---
_id: '59258'
article_number: '44'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters. Applied Mathematics & Optimization.
2025;91(2). doi:10.1007/s00245-025-10243-9
apa: Winkler, M. (2025). Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters. Applied Mathematics & Optimization,
91(2), Article 44. https://doi.org/10.1007/s00245-025-10243-9
bibtex: '@article{Winkler_2025, title={Rough Data in an Evolution System Generalizing
1D Thermoviscoelasticity with Temperature-Dependent Parameters}, volume={91},
DOI={10.1007/s00245-025-10243-9},
number={244}, journal={Applied Mathematics & Optimization}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }'
chicago: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters.” Applied Mathematics & Optimization
91, no. 2 (2025). https://doi.org/10.1007/s00245-025-10243-9.
ieee: 'M. Winkler, “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters,” Applied Mathematics & Optimization,
vol. 91, no. 2, Art. no. 44, 2025, doi: 10.1007/s00245-025-10243-9.'
mla: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters.” Applied Mathematics & Optimization,
vol. 91, no. 2, 44, Springer Science and Business Media LLC, 2025, doi:10.1007/s00245-025-10243-9.
short: M. Winkler, Applied Mathematics & Optimization 91 (2025).
date_created: 2025-04-02T11:23:25Z
date_updated: 2026-02-26T15:59:30Z
department:
- _id: '90'
doi: 10.1007/s00245-025-10243-9
intvolume: ' 91'
issue: '2'
language:
- iso: eng
project:
- _id: '245'
name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken
für Leistungsschallanwendungen (NEPTUN)'
publication: Applied Mathematics & Optimization
publication_identifier:
issn:
- 0095-4616
- 1432-0606
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with
Temperature-Dependent Parameters
type: journal_article
user_id: '11829'
volume: 91
year: '2025'
...
---
_id: '63344'
abstract:
- lang: eng
text: "Abstract\r\n A Neumann-type initial-boundary
value problem for \r\n \r\n
\ $$\\begin{aligned} \\left\\{ \\begin{array}{l}
u_{tt} = \\nabla \\cdot (\\gamma (\\Theta ) \\nabla u_t) + a \\nabla \\cdot (\\gamma
(\\Theta ) \\nabla u) + \\nabla \\cdot f(\\Theta ), \\\\ \\Theta _t = D\\Delta
\\Theta + \\Gamma (\\Theta ) |\\nabla u_t|^2 + F(\\Theta )\\cdot \\nabla u_t,
\\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n u\r\n
\ \r\n tt\r\n
\ \r\n \r\n
\ =\r\n ∇\r\n
\ ·\r\n \r\n
\ (\r\n γ\r\n
\ \r\n (\r\n
\ Θ\r\n )\r\n
\ \r\n ∇\r\n
\ \r\n u\r\n
\ t\r\n \r\n
\ )\r\n \r\n
\ +\r\n a\r\n
\ ∇\r\n ·\r\n
\ \r\n (\r\n
\ γ\r\n \r\n
\ (\r\n Θ\r\n
\ )\r\n \r\n
\ ∇\r\n u\r\n
\ )\r\n \r\n
\ +\r\n ∇\r\n
\ ·\r\n f\r\n
\ \r\n (\r\n
\ Θ\r\n )\r\n
\ \r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ Θ\r\n t\r\n
\ \r\n =\r\n
\ D\r\n Δ\r\n
\ Θ\r\n +\r\n
\ Γ\r\n \r\n
\ (\r\n Θ\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ |\r\n ∇\r\n
\ \r\n u\r\n
\ t\r\n \r\n
\ |\r\n \r\n
\ 2\r\n \r\n
\ +\r\n F\r\n
\ \r\n (\r\n
\ Θ\r\n )\r\n
\ \r\n ·\r\n
\ ∇\r\n \r\n
\ u\r\n t\r\n
\ \r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n \r\n
\ \r\n \r\n \r\n
\ is considered in a smoothly bounded domain \r\n
\ \r\n $$\\Omega
\\subset \\mathbb {R}^n$$\r\n \r\n
\ \r\n Ω\r\n ⊂\r\n
\ \r\n \r\n R\r\n
\ \r\n n\r\n
\ \r\n \r\n \r\n
\ \r\n , \r\n
\ \r\n $$n\\ge 1$$\r\n
\ \r\n
\ \r\n n\r\n ≥\r\n
\ 1\r\n \r\n \r\n
\ \r\n . In the
case when \r\n \r\n $$n=1$$\r\n
\ \r\n
\ \r\n n\r\n =\r\n
\ 1\r\n \r\n \r\n
