Kinetic and Related Models (KRM)

Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval

Pages: 761 - 787, Volume 6, Issue 4, December 2013      doi:10.3934/krm.2013.6.761

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Raffaele Esposito - International Research Center M&MOCS, Università di L'Aquila, Cisterna di Latina, 04012, Italy (email)
Yan Guo - Division of Applied Mathematics, Brown University, Providence, RI 02812, United States (email)
Rossana Marra - Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma, Italy (email)

Abstract: We consider a kinetic model for a system of two species of particles on a sufficiently large periodic interval, interacting through a long range repulsive potential and by collisions. The model is described by a set of two coupled Vlasov-Boltzmann equations. For temperatures below the critical value and suitably prescribed masses, there is a non homogeneous solution, the double soliton, which is a minimizer of the entropy functional. We prove the stability, up to translations, of the double soliton under small perturbations. The same arguments imply the stability of the pure phases, as well as the stability of the mixed phase above the critical temperature. The mixed phase is proved to be unstable below the critical temperature.

Keywords:  Phase transitions, Vlasov-Boltzmann equation, stability.
Mathematics Subject Classification:  Primary: 82B26, 82C40; Secondary: 76P05.

Received: July 2013;      Revised: August 2013;      Available Online: November 2013.