On the inviscid limit of the 2D Euler equations with vorticity along the $(LMO^\alpha)_\alpha$ scale

In a recent paper [5], the global well-posedness of the two-dimensional Euler equation with vorticity in \mbox{$L^1\cap LBMO$} was proved, where $LBMO$ is a Banach space which is strictly imbricated between \mbox{$L^\infty$} and $BMO$. In the present paper we prove a global result of inviscid limit of the Navier-stokes system with data in this space and other spaces with the same BMO flavor. Some results of local uniform estimates on solutions of the Navier-Stokes equations, independent of the viscosity, are also obtained.