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

TL;DR

This paper proves the global solvability and stabilization of solutions for a multi-dimensional small-strain thermoviscoelasticity model, extending previous results to arbitrary initial data size and more general heat capacity dependencies.

Contribution

It establishes the first global existence results for the model in multiple dimensions without data size restrictions and introduces novel entropy-based analytical techniques.

Findings

Global solutions exist for arbitrary initial data size.

Temperature stabilizes to a spatially homogeneous state over time.

New entropy functional methods are developed for analysis.

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 \[ u_{tt} = \mu \Delta u_t + (\lambda+\mu)\nabla\nabla\cdot u_t + \hat{\mu} \Delta u + (\hat{\lambda}+\hat{\mu}) \nabla\nabla\cdot u - B\nabla\Theta, \qquad \qquad \kappa \Theta_t = D\Delta\Theta + \mu |\nabla u_t|^2 + (\lambda+\mu) |{\rm div} \, u_t|^2 - B\Theta {\rm div} \, u_t, \qquad \qquad \qquad (\star) \] 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 $\Theta$. 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 $\kappa$ to depend on $\Theta$. Apart from that, results on large time behavior are derived which particularly assert stabilization of $\Theta$ 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.