Seminar Analysis: Sara Daneri (Max Planck Institute for Mathematics in the Sciences, Leipzig)

Dissipative Hölder solutions to the incompressible Euler equations

We consider the Cauchy problem for the incompressible Euler equations on the three-dimensional torus. According to a conjecture due to Onsager, which is well known in turbulence theory, while all the solutions which are uniformly α-Hölder continuous in space for any α>1/3 must conserve the total kinetic energy, for any α<1/3 there can be uniformly α-Hölder solutions which are strictly dissipative. While the first part of the conjecture is well established since a long time, the second part is still open in its full generality. In the result that we present we show that, for any α<1/5, there exist C^α vector fields being the initial data of infinitely many C^α solutions of the Euler equations which dissipate the total kinetic energy.

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