Estimates on higher derivatives for the Navier-Stokes equations and Hölder continuity for integro-differential equations
dc.contributor.advisor | Vasseur, Alexis F. | en |
dc.contributor.committeeMember | Caffarelli, Luis | en |
dc.contributor.committeeMember | Figalli, Alessio | en |
dc.contributor.committeeMember | Gamba, Irene | en |
dc.contributor.committeeMember | Morrison, Philip | en |
dc.contributor.committeeMember | Pavlovic, Natasa | en |
dc.creator | Choi, Kyudong | en |
dc.date.accessioned | 2012-10-26T14:02:16Z | en |
dc.date.accessioned | 2017-05-11T22:29:05Z | |
dc.date.available | 2012-10-26T14:02:16Z | en |
dc.date.available | 2017-05-11T22:29:05Z | |
dc.date.issued | 2012-08 | en |
dc.date.submitted | August 2012 | en |
dc.date.updated | 2012-10-26T14:02:24Z | en |
dc.description | text | en |
dc.description.abstract | This 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.description.department | Mathematics | en |
dc.format.mimetype | application/pdf | en |
dc.identifier.slug | 2152/ETD-UT-2012-08-5907 | en |
dc.identifier.uri | http://hdl.handle.net/2152/ETD-UT-2012-08-5907 | en |
dc.language.iso | eng | en |
dc.subject | Navier-Stokes | en |
dc.subject | Higher derivatives | en |
dc.subject | De Giorgi techniques | en |
dc.subject | Integro-differential equations | en |
dc.subject | Persistence of Hölder continuity | en |
dc.subject | Dual class of test functions | en |
dc.title | Estimates on higher derivatives for the Navier-Stokes equations and Hölder continuity for integro-differential equations | en |
dc.type.genre | thesis | en |