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For any \theta<1/10 we construct periodic weak solutions of theincompressible Euler equations which dissipate the total kinetic energy and areH\"older-continuous with exponent \theta. A famous conjecture of Onsager statesthe existence of such dissipative solutions with any H\"older exponent\theta<1/3. Our theorem is the first result in this direction.

Dissipative Euler Flows and Onsager's Conjecture