Onsager's conjecture for admissible weak solutions

We prove that given any $\beta <1\mathrm{/}3$, a time interval $[0,T]$, and given any smooth energy profile $e\text{\hspace{0.17em}⁣}:[0,T]\to (0,\mathrm{\infty })$, 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$forall$t\in [0,T]$\mathrm{.}Moreover,weshowthatasuitable$h$-principleholdsintheregularityclass$C^\beta_{t,x}$,forany$\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.