Institut für Mathematik


Modul:   MAT675  PDE and Mathematical Physics

Non selection of vanishing viscosity solutions to the advection equation and anomalous dissipation

Vortrag von Massimo Sorella

Sprecher eingeladen von: Prof. Dr. Klaus Widmayer

Datum: 02.03.23  Zeit: 16.15 - 18.00  Raum: ETH HG G 19.1

In this seminar we outline a recent example of a turbulent divergence free velocity field \(u \in C^\alpha ([0,1 ] \times \T^2)\), with \(\alpha < 1\), having the \textit{non-selection} property. The latter is defined as follows: consider the sequence \(\{ \theta_\nu \}_{\nu >0}\) of solutions to the associated advection diffusion equation with viscosity parameter \(\nu>0\) and fixed initial datum \(\theta_{\text{in}} \in C^\infty\). Then, at least two distinct limiting solutions of the advection equation in the weak* topology arise from the sequence \(\{\theta_\nu\}_{\nu >0}\) as \(\nu \to 0\). Finally, we also mention a recent result of anomalous dissipation, at the level of the forced Navier--Stokes equations in the sharp regularity class \(L^3_t C^{1/3-}_x\) based on the previous turbulent} velocity field, which in particular implies the failure of the energy balance in the forced Euler equations. These are joint works with Elia Bru\'e, Maria Colombo, Gianluca Crippa and Camillo De Lellis.