Isett, Philip (2017) Hölder continuous Euler flows in three dimensions with compact support in time. Annals of Mathematics Studies. Vol.196. Princeton University Press , Princeton, NJ. ISBN 978-0-691-17483-9. https://resolver.caltech.edu/CaltechAUTHORS:20180702-123946440
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Abstract
Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. In this book, Philip Isett uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations. The construction itself—an intricate algorithm with hidden symmetries—mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful "Main Lemma"—used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem—has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.
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| Additional Information: | © 2017 Princeton University Press. The author is endebted to his advisor Sergiu Klainerman for his guidance, for introducing him to the work of DeLellis and Széekelyhidi and for introducing him to many great opportunities during his graduate studies. The author is thankful to professors C. Fefferman, A. Ionescu, V. Vicol and P. Constantin, and to his colleagues S-J. Oh and J. Luk for all of their help. The author also thanks S-J. Oh for suggestions regarding the proofs in Section (27), and thanks Vlad Vicol and Ting Zhang for corrections to earlier drafts of the paper. The author is deeply grateful to C. DeLellis and L. Széekelyhidi for discussions in 2011 about basic principles of convex integration and difficulties related to producing continuous Euler flows which helped to start the author’s investigations in the subject. The author also thanks M. Gromov and C. Villani for insightful conversations about convex integration. This work was supported by the NSF Graduate Research Fellowship Grant DGE-1148900. | |||||||||
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| Series Name: | Annals of Mathematics Studies | |||||||||
| DOI: | 10.48550/arXiv.1211.4065 | |||||||||
| Record Number: | CaltechAUTHORS:20180702-123946440 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20180702-123946440 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 87520 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | Tony Diaz | |||||||||
| Deposited On: | 02 Jul 2018 22:36 | |||||||||
| Last Modified: | 02 Jun 2023 00:17 |
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