  We prove that given initial data $\omega_0\in L^\infty(\mathbb{T}^2)$,
forcing $g\in L^\infty(0,T; L^\infty(\mathbb{T}^2))$, and any $T>0$, the
solutions $u^\nu$ of Navier-Stokes converge strongly in
$L^\infty(0,T;W^{1,p}(\mathbb{T}^2))$ for any $p\in [1,\infty)$ to the unique
Yudovich weak solution $u$ of the Euler equations. A consequence is that
vorticity distribution functions converge to their inviscid counterparts. As a
byproduct of the proof, we establish continuity of the Euler solution map for
Yudovich solutions in the $L^p$ vorticity topology. The main tool in these
proofs is a uniformly controlled loss of regularity property of the linear
transport by Yudovich solutions. Our results provide a partial foundation for
the Miller--Robert statistical equilibrium theory of vortices as it applies to
slightly viscous fluids.
