The semiclassical limit in the quantum drift-diffusion equations with isentropic pressure

Abstract

The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditions. It is shown that the semiclassical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.

Keywords

Quantum drift-diffusion / Weak solution / Semiclassical limit / Isentropic

References

[1] Adams R.. Sobolev Spaces, 1975, New York: Academic Press

[2] Ben Abdallah N., Unterreiter A.. On the stationary quantum drift diffusion model. Z. Angew. Math. Phys., 1998, 49: 251-275

[3] Brennan K.. The Physics of Semiconductors, 1999, Singapore: World Scientific

[4] Chen L., Ju Q. C.. Existence of weak solution and semiclassical limit for quantum drift-diffusion model. Z. Angew. Math. Phys., 2007, 58: 1-15

[5] Chen, X. Q., Two models including transport equation in meteorlogy and semiconductors, Dissertation, Tsinghua University, submitted.

[6] Ju, Q. C. and Chen, L., Semiclassical limit for bipolar quantum drift-diffusion model, Acta Math. Sci., to appear.

[7] Jüngel A.. Nonlinear problems on quantum semiconductor modeling. Nonlin. Anal., 2001, 47: 5873-5884

[8] Jüngel A.. Quasi-hydrodynamic Semiconductor Equations, 2001, Basel: Birkhäuser

[9] Jüngel A., Pinnau R.. A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system. SIAM J. Num. Anal., 2001, 39(2): 385-406

[10] Jüngel A., Pinnau R.. Convergent semidiscretization of a nonlinear fourth order parabolic system. Math. Mod. Num. Anal., 2003, 37: 277-289

[11] Jüngel A., Violet I.. The quasineutral limit in the quantum drift-diffusion equation. Asymp. Anal., 2007, 53: 139-157

[12] Markowich P., Ringhofer C., Schmeiser C.. Semiconductor Equations, 1990, Vienna: Springer

[13] Pinnau R.. A Review on the Quantum Drift-Diffusion Model. Fachbereich Mathematik, 2001, Darmstadt: Darmstadt Techn. Univ.

[14] Sze S. M.. Semiconductor Devices, Physics and Technology, 2002 2 New York: John Wiley & Sons

[15] Unterreiter A.. The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model. Comm. Math. Phys., 1997, 188: 69-88