The Inviscid Limit in Uniformly Local Yudovich Spaces

We consider solutions to the two-dimensional Navier-Stokes and Euler equations with the velocity in $L^{\mathrm{\infty }}({\mathbb{R}}^2)$ and the vorticity in the uniformly local Yudovich space $Y_{ul}^{\theta }({\mathbb{R}}^2)$. We show that if $\theta$ grows sufficiently slowly, then for every $T>0$, solutions of the Navier-Stokes equation converge to solutions of the Euler equation in $L^{\mathrm{\infty }}([0,T];L_{ul}^2({\mathbb{R}}^2))$ as $\nu \to 0^{+}$, provided they are generated from the same initial data.