Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface

This work is dedicated to the study of a linear model arising in fluid-structure interaction and introduced by Rauch, Zhang and Zuazua. The system is formed of a heat and a wave equation, taking place in two distinct domains, and coupled by transmission conditions at the interface of the domains. Two different transmission conditions are considered. In both cases, when the interface geometrically controls the wave domain, we show the quick polynomial decay of the energy for solutions with smooth initial data, improving the rate of decay obtained by the previous authors. The polynomial stability is deduced from an optimal observability inequality conjectured in their work. The proof of this estimate mainly relies on a known generalized trace lemma for solutions of partial differential equations and the results of Bardos, Lebeau and Rauch on the control of the wave equation. Without the geometric condition, we show, using a Carleman inequality of Lebeau and Robbiano and an abstract theorem of N. Burq, a logarithmic decay for solution of the system with one of the two transmission conditions. This result improves the speed of decay obtained by Zhang and Zuazua, and is also optimal in some geometries.