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

Abstract

This paper is concerned with the Cauchy problem of the three-dimensional viscous liquid-gas two-phase flow model. We prove the global existence of a strong solution when H 2 -norm of the initial perturbation around a constant state is sufficiently small and its L 1 -norm is bounded. Moreover, the optimal convergence rates are also obtained for such a solution.

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

“…Under the framework of Besov spaces, Hao-Li [14] obtained the existence and uniqueness of the global strong solution to the Cauchy problem of model (1), provided the initial value are close to a constant equilibrium state. The global existence and long time behavior for the Cauchy problem was investigated for the model (1) by Zhang-Zhu [28], where the global strong solution and optimal convergence rates were obtained. We refer to [15,21,25,27] for the different types of blow-up criteria for the local strong solution to the viscous liquid-gas two-phase flow model.…”

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

“…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].…”

“…Hence, it is a meaningful and interesting for us to directly discuss and study some properties about the viscous liquid-gas two-phase flow model in Eulerian coordinates. Zhang and Zhu in [37] have studied the asymptotic behaviors of the strong solutions in Eulerian coordinates to the Cauchy problem of a three-dimensional viscous liquidgas two-phase flow model and get the optimal convergence rates when H 2 -norm of the initial perturbation around a constant state is suciently small and its L 1 -norm is bounded. However, so far there is no result on the asymptotic behavior of a solution to the initial boundary value problem (4)- (7).…”

“…However, it is non-trivial to apply directly the ideas used in single-phase models into the two-phase models since the momentum equation is given only for the mixture and that the pressure involves the masses of two phases in a nonlinear way, which makes it rather difficult to get the uniform time-independent estimates of the masses and the mixed velocity (n, m, u). To overcome these difficulties, we first take a nonlinear variable transformation as in [18,19,37] so as to separate the two mass variables from each other, which enables us to decompose the original system into a homogenous transport equation and a coupled hyperbolic-parabolic system. More precisely, we introduce a new variable…”

“…However, we can not work directly on the system of the variable (ρ, m, u) as in [19]. Indeed it seems impossible to obtain the uniform time-independent estimates of the variables m. The key idea here is that instead of the variables (ρ, m, u), we study the system of the variables (ρ, p, u) as in [37]. To see this, we rewrite the pressure in (3) in terms of the variables (ρ, m) as follows:…”