On the inviscid limit of the compressible Navier-Stokes equations near Onsager's regularity in bounded domains

The viscous dissipation limit of weak solutions is considered for the Navier-Stokes equations of compressible isentropic flows confined in a bounded domain. We establish a Kato-type criterion for the validity of the inviscid limit for the weak solutions of the Navier-Stokes equations in a function space with the regularity index close to Onsager's critical threshold. In particular, we prove that under such a regularity assumption, if the viscous energy dissipation rate vanishes in a boundary layer of thickness in the order of the viscosity, then the weak solutions of the Navier-Stokes equations converge to a weak admissible solution of the Euler equations. Our approach is based on the commutator estimates and a subtle foliation technique near the boundary of the domain.