Convex integration above the Onsager exponent for the forced Euler equations

TL;DR

This paper introduces a new convex integration method to construct non-unique, smooth solutions to the forced Euler equations with regularity above the Onsager threshold, demonstrating non-uniqueness in higher dimensions.

Contribution

It presents the first convex integration scheme capable of producing solutions with Hölder regularity above 1/3 for the Euler equations, extending non-uniqueness results beyond previous limits.

Findings

Constructed non-unique solutions with regularity just below 1/2

Solutions are genuinely d-dimensional for d ≥ 3

Results apply to any smooth external force data

Abstract

We establish new non-uniqueness results for the Euler equations with external force on $\mathbb{T}^{d}$ $(d\geq3)$. By introducing a novel alternating convex integration scheme, we construct non-unique, almost-everywhere smooth, H\"older-continuous solutions with regularity $\frac{1}{2}-$, which is notably above the Onsager threshold of $\frac{1}{3}$. The solutions we construct differ significantly in nature from those which arise from the recent unstable vortex construction of Vishik; in particular, our solutions are genuinely $d$-dimensional ($d\geq3$), and give non-uniqueness results for any smooth data. To the best of our knowledge, this is the first instance of a convex integration construction above the Onsager exponent.