# Transporting microstructure and dissipative Euler flows

created by delellis on 02 Jul 2013
modified on 10 Dec 2013

[BibTeX]

Preprint

Inserted: 2 jul 2013
Last Updated: 10 dec 2013

Year: 2013

Abstract:

Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in H\"older spaces in {\it arXiv:1202.1751} and {\it arXiv:1205.3626} (2012)). 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 P.~Isett ({\it arXiv:1211.4065}) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better H\"older exponent albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett's final result, adhering more to the original scheme 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 H\"older class $C^{1/5-\varepsilon}$.