June  2021, 14(3): 541-570. doi: 10.3934/krm.2021015

Macroscopic limit of the kinetic Bloch equation

1. 

Léonard de Vinci Pôle Universitaire, Research Center, 92 916 Paris La Défense Cedex, France

2. 

Laboratoire AMNEDP, Faculté de Mathématiques, Université USTHB, B. P. 32 ElAlia, Bab Ezzouar, 16111 Alger, Algérie

* Corresponding author: Djamila Hamroun

Received  October 2020 Revised  February 2021 Published  June 2021 Early access  March 2021

This work concerns the existence of solution of the kinetic spinor Boltzmann equation as well as the asymptotic behavior of such solution when $ \varepsilon \to 0 $, that is when the time relaxation of the spin-flip collisions is very small in comparison to the time relaxation parameter of the collisions with no spin reversal. Due to the lack of regularity of the weak solution, the switching term $ H_\varepsilon\times M_\varepsilon $ is not stable under the weak convergences. Hence we establish new estimates of the solutions in a weighted Sobolev space of order 3.

Citation: Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic and Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015
References:
[1]

L. Arkeryd, A kinetic equation for spin polarized Fermi systems, Kinet. Relat. Models, 7 (2014), 1-8.  doi: 10.3934/krm.2014.7.1.

[2]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.

[3]

R. El Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation, Commun. Math. Sci., 12 (2014), 565-592.  doi: 10.4310/CMS.2014.v12.n3.a9.

[4]

L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/cbms/074.

[5]

G. D. Gaspari, Bloch equation for conduction-electron spin resonance, Phys. Rev., 131 (1966), 215-219.  doi: 10.1103/PhysRev.151.215.

[6]

K. Hamdache and L. Tartar, The appearance of memory effects for a conservative system, Nonlinear Anal., 177 (2018), 532-542.  doi: 10.1016/j.na.2018.04.017.

[7]

A. JüngelC. Negulescu and P. Shpartko, Bounded weak solutions to a matrix drift-diffusion for spin-coherent electron transport in semiconductors, Math. Models Methods Appl. Sci., 25 (2015), 929-958.  doi: 10.1142/S0218202515500232.

[8]

S. Possanner and C. Negulescu, Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport, Kinet. Relat. Models, 4 (2011), 1159-1191.  doi: 10.3934/krm.2011.4.1159.

[9]

Y. Qi and S. Zhang, Spin diffusion at finite electric and magnetic field, Phys. Rev. B, 67 (2003). doi: 10.1103/PhysRevB.67.052407.

[10]

H. C. Torrey, Bloch equation with diffusion term, Phys. Rev., 104 (1956), 563-565.  doi: 10.1103/PhysRev.104.563.

show all references

References:
[1]

L. Arkeryd, A kinetic equation for spin polarized Fermi systems, Kinet. Relat. Models, 7 (2014), 1-8.  doi: 10.3934/krm.2014.7.1.

[2]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.

[3]

R. El Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation, Commun. Math. Sci., 12 (2014), 565-592.  doi: 10.4310/CMS.2014.v12.n3.a9.

[4]

L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/cbms/074.

[5]

G. D. Gaspari, Bloch equation for conduction-electron spin resonance, Phys. Rev., 131 (1966), 215-219.  doi: 10.1103/PhysRev.151.215.

[6]

K. Hamdache and L. Tartar, The appearance of memory effects for a conservative system, Nonlinear Anal., 177 (2018), 532-542.  doi: 10.1016/j.na.2018.04.017.

[7]

A. JüngelC. Negulescu and P. Shpartko, Bounded weak solutions to a matrix drift-diffusion for spin-coherent electron transport in semiconductors, Math. Models Methods Appl. Sci., 25 (2015), 929-958.  doi: 10.1142/S0218202515500232.

[8]

S. Possanner and C. Negulescu, Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport, Kinet. Relat. Models, 4 (2011), 1159-1191.  doi: 10.3934/krm.2011.4.1159.

[9]

Y. Qi and S. Zhang, Spin diffusion at finite electric and magnetic field, Phys. Rev. B, 67 (2003). doi: 10.1103/PhysRevB.67.052407.

[10]

H. C. Torrey, Bloch equation with diffusion term, Phys. Rev., 104 (1956), 563-565.  doi: 10.1103/PhysRev.104.563.

[1]

Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic and Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036

[2]

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

[3]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic and Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443

[4]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[5]

Giada Basile, Tomasz Komorowski, Stefano Olla. Diffusion limit for a kinetic equation with a thermostatted interface. Kinetic and Related Models, 2019, 12 (5) : 1185-1196. doi: 10.3934/krm.2019045

[6]

Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895

[7]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[8]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[9]

Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic and Related Models, 2017, 10 (3) : 541-551. doi: 10.3934/krm.2017021

[10]

Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic and Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019

[11]

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

[12]

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

[13]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793

[14]

Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230

[15]

Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic and Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007

[16]

Kazuki Himoto, Hideaki Matsunaga. The limits of solutions of a linear delay integral equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3033-3048. doi: 10.3934/dcdsb.2020050

[17]

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

[18]

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

[19]

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

[20]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (237)
  • HTML views (162)
  • Cited by (0)

Other articles
by authors

[Back to Top]