On Kato's Conditions for Vanishing Viscosity

• Published in: Indiana University mathematics journal 2007-01, Vol.56 (4), p.1711-1721
• Main Author: Kelliher, James P.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Boundary layers; College mathematics; Euler equations; Exact sciences and technology; General mathematics; General, history and biography; Greens theorem; Mathematical analysis; Mathematical constants; Mathematical theorems; Mathematics; Navier Stokes equation; Partial differential equations; Sciences and techniques of general use; Viscosity; Vorticity
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2007.56.3080

Let u be a solution to the Navier-Stokes equations with viscosity v in a bounded domain Ω in ℝd, d ≥ 2, and let ū be the solution to the Euler equations in Ω. In 1983 Tosio Kato showed that for sufficiently regular solutions, u → ū in L∞([0,T]; L2(Ω)) as v → 0 if and only if $\mathrm{v}{\Vert \nabla \mathrm{u}\Vert }_{\mathrm{X}}^{2}\to 0$ as v → 0, where X = L2([0,T] × Γcv), Γcv being a layer of thickness cv near the boundary. We show that Kato's condition is equivalent to $\mathrm{v}{\Vert \mathrm{\omega }\left(\mathrm{u}\right)\Vert }_{\mathrm{X}}^{2}\to 0$ as v → 0, where ω(u) is the vorticity (curl) of u, and is also equivalent to ${\mathrm{v}}^{-1}{\Vert \mathrm{u}\Vert }_{\mathrm{X}}^{2}\to 0$ as v → 0.