Non-conservative solutions of the Euler-$\alpha$ equations

TL;DR

This paper demonstrates non-uniqueness and non-conservation of energy for weak solutions of the Euler-$ alpha$ equations in certain regularity classes, using convex integration methods, and proposes an Onsager-type conjecture for these equations.

Contribution

It establishes the existence of non-conservative weak solutions in specific regularity classes for Euler-$\alpha$ equations and formulates an Onsager-type conjecture for these models.

Findings

Weak solutions are not unique in certain regularity classes.

Solutions may fail to conserve the Hamiltonian.

A threshold regularity class is proposed for rigidity versus flexibility.

Abstract

The Euler-$\alpha$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $\alpha$. We show that there exists $\beta>1$ such that weak solutions to the two and three dimensional Euler-$\alpha$ equations in the class $C^0_t H^\beta_x$ are not unique and may not conserve the Hamiltonian of the system, thus demonstrating flexibility in this regularity class. The construction utilizes a Nash-style intermittent convex integration scheme. We also formulate an appropriate version of the Onsager conjecture for Euler-$\alpha$, postulating that the threshold between rigidity and flexibility is the regularity class $L^3_t B^{\frac{1}{3}}_{3,\infty,x}$.