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

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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

Raffaele Esposito ^1, , Yan Guo ^2, and Rossana Marra ^3,

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

Abstract
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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: Vlasov-Boltzmann equation, Phase transitions, stability..

Mathematics Subject Classification: Primary: 82B26, 82C40; Secondary: 76P0.

Citation: Raffaele Esposito, Yan Guo, Rossana Marra. Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval. Kinetic and Related Models, 2013, 6 (4) : 761-787. doi: 10.3934/krm.2013.6.761

References:

[1]	S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction I: Derivation of kinetic and hydrodynamic equation, J. Statist. Phys, 101 (2000), 1087-1136. doi: 10.1023/A:1026481706240.
[2]	S. Bastea and J. L. Lebowitz, Spinodal decomposition in binary gases, Phys. Rev. Lett., 78 (1997), 3499-3502. doi: 10.1103/PhysRevLett.78.3499.
[3]	E. A. Carlen, M. Carvalho, R. Esposito, J. L. Lebowitz and R. Marra, Free energy minimizers for a two-species model with segregation and liquid-vapor transition, Nonlinearity, 16 (2003), 1075-1105. doi: 10.1088/0951-7715/16/3/316.
[4]	E. A. Carlen, M. C. Carvalho, R. Esposito, J. L. Lebowitz and R. Marra, Displacement convexity and minimal fronts at phase boundaries, Arch. Rat. Mech. Anal., 194 (2009), 823-847. doi: 10.1007/s00205-008-0190-9.
[5]	A. De Masi, E. Olivieri and E. Presutti, Spectral properties of integral operators in problems of interface dynamics and metastability, Markov Processes and Related Fields, 4 (1998), 27-112.
[6]	R. Esposito , Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics, Arch. Rational Mech. Anal., 195 (2010), 75-116. doi: 10.1007/s00205-008-0184-7.
[7]	R. Esposito, Y. Guo and R. Marra, Phase transition in a Vlasov-Boltzmann binary mixture, Commun. Math. Phys., 296 (2010), 1-33. doi: 10.1007/s00220-010-1009-8.
[8]	Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.
[9]	Y. Guo and W. A. Strauss, Unstable oscillatory-tail waves in collisionless plasmas, SIAM J. Math. Anal., 30 (1999), 1076-1114 (electronic). doi: 10.1137/S0036131098333918.
[10]	T. Kato, Perturbation Theory for Linear Operators, Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976.
[11]	O. Penrose, Electrostatic instability of a non-Maxwellian plasma, Phys. Fluids., 3 (1960), 258-265. doi: 10.1063/1.1706024.
[12]	E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics, Theoretical and Mathematical Physics. Springer, Berlin, 2009.

show all references

References:

[1]	S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction I: Derivation of kinetic and hydrodynamic equation, J. Statist. Phys, 101 (2000), 1087-1136. doi: 10.1023/A:1026481706240.
[2]	S. Bastea and J. L. Lebowitz, Spinodal decomposition in binary gases, Phys. Rev. Lett., 78 (1997), 3499-3502. doi: 10.1103/PhysRevLett.78.3499.
[3]	E. A. Carlen, M. Carvalho, R. Esposito, J. L. Lebowitz and R. Marra, Free energy minimizers for a two-species model with segregation and liquid-vapor transition, Nonlinearity, 16 (2003), 1075-1105. doi: 10.1088/0951-7715/16/3/316.
[4]	E. A. Carlen, M. C. Carvalho, R. Esposito, J. L. Lebowitz and R. Marra, Displacement convexity and minimal fronts at phase boundaries, Arch. Rat. Mech. Anal., 194 (2009), 823-847. doi: 10.1007/s00205-008-0190-9.
[5]	A. De Masi, E. Olivieri and E. Presutti, Spectral properties of integral operators in problems of interface dynamics and metastability, Markov Processes and Related Fields, 4 (1998), 27-112.
[6]	R. Esposito , Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics, Arch. Rational Mech. Anal., 195 (2010), 75-116. doi: 10.1007/s00205-008-0184-7.
[7]	R. Esposito, Y. Guo and R. Marra, Phase transition in a Vlasov-Boltzmann binary mixture, Commun. Math. Phys., 296 (2010), 1-33. doi: 10.1007/s00220-010-1009-8.
[8]	Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.
[9]	Y. Guo and W. A. Strauss, Unstable oscillatory-tail waves in collisionless plasmas, SIAM J. Math. Anal., 30 (1999), 1076-1114 (electronic). doi: 10.1137/S0036131098333918.
[10]	T. Kato, Perturbation Theory for Linear Operators, Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976.
[11]	O. Penrose, Electrostatic instability of a non-Maxwellian plasma, Phys. Fluids., 3 (1960), 258-265. doi: 10.1063/1.1706024.
[12]	E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics, Theoretical and Mathematical Physics. Springer, Berlin, 2009.

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