#### Document Type

Dissertation

#### Date of Degree

Fall 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Gerhard O. Ströhmer

#### Abstract

We consider the flow of an ideal gas with internal friction and heat conduction in a layer between a fixed plane and an upper free boundary. We describe the top free surface as the graph of a time dependent function. This forces us to exclude breaking waves on the surface. For this and other reasons we need to confine ourselves to flow close to a motionless equilibrium state which is fairly easy to compute. The full equations of motion, in contrast to that, are quite difficult to solve. As we are close to an equilibrium, a linear system of equations can be used to approximate the behavior of the nonlinear system.

Analytic, strongly continuous semigroups defined on a suitable Banach space X are used to determine the behavior of the linear problem. A strongly continuous semigroup is a family of bounded linear operators {T(t)} on X where 0 ≤ t < infinity satisfying the following conditions.

1. T(s+t)=T(s)T(t) for all s,t ≥ 0

2. T(0)=E, the identity mapping.

3. For each x ∈ X, T(t)x is continuous in t on [0,infinity).

Then there exists an operator A known as the infinitesimal generator of such that T(t)=exp (tA). Thus, an analytic semigroup can be viewed as a generalization of the exponential function.

Some estimates about the decay rates are derived using this theory. We then prove the existence of long term solutions for small initial values. It ought to be emphasized that the decay is not an exponential one which engenders significant difficulties in the transition to nonlinear stability.

#### Keywords

Nonlinear Dynamics, Partial Differential Equations, Semigroup Theory

#### Pages

ix, 139

#### Bibliography

138-139

#### Copyright

Copyright © 2016 Dana Michelle Bates