Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line

Haiyan  Yin; Changjiang  Zhu

doi:10.3934/cpaa.2015.14.2021

Article Contents

2015, Volume 14, Issue 5: 2021-2042. Doi: 10.3934/cpaa.2015.14.2021

This issue Previous Article Multiplicity of solutions for a fractional Kirchhoff type problem Next Article A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains

Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line

Haiyan Yin^1, and
Changjiang Zhu^2,

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079
2.
School of Mathematics, South China University of Technology, Guangzhou 510641

Received: August 31, 2014

Revised: December 31, 2014

Published: August 2015

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Abstract

In this paper we study an asymptotic behavior of a solution to the initial boundary value problem for a viscous liquid-gas two-phase flow model in a half line $R_+:=(0,\infty).$ Our idea mainly comes from [23] and [29] which describe an isothermal Navier-Stokes equation in a half line. We obtain the convergence rate of the time global solution towards corresponding stationary solution in Eulerian coordinates. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. These theorems are proved by a weighted energy method.

Keywords:

Viscous liquid-gas two-phase flow model,
stationary solution,
initial boundary value problem,
convergence rate,
weighted energy method.

Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 76T10.

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