Onsager's conjecture for admissible weak solutions

Abstract

We prove that given any $\beta <1/3$, a time interval $[0,T]$, and given any smooth energy profile $e:[0,T]\to (0,\infty )$, there exists a weak solution $v$ of the three-dimensional Euler equations such that $v\in C^{\beta }([0,T]\times {\mathbb{T}}^3)$, with $e(t)={\int }_{{\mathbb{T}}^3}|v(x,t){|}^{2}dx$ for all $t\in [0,T]$. Moreover, we show that a suitable $h$-principle holds in the regularity class $C_{t,x}^{\beta }$, for any $\beta <1/3$. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.

36 pages, 1 figure