On Kato's conditions for vanishing viscosity

Let u be a solution to the Navier-Stokes equations with viscosity v in a bounded domain Q in R d , d > 2, and let Ū be the solution to the Euler equations in Q. In 1983 Tosio Kato showed that for sufficiently regular solutions, u → Ū in L ∞ ([0, T]; L 2 (Ω)) as v → 0 if and only if ν ∥∇u∥ 2X → 0 as ν→ 0, where X = L 2 ([0, T] x Γ cv ), Γ cν being a layer of thickness cv near the boundary. We show that Kato's condition is equivalent to v ∥ω(u)∥2X→ 0 as ν → 0, where ω(u) is the vorticity (curl) ofu, and is also equivalent to ν -1 ∥u∥2X → 0 as v - 0.