Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation

Previous Article
KRM Home
This Issue
Next Article
Identification of photon sources, stochastically embedded in an interstellar cloud

September 2009, 2(3): 403-424. doi: 10.3934/krm.2009.2.403

Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation

Naoufel Ben Abdallah ^1, and Hédia Chaker ^2,

1.	Institut de Mathématiques de Toulouse, Université de Toulouse and CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 9, France
2.	Laboratoire de Modélisation et Simulation Numérique dans les Sciences de l'INgénieurs, Ecole Nationale d'Ingénieurs de Tunis, BP 37, Campus Universitaire, Le Belvédère, 1002, Tunis, Tunisia

Received June 2009 Revised June 2009 Published July 2009

Abstract
Related Papers

In a previous work [J. of Hyperbolic Diff. Eq. 4, pp. 679-704 (2007)], the high field asymptotics of the fermionic Boltzmann equation has been proven to lead to a nonlinear conservation law for the particle density. Under symmetry conditions on the collission cross section, the nonlinear limiting flux is parallel to the force field. In the present work, we investigate the orthogonal direction to the electric field, and prove after a suitable rescaling that the behaviour is governed by the original conservation law with an additional nonlinear diffusion in the orthogonal (to the force field) direction. The main tool used in the convergence proof is a new estimate for the dissipation of the family of entropies introduced in the above cited work. While the entropy dissipation is usually estimated by quantities of the type dist $(f, F_{eq})$ representing the distance of the distribution function to the equilibrium set, the new estimate involves a quantity of the form $\int$ dist $(f, F_{eq}(\u)) d\mu(\u)$, where the macroscopic equilibria depend on a velocity variable $\u$ and $\mu$ is a probability measure. This estimate allows to control high velocities, pass to the limit in the diffusion current and shows the convergence to the entropy solution of the limiting equation.

Keywords: High field, entropy inequality, entropy solution., diffusion equation.

Mathematics Subject Classification: Primary: 35B25, 35B40, 82C40, 82C70, Secondary: 82D1.

Citation: Naoufel Ben Abdallah, Hédia Chaker. Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation. Kinetic & Related Models, 2009, 2 (3) : 403-424. doi: 10.3934/krm.2009.2.403

[1]	Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[2]	Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[3]	Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304
[4]	Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
[5]	Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[6]	Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237
[7]	Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[8]	Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701
[9]	Liming Ling. The algebraic representation for high order solution of Sasa-Satsuma equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1975-2010. doi: 10.3934/dcdss.2016081
[10]	Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic & Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479
[11]	Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783
[12]	N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[13]	José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301
[14]	Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226
[15]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[16]	Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395
[17]	Nicolas Bedaride. Entropy of polyhedral billiard. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89
[18]	Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
[19]	Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237
[20]	Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555

2017 Impact Factor: 1.219

American Institute of Mathematical Sciences