Remarks on the solution map for Yudovich solutions of the Euler equations

Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in ${\mathbb{R}}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $p\in [1,\mathrm{\infty })$ and is weakly-$\ast$ continuous in $L^{\mathrm{\infty }}$.