article

Title: On Kato's conditions for vanishing viscosity

Authors: James P. Kelliher

Issue: Volume 56 (2007), Issue 4, 1711-1721

Abstract:

$Let $u$ be a solution to the Navier-Stokes equations with viscosity $\nu$ in a bounded domain $\Omega$ in $\mathbb{R}^d$, $d \ge 2$, and let $\bar{u}$ be the solution to the Euler equations in $\Omega$. In 1983 Tosio Kato showed that for sufficiently regular solutions, $u \to \bar{u}$ in $L^{\infty}([0, T] ; L^2(\Omega))$ as $\nu \to 0$ if and only if $\nu \| \nabla u \|_X^2 \to 0$ as $\nu \to 0$, where $X = L^2([0, T] \times \Gamma_{c \nu})$, $\Gamma_{c \nu}$ being a layer of thickness $c \nu$ near the boundary. We show that Kato's condition is equivalent to $\nu \| \omega(u) \|_X^2 \to 0$ as $\nu \to 0$, where $\omega(u)$ is the vorticity (curl) of $u$, and is also equivalent to $\nu^{-1} \| u \|_X^2 \to 0$ as $\nu \to 0$.$