A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912881886363648 |
|---|---|
| author | Winkler, Michael |
| author_facet | Winkler, Michael |
| contents | In an open bounded interval $Ω$, the problem \[
u_{tt} = u_{xx} - \big(f(Θ)\big)_x,
Θ_t = Θ_{xx} - f(Θ) u_{xt}, \] is considered under the boundary conditions $u|_{\partialΩ}=Θ_x|_{\partialΩ}=0$, and for $f\in C^2([0,\infty))$ satisfying $f(0)=0$, $f'>0$ on $[0,\infty)$ and $f'\in W^{1,\infty}((0,\infty))$.
In the sense of unconditional global existence, uniqueness and continuous dependence, this problem is shown to be well-posed within ranges of initial data merely satisfying \[
u_0\in W_0^{1,2}(Ω),
\quad
u_{0t} \in L^2(Ω)
\quad \mbox{and} \quad
Θ_0 \in L^2(Ω)
\mbox{ with $Θ\ge 0$ a.e.~in $Ω$,} \] and in classes of solutions fulfilling \[
u\in C^0([0,\infty);W_0^{1,2}(Ω)),
\qquad
u_t \in C^0([0,\infty);L^2(Ω))
\qquad \mbox{and} \qquad
Θ\in C^0([0,\infty);L^2(Ω)) \cap L^2_{loc}([0,\infty);W^{1,2}(Ω)). \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05963 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks Winkler, Michael Analysis of PDEs 74H20, 74F05, 35L05, 35D30 In an open bounded interval $Ω$, the problem \[ u_{tt} = u_{xx} - \big(f(Θ)\big)_x, Θ_t = Θ_{xx} - f(Θ) u_{xt}, \] is considered under the boundary conditions $u|_{\partialΩ}=Θ_x|_{\partialΩ}=0$, and for $f\in C^2([0,\infty))$ satisfying $f(0)=0$, $f'>0$ on $[0,\infty)$ and $f'\in W^{1,\infty}((0,\infty))$. In the sense of unconditional global existence, uniqueness and continuous dependence, this problem is shown to be well-posed within ranges of initial data merely satisfying \[ u_0\in W_0^{1,2}(Ω), \quad u_{0t} \in L^2(Ω) \quad \mbox{and} \quad Θ_0 \in L^2(Ω) \mbox{ with $Θ\ge 0$ a.e.~in $Ω$,} \] and in classes of solutions fulfilling \[ u\in C^0([0,\infty);W_0^{1,2}(Ω)), \qquad u_t \in C^0([0,\infty);L^2(Ω)) \qquad \mbox{and} \qquad Θ\in C^0([0,\infty);L^2(Ω)) \cap L^2_{loc}([0,\infty);W^{1,2}(Ω)). \] |
| title | A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks |
| topic | Analysis of PDEs 74H20, 74F05, 35L05, 35D30 |
| url | https://arxiv.org/abs/2602.05963 |