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Published May 2017 | Submitted
Journal Article Open

On the kinetic energy profile of Hölder continuous Euler flows

Abstract

In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L_t^∞(B_(3,∞)^(1/3) due to low regularity of the energy profile. The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class C_(t,x)^(1/5 − ϵ). The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [1].

Additional Information

© 2016 Elsevier Masson SAS. Received 22 October 2015, Revised 22 March 2016, Accepted 13 May 2016, Available online 1 June 2016. The work of P. Isett is supported by the National Science Foundation under Award No. DMS-1402370. S.-J. Oh is a Miller Research Fellow, and would like to thank the Miller Institute at UC Berkeley for support. We thank Emil Wiedemann and Camillo De Lellis for encouraging our pursuit of Theorem 1.1. We also thank the Institut Henri Poincaré for its hospitality, where part of this work was done.

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August 21, 2023
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