Dynamics and regularity in periodic dispersive PDES: A study of nonlinear smoothing & well-posedness

This thesis will cover many properties related to solutions to well-studied dispersive partial differential equations (PDEs). This is the culmination of most of my graduate work, and covers results ranging from local well-posedness to fractal dimension bounds. It is not necessarily meant to be read linearly, as the first (non-background) chapters of both Part II and Part III contain the hardest material. The general outline of the thesis will be as follows: Part I is dedicated to the background material necessary for the later chapters; Part II is dedicated to results pertaining to Nonlinear Schrödinger (NLS) type equations; while Part III is dedicated to results for Korteweg-de Vries (KdV) type equations. While Part I is perhaps too detailed of a background, it served as an opportunity for me to collect basic results and their proofs. Both Part II and Part III contain background sections of their own, Chapters 8 and 12 respectively, detailing past results and sketches of important and relevant results (so long as the methods are important, that is). Past the backgrounds to Parts II and III, the chapters within the respective parts contain the statements and proofs of my contributions. Part II progresses naturally, beginning with establishing an improved global well-posedness result for the cubic NLS equation on T – results that will require the preliminary results of Section 3.4. From there, we study fractal dimension bounds associated to both the linear and nonlinear Schrödinger equation posed on Sd, which will require the background work of Chapter 4, Chapter 5, and Chapter 7. We conclude Part II with a study of nonlinear smoothing (and also global attractors) for the p-NLS family, requiring the preliminary work of Chapter 6. Part III progresses similarly, beginning with establishing a new local well-posedness result for the KdV equation (without the theory of Complete Integrability) in Chapter 13, utilizing the tools of Section 3.2 and Section 4.0.1. We then progress to the study of higher order members of the KdV hierarchy (as discussed in the background section of Part III), where we’ll establish local well-posedness, nonlinear smoothing, and unconditional well-posedness. We then conclude in a similar manner as Chapter 11, extending nonlinear smoothing results to global attractor results. My original contributions are contained within Part II and Part III, while many of the proofs within Part I are my own attempts at (by now) rather classical results. While the most interesting results are contained within Chapter 9, Chapter 10, Chapter 13, and Chapter 14, the results of Chapter 11 and Chapter 15 are probably best understood together (although, one should only study the proofs of Chapter 11 to understand both). Overall, the material of this manuscript can best be understood as a study of normal form procedures and of resonance, both of which are timeless topics.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• McConnell, Ryan

Contributors dc:contributor

• Erdogan, Mehmet B
• Tzirakis, Nokolaos
• Laugesen, Richard
• Li, Xiaochun

• Harmonic Analysis
• PDE
• Dispersive
• Smoothing
• Nonlinear Smoothing
• Wellposedness

Statement dc:rights

• Copyright 2025 Ryan McConnell

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129419

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129419