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