# Uniform regularity for the Navier-Stokes equation with Navier boundary condition - Mathematics > Analysis of PDEs

Abstract: We prove that there exists an interval of time which is uniform in thevanishing viscosity limit and for which the Navier-Stokes equation with Navierboundary condition has a strong solution. This solution is uniformly bounded ina conormal Sobolev space and has only one normal derivative bounded in$L^\infty$. This allows to get the vanishing viscosity limit to theincompressible Euler system from a strong compactness argument.