# American Institute of Mathematical Sciences

December  2011, 4(4): 1063-1079. doi: 10.3934/krm.2011.4.1063

## The Spherical Harmonics Expansion model coupled to the Poisson equation

 1 RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, Linz, A-4040, Austria 2 Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, 10012, United States 3 Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, A-1090, Austria 4 Department of Mathematics, University of Tunis ElManar, Faculty of Sciences of Tunis, 2092 El-Manar, Tunisia

Received  June 2011 Revised  October 2011 Published  November 2011

The Spherical Harmonics Expansion (SHE) assumes a momentum distribution function only depending on the microscopic kinetic energy. The SHE-Poisson system describes carrier transport in semiconductors with self-induced electrostatic potential. Existence of weak solutions to the SHE-Poisson system subject to periodic boundary conditions is established, based on appropriate a priori estimates and a Schauder fixed point procedure. The long time behavior of the one-dimensional Dirichlet problem with well prepared boundary data is studied by an entropy-entropy dissipation method. Strong convergence to equilibrium is proven. In contrast to earlier work, the analysis is carried out without the use of the derivation from a kinetic problem.
Citation: Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic and Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063
##### References:
 [1] N. Ben Abdallah, Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system, Math. Meth. in the Appl. Sci., 17 (1994), 451-476. doi: 10.1002/mma.1670170604. [2] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567. [3] N. Ben Abdallah and J. Dolbeault, Relative entropies for the Vlasov-Poisson system in bounded domains, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 867-872. doi: 10.1016/S0764-4442(00)00268-8. [4] N. Ben Abdallah, P. Degond, P. Markowich and C. Schmeiser, High field approximations of the spherical harmonics expansion model for semiconductors, ZAMP, 52 (2001), 201-230. doi: 10.1007/PL00001544. [5] N. Ben Abdallah and M. L. Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system, DCDS-B, 4 (2004), 1129-1142. doi: 10.3934/dcdsb.2004.4.1129. [6] H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,'' Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [7] D. Chen, E. C. Kan, U. Ravaioli, C. Shu and R. W. Dutton, An improved energy transport model including non-parabolicity and non-Maxwellian distribution effects, IEEE Electron Dev. Lett., 13 (1992), 235-239. [8] C. Cercignani, R. Illner, M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'' Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. [9] P. Degond and S. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), 4634-4663. doi: 10.1063/1.532528. [10] L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998. [11] H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181. doi: 10.1063/1.1706716. [12] H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik" (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294. [13] E. Lyumkis, B. Polsky, A. Shur and P. Visocky, Transient semiconductor device simulation including energy balance equation, COMPEL, 11 (1992), 311-325. [14] P. A. Markowich, F. Popaud and C. Schmeiser, Diffusion approximation of nonlinear electron phonon collision mechanisms, RAIRO Modél. Math. Anal. Num., 29 (1995), 857-869. [15] P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. [16] N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal., 38 (2007), 1788-1807. doi: 10.1137/050630763. [17] M. L. Tayeb, From Boltzmann equation to spherical harmonics expansion model: Diffusion limit and Poisson coupling, Comm. Math. Sci., 9 (2011), 255-275.

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##### References:
 [1] N. Ben Abdallah, Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system, Math. Meth. in the Appl. Sci., 17 (1994), 451-476. doi: 10.1002/mma.1670170604. [2] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567. [3] N. Ben Abdallah and J. Dolbeault, Relative entropies for the Vlasov-Poisson system in bounded domains, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 867-872. doi: 10.1016/S0764-4442(00)00268-8. [4] N. Ben Abdallah, P. Degond, P. Markowich and C. Schmeiser, High field approximations of the spherical harmonics expansion model for semiconductors, ZAMP, 52 (2001), 201-230. doi: 10.1007/PL00001544. [5] N. Ben Abdallah and M. L. Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system, DCDS-B, 4 (2004), 1129-1142. doi: 10.3934/dcdsb.2004.4.1129. [6] H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,'' Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [7] D. Chen, E. C. Kan, U. Ravaioli, C. Shu and R. W. Dutton, An improved energy transport model including non-parabolicity and non-Maxwellian distribution effects, IEEE Electron Dev. Lett., 13 (1992), 235-239. [8] C. Cercignani, R. Illner, M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'' Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. [9] P. Degond and S. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), 4634-4663. doi: 10.1063/1.532528. [10] L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998. [11] H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181. doi: 10.1063/1.1706716. [12] H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik" (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294. [13] E. Lyumkis, B. Polsky, A. Shur and P. Visocky, Transient semiconductor device simulation including energy balance equation, COMPEL, 11 (1992), 311-325. [14] P. A. Markowich, F. Popaud and C. Schmeiser, Diffusion approximation of nonlinear electron phonon collision mechanisms, RAIRO Modél. Math. Anal. Num., 29 (1995), 857-869. [15] P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. [16] N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal., 38 (2007), 1788-1807. doi: 10.1137/050630763. [17] M. L. Tayeb, From Boltzmann equation to spherical harmonics expansion model: Diffusion limit and Poisson coupling, Comm. Math. Sci., 9 (2011), 255-275.
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