On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions

Abstract

We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluid and the momentum equations for the mixture. This model can be derived from the compressible two-fluid model with equal velocities (Bresch et al., in Arch Rational Mech Anal 196:599–629, 2010) and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes (Mellet and Vasseur, Commun Math Phys 281(3):573–596, 2008) (see also the compressible Oldroyd-B model with stress diffusion (Barrett et al., Commun Math Sci 15:1265–1323, 2017). Another interesting connection is that it is formally the equations of compressible magnetohydrodynamic (MHD) flows without resistivity in two dimensions under the action of vertical magnetic field (Li and Sun, J Differ Equ 267(6): 3827–3851, 2019). Under weak assumptions on the initial data which can be discontinuous, unbounded and large as well as involve transition to pure single-phase points or regions, we show existence of global weak solutions with finite energy. The essential novelty of this work, compared with previous works on the same model, is that transition to each single-phase flow is allowed without any constraints between adiabatic constants or two densities. It means that one of the phases can vanish in a point while the other can persist. The lack of enough regularity for each densities brings up essential difficulties in the two-component pressure compared with the single-phase model, i.e., compressible Navier-Stokes equations. The key points to achieve the main result rely on the variables reduction technique for the pressure function, domain separation, and some new estimates. As a byproduct, we obtain the existence of global weak solutions to the compressible MHD system without resistivity in two dimensions under the action of non-negatively vertical magnetic field, which represents a step forward to the study of the global large solution to the compressible MHD system without resistivity.