From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

Abstract

We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) 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.

Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation

“…The existence of solutions and their regularity is standard for the Cahn-Hilliard equation, see [28,29]. Thus, we admit here the first part of Theorem 1 and refer to an extended version of the present paper in [30] for details. Weak solutions are defined as follows: Definition 4 (Weak solutions).…”

“…Before proving this proposition, we first state a useful lemma that we admit, and refer the reader to [30] for the details.…”

“…Equation for R 0 . Because R 0 = R cannot fit (29), from (28) we immediately deduce the formmula for R 0 in (30).…”