A Proof of Onsager's Conjecture
Philip Isett
TL;DR
<3-5 sentence high-level summary> The paper proves Onsager's conjecture by constructing nontrivial weak solutions to the 3D incompressible Euler equations with Hölder regularity α<1/3 that do not conserve energy. The authors develop a refined convex integration framework augmented by a novel gluing technique that partitions the stress into time-local components, allowing long-time interaction control via Mikado flows. The main technical advance is the gluing approximation, together with a robust transport-elliptic anti-divergence construction, which together yield a convergent iteration producing compact-time-support solutions. The result completes the negative direction of Onsager’s conjecture and opens avenues for extensions to other domains and related fluid equations.
Abstract
For any $α< 1/3$, we construct weak solutions to the $3D$ incompressible Euler equations in the class $C_tC_x^α$ that have nonempty, compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for $α> 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent $α= 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^α$. The previous best results were solutions in the class $C_tC_x^α$ for $α< 1/5$, due to the author, and in the class $L_t^1 C_x^α$ for $α< 1/3$ due to Buckmaster, De Lellis and Székelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis and Székelyhidi. The present proof combines the method of convex integration and a new "gluing approximation" technique. The convex integration part of the proof relies on the "Mikado flows" introduced by [Daneri, Székelyhidi] and the framework of estimates developed in the author's previous work.