Global well-posedness for a two-dimensional Navier-Stokes-Cahn-Hilliard-Boussinesq system with singular potential

Abstract: We study a general Navier-Stokes-Cahn-Hilliard-Boussinesq system that describes the motion of a mixture of two incompressible Newtonian fluids with thermo-induced Marangoni effects. The Cahn-Hilliard dynamics of the binary mixture is governed by aggregation/diffusion competition of the free energy with a physically-relevant logarithmic potential. The coupled system is studied in a bounded smooth domain $\Omega\subset \mathbb{R}^2$ and is supplemented with a no-slip condition for the fluid velocity, homogeneous Neumann boundary conditions for the order parameter and the chemical potential, homogeneous Dirichlet boundary condition for the relative temperature, and suitable initial conditions. For the corresponding initial boundary value problem, we first prove the existence of global weak solutions and their continuous dependence with respect to the initial data. Under additional assumptions on the initial data, we prove the existence and uniqueness of a global strong solution and the validity of the strict separation property.