A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks

TL;DR

This paper establishes the well-posedness of a one-dimensional nonlinear thermoelasticity model with rough initial data, proving global existence, uniqueness, and continuous dependence of solutions in minimal regularity frameworks.

Contribution

It introduces a simple model for nonlinear thermoelasticity and proves its well-posedness in low-regularity data settings, extending previous results to rough-data frameworks.

Findings

Global existence and uniqueness of solutions

Solutions depend continuously on initial data

Well-posedness holds under minimal regularity assumptions

Abstract

In an open bounded interval $\Omega$, the problem \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered under the boundary conditions $u|_{\partial\Omega}=\Theta_x|_{\partial\Omega}=0$, 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 \[ u_0\in W_0^{1,2}(\Omega), \quad u_{0t} \in L^2(\Omega) \quad \mbox{and} \quad \Theta_0 \in L^2(\Omega) \mbox{ with $\Theta\ge 0$ a.e.~in $\Omega$,} \] and in classes of solutions fulfilling \[ u\in C^0([0,\infty);W_0^{1,2}(\Omega)), \qquad u_t \in C^0([0,\infty);L^2(\Omega)) \qquad \mbox{and} \qquad \Theta\in C^0([0,\infty);L^2(\Omega)) \cap L^2_{loc}([0,\infty);W^{1,2}(\Omega)). \]