Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems

SIAM Journal on Mathematical Analysis

Volumn 32, Issue 4, 2000, Pages 760-777

  Jüngel, Ansgar ^a   Pinnau, René ^b  

^a UNIVERSITY OF KONSTANZ   (Germany)

^b DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Entropy; Existence; Global solution; Higher order parabolic PDE; Positivity; Uniqueness

Indexed keywords

EID: 0034353705     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141099360269     Document Type: Article

References (23)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, 1st ed., New York, 1975.
2. M. G. ANCONA AND G. J. IAFRATE, Quantum correction of the equation of state of an electron gas in a semiconductor, Phys. Rev. B, 39 (1989), pp. 9536-9540.
3. A. L. BERTOZZI, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), pp. 689-697.
4. F. BERNIS AND A. FRIEDMAN, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), pp. 179-206.
5. F. BREZZI, I. GASSER, P. A. MARKOWICH, AND CH. SCHMEISER, Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension, Appl. Math. Lett., 8 (1995), pp. 47-52.
6. P. M. BLEHER, J. L. LEBOWITZ, AND E. R. SPEER, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Comm. Pure Appl. Math., 47 (1994), pp. 923-942.
7. A. L. BERTOZZI AND M. C. PUGH, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), pp. 625-661.
8. B. DERRIDA, J. L. LEBOWITZ, E. SPEER, AND H. SPOHN, Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett., 67 (1991), pp. 165-168.
9. R. DAL PASSO, H. GARCKE, AND G. GRÜN, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), pp. 321-342.
10. C. L. GARDNER, The quantum, hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), pp. 409-427.
11. I. GAMBA AND A. JUNGEL, Positive solutions to singular second and third order differential equations for quantum fluids, Arch. Rational Mech. Anal., to appear.
12. M. T. GYI AND A. JUNGEL, A quantum regularization of the one-dimensional hydrodynamic model for semiconductors, Adv. Differential Equations, 5 (2000), pp. 773-800.
13. [Grü95] G. GRÜN, Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening, Z. Anal. Anwendungen, 14 (1995), pp. 541-574.
14. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 1st ed., Springer-Verlag, Berlin, 1983.
15. A. JÜNGEL AND R. PINNAU, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM J. Numer. Anal, submitted.
16. [Jün97] A. JÜNGEL, A note on current-voltage characteristics from the quantum hydrodynamic equations for semiconductors, Appl. Math. Lett., 10 (1997), pp. 29-34.
17. [Jün98] A. JÜNGEL, A steady-state potential flow Euler-Poisson system for charged quantum fluids, Comm. Math. Phys., 194 (1998), pp. 463-479.
18. R. PINNAU, The linearized transient quantum drift diffusion model - stability of stationary states, ZAMM Z. Angew. Math. Mech., 80 (2000), pp. 327-344.
19. R. PINNAU, A note on boundary conditions for quantum hydrodynamic models, Appl. Math. Lett., 12 (1999), pp. 77-82.
20. R. PINNAU AND A. UNTERREITER, The stationary current-voltage characteristics of the quantum drift-diffusion model, SIAM J. Numer. Anal., 37 (1999), pp. 211-245.
21. K. REKTORYS, The Method of Discretization in Time and Partial Differential Equations, 1st ed., Dordrecht, Boston, 1982.
22. p(0,T;B), Ann. Math. Pura Appl. (4), 146 (1987), pp. 65-96.
23. E. ZEIDLER, Nonlinear Functional Analysis and its Applications, 1st ed., Volume II/A and II/B, Springer-Verlag, Berlin, 1990.