A gradient flow scheme for nonlinear fourth order equations

Discrete and Continuous Dynamical Systems - Series B

Volumn 14, Issue 3, 2010, Pages 935-959

  Düring, Bertram ^a   Matthes, Daniel ^a   Milišić, Josipa Pina ^b  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b UNIVERSITY OF ZAGREB   (Croatia)

Author keywords

Higher order diffusion equation; Numerical solution; Wasserstein gradient flow

Indexed keywords

EID: 77957762117     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2010.14.935     Document Type: Article

References (32)

1. L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Birkhäuser, Basel, 2005.
2. L. Ambrosio and G. Savaré, Gradient flows of probability measures, Handbook of Differential Equations: Evolutionary Equations. Vol. III, 1-136, Handb. Differ. Equ., Elsevier/NorthHolland, Amsterdam, 2007.
3. J. W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation, Numer. Math., 80 (1998), 525-556.
4. J. Becker, G. Grún, M. Lenz and M. Rumpf, Numerical methods for fourth order nonlinear degenerate diffusion problems, Mathematical theory in fluid mechanics (Paseky, 2001). Appl. Math., 47 (2002), 517-543.
5. F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206.
6. A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. (Pubitemid 126353359)
7. A. L. Bertozzi and L. Zhornitskaya, Positivity-preserving numerical schemes for lubricationtype equations, SIAM J. Numer. Anal., 37 (2000), 523-555.
8. P. Bleher, J. Lebowitz and E. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., 47 (1994), 923-942.
9. M. Cáceres, J. A. Carrillo and G. Toscani, Long-time behavior for a nonlinear fourth-order parabolic equation, Trans. Amer. Math. Soc., 357 (2005), 1161-1175.
10. J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), 75-198.
11. J. W. Cahn and J. E. Hilliard, Spinodal decomposition: A reprise, Acta Metallurgica, 19 (1971), 151-161.
12. J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal., 179 (2006), 217-263.
13. J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009/10), 4305-4329.
14. J. A. Carrillo, A. Júngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation, Discrete Contin. Dynam. Sys. B, 3 (2003), 1-20.
15. R. Dal Passo, H. Garcke and G. Grún, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.
16. P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.
17. B. Derrida, J. Lebowitz, E. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett., 67 (1991), 165-168.
18. C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
19. U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.
20. M. P. Gualdani, A. Júngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions, SIAM J. Math. Anal., 37 (2006), 1761-1779.
21. R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
22. A. Júngel, "Transport Equations for Semiconductors," Lecture Notes in Physics No. 773. Springer, Berlin, 2009.
23. A. Júngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659.
24. A. Júngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 39 (2008), 1996-2015.
25. A. Júngel and D.Matthes, A review on results for the Derrida-Lebowitz-Speer-Spohn equation, To appear in Proceedings of the Conference Equadiff 2007, 2010.
26. A. Júngel and J. P. Milisic, A sixth-order nonlinear parabolic equation for quantum systems, SIAM J. Math. Anal., 41 (2009), 1472-1490.
27. A. Júngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM J. Math. Anal., 32 (2000), 760-777.
28. A. Júngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourthorder parabolic equation, SIAM J. Numer. Anal., 39 (2001), 385-406.
29. A. Júngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 861-877.
30. R. S. Laugesen, New dissipated energies for the thin fluid film equation, Commun. Pure Appl. Anal., 4 (2005), 613-634.
31. F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
32. C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.