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UBC Theses and Dissertations
On problems of regularity and existence for critical drift elliptic equations and Navier-Stokes equations Chernobai, Mikhail
Abstract
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.
Item Metadata
| Title |
On problems of regularity and existence for critical drift elliptic equations and Navier-Stokes equations
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| Creator | |
| Supervisor | |
| Publisher |
University of British Columbia
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| Date Issued |
2024
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| 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.
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| Genre | |
| Type | |
| Language |
eng
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| Date Available |
2024-07-25
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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| DOI |
10.14288/1.0444815
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| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
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| Graduation Date |
2024-11
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| Campus | |
| Scholarly Level |
Graduate
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| Rights URI | |
| Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International