Inviscid limit on $L^p$-based Sobolev conormal spaces for the 3D Navier-Stokes equations with the Navier boundary conditions

Abstract: We establish uniform bounds and the inviscid limit in $L^p$-based Sobolev conormal spaces for the solutions of the Navier-Stokes equations with the Navier boundary conditions in the half-space. We extend the vanishing viscosity results of~\cite{BdVC1} and~\cite{AK1} by weakening the normal and the conormal regularity assumptions, respectively. We require the initial data to be Lipschitz with three integrable conormal derivatives. We also assume that the initial normal derivative has one or two integrable conormal derivative depending on the sign of the friction coefficient. Finally, we establish the existence and uniqueness of the Euler equations with a bounded normal derivate, two bounded conormal derivatives, and three integrable conormal derivatives.