From Vlasov equation to degenerate nonlocal Cahn-Hilliard equation. (English) Zbl 1522.35506
In this paper, the authors provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced by S. Takata and 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.
Reviewer: Changxing Miao (Beijing)
MSC:
| 35Q83 | Vlasov equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q84 | Fokker-Planck equations |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 82D05 | Statistical mechanics of gases |
| 76N15 | Gas dynamics (general theory) |
Citations:
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