## Dissertations, Theses, and Student Research Papers in Mathematics

5-2014

#### Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor George Avalos. Lincoln, Nebraska: May, 2014

Copyright (c) 2014 Thomas J. Clark

#### Abstract

We will present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. In Chapter \ref{ChWellposedness}, the wellposedness of this PDE model is established by means of constructing for it a nonstandard semigroup generator representation; this representation is essentially accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\mathcal{O}$ being coupled to a fourth order plate equation, possibly with rotational inertia parameter $\rho >0$, which evolves on a flat portion $\Omega$ of the boundary of $\mathcal{O}$. The coupling on $\Omega$ is implemented via the Dirichlet trace of the Stokes system fluid variable -- and so the no-slip condition is necessarily not in play -- and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on this elastic portion of the boundary. We note here that inasmuch as the Stokes fluid velocity does not vanish on $\Omega$, the pressure variable cannot be eliminated by the classic Leray projector; instead, the pressure is identified as the solution of a certain elliptic boundary value problem. Eventually, wellposedness of this fluid-structure dynamics is attained through a certain nonstandard variational (inf-sup") formulation. Chapter 1 also includes two abstract results. The first qualitative result shows that zero is in the resolvent set of the operator which generates the $C_0$-semigroup in the wellposedness argument. The second establishes the backward uniqueness property for the fluid-structure system.

Subsequently, in Chapter 2 we show how our constructive proof of wellposedness naturally gives rise to a certain mixed finite element method for numerically approximating solutions of this fluid-structure dynamics. This method is demonstrated for a certain test problem in the $\rho=0$ case. In addition, error estimates for the rate of convergence of the numerical method are provided and a test problem is solved to demonstrate the efficacy of the numerical code.

Adviser: George Avalos

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