On the kinetic energy profile of Hölder continuous Euler flows

The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class Ct,x1/5−ϵ. The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster-De Lellis-Székelyhidi [1].