Non-conservative solutions of the Euler-α equations

2021/11/01 by Beekie, Rajendra, Novack, Matthew
#35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.01027

Abstract

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 C⁰_t H^β_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-α, postulating that the threshold between rigidity and flexibility is the regularity class L³_t B^(1)/(3)_3,∞,x.

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