H. Abels, H. Garcke, and G. Grün (2012) proposed a diffuse interface model to describe liquid-liquid phase separation in incompressible binary fluids of different densities. This model consists of the Navier-Stokes system which is non-linearly coupled with an advective Cahn-Hilliard equation. In this talk, however, instead of taking the usual free energy functional, we consider a non-local version. Therefore, the resulting Cahn-Hilliard equation is a second-order (spatially) non-local equation. This system was already analysed by S. Frigeri (2016) who established the existence of a global weak solution. I intend to present some further results in dimension two which I have obtained jointly with C.G. Gal, A. Giorgini, and A. Poiatti (2023). These results are mainly concerned with strong solutions, uniqueness, and convergence to a single equilibrium. Some related open issues will also be discussed.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
