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September  2016, 9(3): 571-585. doi: 10.3934/krm.2016007

A kinetic reaction model: Decay to equilibrium and macroscopic limit

1. 

Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria

2. 

Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received  March 2015 Revised  September 2015 Published  May 2016

We propose a kinetic relaxation-model to describe a generation-recombination reaction of two species. The decay to equilibrium is studied by two recent methods [9,13] for proving hypocoercivity of the linearized equations. Exponential decay of small perturbations can be shown for the full nonlinear problem. The macroscopic/fast-reaction limit is derived rigorously employing entropy decay, resulting in a nonlinear diffusion equation for the difference of the position densities.
Citation: Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic & Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007
References:
[1]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 124 (2006), 881-912. doi: 10.1007/s10955-005-8075-x.  Google Scholar

[2]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135. doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar

[3]

J. Carrillo, L. Desvillettes and K. Fellner, Fast-reaction limit for the inhomogeneous Aizenman-Bak model, Kinetic and Related Models, 1 (2008), 127-137. doi: 10.3934/krm.2008.1.127.  Google Scholar

[4]

J. Carrillo, L. Desvillettes and K. Fellner, Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion, Comm. Part. Diff. Eq., 34 (2009), 1338-1351. doi: 10.1080/03605300903225396.  Google Scholar

[5]

I. Choquet, P. Degond and C. Schmeiser, Energy-transport models for charge carriers involving impact ionization in semiconductors, Transport Theory and Statistical Physics, 32 (2003), 99-132.  Google Scholar

[6]

P. Degond, A. Nouri and C. Schmeiser, Macroscopic models for the ionization in the presence of strong electric fields, Transport Theory and Stat. Phys., 29 (2000), 551-561. doi: 10.1080/00411450008205891.  Google Scholar

[7]

L. Desvillettes and A. Trescases, New results for triangular reaction cross diffusion system, J. Math. Anal. Appl., 430 (2015), 32-59, arXiv:1408.5814. doi: 10.1016/j.jmaa.2015.03.078.  Google Scholar

[8]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C.R. Acad. Sci. Paris, 347 (2009), 511-516. doi: 10.1016/j.crma.2009.02.025.  Google Scholar

[9]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS, 367 (2015), 3807-3828. doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[10]

F. Golse, From kinetic to macroscopic models, in Kinetic Equations and Asymptotic Theory, (eds. B. Perthame and L. Desvillettes), Series in Appl. Math. 4, Gauthier, Villars, (2000), 41-126. Google Scholar

[11]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.  Google Scholar

[12]

D. Hilhorst, R. van der Hout and L. A. Peletier, Nonlinear diffusion in the presence of fast reaction, Nonlinear Anal.: Theory, Meth. & Appl., 41 (2000), 803-823. doi: 10.1016/S0362-546X(98)00311-3.  Google Scholar

[13]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.  Google Scholar

[14]

J. Polewczak, The kinetic theory of simple reacting spheres: I. Global existence result in a dilute-gas case, J. Stat. Phys., 100 (2000), 327-362. doi: 10.1023/A:1018608216136.  Google Scholar

[15]

F. Poupaud and C. Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Meth. in the Appl. Sci., 14 (1991), 301-318. doi: 10.1002/mma.1670140503.  Google Scholar

[16]

C. Villani, Hypocoercivity, Memoirs of the AMS 950, 2009. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

show all references

References:
[1]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 124 (2006), 881-912. doi: 10.1007/s10955-005-8075-x.  Google Scholar

[2]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135. doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar

[3]

J. Carrillo, L. Desvillettes and K. Fellner, Fast-reaction limit for the inhomogeneous Aizenman-Bak model, Kinetic and Related Models, 1 (2008), 127-137. doi: 10.3934/krm.2008.1.127.  Google Scholar

[4]

J. Carrillo, L. Desvillettes and K. Fellner, Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion, Comm. Part. Diff. Eq., 34 (2009), 1338-1351. doi: 10.1080/03605300903225396.  Google Scholar

[5]

I. Choquet, P. Degond and C. Schmeiser, Energy-transport models for charge carriers involving impact ionization in semiconductors, Transport Theory and Statistical Physics, 32 (2003), 99-132.  Google Scholar

[6]

P. Degond, A. Nouri and C. Schmeiser, Macroscopic models for the ionization in the presence of strong electric fields, Transport Theory and Stat. Phys., 29 (2000), 551-561. doi: 10.1080/00411450008205891.  Google Scholar

[7]

L. Desvillettes and A. Trescases, New results for triangular reaction cross diffusion system, J. Math. Anal. Appl., 430 (2015), 32-59, arXiv:1408.5814. doi: 10.1016/j.jmaa.2015.03.078.  Google Scholar

[8]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C.R. Acad. Sci. Paris, 347 (2009), 511-516. doi: 10.1016/j.crma.2009.02.025.  Google Scholar

[9]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS, 367 (2015), 3807-3828. doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[10]

F. Golse, From kinetic to macroscopic models, in Kinetic Equations and Asymptotic Theory, (eds. B. Perthame and L. Desvillettes), Series in Appl. Math. 4, Gauthier, Villars, (2000), 41-126. Google Scholar

[11]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.  Google Scholar

[12]

D. Hilhorst, R. van der Hout and L. A. Peletier, Nonlinear diffusion in the presence of fast reaction, Nonlinear Anal.: Theory, Meth. & Appl., 41 (2000), 803-823. doi: 10.1016/S0362-546X(98)00311-3.  Google Scholar

[13]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.  Google Scholar

[14]

J. Polewczak, The kinetic theory of simple reacting spheres: I. Global existence result in a dilute-gas case, J. Stat. Phys., 100 (2000), 327-362. doi: 10.1023/A:1018608216136.  Google Scholar

[15]

F. Poupaud and C. Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Meth. in the Appl. Sci., 14 (1991), 301-318. doi: 10.1002/mma.1670140503.  Google Scholar

[16]

C. Villani, Hypocoercivity, Memoirs of the AMS 950, 2009. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

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