Optimal decay rates for the viscous two-phase model without constraints on transition to single-phase flow

Abstract

In this paper, we study a free-boundary problem of the liquid-gas two-phase drift-flux model with constant viscosity, where the initial liquid and gas masses are assumed to be connected with vacuum continuously, and the pressure function takes a singular form. We utilize the flow map to reformulate the problem and prove the global existence and decay rates of strong solutions with general initial data that involve transition to pure single-phase points or regions. In particular, both the optimal decay rates of the mass functions and the decay estimates of the velocity towards its asymptotic profile as time goes to infinity are studied. As far as we are concerned, this is the first result on the global existence of strong solutions to the two-phase drift-flux model with singular pressure function and with general initial data that allow for transition to single-phase flow. It is also worth noting that our method in recovering the non-weighted estimates of the solution by utilizing the Poincaré type inequality can apply to the single-phase problem studied in Zeng (2015) [37], so as to derive the decay estimates for the velocity and relax the constraints on the initial data.