Admissible solutions of the 2D Onsager's conjecture

TL;DR

This paper constructs energy-dissipating weak solutions to the 2D Euler equations with Hölder continuity below the Onsager critical exponent, using innovative traveling waves and a new iterative scheme.

Contribution

It introduces a new class of traveling waves and a multiple iteration scheme to construct dissipative solutions below the Onsager critical exponent in any dimension.

Findings

Existence of Hölder continuous dissipative solutions for γ<1/3

Initial data of solutions are dense in certain Besov spaces

Framework applicable in any dimension d ≥ 2

Abstract

We show that for any $\gamma < \frac{1}{3}$ there exist H\"{o}lder continuous weak solutions $v \in C^{\gamma}([0,T] \times \mathbb{T}^2)$ of the two-dimensional incompressible Euler equations that strictly dissipate the total kinetic energy, improving upon the elegant work of Giri and Radu [Invent. Math., 238 (2), 2024]. Furthermore, we prove that the initial data of these \textit{admissible} solutions are dense in $B^{\gamma}_{\infty,r<\infty}$. Our approach introduces a new class of traveling waves, refining the traditional temporal oscillation function first proposed by Cheskidov and Luo [Invent. Math., 229(3), 2022], to effectively modulate energy on any time intervals. Additionally, we propose a novel ``multiple iteration scheme'' combining Newton-Nash iteration with a Picard-type iteration to generate an energy corrector for controlling total kinetic energy during the perturbation step. This framework enables us to construct dissipative weak solutions below the Onsager critical exponent in any dimension $d \geq 2$.