The Dirichlet problem of the quantum drift-diffusion model

Nonlinear Analysis, Theory, Methods and Applications

Volumn 69, Issue 9, 2008, Pages 3084-3092

  Chen, Xiuqing ^a   Chen, Li ^b   Jian, Huaiyu ^b  

^a BEIJING UNIVERSITY OF POSTS AND TELECOMMUNICATIONS   (China)

^b TSINGHUA UNIVERSITY   (China)

Author keywords

Fourth order parabolic system; Quantum drift diffusion model; Weak solution

Indexed keywords

BOUNDARY CONDITIONS;

APPROXIMATE METHODS; DIRICHLET BOUNDARY CONDITION; DIRICHLET PROBLEM; ENTROPY ESTIMATE; PARABOLIC SYSTEM; QUANTUM DRIFT-DIFFUSION MODELS; SEMIDISCRETIZATION; WEAK SOLUTION;

DIFFUSION;

EID: 77951490051     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2007.09.003     Document Type: Article

References (20)

1. N.B. Abdallah, A. Unterreiter, On the stationary quantum drift diffusion model, Z. Angew. Math. Phys. 49 (2) (1998) 251-275.
2. P.M. Bleher, J.L. Lebowitz, E.R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Comm. Pure Appl. Math. 47 (1994) 923-942.
3. M.J. C'aceres, J.A. Carrillo, G. Toscani, Long-time behaviour for a nonlinear fourth-order parabolic equation, Trans. Amer. Math. Soc. 357 (3) (2004) 1161-1175.
4. L. Chen, M. Dreher, The viscous model of quantum hydrodynamics in several dimensions, preprint.
5. Q.C. Ju, L. Chen, Semiclassical limit for bipolar quantum drift-diffusion model (submitted for publication).
6. L. Chen, Q.C. Ju, Existence of weak solution and semiclassical limit for quantum drift-diffusion model, Z. Angew. Math. Phys. 58 (1) (2007) 1-15.
7. L. Chen, Q.C. Ju, The semiclassical limit in the quantum drift-diffusion equations with isentropic pressure (submitted for publication).
8. X.Q. Chen, L. Chen, H.Y. Jian, Existence, semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model (submitted for publication).
9. J. Dolbeault, I. Gentil, A. Jüngel, A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities, 2004, preprint.
10. P. Degond, F. M'ehats, C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys. 118 (2005) 625-665.
11. M. Gualdani, A. Jüngel, G. Toscani, A nonlinear fourth-order parabolic equation with non-homogeneous boundary conditions, 2004, preprint.
12. A. Jüngel, Nonlinear problems on quantum semiconductor modeling, Nonlinear Anal. 47 (2001) 5873-5884.
13. A. Jüngel, Transport equation for semiconductors, in: Preliminary Lecture Notes, 2005.
14. A. Jüngel, H.L. Li, Quantum Euler-Poisson systems: Global existence and exponential decay, Quart. Appl. Math., 2004, preprint.
15. A. Jüngel, R. Pinnau, Global non-negative solutions of a nonlinear fourth order parabolic equation for quantum systems, SIAM J. Math. Anal. 32 (4) (2000) 760-777.
16. A. Jüngel, R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system, SIAM J. Numer. Anal. 39 (2) (2001) 385-406.
17. A. Jüngel, R. Pinnau, Convergent semidiscretization of a nonlinear fourth order parabolic system, Math. Modelling Numer. Anal. 37 (2) (2003) 277-289.
18. A. Jüngel, G. Toscani, Exponential decay in time of solutions to a nonlinear fourth-order parabolic equation, Z. Angew. Math. Phys. 54 (2003) 377-386.
19. R. Pinnau, A review on the quantum drift diffusion model, Transport Theory Statist. Phys. 31 (4-6) (2002) 367-395.
20. J. Simon, Compact sets in the space L p(0, T; B), Ann. Mat. Pura Appl. 146 (1987) 65-96.