# Vanishing viscosity limit to vortex sheet for the isentropic compressible circularly symmetric 2D flow

## TL;DR

This paper rigorously analyzes the vanishing viscosity limit of 2D isentropic compressible flow with vortex sheets, showing approximation by inviscid flow away from boundaries and identifying vortex layers near boundaries.

## Contribution

It provides a rigorous justification of the asymptotic behavior of solutions in the small viscosity limit for compressible flows with vortex sheets, including boundary layer analysis.

## Key findings

- Inviscid flow approximates viscous flow away from boundaries.
- Vortex layers form near the boundary for angular velocity.
- Radial velocity and pressure do not develop boundary layers.

## Abstract

In this paper, we consider the small viscosity limit problem for the isentropic compressible Navier-Stokes equations in a 2D exterior domain with impermeable boundary conditions , and the corresponding Euler equations have vortex sheet solutions.We obtain that away from the boundary and the contact discontinuous the isentropic compressible viscous flow can be approximated by the corresponding inviscid flow, near the boundary (the contact discontinuous) there is a boundary layer (vortex layer)for the angular velocity in the leading order expansion of solution, while the radial velocity and the pressure do not have boundary layers (vortex layers) in the leading order. We rigorously justify the asymptotic behavior of solutions in the $L^{\infty}$ space for the small viscosities limit in the Lagrangian coordinates.