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January  2017, 16(1): 209-242. doi: 10.3934/cpaa.2017010

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

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

Received  March 2016 Revised  July 2016 Published  November 2016

Fund Project: 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

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.

Citation: Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010
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