On the Aubry-Mather theory for partial differential equations and the stability of stochastically forced ordinary differential equations
| dc.contributor.advisor | Llave, Rafael de la | en |
| dc.contributor.committeeMember | Caffarelli, Luis | en |
| dc.contributor.committeeMember | Koch, Hans | en |
| dc.contributor.committeeMember | Radin, Charles | en |
| dc.contributor.committeeMember | Rodin, Greg | en |
| dc.contributor.committeeMember | Ying, Lexing | en |
| dc.creator | Blass, Timothy James | en |
| dc.date.accessioned | 2011-06-01T13:51:55Z | en |
| dc.date.accessioned | 2011-06-01T13:52:06Z | en |
| dc.date.accessioned | 2017-05-11T22:22:02Z | |
| dc.date.available | 2011-06-01T13:51:55Z | en |
| dc.date.available | 2011-06-01T13:52:06Z | en |
| dc.date.available | 2017-05-11T22:22:02Z | |
| dc.date.issued | 2011-05 | en |
| dc.date.submitted | May 2011 | en |
| dc.date.updated | 2011-06-01T13:52:06Z | en |
| dc.description | text | en |
| dc.description.abstract | This dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three separate papers. In Chapter 2 we consider gradient descent equations for energy functionals of the type [mathematical equation] where A is a second-order uniformly elliptic operator with smooth coefficients. We consider the gradient descent equation for S, where the gradient is an element of the Sobolev space H[superscipt beta], [beta is an element of](0, 1), with a metric that depends on A and a positive number [gamma] > sup |Vāā|. The main result of Chapter 2 is a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator, and we provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional. In Chapter 3 we investigate the differentiability of the minimal average energy associated to the functionals [mathematical equation] using numerical and perturbation methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the minimal average energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter [epsilon], and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series. In Chapter 4 we present a method for determining the stability of a class of stochastically forced ordinary differential equations, where the forcing term can be obtained by passing white noise through a filter of arbitrarily high degree. We use the Fokker-Planck equation to write a partial differential equation for the second moments, which we turn into an eigenvalue problem for a second-order differential operator. We develop ladder operators to determine analytic expressions for the eigenvalues and eigenfunctions of this differential operator, and thus determine the stability. | en |
| dc.description.department | Mathematics | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | http://hdl.handle.net/2152/ETD-UT-2011-05-2798 | en |
| dc.language.iso | eng | en |
| dc.subject | Partial differential equations | en |
| dc.subject | Aubry-Mather theory | en |
| dc.subject | Comparison principle | en |
| dc.subject | Lindstedt series | en |
| dc.subject | Cell problem | en |
| dc.subject | Stability | en |
| dc.subject | Stochastically forced ordinary differential equations | en |
| dc.title | On the Aubry-Mather theory for partial differential equations and the stability of stochastically forced ordinary differential equations | en |
| dc.type.genre | thesis | en |