Transporting microstructure and dissipative Euler flows

TL;DR

This paper presents a simplified proof demonstrating the existence of energy-dissipating solutions to the 3D Euler equations with Hölder regularity just below the Onsager conjecture threshold, advancing understanding of turbulence and fluid dynamics.

Contribution

The authors provide a shorter, more direct proof of Isett's result on Hölder continuous solutions dissipating energy, refining the iterative scheme for Euler equations.

Findings

Existence of energy-dissipating solutions in C^{1/5 - ε} regularity class.

Improved proof technique adhering to the original iterative scheme.

Solutions demonstrate Onsager's conjecture threshold is nearly attainable.

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 (arXiv:1202.1751 and 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 (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 $\epsilon$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the H\"older class $C^{\frac{1}{5}-\epsilon}$.