“Lagrangian and conservative solutions of the 2D Euler equations”
Mercoledì 17 Giugno 2020, ore 15:00 - Zoom - Gennaro Ciampa (University of Basel, CH)
Smooth solutions of the 2D incompressible Euler equations enjoy two very natural properties: the first one is that they are Lagrangian, namely the vorticity is advected by the flow of the velocity; the second property is that smooth solutions conserve the kinetic energy. When we consider solutions in weaker classes, precisely when the initial vorticity is in $L^p$ with $1\leq p \le \infty$, the existence of Lagrangian solutions and the conservation of the energy may depend in general on the approximation scheme. Furthermore, a reasonable question is whether solutions constructed by a given method are unique. In this talk we prove the existence of solutions which enjoy the above properties constructed via different methods. Moreover, we will show that already in the linear case a smooth approximation of the velocity field can produce different solutions in the limit.
Based on joint works with G. Crippa (University of Basel) and S. Spirito (University of L’Aquila).