Dissipative Euler Flows and Onsager's Conjecture

TL;DR

This paper constructs periodic weak solutions to the incompressible Euler equations that dissipate energy and are H"older-continuous with exponent less than 1/10, providing a first step towards Onsager's conjecture.

Contribution

It presents the first construction of dissipative Euler solutions with H"older exponent below 1/10, advancing understanding of Onsager's conjecture.

Findings

Constructed dissipative solutions with ta<1/10

Demonstrated energy dissipation in weak solutions

Progressed towards Onsager's conjecture

Abstract

For any \theta<1/10 we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are H\"older-continuous with exponent \theta. A famous conjecture of Onsager states the existence of such dissipative solutions with any H\"older exponent \theta<1/3. Our theorem is the first result in this direction.