Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Anomalous dissipation in fluid dynamics

Talk by Prof. Dr. Gianluca Crippa

Speaker invited by: Prof. Dr. Rémi Abgrall

**Date:** 25.10.23 **Time:** 16.30 - 17.30 **Room:** ETH HG E 1.2

Kolmogorov's K41 theory of fully developed turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids: although smooth solutions of the Euler equations conserve the energy, in a turbulent regime information is transferred to small scales and dissipation can happen even without the effect of viscosity, and it is rather due to the limited regularity of the solutions. In rigorous mathematical terms, however, very little is known. In a recent work in collaboration with M.~Colombo and M.~Sorella we consider the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. In my talk, I will present the general context and illustrate the main ideas behind our construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. I will also describe how the same techniques provide an example of lack of selection for passive-scalar advection under vanishing diffusivity, and an example of anomalous dissipation for the forced Euler equations in the supercritical Onsager regularity regime (this last result has been obtained in collaboration with E.~Bru\`e, M.~Colombo, C.~De Lellis, and M.~Sorella).''