Non-conservative Solutions of the Euler- α Equations

The Euler-α equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than α. We show that there exists β> 1 such that weak solutions to the two and three dimensional Euler-α equations in the class Ct0Hxβ 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-α, postulating that the threshold between rigidity and flexibility is the regularity class [InlineEquation not available: see fulltext.].