Vanishing Viscosity Limit For the 3D Nonhomogeneous Incompressible Navier-Stokes Equations With a Slip Boundary Condition

TL;DR

This paper studies the behavior of solutions to 3D nonhomogeneous incompressible Navier-Stokes equations with slip boundary conditions as viscosity approaches zero, establishing well-posedness and convergence rates.

Contribution

It proves local well-posedness and the vanishing viscosity limit with convergence rates for these equations under slip boundary conditions.

Findings

Established local well-posedness of strong solutions.

Proved vanishing viscosity limit with a strong convergence rate.

Identified additional density conditions needed for boundary compatibility.

Abstract

In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition.