Dissipative Euler Flows and Onsager's Conjecture

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