Existence, uniqueness, and time-asymptotics of regular solutions in multidimensional thermoelasticity on domains with boundary
Piotr Michał Bies
TL;DR
The paper analyzes a nonlinear multidimensional thermoelasticity model on convex, bounded domains with novel displacement boundary conditions and a Neumann temperature condition. It introduces a Fisher-information–based energy functional and employs the Helmholtz decomposition to establish global existence and uniqueness for small initial data, along with positivity of the temperature. The authors then derive precise time-asymptotics: the divergence-free part of the displacement exhibits oscillations, while the curl-free part and the temperature are strongly coupled and decay, with the temperature tending to a spatially uniform limit. This work provides a rigorous energy-framework treatment of boundary-driven thermoelastic systems and clarifies the long-time dissipation mechanisms in the presence of boundary effects.
Abstract
In the paper, we investigate the nonlinear thermoelasticity model in two- and three-dimensional convex and bounded domains. We propose new boundary conditions for the displacement. These conditions are not usual in thermoelasticity. Whereas, we posit the Neumann boundary condition for the temperature. We prove the existence of global, unique solutions for small initial data. The temperature positivity is also shown. Next, we investigate the long-time behavior of solutions. We show that the divergence-free part of the displacement oscillates. On the other hand, we prove that the potential part and the temperature are strongly coupled. The non-rotation part is strongly affected by heat propagation. It turns out that it tends to $0$ as $x$ approaches infinity. Additionally, the temperature converges to a constant function. Our techniques are firmly based on the functional $\F$ adopted from {\sc Bies, P. M., Cieślak, T., Fuest, M., Lankeit, J., Muha, B., and Trifunović, S.}, \emph{Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity}, arXiv: 2507.20794, 2025. The functional is based on the Fisher information and higher-order derivatives of the displacement. It responds well to the new boundary conditions. It allows us to close a priori estimates. We also need $L^{\infty}$-estimates for the temperature here. The Moser iterative procedure ensures it. The Helmholtz decomposition is applied to the displacement. The boundary conditions are crucial here. The boundary integrals that appear in the calculations at this point disappear thanks to these conditions. This allows us to split the problem into two separate ones. Each of them is associated with one part of the Helmholtz decomposition.