Abstract:

Kolmogorov's K41 theory of turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids. This phenomenon can be described as follows: although smooth solutions of the Euler equations conserve the kinetic energy, in turbulent fluids the energy can be transferred  to high frequencies and anomalously dissipated. Hence turbulent solutions of the Navier-Stokes equations are expected to converge, in the vanishing viscosity limit, to irregular solutions of the Euler equations, with decreasing kinetic energy.

In rigorous analytical terms, however, this phenomenon is little understood. In this talk, I will present the recent developments on this topic and focus on a joint work with G. Crippa and M. Sorella which considers the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. I will discuss the construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. The techniques developed in this context allow also to answer the question of (lack of) selection for passive-scalar advection under vanishing diffusivity. Finally, I will present a joint work with E. Brue’, G. Crippa, C. De Lellis, and M. Sorella, where we use the previous construction to give example of anomalous dissipation for the forced Navier-Stokes equations in the supercritical Onsager regularity regime.

For further information please contact elisur.magrini@unibocconi.it