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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
Abstract

We show that for any α < 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