# American Institute of Mathematical Sciences

December  2013, 6(4): 761-787. doi: 10.3934/krm.2013.6.761

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

 1 International Research Center M&MOCS, Università di L'Aquila, Cisterna di Latina, 04012, Italy 2 Division of Applied Mathematics, Brown University, Providence, RI 02812 3 Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma

Received  July 2013 Revised  August 2013 Published  November 2013

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.
Citation: Raffaele Esposito, Yan Guo, Rossana Marra. Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval. Kinetic & Related Models, 2013, 6 (4) : 761-787. doi: 10.3934/krm.2013.6.761
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##### References:
 [1] Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545 [2] Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13 [3] Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 [4] El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401 [5] Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic & Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269 [6] Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253 [7] Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883 [8] Renjun Duan, Shuangqian Liu, Tong Yang, Huijiang Zhao. Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. Kinetic & Related Models, 2013, 6 (1) : 159-204. doi: 10.3934/krm.2013.6.159 [9] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 [10] Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139 [11] Zhigang Wu, Wenjun Wang. Uniform stability of the Boltzmann equation with an external force near vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 811-823. doi: 10.3934/cpaa.2015.14.811 [12] Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141 [13] Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025 [14] Steffen Arnrich. Modelling phase transitions via Young measures. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29 [15] Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287 [16] Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163 [17] Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565 [18] Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1965-1973. doi: 10.3934/dcds.2013.33.1965 [19] Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997 [20] Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317

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