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Localized Non-blowup Conditions for 3D Incompressible Euler Flows and Related Equations

Citation

Yu, Xinwei (2005) Localized Non-blowup Conditions for 3D Incompressible Euler Flows and Related Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K5E1-W938. https://resolver.caltech.edu/CaltechETD:etd-05302005-161405

Abstract

In this thesis, new results excluding finite time singularities with localized assumptions/conditions are obtained for the 3D incompressible Euler equations.

The 3D incompressible Euler equations are some of the most important nonlinear equations in mathematics. They govern the motion of ideal fluids. After hundreds of years of study, they are still far from being well-understood. In particular, a long-outstanding open problem asks whether finite time singularities would develop for smooth initial values. Much theoretical and numerical study on this problem has been carried out, but no conclusion can be drawn so far.

In recent years, several numerical experiments have been carried out by various authors, with results indicating possible breakdowns of smooth solutions in finite time. In these numerical experiments, certain properties of the velocity and vorticity field are observed in near-singular flows. These properties violate the assumptions of existing theoretical theorems which exclude finite time singularities. Thus there is a gap between current theoretical and numerical results. To narrow this gap is the main purpose of the work presented in this thesis.

In this thesis, a new framework of investigating flows carried by divergence-free velocity fields is developed. Using this new framework, new, localized sufficient conditions for the flow to remain smooth are obtained rigorously. These new results can deal with fast shrinking large vorticity regions and are applicable to recent numerical experiments. The application of the theorems in this thesis reveals new subtleties, and yields new understandings of the 3D incompressible Euler flow.

This new framework is then further applied to a two-dimensional model equation, the 2D quasi-geostrophic equation, for which global existence is still unproved. Under certain assumptions, we obtain new non-blowup results for the 2D quasi-geostrophic equation.

Finally, future plans of applying this new framework to some other PDEs as well as other possibilities of attacking the 3D Euler and 2D quasi-geostrophic singularity problems are discussed.

Item Type: Thesis (Dissertation (Ph.D.)) classical solution; incompressible Euler equations; localized non-blowup conditions; regularity California Institute of Technology Engineering and Applied Science Applied And Computational Mathematics Public (worldwide access) Hou, Thomas Y. Hou, Thomas Y. (chair)Meiron, Daniel I.Marsden, Jerrold E.Bruno, Oscar P. 18 May 2005 xinwei.yu (AT) gmail.com CaltechETD:etd-05302005-161405 https://resolver.caltech.edu/CaltechETD:etd-05302005-161405 10.7907/K5E1-W938 No commercial reproduction, distribution, display or performance rights in this work are provided. 2302 CaltechTHESIS Imported from ETD-db 01 Jun 2005 17 Dec 2020 00:48

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