# Anomalous dissipation for 1/5-Hölder Euler flows

Buckmaster, Tristan; De Lellis, Camillo; et al (2015). Anomalous dissipation for 1/5-Hölder Euler flows. Annals of Mathematics. Second Series, 128(1):127-172.

## Abstract

Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent-albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive ε, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class $C^{1/5-\epsilon}$.

Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent-albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive ε, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class $C^{1/5-\epsilon}$.

## Citations

4 citations in Web of Science®
3 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English July 2015 27 Jan 2016 11:21 05 Apr 2016 19:41 Mathematical Sciences Publishers 0003-486X https://doi.org/10.4007/annals.2015.182.1.3
Permanent URL: https://doi.org/10.5167/uzh-116667

Content: Accepted Version
Language: English
Filetype: PDF - Registered users only
Size: 646kB
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