A Besicovitch set is defined as a compact subset of \u211d\u207f which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to n. This thesis gives an improved Hausdorff dimension estimate, d \u2a7e 0.60376707287 n + O(1), for Besicovitch sets displaying a special structural property called \"stickiness.\" The improved estimate comes from using an incidence geometry argument called a \"k-planebrush,\" which is a higher dimensional analogue of Wolff's \"hairbrush\" argument from 1995.

\r\n \r\nIn addition, an x-ray transform estimate is obtained as a corollary of Zahl's k-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in \u211d\u207f must have Minkowski dimension greater than (2 - \u221a2 + \u03b5)n. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao(2000), it provides a new proof of the same result.

", "doi": "10.7907/japt-b214", "publication_date": "2024", "thesis_type": "phd", "thesis_year": "2024" }, { "id": "thesis:14584", "collection": "thesis", "collection_id": "14584", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05172022-223804694", "primary_object_url": { "basename": "Jiajie_Chen_2022.pdf", "content": "final", "filesize": 2535177, "license": "other", "mime_type": "application/pdf", "url": "/14584/23/Jiajie_Chen_2022.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "Singularity Formation in Incompressible Fluids and Related Models", "author": [ { "family_name": "Chen", "given_name": "Jiajie", "orcid": "0000-0002-0194-1975", "clpid": "Chen-Jiajie" } ], "thesis_advisor": [ { "family_name": "Hou", "given_name": "Thomas Y.", "orcid": "0000-0001-6287-1133", "clpid": "Hou-T-Y" } ], "thesis_committee": [ { "family_name": "Stuart", "given_name": "Andrew M.", "orcid": "0000-0001-9091-7266", "clpid": "Stuart-Andrew-M" }, { "family_name": "Hou", "given_name": "Thomas Y.", "orcid": "0000-0001-6287-1133", "clpid": "Hou-T-Y" }, { "family_name": "Isett", "given_name": "Philip", "orcid": "0000-0001-9038-5546", "clpid": "Isett-Phlip" }, { "family_name": "Owhadi", "given_name": "Houman", "orcid": "0000-0002-5677-1600", "clpid": "Owhadi-H" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Whether the three-dimensional (3D) incompressible Euler equations can develop a finite-time singularity from smooth initial data with finite energy is a major open problem in partial differential equations. A few years ago, Tom Hou and Guo Luo obtained strong numerical evidence of a potential finite time singularity of the 3D axisymmetric Euler equations with boundary from smooth initial data. So far, there is no rigorous justification. In this thesis, we develop a framework to study the Hou-Luo blowup scenario and singularity formation in related equations and models. In addition, we analyze the obstacle to singularity formation.

\r\n\r\nIn the first part, we propose a novel framework of analysis based on the dynamic rescaling formulation to study singularity formation. Our strategy is to reformulate the problem of proving finite time blowup into the problem of establishing the nonlinear stability of an approximate self-similar blowup profile using the dynamic rescaling equations. Then we prove finite time blowup of the 2D Boussinesq and the 3D Euler equations with C^{1,\u03b1} velocity and boundary. This result provides the first rigorous justification of the Hou-Luo scenario using C^{1,\u03b1} velocity.

In the second part, we further develop the framework for smooth data. The method in the first part relies crucially on the low regularity of the data, and there are several essential difficulties to generalize it to study the Hou-Luo scenario with smooth data. We demonstrate that some of the challenges can be overcome by proving the asymptotically self-similar blowup of the Hou-Luo model. Applying this framework, we establish the finite time blowup of the De Gregorio (DG) model on the real line (\u211d) with smooth data. Our result resolves the open problem on the regularity of this model on \u211d that has been open for quite a long time.

\r\n\r\nIn the third part, we investigate the competition between advection and vortex stretching, an essential difficulty in studying the regularity of the 3D Euler equations. This competition can be modeled by the DG model on S^{1}. We consider odd initial data with a specific sign property and show that the regularity of the initial data in this class determines the competition between advection and vortex stretching. For any 0 < \u03b1 < 1, we construct a finite time blowup solution from some C^{\u03b1} initial data. On the other hand, we prove that the solution exists globally for C^{1} initial data. Our results resolve some conjecture on the finite time blowup of this model and imply that singularities developed in the DG model and the generalized Constantin-Lax-Majda model on S^{1} can be prevented by stronger advection.