The Inviscid Limit in Uniformly Local Yudovich Spaces

Abstract: We consider solutions to the two-dimensional Navier-Stokes and Euler equations with the velocity in $L<sup>{\infty}(\mathbb{R}<sup>2)$ and the vorticity in the uniformly local Yudovich space $Y<sup>θ_{ul}(\mathbb{R}<sup>2)$. We show that if $θ$ grows sufficiently slowly, then for every $T&gt;0$, solutions of the Navier-Stokes equation converge to solutions of the Euler equation in $$L<sup>{\infty}([0,T];L<sup>2_{ul}(\mathbb{R}<sup>2))$$ as $ν\to 0<sup>+$, provided they are generated from the same initial data.