Semester
Spring
Date of Graduation
2026
Document Type
Dissertation
Degree Type
PhD
College
Eberly College of Arts and Sciences
Department
Mathematics
Committee Chair
Qingtian Zhang
Committee Co-Chair
Charis Tsikkou
Committee Member
Dening Li
Committee Member
Adrian Tudorascu
Committee Member
Weichao Tu
Abstract
ABSTRACT
Global Weak Solutions of Optical Variational Wave System
Shahrazad Hamed Mahal Alnafie
The coupling of a variational wave equation with Maxwell’s equations gives rise to the optical variational wave system, a hyperbolic PDE system that models the director field of the nematic liquid crystals. This system presents unique analytical challenges that have not been addressed in the existing literature. In this dissertation, we study the one-dimensional case of this system.
We establish the global existence of conservative weak solutions to the associated Cauchy problem. The hyperbolic system is derived using the energy variational method. Through a sequence of suitable coordinate transformations, we reformulate the system while preserving its hyperbolicity. Due to degeneracy arising from the change of variables, we construct a non-degenerate approximate sequence of solutions, prove local well-posedness, and establish pre-compactness of the approximate solution sequence. This allows us to pass to the limit in the original coordinates and prove the global existence of conservative weak solutions.
This result highlights the analytical significance of the optical variational wave system in liquid crystal theory and provides a rigorous foundation for future analytical work on this model in higher spatial dimensions.
Recommended Citation
Alnafie, Shahrazad Hamed, "Global Weak Solutions of Optical Variational Wave System" (2026). Graduate Theses, Dissertations, and Problem Reports. 13367.
https://researchrepository.wvu.edu/etd/13367