Global weak solution to the viscous two-phase model with finite energy

In this paper, we prove the existence of global weak solutions to the compressible NavierStokes equations when the pressure law is in two variables. The method is based on the Lions argument and the Feireisl-Novotny-Petzeltova method. The main contribution of this paper is to develop a new argument for handling a nonlinear pressure law P (ρ, n) = ρ + n where ρ, n satisfy the mass equations. This yields the strong convergence of the densities, and provides the existence of global solutions in time, for the compressible barotropic Navier-Stokes equations, with large data. The result holds in three space dimensions on condition that the adiabatic constants γ > 95 and α ≥ 1. Our result is the first global existence theorem on the viscous compressible two-phase model with pressure law in two variables, for large initial data, in the multidimensional space. keywords. two-phase model, compressible Navier-Stokes equations, global weak solutions. AMS Subject Classifications (2010). 76T10, 35Q30, 35D30