## Topological Methods in Nonlinear Analysis

### Three-dimensional thermo-visco-elasticity with the Einstein-Debye $(\theta^3+\theta)$-law for the specific heat. Global regular solvability

#### Abstract

A three-dimensional thermo-visco-elastic system for the Kelvin-Voigt type material at small strain is considered. The system involves the constant heat conductivity and the specific heat satisfying the Einstein-Debye $(\theta^3+\theta)$-law. Such a nonlinear law, relevant at relatively low temperatures, represents the main novelty of the paper. The existence of global regular solutions is proved without the small data assumption. The crucial part of the proof is the strictly positive lower bound on the absolute temperature $\theta$. The problem remains open in the case of the Debye $\theta^3$-law. The existence of local in time solutions is proved by the Banach successive approximations method. The global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 1 (2018), 161-193.

Dates
First available in Project Euclid: 18 August 2018

https://projecteuclid.org/euclid.tmna/1534557630

Digital Object Identifier
doi:10.12775/TMNA.2018.016

Mathematical Reviews number (MathSciNet)
MR3867984

Zentralblatt MATH identifier
07029866

#### Citation

Pawłow, Irena; Zajączkowski, Wojciech M. Three-dimensional thermo-visco-elasticity with the Einstein-Debye $(\theta^3+\theta)$-law for the specific heat. Global regular solvability. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 161--193. doi:10.12775/TMNA.2018.016. https://projecteuclid.org/euclid.tmna/1534557630

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