Item Details

Well-Posedness and Stability for Nonlinear Schroedinger Equations With Dynamic/Wentzell Boundary Conditions

Lefler, Christopher
Format
Thesis/Dissertation; Online
Author
Lefler, Christopher
Advisor
Lasiecka, Irena
Abstract
Semilinear Schroedinger equations with cubic nonlinear part on bounded domains subject to a Wentzell (dynamic) boundary condition are studied in dimensions 2 and 3. We prove well-posedness on the Sobolev space H^2 as well as exponentially stable. The former result relies on first proving well-posedness of the linear model through treating the problem as a Wentzell problem, to which semigroup methods are applied. Obtaining well-posedness of the nonlinear model requires reformulating the problem as having a dynamic boundary condition, to which a fixed point argument is applied. Global well-posedness of the nonlinear model can only be achieved in 2 dimensions. In 3 dimensions we are able to prove global existence of weak solutions on the Sobolev space H^1 by the Galerkin approach, but we are not able to obtain uniqueness or continuous dependence on the initial data. Exponential stability of the linear model has been established previously in the literature. We adapt techniques from the linear model to achieve exponential stability of the nonlinear model.
Language
English
Date Received
20140502
Published
University of Virginia, Department of Mathematics, PHD (Doctor of Philosophy), 2014
Published Date
2014-05-01
Degree
PHD (Doctor of Philosophy)
Collection
Libra ETD Repository
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