Long-time asymptotics for reaction-diffusion and stochastic Burgers equations
Abstract/Contents
- Abstract
- This dissertation consists of two parts. In the first, we study the long-time behavior of various reaction-diffusion equations. We begin by describing a complete asymptotic expansion for solutions to the classical Fisher—KPP equation in the long-time limit. We then take up a version of Fisher—KPP with nonlocal diffusion. We show that typical nonlocal equations resemble the local model at long times; however, strongly asymmetric nonlocal diffusion can disrupt this behavior. Next, we use a connection between the Fisher—KPP equation and branching Brownian motion (BBM) to study the probability that the leading two particles in BBM are unusually far apart. This is joint work with J. Berestycki, Brunet, Mytnik, Roquejoffre, and Ryzhik. Finally, in collaboration with H. Berestycki, we study steady states, propagation, and traveling waves for a much wider class of reaction-diffusion equations in half-spaces. The second part of this dissertation explores the long-time behavior of the stochastic Burgers equation; it consists of joint work with Dunlap and Ryzhik. We draw on the theory of parabolic partial differential equations to construct stationary solutions to the stochastic Burgers equation on the line. Moreover, we show that the extremal stationary solutions are parameterized by their means and arise as stable long-time limits of constant initial data.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2021; ©2021 |
| Publication date | 2021; 2021 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Graham, Cole A |
|---|---|
| Degree supervisor | Ryzhik, Leonid |
| Thesis advisor | Ryzhik, Leonid |
| Thesis advisor | Papanicolaou, George |
| Thesis advisor | Ying, Lexing |
| Degree committee member | Papanicolaou, George |
| Degree committee member | Ying, Lexing |
| Associated with | Stanford University, Department of Mathematics |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Cole Graham. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis Ph.D. Stanford University 2021. |
| Location | https://purl.stanford.edu/bb869hj6992 |
Access conditions
- Copyright
- © 2021 by Cole A Graham
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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