The Role of Dissipation in the Existence of Time-Periodic Solutions to PDE Systems
By Boris Muha
Appears in collection : Mathematics of fluids in motion: Recent results and trends / Fluides en mouvement : résultats récents et perspectives
In many mechanical systems where energy is conserved, the phenomenon of resonance can occur, meaning that for certain time-periodic forces, the solution of the system becomes unbounded. Examples of partial differential equations describing such systems include the wave equation and equations of linearized elasticity (Lamé system). On the other hand, resonance does not occur in systems with strong dissipation, such as systems described by the heat equation. More precisely, in such a system, there exists a unique time-periodic solution for each time-periodic right-hand side. In this lecture, we will address the question "how much dissipation is necessary to prevent the occurrence of resonance?". We will analyze periodic solutions to the so-called heat-wave system, where the wave equation is coupled with the heat conduction equation via a common boundary. In this system, dissipation only exists in the heat component, and the system can be viewed as a simplified model of fluid-structure interaction. We will demonstrate that in certain geometric configurations, there exists a unique time-periodic solution for each time-periodic right-hand side, assuming sufficient regularity of the forcing term. A counterexample illustrates that this regularity requirement is stronger than in the case of the Cauchy problem. Finally, we will discuss the open question of whether the result is valid for arbitrary geometry or if there exists a geometry where resonance can occur.
