Date of Award

2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Robert Strain

Abstract

We consider the Muskat Problem with surface tension in two dimensions over the real line, with $H^{s}$ initial data and allowing the two fluids to have different constant densities and viscosities. We take the angle between the interface and the horizontal, and derive an evolution equation for it. We use energy methods to prove that a solution $\theta$ exists locally and can be continued while $||\theta||_{s}$ remains bounded and the arc chord condition holds. Furthermore, the resulting solution is unique, and depends continuously on the initial data. Additionally, when both fluids have the same viscosity and the initial data is sufficiently small, we show the energy is non-increasing, and that the solution $\theta$ exists globally in time.

Available for download on Friday, April 26, 2019

Included in

Mathematics Commons

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