Operator Theoretic Methods for Coupled PDE in General Geometries

Authors

• Yuhao Mu, University of Nebraska-LincolnFollow

First Advisor

George Avalos

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

Date of this Version

7-13-2026

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor George Avalos

Lincoln, Nebraska, May 2026

Comments

Copyright 2026, Yuhao Mu. Used by permission

Abstract

This thesis develops and applies operator theoretic methods applied to coupled PDE analysis in general geometries, in which the coupling involves interchange across a boundary. The general geometries refer to domains with merely Lipschitz continuous boundaries (e.g. any non-convex polygon), as opposed to those with any further smoothness conditions on the boundary. The key results include a new pressure elimination method for coupled fluid–structure interaction (FSI) PDEs with boundary interchange, which allows an explicit semigroup representation of the PDE in terms of just the fluid and structure variables. This novel technique, valid over arbitrary bounded Lipschitz domains, leads to a new proof of well-posedness for a linear Stokes-Lamé fluid–elasticity PDE (a well-known physical model for low Reynolds numbers), that is now valid over such general geometries. Immediate applications include the rigorous justification of convergence rates for the associated finite element method (FEM) for the static resolvent PDE over polygonal domains. We extend our methods to the associated nonlinear version of the coupled PDE system, with Navier–Stokes instead of Stokes flow. By applying our new pressure elimination technique along with techniques of truncation, nonlinear mixed variational forms, and nonlinear semigroups, we prove several key results relevant to the well-posedness of the nonlinear coupled system over such general geometries.

Recommended Citation

Mu, Yuhao, "Operator Theoretic Methods for Coupled PDE in General Geometries" (2026). Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–. 519.
https://digitalcommons.unl.edu/dissunl/519