Continuous dissipative Euler flows and a conjecture of Onsager

It is known since the pioneering works of Scheffer and Shnirelman that there are nontrivial distributional solutions to the Euler equations which are compactly supported in space and time. Obviously these solutions do not respect the classical conservation law for the total kinetic energy and they are therefore very irregular. In recent joint works we have proved the existence of continuous and even Hölder continuous solutions which dissipate the kinetic energy. Our theorem might be regarded as a first step towards a conjecture of Lars Onsager, which in 1949 asserted the existence of dissipative Hölder solutions for any Hölder exponent smaller than 1 3 . 2010 Mathematics Subject Classification. 35D30, 76F05, 34A60, 53B20