DOI

10.17077/etd.aetzxole

Document Type

Dissertation

Date of Degree

Fall 2016

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Gerhard O. Ströhmer

First Committee Member

Tong Li

Second Committee Member

Paul S Muhly

Third Committee Member

Charles D Frohman

Fourth Committee Member

Victor P Camillo

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.

Public Abstract

The subject of this thesis is about the motion of a compressible fluid with internal friction and heat conduction in a layer between a fixed lower boundary and a free upper boundary. We describe the top boundary as the graph of a 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. We prove the existence of long term solutions for initial values close to the equilibrium. These solutions approach the equilibrium over time. This result is obtained by considering the linearization of the problem using the theory of analytic semigroups and deriving some estimates about the decay rates of some of the variables. 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 pages

Bibliography

Includes bibliographical references (pages 138-139).

Copyright

Copyright © 2016 Dana Michelle Bates

Included in

Mathematics Commons

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