Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid

Li Fang,Zhenhua Guo

doi:10.3934/cpaa.2017010

Abstract

In this paper, we study the zero dissipation limit toward rarefaction waves for solutions to a one-dimensional compressible non-Newtonian fluid for general initial data, whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we construct a sequence of solutions to the one-dimensional compressible non-Newtonian fluid which converge to the above rarefaction wave with vacuum as the viscosity coefficient $\epsilon$ tends to zero. Moreover, the uniform convergence rate is obtained, based on one fact that the viscosity constant can control the degeneracies caused by the vacuum in rarefaction waves and another fact that the energy estimates are obtained under some a priori assumption.

Highlights

Compressible non-Newtonian fluid, zero dissipation limit, rarefaction wave, vacuum
The present work is concerned with the following system about a one-dimensional compressible non-Newtonian fluid ρt +x = 0, x ∈ R, t > 0, ρut + ρuux + px(ρ) = ((μ0 + u2x) α−2 2 ux)x, (1)where μ0 > 0, > 0 and α > 2 are given constants
Let the pressure p be given by the γ−law ργ p(ρ) =

Summary

Introduction

Compressible non-Newtonian fluid, zero dissipation limit, rarefaction wave, vacuum. Yuan and his cooperators obtained the existence result on the local strong solutions of initial-boundary-value problem in one dimensional non-Newtonian fluids see [28, 29] and the references therein. It is interesting to study the zero dissipation limit of compressible non-Newtonian system (1) in the case when the Euler system (2) has two rarefaction waves with vacuum in middle.

Results

Conclusion