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dc.contributor.advisorVasseur, Alexis F.en
dc.creatorChoi, Kyudongen
dc.date.accessioned2012-10-26T14:02:16Zen
dc.date.available2012-10-26T14:02:16Zen
dc.date.issued2012-08en
dc.date.submittedAugust 2012en
dc.identifier.urihttp://hdl.handle.net/2152/ETD-UT-2012-08-5907en
dc.descriptiontexten
dc.description.abstractThis thesis is divided into two independent parts. The first part concerns the 3D Navier-Stokes equations. The second part deals with regularity issues for a family of integro-differential equations. In the first part of this thesis, we consider weak solutions of the 3D Navier-Stokes equations with L² initial data. We prove that ([Nabla superscript alpha])u is locally integrable in space-time for any real [alpha] such that 1 < [alpha] < 3. Up to now, only the second derivative ([Nabla]²)u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in local weak-L[superscript (4/([alpha]+1))]. These estimates depend only on the L² norm of the initial data and on the domain of integration. Moreover, they are valid even for [alpha] ≥ 3 as long as u is smooth. The proof uses a standard approximation of Navier-Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. In the second part of this thesis, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in C[superscript beta] for all time if its initial data lies in C[superscript beta]. Also, we prove a C[superscript beta]-regularization effect from [mathematical equation] initial data. It provides an alternative proof to the result of Caffarelli, Chan and Vasseur [10], which was based on De Giorgi techniques. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption [mathematical equation]on the divergent-free drift velocity. The proof is in the spirit of Kiselev and Nazarov [48] where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation by observing the evolution of a dual class of test functions.en
dc.format.mimetypeapplication/pdfen
dc.language.isoengen
dc.subjectNavier-Stokesen
dc.subjectHigher derivativesen
dc.subjectDe Giorgi techniquesen
dc.subjectIntegro-differential equationsen
dc.subjectPersistence of Hölder continuityen
dc.subjectDual class of test functionsen
dc.titleEstimates on higher derivatives for the Navier-Stokes equations and Hölder continuity for integro-differential equationsen
dc.date.updated2012-10-26T14:02:24Zen
dc.identifier.slug2152/ETD-UT-2012-08-5907en
dc.contributor.committeeMemberCaffarelli, Luisen
dc.contributor.committeeMemberFigalli, Alessioen
dc.contributor.committeeMemberGamba, Ireneen
dc.contributor.committeeMemberMorrison, Philipen
dc.contributor.committeeMemberPavlovic, Natasaen
dc.description.departmentMathematicsen
dc.type.genrethesisen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Texas at Austinen
thesis.degree.levelDoctoralen
thesis.degree.nameDoctor of Philosophyen


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