From a multidimensional quantum hydrodynamic model to the classical drift-diffusion equation

Li, Yeping

doi:10.1090/S0033-569X-09-01156-7

Author: Yeping Li
Journal: Quart. Appl. Math. 67 (2009), 489-502
MSC (2000): Primary 35B25, 35M20, 35L45
DOI: https://doi.org/10.1090/S0033-569X-09-01156-7
Published electronically: May 6, 2009
MathSciNet review: 2547643
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the paper, we discuss the combined semiclassical and relaxation-time limits of a multidimensional isentropic quantum hydrodynamical model for semiconductors with small momentum relaxation time and Planck constant. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohn potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the Planck constant and the relaxation time tend to zero, periodic initial-value problems of a scaled isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion models have smooth solutions.

References

Giuseppe Alì and Ansgar Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations 190 (2003), no. 2, 663–685. MR 1970046, DOI https://doi.org/10.1016/S0022-0396%2802%2900157-2

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in Quantum Fluid Dynamics, to appear in Comm. Math. Phys., 2008.

F. Brezzi, I. Gasser, P. A. Markowich, and C. Schmeiser, Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension, Appl. Math. Lett. 8 (1995), no. 1, 47–52. MR 1355150, DOI https://doi.org/10.1016/0893-9659%2894%2900109-P

Naoufel Ben Abdallah and Andreas Unterreiter, On the stationary quantum drift-diffusion model, Z. Angew. Math. Phys. 49 (1998), no. 2, 251–275. MR 1629183, DOI https://doi.org/10.1007/s000330050218

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Statist. Phys. 112 (2003), no. 3-4, 587–628. MR 1997263, DOI https://doi.org/10.1023/A%3A1023824008525

Carl L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54 (1994), no. 2, 409–427. MR 1265234, DOI https://doi.org/10.1137/S0036139992240425

Ingenuin Gasser, Peter A. Markowich, and Christian A. Ringhofer, Closure conditions for classical and quantum moment hierarchies in the small-temperature limit, Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), 1996, pp. 409–423. MR 1407543, DOI https://doi.org/10.1080/00411459608220710

Ingenuin Gasser and Peter A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal. 14 (1997), no. 2, 97–116. MR 1451208

Irene M. Gamba and Ansgar Jüngel, Positive solutions to singular second and third order differential equations for quantum fluids, Arch. Ration. Mech. Anal. 156 (2001), no. 3, 183–203. MR 1816474, DOI https://doi.org/10.1007/s002050000114

Ingenuin Gasser and Roberto Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math. 57 (1999), no. 2, 269–282. MR 1686190, DOI https://doi.org/10.1090/qam/1686190

Feimin Huang, Hai-Liang Li, and Akitaka Matsumura, Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors, J. Differential Equations 225 (2006), no. 1, 1–25. MR 2228690, DOI https://doi.org/10.1016/j.jde.2006.02.002

Ansgar Jüngel, Hai-Liang Li, and Akitaka Matsumura, The relaxation-time limit in the quantum hydrodynamic equations for semiconductors, J. Differential Equations 225 (2006), no. 2, 440–464. MR 2225796, DOI https://doi.org/10.1016/j.jde.2005.11.007

Ansgar Jüngel, Quasi-hydrodynamic semiconductor equations, Progress in Nonlinear Differential Equations and their Applications, vol. 41, Birkhäuser Verlag, Basel, 2001. MR 1818867

Ansgar Jüngel and Hailiang Li, Quantum Euler-Poisson systems: existence of stationary states, Arch. Math. (Brno) 40 (2004), no. 4, 435–456. MR 2129964

Ansgar Jüngel and Hailiang Li, Quantum Euler-Poisson systems: global existence and exponential decay, Quart. Appl. Math. 62 (2004), no. 3, 569–600. MR 2086047, DOI https://doi.org/10.1090/qam/2086047

Tao Luo, Roberto Natalini, and Zhouping Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math. 59 (1999), no. 3, 810–830. MR 1661255, DOI https://doi.org/10.1137/S0036139996312168

Corrado Lattanzio and Pierangelo Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors, Discrete Contin. Dynam. Systems 5 (1999), no. 2, 449–455. MR 1665756, DOI https://doi.org/10.3934/dcds.1999.5.449

Corrado Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci. 10 (2000), no. 3, 351–360. MR 1753116, DOI https://doi.org/10.1142/S0218202500000215

Hailiang Li and Pierangelo Marcati, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, Comm. Math. Phys. 245 (2004), no. 2, 215–247. MR 2039696, DOI https://doi.org/10.1007/s00220-003-1001-7

H.-L. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson, Electronic J. Differential Equations, 2003(2003), 1-17.

Yeping Li, Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors, Math. Methods Appl. Sci. 30 (2007), no. 17, 2247–2261. MR 2363951, DOI https://doi.org/10.1002/mma.890

Pierangelo Marcati and Roberto Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal. 129 (1995), no. 2, 129–145. MR 1328473, DOI https://doi.org/10.1007/BF00379918

P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852

Andreas Unterreiter, The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model, Comm. Math. Phys. 188 (1997), no. 1, 69–88. MR 1471332, DOI https://doi.org/10.1007/s002200050157

Wen-An Yong, Basic aspects of hyperbolic relaxation systems, Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser Boston, Boston, MA, 2001, pp. 259–305. MR 1842777

Wen-An Yong, Diffusive relaxation limit of multidimensional isentropic hydrodynamical models for semiconductors, SIAM J. Appl. Math. 64 (2004), no. 5, 1737–1748. MR 2084208, DOI https://doi.org/10.1137/S0036139903427404

Bo Zhang and Joseph W. Jerome, On a steady-state quantum hydrodynamic model for semiconductors, Nonlinear Anal. 26 (1996), no. 4, 845–856. MR 1362757, DOI https://doi.org/10.1016/0362-546X%2894%2900326-D

Chen Zhu and Harumi Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations 166 (2000), no. 1, 1–32. MR 1779253, DOI https://doi.org/10.1006/jdeq.2000.3799