Quantitative positivity properties of the temperature in multi-dimensional thermoviscoelastic systems
Abstract
We consider the classical model for small-strain thermoviscoelastic evolution in isotropic Kelvin-Voigt materials with inertia, as given by utt = μΔut + (λ+μ)∇∇· ut + μ̂Δu + (λ̂+μ̂)∇∇· u − B∇Θ, κΘt = DΔΘ+ μ|∇ ut|2 + (λ+μ)|div ut|2 − BΘ div ut. Recently, Winkler (arXiv:2602.05964) established global-in-time solvability for an initial value problem in smoothly bounded multi-dimensional domains Ω under homogeneous Dirichlet boundary conditions for the displacement variable u, and homogeneous Neumann boundary conditions for the temperature Θ, allowing the heat capacity κ to depend on Θ. Moreover, it was shown there that the corresponding temperature is positive a.e. in Ω×(0,∞). In the present work, we strengthen this positivity result by deriving a quantitative lower bound for Θ. Our analysis applies to heat capacities κ with power-law scaling for small arguments, precisely: either κ(ξ)∼ξα for some α≥ 0, or κ(ξ)≥ξ, in a suitable neighborhood of ξ=0. Under these structural assumptions on κ, we show that for every τ>0 and T>τ there exists c(τ,T)>0 such that Θ(x,t)≥ c(τ,T) for a.e. (x,t)∈Ω×(τ,T).
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BibTeX
@misc{fricke2026quantitative,
title = {{Quantitative positivity properties of the temperature in multi-dimensional thermoviscoelastic systems}},
author = {Torben Fricke},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31124},
eprint = {2610.31124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31124},
}
RIS
TY - UNPB
TI - Quantitative positivity properties of the temperature in multi-dimensional thermoviscoelastic systems
AU - Torben Fricke
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31124
UR - https://arxiv.org/abs/2610.31124
AB - We consider the classical model for small-strain thermoviscoelastic evolution in isotropic Kelvin-Voigt materials with inertia, as given by u_tt = μΔu_t + (λ+μ)∇∇· u_t + μ̂Δu + (λ̂+μ̂)∇∇· u − B∇Θ, κΘ_t = DΔΘ+ μ|∇ u_t|^2 + (λ+μ)|div u_t|^2 − BΘ div u_t. Recently, Winkler (arXiv:2602.05964) established global-in-time solvability for an initial value problem in smoothly bounded multi-dimensional domains Ω under homogeneous Dirichlet boundary conditions for the displacement variable u, and homogeneous Neumann boundary conditions for the temperature Θ, allowing the heat capacity κ to depend on Θ. Moreover, it was shown there that the corresponding temperature is positive a.e. in Ω×(0,∞). In the present work, we strengthen this positivity result by deriving a quantitative lower bound for Θ. Our analysis applies to heat capacities κ with power-law scaling for small arguments, precisely: either κ(ξ)∼ξ^α for some α≥ 0, or κ(ξ)≥ξ, in a suitable neighborhood of ξ=0. Under these structural assumptions on κ, we show that for every τ>0 and T>τ there exists c(τ,T)>0 such that Θ(x,t)≥ c(τ,T) for a.e. (x,t)∈Ω×(τ,T).
ER -