Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

In this paper we prove the uniform-in-time Lp convergence in the inviscid limit of a family ων of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution ω of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of ων to ω in Lp. Finally, we show that solutions of the Euler equations with Lp vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.