Onsager's conjecture for admissible weak solutions
Tristan Buckmaster, Camillo De Lellis, László Székelyhidi, Vlad Vicol
TL;DR
The paper settles a sharp aspect of Onsager's conjecture by constructing Hölder continuous weak solutions to the 3D Euler equations that dissipate energy according to an arbitrary smooth, strictly positive profile e(t) for any β<1/3. It achieves this via a refined convex integration scheme built around a three-stage iteration (mollification, gluing, perturbation) and powered by Mikado flows, with careful control of the Reynolds stress through a trace-free gluing step and an energy-preserving perturbation. A key novelty is maintaining a trace-free Reynolds stress during gluing using an elliptic inverse-divergence operator and exploiting spacetime localization to push the Hölder regularity toward the Onsager threshold. The authors also prove an h-principle in C^β_{t,x}, showing that dissipative weak solutions are typical in the subsolution space, thereby strengthening the interpretation of energy-dissipating solutions within the framework of differential inclusions.
Abstract
We prove that given any $β<1/3$, 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 $v \in C^β([0,T]\times \mathbb{T}^3)$, 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 $C^β_{t,x}$, for any $β<1/3$. 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.