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

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 $L^3_t B^{\frac{1}{3}}_{3,\infty,x}$.