Finite-Energy Solutions for Compressible Two-Fluid Stokes System

We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions–Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier–Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via an unconventional approximation using the Lagrangian formulation, truncations, and a stability result of trajectories for rough velocity fields.