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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 $\alpha <\frac{1}{7}$ there exist $\alpha$-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