On the non-homogeneous Navier-Stokes system with Navier friction boundary conditions

TL;DR

This paper investigates the existence of weak solutions for the non-homogeneous Navier-Stokes equations with Navier friction boundary conditions, including vacuum zones, and studies their convergence to Euler solutions as viscosity vanishes.

Contribution

It establishes the existence of weak solutions under rough data conditions and demonstrates convergence to Euler equations with variable density in the zero-viscosity limit.

Findings

Existence of weak solutions with vacuum zones

Convergence of Navier-Stokes solutions to Euler solutions as viscosity approaches zero

Weak solutions exist under rough initial data conditions

Abstract

We address the issue of existence of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions allowing the presence of vacuum zones and assuming rough conditions on the data. We also study the convergence, as the viscosity goes to zero, of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions to the strong solution of the Euler equations with variable density, provided that the initial data converge in $L^{2}$ to a smooth enough limit.