A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks
Michael Winkler
TL;DR
The paper analyzes a one-dimensional nonlinear thermoelastic system with hyperbolic–parabolic coupling, proving unconditional global well-posedness for rough initial data. It develops a weak-solution framework and proves existence, uniqueness, and continuous dependence by combining parabolic regularization, a new short-time higher-order energy estimate with a cancellation identity, and a density argument to extend results to general initial data. Key contributions include precise difference estimates for nonlinear terms, Grönwall-type stability bounds, and a robust construction scheme yielding global solutions whose regularity persists in time. These results establish a solid well-posedness theory for large-data thermoelastic systems in one dimension and provide a groundwork for analyzing long-time behavior via energy identities.
Abstract
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 \[ u\in C^0([0,\infty);W_0^{1,2}(Ω)), \qquad u_t \in C^0([0,\infty);L^2(Ω)) \qquad \mbox{and} \qquad Θ\in C^0([0,\infty);L^2(Ω)) \cap L^2_{loc}([0,\infty);W^{1,2}(Ω)). \]