Speaker
Description
In this talk, I will report a recent work on a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, and viscous Newtonian fluids with unmatched densities. This model is compatible with continuum mixture theory. It adopts a mass-averaged (barycentric) velocity so that the two-phase flow is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation of the chemical potential. For the initial-boundary value problem in $\mathbb T^3$ with a class of physically relevant singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a suitable reduction of the original system to a Korteweg-type fluid model combined with a two-layer approximation, together with delicate estimates for the mass density and the phase-field variable inspired by the celebrated Bresch-Desjardins entropy. A key observation is that capillarity at the free interface provides a damping effect on the density evolution. For the limiting procedure, we derive delicate tail estimates to exclude possible concentrations of the singular potential, since no integrability of the pressure is available a priori. This work appears to be the first existence result for the Navier-Stokes/Cahn-Hilliard type system with unmatched densities and mass-averaged velocity without spatial regularization.