Heteroclinic orbits, mobility parameters and stability for thin film type equations

Electronic Journal of Differential Equations

Volumn 2002, Issue , 2002, Pages

  Laugesen, Richard S ^a   Pugh, Mary C ^b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Heteroclinic orbits; Lubrication theory; Nonlinear pde of parabolic type; Stability problems

Indexed keywords

EID: 52449083199     PISSN: 10726691     EISSN: 10726691     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. J. W. Barrett, J. F. Blowey, and H. Garcke. Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer Math, 80(4):525-556, 1998.
2. A. J. Bernoff, A. L. Bertozzi, and T. P. Witelski. Axisymmetric surface diffusion: dynamics and stability of self-similar pinchoff. J. Statist. Phys., 93(3-4):725-776, 1998.
3. A. L. Bertozzi, M. P. Brenner, T. F. Dupont, and L. P. Kadanoff. Singularities and similarities in interface flow. In L. Sirovich, editor, Trends and Perspectives in Applied Mathematics, volume 100 of Applied Mathematical Sciences, pages 155-208. Springer-Verlag, New York, 1994.
4. A. L. Bertozzi and M. C. Pugh. Long-wave instabilities and saturation in thin film equations. Commun Pur Appl Math, 51:625-661, 1998.
5. A. L. Bertozzi and M. C. Pugh. Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J., 49(4):1323-1366, 2000.
6. Wolf-Jürgen Beyn, Alan Champneys, Eusebius Doedel, Willy Govaerts, Yuri A. Kuznetsov, and Björn Sandstede. Numerical continuation, and computation of normal forms. In Handbook of dynamical systems, Vol. 2, pages 149-219. North-Holland, Amsterdam, 2002.
7. J. Bricmont, A. Kupiainen, and J. Taskinen. Stability of Cahn-Hilliard fronts. Commun Pur Appl Math, 52(7):839-871, 1999.
8. E. A. Carlen, M. C. Carvalho, and E. Orlandi. A simple proof of stability of fronts for the Cahn-Hilliard equation. Commun. Math. Phys., 224(1):323-340, 2001.
9. P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, and S.-M. Zhou. Droplet breakup in a model of the Hele-Shaw cell. Phys Rev E, 47(6):4169-4181, June 1993.
10. E. J. Doedel, M. J. Friedman, and B. I. Kunin. Successive continuation for locating connecting orbits. Numer. Algorithms, 14(1-3):103-124, 1997. Dynamical numerical analysis (Atlanta, GA, 1995).
11. Paul C. Fife. Models for phase separation and their mathematics. Electron. J. Differential Equations, pages No. 48, 26 pp. (electronic), 2000. Originally written in June 1991.
12. Paul C. Fife. Pattern formation in gradient systems. In Handbook of dynamical systems, Vol. 2, pages 677-722. North-Holland, Amsterdam, 2002.
13. R. Goldstein, A. Pesci, and M. Shelley. Instabilities and singularities in Hele-Shaw flow. Phys Fluids, 10(11):2701-2723, 1998.
14. G. Grün and M. Rumpf. Simulation of singularities and instabilities arising in thin film flow. Eur J Appl Math, 12(3) 3:293-320, 2001.
15. Günther Grün and Martin Rumpf. Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math., 87(1):113-152, 2000.
16. A. Iserles. A first course in the numerical analysis of differential equations. Cambridge University Press, Cambridge, 1996.
17. R. S. Laugesen and M. C. Pugh. Properties of steady states for thin film equations. Eur J Appl Math, 11(3):293-351, 2000.
18. R. S. Laugesen and M. C. Pugh. Linear stability of steady states for thin film and Cahn-Hilliard type equations. Arch Ration Mech An, 154(1):3-51, 2000.
19. R. S. Laugesen and M. C. Pugh. Energy levels of steady states for thin film type equations. J Diff Eq, 182(2):377-415, 2002.
20. G. J. Lord, A. R. Champneys, and G. W. Hunt. Computation of homoclinic orbits in partial differential equations: an application to cylindrical shell buckling. SI AM J. Sei. Comput., 21(2):591-619 (electronic), 1999.
21. P. Manneville. Dissipative structures and weak turbulence. Academic Press Inc., Boston, MA, 1990.
22. T. G. Myers. Thin films with high surface tension. SIAM Rev., 40(3):441-462 (electronic), 1998.
23. A. Oron and S. G. Bankoff. Dewetting of a heated surface by an evaporating liquid film under conjoining/disjoining pressures. J Colloid Interface Sei, 218(1):152-166, 1999.
24. A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquid films. Rev Mod Phys, 69(3):931-980, Dec 1997.
25. A. Oron and P. Rosenau. Formation of patterns induced by thermocapillarity and gravity. J Phys II, 2:131-146, 1992.
26. F. Otto. Lubrication approximation with prescribed non-zero contact angle: an existence result. Commun Part Diff Eq, 23:2077-2164, 1998.
27. J. E. Taylor and J. W. Cahn. Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Statist. Phys., 77(1-2):183-197, 1994.
28. M. B. Williams and S. H. Davis. Nonlinear theory of film rupture. J Colloid Interf Sei, 90(1):220-228, 1982.
29. Thomas P. Witelski and Andrew J. Bernoff. Stability of self-similar solutions for van der Waals driven thin film rupture. Phys. Fluids, 11(9):2443-2445, 1999.
30. Thomas P. Witelski and Andrew J. Bernoff. Dynamics of three-dimensional thin film rupture. Phys. D, 147(1-2):155-176, 2000.
31. W. W. Zhang and J. R. Lister. Similarity solutions for van der Waals rupture of a thin film on a solid substrate. Phys Fluids, 11(9):2454-2462, 1999.
32. L. Zhornitskaya and A. L. Bertozzi. Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal., 37(2):523-555 (electronic), 2000.