The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients

Wuming Li; Xiaojun Liu; Quansen Jiu

doi:10.3934/cpaa.2013.12.647

Article Contents

2013, Volume 12, Issue 2: 647-661. Doi: 10.3934/cpaa.2013.12.647

This issue Previous Article Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations Next Article Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$

The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients

1.
School of Mathematical and information Science, Henan Polytechnic University, Jiaozuo Henan, 454003
2.
Beijing No. 19 Middle School, Beijing 100089
3.
School of Mathematical Sciences, Capital Normal University, Beijing 100037

Received: January 31, 2010

Revised: March 31, 2012

Published: February 2013

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Abstract

We consider the Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, we are mainly concerned with the decay estimates of the density and velocity as $t \rightarrow \infty$. Firstly, we obtain the decay estimates of $\rho-\bar{\rho}$ and u in $L^2(R)$ norm, then we obtain the decay estimate of $\rho-\bar{\rho}$ in $L^{\infty}(R)$ norm as $\bar{\rho}>0$. Secondly, we construct a functional and use the energy method to obtain the decay estimate of $\rho$ in $L^{\infty}(R)$ norm as $\bar{\rho}=0$.

Keywords:

Navier-Stokes equations,
decay estimates.

Mathematics Subject Classification: 35Q30, 35B05.

Citation:

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