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dc.contributor.advisor Kiselev, Alexander
dc.creatorDo, Tam
dc.date.accessioned 2017-08-01T18:01:27Z
dc.date.available 2017-08-01T18:01:27Z
dc.date.created 2017-05
dc.date.issued 2017-04-19
dc.date.submitted May 2017
dc.identifier.citation Do, Tam. "Global Regularity and Finite-time Blow-up in Model Fluid Equations." (2017) Diss., Rice University. https://hdl.handle.net/1911/96084.
dc.identifier.urihttps://hdl.handle.net/1911/96084
dc.description.abstract Determining the long time behavior of many partial differential equations modeling fluids has been a challenge for many years. In particular, for many of these equations, the question of whether solutions exist for all time or form singularities is still open. The structure of the nonlinearity and non-locality in these equations makes their analysis difficult using classical methods. In recent years, many models have been proposed to study fluid equations. In this thesis, we will review some new result in regards to these models as well as give insight into the relation between these models and the true equations. First, we analyze a one-dimensional model for the two-dimensional surface quasi-geostrophic equation and vortex sheets. The model gained prominence due to the work of Cordoba, Cordoba, and Fontelos and is often referred to as the CCF model. We will show that solutions are globally regular in the presence of logarithmically supercritcal dissipation and that solutions eventually gain regularity in the presence of supercritical dissipation. Finally, by analyzing a dyadic model of the equation, we will gain insight into how certain possible singularities in the CCF model can be supressed by dissipation. For the second part of this thesis, we study some one-dimensional model equations for the Euler equations. These models are influenced by the recent numerical simulations of Tom Hou and Guo Luo. They observed possible singularity formation for the three-dimensional Euler equation at the boundary of a cylindrical domain under certain symmetry assumptions. Under these assumptions, a singularity was observed numerically and the solution was observe to have hyperbolic structure near the singularity. Hou and Luo proposed a one-dimensional model system to study singularity formation theoritically. We will study a family of one-dimensional models generalizing their model. The results in chapter 2 are the results of joint work with A. Kiselev, V. Hoang, M. Radosz, and X. Xu.
dc.format.mimetype application/pdf
dc.language.iso eng
dc.subjectfluid mechanics
partial differential equations
dc.title Global Regularity and Finite-time Blow-up in Model Fluid Equations
dc.date.updated 2017-08-01T18:01:27Z
dc.type.genre Thesis
dc.type.material Text
thesis.degree.department Mathematics
thesis.degree.discipline Natural Sciences
thesis.degree.grantor Rice University
thesis.degree.level Doctoral
thesis.degree.name Doctor of Philosophy


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