# Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier-Stokes equations

## TL;DR

This paper proves that solutions of the 2D compressible Navier-Stokes equations converge to planar rarefaction waves of the Euler equations as viscosity vanishes, with a uniform convergence rate away from initial time.

## Contribution

It establishes the vanishing viscosity limit for 2D compressible Navier-Stokes equations converging to planar rarefaction waves, introducing hyperbolic waves to rigorously justify the limit.

## Key findings

- Existence of smooth solutions converging to rarefaction waves
- Uniform convergence rate away from initial time
- Use of hyperbolic waves to recover physical viscosities

## Abstract

The vanishing viscosity limit of the two-dimensional (2D) compressible isentropic Navier-Stokes equations is studied in the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. It is proved that there exists a family of smooth solutions for the 2D compressible Navier-Stokes equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficients away from the initial time. In the proof, the hyperbolic wave is crucially introduced to recover the physical viscosities of the inviscid rarefaction wave profile, in order to rigorously justify the vanishing viscosity limit.