*Published Paper*

**Inserted:** 12 mar 2017

**Last Updated:** 1 dec 2020

**Journal:** Comm. Pure Appl. Math.

**Volume:** 72

**Number:** 2

**Pages:** 229-274

**Year:** 2019

**Abstract:**

We prove that given any $\beta<1/3$, a time interval $[0,T]$, and given any smooth energy profile $e \colon [0,T] \to (0,\infty)$, 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 $C^\beta_{t,x}$, 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.

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