{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/129419"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/129419","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Dynamics and regularity in periodic dispersive PDES: A study of nonlinear smoothing & well-posedness","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","abstract_has_math":false,"creators":["McConnell, Ryan"],"institution":"University of Illinois Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Erdogan, Mehmet B","Tzirakis, Nokolaos","Laugesen, Richard","Li, Xiaochun"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-04-27","date_published":"2025-04-27","updated_at":"2026-07-22T22:25:05Z","subjects":["Harmonic Analysis","PDE","Dispersive","Smoothing","Nonlinear Smoothing","Wellposedness"],"languages":["en","eng"],"rights":["Copyright 2025 Ryan McConnell"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/129419","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Erdogan, Mehmet B","Tzirakis, Nokolaos","Laugesen, Richard","Li, Xiaochun"]},{"key":"dc:creator","label":"Author","values":["McConnell, Ryan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-04-27","2025-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Harmonic Analysis","PDE","Dispersive","Smoothing","Nonlinear Smoothing","Wellposedness"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2025 Ryan McConnell"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/129419"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","The student, Ryan McConnell, accepted the attached license on 2025-04-21 at 18:59.","The student, Ryan McConnell, submitted this Dissertation for approval on 2025-04-21 at 19:08.","This Dissertation was approved for publication on 2025-04-27 at 15:04.","DSpace SAF Submission Ingestion Package generated from Vireo submission #21874 on 2025-10-19 at 18:18:33","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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Dynamics and regularity in periodic dispersive PDES: A study of nonlinear smoothing & well-posedness"]}]}],"canonical_facts":{"dc:contributor":["Erdogan, Mehmet B","Tzirakis, Nokolaos","Laugesen, Richard","Li, Xiaochun"],"dc:creator":["McConnell, Ryan"],"dc:date":["2025-04-27","2025-05"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","The student, Ryan McConnell, accepted the attached license on 2025-04-21 at 18:59.","The student, Ryan McConnell, submitted this Dissertation for approval on 2025-04-21 at 19:08.","This Dissertation was approved for publication on 2025-04-27 at 15:04.","DSpace SAF Submission Ingestion Package generated from Vireo submission #21874 on 2025-10-19 at 18:18:33","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."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/129419"],"dc:language":["en","eng"],"dc:rights":["Copyright 2025 Ryan McConnell"],"dc:subject":["Harmonic Analysis","PDE","Dispersive","Smoothing","Nonlinear Smoothing","Wellposedness"],"dc:title":["Dynamics and regularity in periodic dispersive PDES: A study of nonlinear smoothing & well-posedness"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:05Z"}