  The aim of this note is to study the Cauchy problem for the 2D Euler
equations under very low regularity assumptions on the initial datum. We prove
propagation of regularity of logarithmic order in the class of weak solutions
with $L^p$ initial vorticity, provided that $p\geq 4$. We also study the
inviscid limit from the 2D Navier-Stokes equations for vorticity with
logarithmic regularity in the Yudovich class, showing a rate of convergence of
order $|\log\nu|^{-\alpha/2}$ with $\alpha>0$.
