Non-conservative Solutions of the Euler-$$\alpha $$ Equations

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 .