  We show strong convergence of the vorticities in the vanishing viscosity
limit for the incompressible Navier-Stokes equations on the two-dimensional
torus, assuming only that the initial vorticity of the limiting Euler equations
is in $L^p$ for some $p>1$. This substantially extends a recent result of
Constantin, Drivas and Elgindi, who proved strong convergence in the case
$p=\infty$. Our proof, which relies on the classical renormalization theory of
DiPerna-Lions, is surprisingly simple.
