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Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities

Michael Winkler

TL;DR

This work analyzes a one-dimensional thermoviscoelastic evolution problem with temperature-dependent viscosity $\gamma(\Theta)$ and nonlinear coupling $f(\Theta)$. By leveraging maximal Sobolev regularity for the parabolic temperature equation and a careful energy plus interpolation framework, it proves global solvability on bounded domains under a small oscillation condition $\gamma_0 \le γ(ξ) \le γ_0+δ$ and subcritical growth $|f(ξ)| \le K_f (ξ+1)^α$ with $α<\frac{3}{2}$, for suitably regular initial data; it also provides global strong solutions under weaker initial data. A key part of the analysis is a loop-type argument that binds $\int_0^T\int_\Omega v_x^4$ and $\int_0^T\int_\Omega (\Theta+1)^{4α}$, using space-time bounds derived from maximal regularity. The results extend the theory of 1D thermo-viscoelastic models with temperature-dependent coefficients, ensuring global well-posedness and regularity even for large data and rough initial conditions.

Abstract

The model \[ \left\{ \begin{array}{l} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \end{array} \right. \] for thermoviscoelastic evolution in one-dimensional Kelvin-Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if $γ_0>0$ is fixed, then there exists $δ=δ(γ_0)>0$ with the property that for suitably regular initial data of arbitrary size an associated initial-boundary value problem posed in an open bounded interval admits a global classical solution whenever $γ\in C^2([0,\infty))$ and $f\in C^2([0,\infty))$ are such that $f(0)=0$ and $|f(ξ)| \le K_f \cdot (ξ+1)^α$ for all $ξ\ge 0$ and some $K_f>0$ and $α<\frac{3}{2}$, and that \[ γ_0 \le γ(ξ) \le γ_0 + δ \qquad \mbox{for all } ξ\ge 0. \] This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.

Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities

TL;DR

This work analyzes a one-dimensional thermoviscoelastic evolution problem with temperature-dependent viscosity and nonlinear coupling . By leveraging maximal Sobolev regularity for the parabolic temperature equation and a careful energy plus interpolation framework, it proves global solvability on bounded domains under a small oscillation condition and subcritical growth with , for suitably regular initial data; it also provides global strong solutions under weaker initial data. A key part of the analysis is a loop-type argument that binds and , using space-time bounds derived from maximal regularity. The results extend the theory of 1D thermo-viscoelastic models with temperature-dependent coefficients, ensuring global well-posedness and regularity even for large data and rough initial conditions.

Abstract

The model \[ \left\{ \begin{array}{l} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \end{array} \right. \] for thermoviscoelastic evolution in one-dimensional Kelvin-Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if is fixed, then there exists with the property that for suitably regular initial data of arbitrary size an associated initial-boundary value problem posed in an open bounded interval admits a global classical solution whenever and are such that and for all and some and , and that This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.
Paper Structure (7 sections, 15 theorems, 164 equations)

This paper contains 7 sections, 15 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controlling the growth of $\int_\Omega (\Theta+1)^q$ in terms of $\int_\Omega v_x^4$
  4. Space-time bounds for $v_x^4$ and $\Theta^{4\alpha}$ via maximal Sobolev regularity
  5. Higher regularity properties of $\Theta$. Proof of Theorem \ref{['theo8']}
  6. Further regularity features of $u$ and $v$
  7. Coping with rough initial data. Proof of Theorem \ref{['theo15']}

Key Result

Theorem 1.1

Let $L>0$ and $\Omega=(0,L) \subset \mathbb{R}$, and let $\gamma_0>0$. Then there exists $\delta=\delta(\gamma_0)>0$ with the property that if $a>0$, if and if furthermore and then whenever there exist uniquely determined functions fulfilling which are such that $\Theta\ge 0$ in $\Omega\times (0,\infty)$, and that (0) is solved in the classical sense.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • ...and 6 more