In this talk we consider the Cauchy problem for the 2D Euler equations for incompressible inviscid fluids. It is well-known that given a smooth initial datum, the Cauchy problem is well-posed and in particular the energy is conserved and the vorticity is transported by the flow of the velocity. When we consider weak solutions this might not be the case anymore. We will review some recent results obtained in collaboration with Gianluca Crippa and Gennaro Ciampa where we extend those properties to the case of irregular vorticities. In particular, under low integrability assumptions on the vorticity we show that certain approximations important from a physical and a numerical point of view converge to solutions satisfying those properties.
Stefano Spirito is an assistant professor in tenure track at the University of L’Aquila since December 2018. Before, he was researcher at GSSI – Gran Sasso Science Institute in L’Aquila and Post-Doc at the University of Basel. He mainly works on fluid dynamics, both compressible and incompressible fluids, and linear transport and continuity equations with irregular vector fields.