\ \r\n , \r\n
\ \r\n $$\\gamma
\\equiv \\Gamma $$\r\n \r\n
\ \r\n γ\r\n ≡\r\n
\ Γ\r\n \r\n \r\n
\ \r\n and \r\n
\ \r\n $$f\\equiv
F$$\r\n \r\n
\ \r\n f\r\n ≡\r\n
\ F\r\n \r\n \r\n
\ \r\n , this
system coincides with the standard model for heat generation in a viscoelastic
material of Kelvin-Voigt type, well-understood in situations in which \r\n
\ \r\n $$\\gamma
=const$$\r\n \r\n
\ \r\n γ\r\n =\r\n
\ c\r\n o\r\n
\ n\r\n s\r\n
\ t\r\n \r\n \r\n
\ \r\n . Covering
scenarios in which all key ingredients \r\n \r\n
\ $$\\gamma ,\\Gamma ,f$$\r\n \r\n \r\n
\ γ\r\n ,\r\n
\ Γ\r\n ,\r\n
\ f\r\n \r\n \r\n
\ \r\n and F
may depend on the temperature \r\n \r\n
\ $$\\Theta $$\r\n \r\n Θ\r\n
\ \r\n \r\n
here, for initial data which merely satisfy \r\n \r\n
\ $$u_0\\in W^{1,p+2}(\\Omega )$$\r\n
\ \r\n
\ \r\n \r\n u\r\n
\ 0\r\n \r\n
\ ∈\r\n \r\n W\r\n
\ \r\n 1\r\n
\ ,\r\n p\r\n
\ +\r\n 2\r\n
\ \r\n \r\n \r\n
\ (\r\n Ω\r\n
\ )\r\n \r\n
\ \r\n \r\n \r\n
\ , \r\n \r\n
\ $$u_{0t}\\in W^{1,p}(\\Omega )$$\r\n
\ \r\n
\ \r\n \r\n u\r\n
\ \r\n 0\r\n
\ t\r\n \r\n
\ \r\n ∈\r\n
\ \r\n W\r\n
\ \r\n 1\r\n
\ ,\r\n p\r\n
\ \r\n \r\n \r\n
\ (\r\n Ω\r\n
\ )\r\n \r\n
\ \r\n \r\n \r\n
\ and \r\n \r\n
\ $$\\Theta _0\\in W^{1,p}(\\Omega )$$\r\n
\ \r\n
\ \r\n \r\n Θ\r\n
\ 0\r\n \r\n
\ ∈\r\n \r\n W\r\n
\ \r\n 1\r\n
\ ,\r\n p\r\n
\ \r\n \r\n \r\n
\ (\r\n Ω\r\n
\ )\r\n \r\n
\ \r\n \r\n \r\n
\ with some \r\n \r\n
\ $$p\\ge 2$$\r\n \r\n \r\n
\ p\r\n ≥\r\n
\ 2\r\n \r\n \r\n
\ \r\n such
that \r\n \r\n $$p>n$$\r\n
\ \r\n
\ \r\n p\r\n >\r\n
\ n\r\n \r\n \r\n
\ \r\n , a result
on local-in-time existence and uniqueness is derived in a natural framework of
weak solvability."
article_number: '44'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters. Applied Mathematics & Optimization.
2025;91(2). doi:10.1007/s00245-025-10243-9
apa: Winkler, M. (2025). Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters. Applied Mathematics & Optimization,
91(2), Article 44. https://doi.org/10.1007/s00245-025-10243-9
bibtex: '@article{Winkler_2025, title={Rough Data in an Evolution System Generalizing
1D Thermoviscoelasticity with Temperature-Dependent Parameters}, volume={91},
DOI={10.1007/s00245-025-10243-9},
number={244}, journal={Applied Mathematics & Optimization}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }'
chicago: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters.” Applied Mathematics & Optimization
91, no. 2 (2025). https://doi.org/10.1007/s00245-025-10243-9.
ieee: 'M. Winkler, “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters,” Applied Mathematics & Optimization,
vol. 91, no. 2, Art. no. 44, 2025, doi: 10.1007/s00245-025-10243-9.'
mla: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity
with Temperature-Dependent Parameters.” Applied Mathematics & Optimization,
vol. 91, no. 2, 44, Springer Science and Business Media LLC, 2025, doi:10.1007/s00245-025-10243-9.
short: M. Winkler, Applied Mathematics & Optimization 91 (2025).
date_created: 2025-12-18T20:20:06Z
date_updated: 2025-12-18T20:20:16Z
doi: 10.1007/s00245-025-10243-9
intvolume: ' 91'
issue: '2'
language:
- iso: eng
publication: Applied Mathematics & Optimization
publication_identifier:
issn:
- 0095-4616
- 1432-0606
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with
Temperature-Dependent Parameters
type: journal_article
user_id: '31496'
volume: 91
year: '2025'
...