| Abstract: |
| In this talk, I will discuss a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, viscous fluids with unmatched densities. This model adopts a mass-averaged velocity so that the fluid mixture is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation for the chemical potential. For the initial boundary value problem in $\mathbb{T}^3$ with a class of physically relevant and singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a reduction of the model and a two-layer approximation with delicate estimates for the order parameter and the mass density that are inspired by the celebrated BD-entropy introduced by Bresch and Desjardins [Comm. Math. Phys. 2003, 238(1-2): 211--223]. A key observation is that capillarity at the interface provides a damping effect on the density evolution. Moreover, for the limiting passage, delicate tail estimates are derived 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 system with general densities and barycentric velocity without spatial regularization. |
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