Global Weak Solutions of a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase flows with Thermo-induced Marangoni Effects

We study a diffuse-interface model that describes the dynamics of two-phase incompressible flows driven by the thermo-induced Marangoni effect. The hydrodynamic system consists of the Navier-Stokes equations for the fluid velocity, the convective Cahn-Hilliard equation for the phase-field variable, and a convective heat equation for the (relative) temperature. For the initial-boundary value problem in two and three dimensions with variable viscosity, mobility, thermal diffusivity, and a physically relevant singular potential, we establish the existence of global weak solutions. The proof relies on an implicit-explicit time discretization scheme that preserves the $L^\infty$-bounds of both the phase-field variable and the temperature. When the spatial dimension is two, we prove the uniqueness of weak solutions for the case with matched densities under suitable assumptions on the initial temperature, mobility, and thermal diffusivity.