Riesz Transforms on a Sphere

Undergraduate Thesis Or Project

Public Deposited

Citeable URL: https://ir.library.oregonstate.edu/concern/undergraduate_thesis_or_projects/w66349104

Descriptions

Attribute Name	Values
Creator	Tyma, Troy
Abstract	Motivated by the Navier-Stokes equations, which are a set of unsolved equations related to fluid motion in R^3, we explored the incompressibility condition and the Neumann boundary problem. After exploring, we noticed that using iterated Riesz transforms of the boundary data could be used to get information about the velocity field directly from the boundary data. We then used this same logic to look at the same problem in a spherical space, and found no definition for the Riesz transforms. By using the incompressibility condition and solving the Neumann boundary problem on a sphere, using both separation of variables and potential theory, we can define the Riesz transforms of a function on the sphere. This allows us to use boundary data on a sphere to describe the divergence-free field, or in the case of the incompressible Navier-Stokes equation, allows us to solve for the velocity field given initial data.
License	Attribution 4.0 (CC BY 4.0)
Resource Type	Undergraduate Thesis
Date Issued	2019-05
Degree Level	Bachelor's
Degree Name	Bachelor of Science (B.S.)
Degree Field	Physics
Degree Grantor	Oregon State University
Commencement Year	2019
Advisor	Kim, Hoe Woon
Academic Affiliation	Physics
Rights Statement	In Copyright
Publisher	Oregon State University
Peer Reviewed	No
Language	English [eng]

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