×

Diffusion limit of a Boltzmann-Poisson system with nonlinear equilibrium state. (English) Zbl 1428.35263

Summary: The diffusion approximation for a Boltzmann-Poisson system is studied. Nonlinear relaxation type collision operator is considered. A relative entropy is used to prove useful \(L^2\)-estimates for the weak solutions of the scaled Boltzmann equation (coupled to Poisson) and to prove the convergence of the solution toward the solution of a nonlinear diffusion equation coupled to Poisson. In one dimension, a hybrid Hilbert expansion and the contraction property of the operator allow to exhibit a convergence rate.

MSC:

35Q20 Boltzmann equations
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
35D30 Weak solutions to PDEs
35B50 Maximum principles in context of PDEs
78A30 Electro- and magnetostatics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexandre, R., Weak solutions of the Vlasov-Poisson initial-boundary value problem, Math. Methods Appl. Sci.16(8) (1993) 587-607. · Zbl 0786.35014
[2] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. de l’Ec. Norm. Sup. 4e série (3) (1970) 185-233. · Zbl 0202.36903
[3] Ben Abdallah, N., Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system, Math. Methods Appl. Sci.17 (1994) 451-476. · Zbl 0806.35172
[4] Ben Abdallah, N. and Degond, P., On a hierarchy of macroscopic models for semiconductors, J. Math. Phys.37 (1996) 3306-3333. · Zbl 0868.45006
[5] Ben Abdallah, N. and Dolbeault, J., Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness), Arch. Ration. Mech. Anal.168(4) (2003) 253-298. · Zbl 1044.76054
[6] Ben Abdallah, N. and Tayeb, M.-L., Diffusion limit for the one dimensional Boltzmann-Poisson system, Discrete Contin. Dyn. Syst. Ser. B4(4) (2004) 1129-1142. · Zbl 1062.76047
[7] Benachour, S., Analyticité des solutions des équations de Vlasov-Poisson, Ann. Scuola Norm. Sup Pisa Cl. Sci. 4e série16(1) (1989) 83-104. · Zbl 0702.35042
[8] Bensoussan, A., Lions, J. L. and Papanicolaou, G., Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci.15 (1979) 53-157. · Zbl 0408.60100
[9] Berthelin, F., Mauser, N. and Poupaud, F., High field limit from a kinetic equation to multidimensional scalar conservation laws, J. Hyperbolic Differential Equations4(1) (2007) 123-145. · Zbl 1116.35085
[10] Bouchut, F., Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal.111(1) (1993) 239-258. · Zbl 0777.35059
[11] Chavanis, P. H., Laurencot, P. and Lemou, M., Chapman-Enskog derivation of the generalized Smoluchowski equation, Physica A341 (2004) 145-164.
[12] Degond, P. and Jüngel, A., High-field approximations of the energy-transport model for semiconductors with non-parabolic band structure, ZAMP52 (2001) 1053-1070. · Zbl 0991.35043
[13] Degond, P. and Schmeiser, C., Macroscopic models for semiconductor heterostructures, J. Math. Phys.9(39) (1998) 4634-4663. · Zbl 0938.82049
[14] Devore, R. and Petrova, G., The averaging lemma, J. Amer. Math. Soc.14(2) (2000) 279-296. · Zbl 1001.35079
[15] Diperna, R. J. and Lions, P.-L., Global weak solution of Vlasov-Maxwell systems, Comm. Pure Appl. Math.XVII (1989) 729-757. · Zbl 0698.35128
[16] Diperna, R. J., Lions, P.-L. and Meyer, Y., \(L^p\) regularity of velocity averages, Ann. Inst. H. Poincaré Ann. Nonlinéaire8 (1991) 271-281. · Zbl 0763.35014
[17] Dolbeault, J., Markowich, P., Oelz, D. and Schmeiser, C., Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels, Arch. Ration. Mech. Anal.186 (2007) 133-158. · Zbl 1148.76047
[18] Gerard, P. and Golse, F., Averaging regularity results for PDEs under transversality assumptions, Comm. Pure Appl. Math.45 (1992) 1-26. · Zbl 0832.35020
[19] Golse, F. and Poupaud, F., Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asympt. Anal.6 (1992) 135-169. · Zbl 0784.35084
[20] Golse, F. and Saint Raymond, L., Velocity averaging in \(L^1\) for transport equation, C. R. Acad. Sci. Paris, Ser. I334 (2002) 557-562. · Zbl 1154.35326
[21] Haskovek, J., Masmoudi, N., Schmeiser, C. and Tayeb, M. L., The spherical harmonics expansion model coupled to the poisson equation, Kinetic Relat. Models4(4) (2011) 1063-1079. · Zbl 1347.35135
[22] Levermore, C. D. and Masmoudi, N., From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal.196(3) (2010) 753-809. · Zbl 1304.35476
[23] Markowich, P. A., Poupaud, F. and Schmeiser, C., Diffusion approximation of nonlinear electron-phonon scattering mechanisms, RAIRO Models Math. Anal. Numer.29 (1995) 857-869. · Zbl 0840.45011
[24] Masmoudi, N. and Tayeb, M.-L., Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal.38 (2007) 1788-1807. · Zbl 1206.82133
[25] Mellet, A. and Perthane, B., \(L^1\) contraction property for a Boltzmann equation with Pauli statistics, C. R. Math. Acad. Sci. Paris335(4) (2002) 337-340. · Zbl 1003.35119
[26] Mock, M. S., Analysis of Mathematical Models of Semiconductor Devices, , Vol. 3 (Boole Press, 1983). · Zbl 0532.65081
[27] Neunzert, H., Pulvirenti, M. and Triolo, L., On the Vlasov-Fokker-Planck equation, Math. Methods Appl. Sci.6(4) (1984) 527-538. · Zbl 0561.35070
[28] Poupaud, F., On a system of non-linear Boltzmann equations of semiconductor physics, SIAM J. Math. Anal.38 (1990) 1593-1606. · Zbl 0724.35104
[29] Poupaud, F., Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asympt. Anal.4(4) (1991) 293-317. · Zbl 0762.35092
[30] Poupaud, F., Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech.72(8) (1992) 359-372. · Zbl 0785.76067
[31] Ringhofer, C., Schmeiser, C. and Zwirchnayr, A., Moments methods for the semiconductor Boltzmann equation on bounded domains, SIAM J. Math. Anal.39(3) (2001) 1078-1095. · Zbl 0994.82080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.