Calculus of Variations and Geometric Measure Theory

T. Buckmaster - C. De Lellis - P. Isett - L. J. Székelyhidi

Anomalous dissipation for $1/5$-Hölder Euler flows

created by delellis on 02 Sep 2014
modified on 24 May 2017


Published Paper

Inserted: 2 sep 2014
Last Updated: 24 may 2017

Journal: Ann. of Math. (2)
Volume: 182
Pages: 127–172
Year: 2015
Links: PDF


Recently the second and fourth author 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 below 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 author and introducing some new devices. More precisely we show that for any positive $\varepsilon$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class $C^{1/5-\varepsilon}$.