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Abstract
In this work we consider a coupled partial differential equation (PDE) model which has appeared in the literature to model various fluid-structure interactions seen in nature. It has been recently shown that this fluid-structure interactive PDE model admits of an explicit semigroup generator representation. In previous work, however, the argument for the maximality criterion is indirect, and does not provide for an explicit solution of the range condition.
The work in Chapter 1 reconsiders the proof of maximality for the fluid-structure generator A, and gives an explicit method for solving the fluid-structure equation. This methodology involves a nonstandard usage of the Babuska-Brezzi Theorem. Chapter 2 contains a proof of strong stability of the semigroup generated by the fluid-structure operator A. Thence solutions of the fluid-structure interactive PDE are asymptotically stable.
The work in Chapter 3 develops a finite element method for approximating solutions of the fluid-structure system; it is based upon our explicit proof of maximality and does not make use of the divergence-free basis functions usually employed in fluid dynamics. A numerical example involving an eigenfunction of A is used to test the method.
The final chapter contains a nonlinear fluid-dynamics result based upon the methodology developed for the linear fluid-structure model, but utilizing nonlinear semigroup theory. Similar results, though employing a Galerkin method of proof, are common. This result can be considered a step towards tackling the fluid-structure problem with nonlinear fluid dynamics.
