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On the Endpoint Regularity in Onsager's Conjecture

Isett, Philip (2017) On the Endpoint Regularity in Onsager's Conjecture. . (Submitted)

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Onsager's conjecture states that the conservation of energy may fail for 3D incompressible Euler flows with Hölder regularity below 1/3. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the 3D incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents [0,1/3). Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a general convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of [IO16b] to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include the fact that singularites of positive space-time Lebesgue measure are necessary for any energy non-conserving solution to exist while having critical regularity of an integrability exponent greater than three.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Isett, Philip0000-0001-9038-5546
Additional Information:The work of P. Isett is supported by the National Science Foundation under Award No. DMS-1402370.
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Record Number:CaltechAUTHORS:20180626-161143819
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87365
Deposited By: Tony Diaz
Deposited On:26 Jun 2018 23:32
Last Modified:04 Aug 2022 18:44

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