On Kato’s Conditions for Vanishing Viscosity

Abstract. Let u be a solution to the Navier-Stokes equations with viscosity ν in a bounded domain Ω in R, d ≥ 2, and let u be the solution to the Euler equations in Ω. In 1983 Tosio Kato showed that for sufficiently regular solutions, u → u in L∞([0, T ];L(Ω)) as ν → 0 if and only if ν ‖∇u‖2 X → 0 as ν → 0, where X = L([0, T ]× Γcν), Γcν being a layer of thickness cν near the boundary. We show that Kato’s condition is equivalent to ν ‖ω(u)‖2 X → 0 as ν → 0, where ω(u) is the vorticity (curl) of u, and is also equivalent to ν ‖u‖2 X → 0 as ν → 0.