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

Article Contents

2017, Volume 16, Issue 1: 209-242. Doi: 10.3934/cpaa.2017010

This issue Previous Article Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D Next Article Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity

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

Li Fang and
Zhenhua Guo^,

School of Mathematics, CNS, Northwest University, Xi'an 710127, China

Author Bio: E-mail address: fangli@nwu.edu.cn
Zhenhua Guo, E-mail address: zhguo@nwu.edu.cn
Zhenhua Guo, E-mail address: zhguo@nwu.edu.cn

Received: February 29, 2016

Revised: June 30, 2016

Published: January 2018

The first author is supported by the National Natural Science Foundation of China 11501445. The second author is supported by the National Natural Science Foundation of China 11331005 and SRDPC 20136101110015.

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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.

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Mathematics Subject Classification: Primary: 35L65, 35Q30; Secondary: 76N15.

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