On nonuniqueness of Hölder continuous globally dissipative Euler flows

De Lellis, Camillo; Kwon, Hyunju

doi:10.2140/apde.2022.15.2003

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On nonuniqueness of Hölder continuous globally dissipative Euler flows

Camillo De Lellis and Hyunju Kwon

Vol. 15 (2022), No. 8, 2003–2059

DOI: 10.2140/apde.2022.15.2003

Abstract

We show that for any $α < \frac{1}{7}$ there exist $α$ -Hölder continuous weak solutions of the three-dimensional incompressible Euler equation, which satisfy the local energy inequality and strictly dissipate the total kinetic energy. The proof relies on the convex integration scheme and the main building blocks of the solution are various Mikado flows with disjoint supports in space and time.

Keywords

Euler equations, Onsager's conjecture, the local energy inequality, convex integration

Mathematical Subject Classification

Primary: 35Q31

Secondary: 35D30, 76B03

Milestones

Received: 11 September 2020

Revised: 9 February 2021

Accepted: 25 March 2021

Published: 10 February 2023

Authors

Camillo De Lellis
School of Mathematics Institute for Advanced Study Princeton, NJ United States

Hyunju Kwon
School of Mathematics Institute for Advanced Study Princeton, NJ United States

Vol. 15, No. 8, 2022