American Institute of Mathematical Sciences

Advanced
June  2009, 2(2): 333-343. doi: 10.3934/krm.2009.2.333

$\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases

Gilberto M. Kremer 1, , Filipe Oliveira 2, and  Ana Jacinta Soares 2,

1. 

Departamento de Física, Universidade Federal do Paraná, Curitiba, Brazil
2. 

Centro de Matemática e Aplicações, Universidade Nova de Lisboa, Lisboa, Portugal; Departamento de Matemática, Universidade do Minho, Braga, Portugal, Portugal

Received  September 2008 Revised  February 2009 Published  May 2009

The trend to equilibrium of a quaternary mixture of monatomic gases undergoing a reversible reaction of bimolecular type is studied in a quite rigorous mathematical picture within the framework of Boltzmann equation extended to chemically reacting mixtures of gases. The $\mathcal H$-theorem and entropy inequality allow to prove two main results under the assumption of uniformly boundedness and equicontinuity of the distribution functions. One of the results establishes the tendency of a reacting mixture to evolve to an equilibrium state as time becomes large. The other states that the solution of the Boltzmann equation for chemically reacting mixtures of gases converges in strong $L^1$-sense to its equilibrium solution.
Keywords: Chemically reactive gases, Trend to equilibrium, H-Theorem., Boltzmann equation.
Mathematics Subject Classification: Primary: 76P05, 80A32, 82C40; Secondary: 35B3.
Citation: Gilberto M. Kremer, Filipe Oliveira, Ana Jacinta Soares. $\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases. Kinetic & Related Models, 2009, 2 (2) : 333-343. doi: 10.3934/krm.2009.2.333
[1]

Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic & Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020
[2]

Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493
[3]

Marzia Bisi, Laurent Desvillettes. Some remarks about the scaling of systems of reactive Boltzmann equations. Kinetic & Related Models, 2008, 1 (4) : 515-520. doi: 10.3934/krm.2008.1.515
[4]

Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6669-6688. doi: 10.3934/dcds.2016090
[5]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145
[6]

Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014
[7]

Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure & Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003
[8]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039
[9]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205
[10]

Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic & Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499
[11]

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
[12]

Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237
[13]

Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic & Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551
[14]

Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579
[15]

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
[16]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221
[17]

Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[18]

Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405
[19]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939
[20]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences