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

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 ideamainly comes from [23] and [29] which describe anisothermal Navier-Stokes equation in a half line. We obtain theconvergence 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.