Infinitely many non-conservative solutions for the three-dimensional Euler equations with arbitrary initial data in $C^{1/3-\epsilon }$

Abstract

Let $0<\beta <\bar{\beta }<1/3$. We construct infinitely many distributional solutions in $C_{x,t}^{\beta }$ to the three-dimensional Euler equations that do not conserve the energy, for a given initial data in $C^{\bar{\beta }}$. We also show that there is some limited control on the increase in the energy for $t>1$.

41 pages, 2 figures