American Institute of Mathematical Sciences

Advanced
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

[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

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

[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

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences