# Vanishing viscosity of one-dimensional isentropic Navier-Stokes equations with density dependent viscous coefficient

## TL;DR

This paper investigates how solutions to one-dimensional isentropic Navier-Stokes equations with density-dependent viscosity approach shock waves as viscosity vanishes, using an elementary energy method.

## Contribution

It demonstrates the convergence of Navier-Stokes solutions to shock waves in the zero-viscosity limit with density-dependent viscosity.

## Key findings

- Solutions converge to shock waves as viscosity tends to zero
- Elementary energy method used for proof
- Applicable to density-dependent viscous coefficients

## Abstract

In this paper, we study the vanishing viscosity of the isentropic compressible Navier-Stokes equations with density dependent viscous coefficient in the presence of the shock wave. Given a shock wave to the corresponding Euler equations, we can construct a sequence of solutions to one-dimensional compressible isentropic Navier-Stokes equations which converge to the shock wave as the viscosity tends to zero. The proof is given by an elementary energy method.