  We consider a vanishing viscosity sequence of weak solutions of the
three-dimensional Navier--Stokes equations on a bounded domain. In a seminal
paper [25] Kato showed that for sufficiently regular solutions, the vanishing
viscosity limit is equivalent to having vanishing viscous dissipation in a
boundary layer of width proportional to the viscosity. We prove that Kato's
criterion holds for H\"older continuous solutions with the regularity index
arbitrarily close to the Onsager's critical exponent through a new boundary
layer foliation and a global mollification.
