Local in time strong solvability of the non-steady Navier–Stokes equations with Navier's boundary condition and the question of the inviscid limit

Abstract

In this Note, we prove the existence of strong solutions to the Navier–Stokes equations for incompressible viscous fluids in a general regular bounded domain of R 3 on a “short” time interval ( 0 , T 0 ) , independent of the viscosity and of the friction between the fluid and the boundary. The solutions to the Navier–Stokes problem satisfy the inhomogeneous Navier's boundary condition and they reveal a remarkable structure of approximation of the solution to the Euler problem, which enables us to solve completely the question of the inviscid limit of the family of obtained solutions on the time interval ( 0 , T 0 ) .