From Vlasov equation to degenerate nonlocal Cahn-Hilliard equation

TL;DR

This paper rigorously derives a nonlocal degenerate Cahn-Hilliard equation as the hydrodynamic limit of a kinetic Vlasov model for phase transitions, using energy estimates and compactness arguments.

Contribution

It provides a new mathematical framework connecting kinetic Vlasov models to nonlocal Cahn-Hilliard equations with degenerate mobility, including novel energy and compactness results.

Findings

Convergence of Vlasov model to nonlocal Cahn-Hilliard equation as scale parameter tends to 0.

Introduction of specific potentials enabling Helmholtz free energy estimates.

Establishment of a new weak compactness bound on the flux.

Abstract

We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux.