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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

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 & 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.  doi: 10.1023/A:1026481706240.  Google Scholar

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[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.  doi: 10.1088/0951-7715/16/3/316.  Google Scholar

[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.  doi: 10.1007/s00205-008-0190-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00205-008-0184-7.  Google Scholar

[7]

R. Esposito, Y. Guo and R. Marra, Phase transition in a Vlasov-Boltzmann binary mixture,, Commun. Math. Phys., 296 (2010), 1.  doi: 10.1007/s00220-010-1009-8.  Google Scholar

[8]

Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains,, Arch. Rational Mech. Anal., 197 (2010), 713.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[9]

Y. Guo and W. A. Strauss, Unstable oscillatory-tail waves in collisionless plasmas,, SIAM J. Math. Anal., 30 (1999), 1076.  doi: 10.1137/S0036131098333918.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators,, Second edition. Grundlehren der Mathematischen Wissenschaften, (1976).   Google Scholar

[11]

O. Penrose, Electrostatic instability of a non-Maxwellian plasma,, Phys. Fluids., 3 (1960), 258.  doi: 10.1063/1.1706024.  Google Scholar

[12]

E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics,, Theoretical and Mathematical Physics. Springer, (2009).   Google Scholar

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.  doi: 10.1023/A:1026481706240.  Google Scholar

[2]

S. Bastea and J. L. Lebowitz, Spinodal decomposition in binary gases,, Phys. Rev. Lett., 78 (1997), 3499.  doi: 10.1103/PhysRevLett.78.3499.  Google Scholar

[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.  doi: 10.1088/0951-7715/16/3/316.  Google Scholar

[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.  doi: 10.1007/s00205-008-0190-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00205-008-0184-7.  Google Scholar

[7]

R. Esposito, Y. Guo and R. Marra, Phase transition in a Vlasov-Boltzmann binary mixture,, Commun. Math. Phys., 296 (2010), 1.  doi: 10.1007/s00220-010-1009-8.  Google Scholar

[8]

Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains,, Arch. Rational Mech. Anal., 197 (2010), 713.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[9]

Y. Guo and W. A. Strauss, Unstable oscillatory-tail waves in collisionless plasmas,, SIAM J. Math. Anal., 30 (1999), 1076.  doi: 10.1137/S0036131098333918.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators,, Second edition. Grundlehren der Mathematischen Wissenschaften, (1976).   Google Scholar

[11]

O. Penrose, Electrostatic instability of a non-Maxwellian plasma,, Phys. Fluids., 3 (1960), 258.  doi: 10.1063/1.1706024.  Google Scholar

[12]

E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics,, Theoretical and Mathematical Physics. Springer, (2009).   Google Scholar

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