Non Selection Of Vanishing Viscosity Solutions To The Advection Equation And Anomalous Dissipation

In this seminar we outline an example of a divergence free velocity field u ∈ Cα([0, 1] × T2), with α < 1, for which there is lack of selection of solutions to the transport equation via vanishing viscosity. With the same construction of the velocity field, rescaled in time and enjoying different regularity, we prove anomalous dissipation in the full Obukhov–Corssin supercritical regime for the advection diffusion equation. If the time permits we will also mention some new results about anomalous dissipation for the forced Navier–Stokes equations.

These are joint works with Elia Bru´e, Maria Colombo, Gianluca Crippa and Camillo De Lellis.