From Vlasov equation to degenerate nonlocal Cahn-Hilliard equation (Q6160275)

In this paper, the authors provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced by \textit{S. Takata} and \textit{T. Noguchi} [J. Stat. Phys. 172, No. 3, 880--903 (2018; Zbl 1400.82223)] in order to describe phase transition of fluids by kinetic equations. More specifically, they proved that, when the scale parameter tends to \(0\), this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For this purpose, the authors introduced apropriate forms of the short and long range potentials, and derived Helmhotlz free energy estimates. In particular, they proved a new weak compactness bound on the flux.