Convergence rates to stationary solutions of a gas–liquid model with external forces

Abstract

Abstract:In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and, we prove the weak solution (cQ(x, t), u(x, t)) behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if θ ∈ (0,

Global existence and optimal convergence rates for the strong solutions in H2 to the 3D viscous liquid–gas two-phase flow model

“…For the one dimensional version of the model (1.1), the existence, uniqueness, regularity, asymptotic behavior and decay rate estimates of (weak, strong or smooth) solutions to the free boundary problem have been investigated in [12,14,[16][17][18]29,38,39] where the liquid is incompressible and the gas is polytropic, and in [13] where both of two fluids are compressible. As a generalization of the results in [13] from 1D to 2D, Yao, Zhang, and Zhu in [40] obtained the existence of the global solution to the 2D model when the initial energy is small enough and there is no initial vacuum.…”

Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain

“…The 1D version of (1) with various initial and initialboundary conditions has been investigated intensively during the past decades, both classical and weak solutions have been constructed, and long time behaviors of different solutions have been investigated. When the liquid is incompressible and the gas is polytropic, the existence, uniqueness, regularity, asymptotic behavior and decay rate estimates of (weak or classical) solutions to the free boundary problem have been studied in [5,7,9,10,11,18,22,23]. If both of two fluids are compressible, Evje-Flåtten [6] obtained the global existence of weak solutions.…”

“…Then, due to (1), (10) and (11), the system (1) can be rewritten in terms of the variables (P, u, s) as follows:…”

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

“…The existence, uniqueness, regularity, asymptotic behavior and decay rate estimates and blow-up phenomena of (weak, strong or smooth) solutions to all kinds of the one-dimensional or multi-dimensional viscous liquid-gas two-phase flow model have been investigated, cf. [10,11,12,13,14,16,19,26,32,33,34,35,36]. There are also many numerical results about viscous liquid-gas two-phase flow model or relevant model during the last decade, see for instance [1,2,3,5,4,8,9,17,27].…”

“…To our knowledge, there are many results about the asymptotic behavior of solutions to all kinds of the onedimensional or multi-dimensional viscous liquid-gas two-phase flow model, please refer to [15,18,16,26,37,33]. As we all know it is in Lagrangian coordinates that one can obtain the large time behavior of solutions in [15,18,16,26,33]. However, in our paper it is in Eulerian coordinates that we consider an asymptotic behavior of a solution to the initial boundary value problem (4)-(7), whereas in Lagrangian coordinates.…”