A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity

TL;DR

This paper extends Kato's criterion for vanishing viscosity limits to less regular solutions of the 3D Navier-Stokes equations, approaching Onsager's critical regularity, using new boundary layer techniques.

Contribution

It proves Kato's criterion applies to Hölder continuous solutions near Onsager's critical exponent with novel boundary layer foliation and mollification methods.

Findings

Kato's criterion holds for Hölder solutions close to Onsager's critical regularity.

Introduces a new boundary layer foliation technique.

Establishes the vanishing viscosity limit under weaker regularity assumptions.

Abstract

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.