# Onsager's conjecture for admissible weak solutions

## TL;DR

This paper proves the existence of weak solutions to the 3D Euler equations with prescribed energy profiles and regularity below 1/3, demonstrating a form of the Onsager conjecture and an h-principle in this regime.

## Contribution

It establishes the existence of energy-prescribed weak solutions with Hölder regularity less than 1/3 and shows a corresponding h-principle, extending Onsager's conjecture.

## Key findings

- Existence of weak solutions with Hölder regularity < 1/3
- Solutions can match arbitrary smooth energy profiles
- A form of the h-principle holds in the subcritical regularity class

## 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.