thermoelastic models with nonlinear heat production and temperature-dependent heat capacity, with implications for the mathematical understanding of large-data global solvability and long-time behavior in complex continua.

Abstract

Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic evolution in Kelvin-Voigt materials with inertia, as given by is globally solvable in multi-dimensional settings and for initial data of arbitrary size. The present manuscript addresses this in the context of an initial value problem in smoothly bounded -dimensional domains with , posed under homogeneous boundary conditions of Dirichlet type for the displacement variable , and of Neumann type for the temperature . Within suitably generalized concepts of solvability, global existence of solutions is shown without any size restrictions on the data, and for a system actually more general than () by, inter alia, allowing the heat capacity to depend on . Apart from that, results on large time behavior are derived which particularly assert stabilization of toward a spatially homogeneous limit. Besides on standard features related to energy conservation and entropy production, in its core parts the analysis relies on an evolution property of certain logarithmic refinements of classical entropy functionals, to the best of our knowledge undiscovered in precedent literature and possibly of independent interest.
Paper Structure (23 sections, 52 theorems, 465 equations)

This paper contains 23 sections, 52 theorems, 465 equations.

Table of Contents

  1. Introduction
  2. Generalized solutions and integrable solutions
  3. Energy control in regularized problems
  4. Global existence in the approximate problems
  5. The test function $\frac{\ln^2(\Theta_\varepsilon+M)}{\Theta_\varepsilon+M}$ in the thermal part
  6. $L^1$ boundedness and uniform integrability properties of $\Theta_\varepsilon$
  7. A time-independent $L^1$ bound for $\Theta_\varepsilon$ when $\liminf_{\xi\to\infty} \kappa(\xi) >0$
  8. Uniform integrability properties of $\Theta_\varepsilon$ when $\lim_{\xi\to\infty} \kappa(\xi) \ln^{(3-n)_+} \xi = +\infty$
  9. An estimate for $\nabla^sv_\varepsilon$ in $L\log L$
  10. First-order estimates for renormalized versions of $\Theta_\varepsilon$
  11. Passing to the limit. Existence of solutions
  12. Extracting subsequences and deriving (\ref{['wu']})
  13. The inequality (\ref{['wt']})
  14. The energy inequality (\ref{['wF']})
  15. Entropy and positivity. Proofs of Theorems \ref{['theo37']} and \ref{['theo38']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $n\ge 2$ and $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, let $D>0$, and suppose that are such that and that with some $k_{\mathbb{D}}>0$ and $k_{\mathbb{C}}>0$ we have Moreover, assume that and suppose that and that Then whenever with and there exist such that $\Theta>0$ a.e. in $\Omega\times (0,\infty)$, and that $(u,\Theta)$ forms a generalized solution of

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 43 more