Dynamics of Essentially Unstable Nonlinear Waves

c 2019 Connor Smith B.S. in Mathematics with Minor in Chemistry, Oregon State University

Submitted to the graduate degree program in Department of Mathematics and the Graduate Faculty of the University of Kansas in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Mathew Johnson, Chair

Milena Stanislavova

Committee members Weishi Liu

Dionyssios Mantzavinos

Jonathan P. Lamb, External Reviewer

Date defended: May 06, 2019 The Thesis Committee for Connor Smith certifies that this is the approved version of the following thesis :

Dynamics of Essentially Unstable Nonlinear Waves

Mathew Johnson, Chair

Date approved: May 13, 2019

2 Abstract

In this thesis we primarily consider the stability of traveling wave solutions to a modified Kuramoto-Sivashinsky

Equation equation modeling nanoscale pattern formation and the St. Venant equations modeling shallow water flow down an inclined plane. Numerical evidence suggests that these equations have no unstable spectrum other than λ = 0, however they both have unstable essential spectrum. This unstable essential spectrum manifests as a convecting, oscillating perturbation which grows to a certain size independent on the initial perturbation — precluding stability in the regular L2 (R) . Exponentially weighted spaces are typically used to handle such instabilities, and in Theorem 5.7 we prove asymptotic orbital linear stability in such an exponentially weighted space. We also discuss difficulties with extending this to a nonlinear stability result. In Section 5.5 we discuss another way of obtaining stability, through ad-hoc periodic wave trains.

Chapter6 concerns the general problem of creating a spectral projection to project away unstable essential spectrum.

We consider this problem in the context of spatially periodic-coefficient PDE by proposing a candidate spectral pro- jection defined via the Bloch transform and showing that initial perturbations which activate a sufficiently unstable part of the essential spectrum lead to solutions which are not Lyapunov stable. We also extend these results to dissipative systems of conservation laws.

Additional chapters of interest are Chapter3 where we address finding the spectrum and Chapter4 where we discuss the numerics which lead to many of the figures in this thesis.

3 Acknowledgements

I would like to thank my advisor, Mathew Johnson. Initially it was for providing a problem whose statement is simple enough to illustrate with pictures, but is rich enough mathematically to confound to this very day. But also for the mentorship, opportunities, and patience while working on said problem. In particular I would like to thank him for going out of his way to teach me what it would mean to be a professor, both in terms of day to day duties, but also in terms of their duty to society.

I would also like to thank the members of my oral exam and thesis defense committee — Milena Stanislavova,

Weishi Liu, Geng Chen, Dionyssios Mantzavinos, and Jonathan P. Lamb — for their time, comments, constructive criticism, and questions.

I would like to thank everyone who either invited me to give a talk, helped with a paper, or wrote a letter of recom- mendation: Blake Barker, Margaret Beck, Stephane Lafortune, Greg Lyng, Melissa Shabazz, and Kevin Zumbrun.

I would like to thank the office support — Kerrie Brecheisen, Debbie Garcia, Gloria Prothe, Lori Springs — for everything they’ve organized, forgiving me for all the deadlines I may have missed, and for providing cookies every

Tuesday and Thursday without fail.

I would like to thank every professor who taught a class I took for making KU an enriching experience. I would like to thank all of the math graduate students for making KU a fun experience.

And finally I would like to thank you, the reader.