Off-campus WSU users: To download campus access dissertations, please use the following link to log into our proxy server with your WSU access ID and password, then click the "Off-campus Download" button below.
Non-WSU users: Please talk to your librarian about requesting this dissertation through interlibrary loan.
Access Type
WSU Access
Date of Award
January 2025
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
Pei-Yong Wang
Abstract
This thesis investigates a two-phase free boundary problem arising from thestudy of relativistic Brillouin electron flow in high-powered devices such as the Radiographic Integrated Test Stand (RITS). The physical model describes electron motion along a Magnetically Insulated Transmission Line (MITL), where electrons are emitted from a cathode and propagate toward an anode under the influence of strong electric and magnetic fields. Two distinct regions—namely, the sheath region and the vacuum—are separated by a moving interface. The location of this interface is not known a priori, leading naturally to the formulation of a free boundary problem. Mathematically, the model is governed by a nonlinear elliptic partial differential equation for the relativistic factor u, derived from Maxwell’s equations and special relativity. In the sheath region, where charge density is significant, u satisfies a nonlinear PDE involving the square of its gradient. In contrast, in the vacuum region, the charge density is negligible, and the equation reduces to a homogeneous elliptic PDE. The two phases are coupled through transmission conditions at the free boundary, which enforce continuity of the potential and a nonlinear jump 80 condition on the normal derivatives. The primary aim of this work is to establish the existence, nondegeneracy, and regularity of solutions to this two-phase problem. To achieve this, we first formu- late the problem within the viscosity solution framework, which weakens regularity requirements and enables a rigorous treatment of the discontinuous interface. A generalized monotonicity formula is then developed to capture the interplay be- tween the sheath and vacuum regions. This formula serves as a central analytical tool, providing quantitative control over the solution near the free boundary. Through a combination of energy estimates, barrier arguments, and scaling techniques, we demonstrate the nondegeneracy and local Lipschitz continuity of the solution. Blow-up analysis is used to examine the asymptotic behavior near regular free boundary points, revealing linear expansion profiles. Finally, we em- ploy a Perron-type construction to prove the existence of solutions, defining the solution as the infimum of a family of admissible supersolutions. This study bridges the physical modeling of relativistic electron flows with modern mathematical techniques in free boundary analysis, contributing new in- sights to the field of nonlinear elliptic PDEs and advancing the understanding of two-phase problems with nonhomogeneous structure.
Recommended Citation
Patwin, David Malik-Mitchell, "An Elementary Approach To A New Monotonicity Formula With Application To The Nonhomogeneous Relativistic Brillouin Electron Flow" (2025). Wayne State University Dissertations. 4264.
https://digitalcommons.wayne.edu/oa_dissertations/4264