On global solutions to the viscous liquid–gas model with unconstrained transition to single-phase flow

Abstract

In this paper, we consider the Dirichlet problem of a one-velocity viscous drift-flux model. One of the phases is compressible, the other one is weakly compressible. Under weak assumptions on the initial data, which can be discontinuous and large as well as involve transition to pure single-phase points or regions, we show existence of global bounded weak solutions. One main ingredient is that we employ a decomposition of the pressure term appearing in the mixture momentum equation into two components, one for each of the two phases. This paves the way for deriving a basic energy equality. In particular, upper bounds on the masses are extracted from the estimates provided by the energy equality. By relying on weak compactness tools we obtain an existence result within the class of weak solutions. An essential novel aspect of this analysis, compared to previous works on the same model, is that the solution space is large enough to allow for transition to single-phase flow without any constraints. In particular, one of the phases can vanish in a point while the other phase can persist. The key to achieve this result, which represents a major step forward compared to previous results for this model, is that we do not rely on any higher-order (i.e. derivatives in space) estimates on the masses or pressure, only low-order estimates provided by the energy equality and the uniform upper bounds on the liquid and gas mass.