  In this paper we prove the uniform-in-time $L^p$ convergence in the inviscid
limit of a family $\omega^\nu$ of solutions of the $2D$ Navier-Stokes equations
towards a renormalized/Lagrangian solution $\omega$ 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 $\omega^\nu$ to $\omega$ in
$L^p$. Finally, we show that solutions of the Euler equations with $L^p$
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.
