A Kato-Type Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity

Abstract

We consider a vanishing viscosity sequence of weak solutions for the three-dimensional Navier–Stokes equations of incompressible fluids in a bounded domain. In a seminal paper (Kato in Seminar on nonlinear partial differential equations, Springer, New York, 1983), 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 the Hölder continuous solutions with the regularity index arbitrarily close to Onsager’s critical exponent through a new boundary layer foliation and a global mollification technique.