A note on the vanishing viscosity limit in the Yudovich class

We consider the inviscid limit for the two-dimensional Navier–Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier–Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $\nu$. Here, we provide an order ${(\nu \mathrm{/}\mathrm{\mid }\log \nu \mathrm{\mid })}^{\frac{1}{2}\exp (-Ct)}$ bound, which slightly improves upon earlier results by Chemin.