*Accepted Paper*

**Inserted:** 1 dec 2020

**Last Updated:** 3 aug 2021

**Journal:** To appear in Analysis and PDEs

**Year:** 2020

**Abstract:**

We show that for any $\alpha<\frac{1}{7}$ there exist $\alpha$-HÃ¶lder 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.

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