On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction

We investigate the stability of three thermoelastic beam systems with hyperbolic heat conduction. First, we study the Bresse-Gurtin-Pipkin system, providing a necessary and sufficient condition for the exponential stability and the optimal polynomial decay rate when the condition is violated. Second, we obtain analogous results for the Bresse-Maxwell-Cattaneo system, completing an analysis recently initiated in the literature. Finally, we consider the Timoshenko-Gurtin-Pipkin system and we find the optimal polynomial decay rate when the known exponential stability condition does not hold. As a byproduct, we fully recover the stability characterization of the Timoshenko-Maxwell-Cattaneo system. The classical "equal wave speeds" conditions are also recovered through singular limit procedures. Our conditions are compatible with some physical constraints on the coefficients as the positivity of the Poisson's ratio of the material. The analysis faces several challenges connected with the thermal damping, whose resolution rests on recently developed mathematical tools such as quantitative Riemann-Lebesgue lemmas.Comment: Abstract shortened and few typos correcte

Non-equilibrium thermodynamics of damped Timoshenko and damped Bresse systems

In this paper, we cast damped Timoshenko and damped Bresse systems into a general framework for non-equilibrium thermodynamics, namely the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework. The main ingredients of GENERIC consist of five building blocks: a state space, a Poisson operator, a dissipative operator, an energy functional, and an entropy functional. The GENERIC formulation of damped Timoshenko and damped Bresse systems brings several benefits. First, it provides alternative ways to derive thermodynamically consistent models of these systems by construct- ing building blocks instead of invoking conservation laws and constitutive relations. Second, it reveals clear physical and geometrical structures of these systems, e.g., the role of the energy and the entropy as the driving forces for the reversible and irreversible dynamics respectively. Third, it allows us to introduce a new GENERIC model for damped Timoshenko systems that is not existing in the literature.Comment: 22 pages, revise