On problems of regularity and existence for critical drift elliptic equations and Navier-Stokes equations
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Chernobai, Mikhail
issued: 2024-07-25T20:02:02Z
issued: 2024
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: This thesis contains two separate parts. The first part is for the three dimensional incompressible Navier-Stokes equations where we obtain a global existence result and eventual regularity for local energy weak solutions with a large class of initial data that allows growth at spatial infinity. Work is motivated by results of Bradshaw, Kukavica and Tsai, where was proven global existence of solutions for initial data in the Herz-type spaces which involve weighted integrals centered at the origin. Our results bridge theexistence theorems of Lemarié-Rieusset in the uniformly local L² space and of Bradshaw, Kukavica and Tsai. The second part of the thesis is about weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation Δu +b(𝝰) ·∇u = f in a bounded domain Ω ⊂ ℝⁿ, n = 2, 3, containing the origin. We focus on the case of critical drift b(𝝰) := b − 𝝰x/|x|² in case n = 2 and b(𝝰) := b − 𝝰x′/x′|² , where x′ = (x1, x2, 0) for the three dimensional case, with div b = 0. The parameter α can be positive or negative.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/88698
note: On problems of regularity andexistence for critical drift ellipticequations and Navier-Stokes equationsbyMikhail ChernobaiSpecialist, Saint Petersburg State University, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2024© Mikhail Chernobai 2024The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:On problems of regularity and existence for critical drift ellipticequations and Navier-Stokes equationssubmitted byMikhail Chernobai in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics.Examining Committee:Tai-Peng Tsai, Professor, Mathematics, UBCSupervisorSven Bachmann, Associate Professor, Mathematics, UBCSupervisory Committee MemberStephen Gustafson, Professor, Mathematics, UBCSupervisory Committee MemberMatthew W Choptuik, Professor, Physics and Astronomy, UBCUniversity ExaminerJingyi Chen, Professor, Mathematics, UBCUniversity ExaminerDoyoon Kim, Professor, Mathematics, Korea UniversityExternal ExamineriiAbstractThis thesis contains two separate parts. The first part is for the three di-mensional incompressible Navier-Stokes equations where we obtain a globalexistence result and eventual regularity for local energy weak solutions witha large class of initial data that allows growth at spatial infinity. Work ismotivated by results of Bradshaw, Kukavica and Tsai, where was provenglobal existence of solutions for initial data in the Herz-type spaces whichinvolve weighted integrals centered at the origin. Our results bridge theexistence theorems of Lemarie´-Rieusset in the uniformly local L2 space andof Bradshaw, Kukavica and Tsai.The second part of the thesis is about weak solvability and properties ofweak solutions to the Dirichlet problem for a scalar elliptic e…
has type: Thesis/Dissertation
date issued: 2024-11
is shown at: 10.14288/1.0444815
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Tsai, Tai-Peng, 1969-
type: Text
identifier URI: http://hdl.handle.net/2429/88698