# Rational Decay Estimates for Fluid-Structure PDE Models

George Avalos^{*}

In this talk, we shall demonstrate how delicate frequency domain relations and estimates, associated with coupled systems of partial differential equation models (PDE's), may be exploited so as to establish results of uniform and rational decay. In particular, our focus will be upon decay properties of coupled PDE systems of different characteristics; e.g., hyperbolic versus parabolic characteristics. For such PDE systems of contrasting dynamics, the attainment of explicit decay rates is known to be a difficult problem, inasmuch as there has not been an established methodology to handle hyperbolic-parabolic systems. For uncoupled wave equations or uncoupled heat equations, there are specific Carleman's multiplier methods in the time domain, wherein the exponential weights in each Carleman's multiplier carefully take into account the particular dynamics involved, be it hyperbolic or parabolic. But for coupled PDE systems which involve hyperbolic dynamics interacting with parabolic dynamics, typically across some boundary interface, Carleman's multipliers are readily applicable. Given that such coupled PDE systems occur frequently in nature and in engineering applications; e.g., fluid-structure and structural acoustic interactions, there is a patent need to devise broadly implementable techniques by which one can infer uniform decay for a given PDE system. As one particular example, we shall work to conclude uniform decays for structural acoustic dynamics. In these PDE models, the structural component is subjected to a structural damping ranging from viscous (weak) to strong (Kelvin-Voight). The rational decay rates we derive for this problem explicitly reflect the extent of the damping which is in play. Since the damped elastic component of the coupled dynamics is present on only a portion of the boundary, there will necessarily be assumptions imposed upon the geometry. This is joint work with Francesca Bucci.

Mathematics Subject Classification: 35M30 74F10

Keywords: fluid-solid; structure-acoustics; stability; rational decay;

Minisymposion: Mathematical Modeling of Physical Phenomena