Doubly nonlinear thin-film equations in one space dimension

Archive for Rational Mechanics and Analysis

Volumn 173, Issue 1, 2004, Pages 89-131

  Ansini, Lidia ^a   Giacomelli, Lorenzo ^a  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 3142757012     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00205-004-0313-x     Document Type: Article

References (48)

1. ALMGREN, R.: Singularity formation in Hele-Shaw bubbles. Phys. Fluids 8, 344-352 (1996)
2. ANDREUCCI, D., TEDEEV, A.F.: Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity. Interfaces Free Bound. 3, 233-264 (2001)
3. ANSINI, L., GIACOMELLI, L.: Shear-thinning liquid films: Macroscopic and asymptotic behavior by quasi self-similar solutions. Nonlinearity 15, 2147-2164 (2002)
4. BERETTA, E., BERTSCH, M, DAL PASSO, R.: Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal. 129, 175-200 (1995)
5. BERNIS, F.: Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. Math. Ann. 279, 373-394 (1988)
6. BERNIS, F., FRIEDMAN, A.: Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83, 179-206 (1990)
7. BERNIS, F.: Integral inequalities with applications to nonlinear degenerate parabolic equations. In: ANGELL, T.S., COOK, L.P., KLEINMAN, R.E. OLMSTEAD, W.E. Nonlinear Boundary Value Problems. SIAM, Philadelphia, 1996
8. BERNIS, F.: Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differential Equations 1, 337-368 (1996)
9. BERNIS, F.: Finite speed of propagation for thin viscous flows when 2 ≤ n < 3. C.R. Acad. Sci. Paris, Sér. I Math. 322, 1169-1174 (1996)
10. BERTOZZI, A.L., PUGH, M.: The lubrication approximation for thin viscous films: The moving contact line with a "porous media" cut-off of van der Waals interactions. Nonlinearity 7, 1535-1564 (1994)
11. BERTOZZI, A.L., PUGH, M.: The lubrication approximation for thin viscous films: regularity and long time behavior of weak solutions. Comm. Pure Appl. Math. 49, 85-123 (1996)
12. BERTOZZI, A.L., PUGH, M.: Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51, 625-661 (1998)
13. BERTSCH, M., DAL PASSO, R., GARCKE, H., GRÜN, G.: The thin viscous flow equation in higher space dimension. Adv. Differential Equations 3, 417-440 (1998)
14. BETELÚ, S.I., FONTELOS, M.A.: Capillarity driven spreading of power-law fluids. Appl. Math. Lett. 16, 1315-1320 (2003)
15. BIRD, R.B., ARMSTRONG, R.C., HASSAGER, O.: Dynamics of polymeric liquids. John Wiley and Sons, 1977
16. CARRÉ, A., EUSTACHE, F.: Spreading kinetics of shear-thinning fluids in wetting and dewetting models. Langmuir 16, 2936-2941 (2000)
17. DAL PASSO, R., GARCKE, H.: Solutions of a fourth-order degenerate parabolic equation with weak initial trace Ann. Sc. Norm. Sup. Pisa Cl. Sci. 28, 153-181 (1999)
18. DAL PASSO, R., GARCKE, H., GRÜN, G.: On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29, 321-342 (1998)
19. DAL PASSO, R., GIACOMELLI, L., GRÜN, G.: A waiting time phenomenon for thin film equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30, 437-463 (2001)
20. DAL PASSO, R., GIACOMELLI, L., GRÜN, G.: Waiting time phenomena for degenerate parabolic equations - a unifying approach. In Geometric Analysis and Nonlinear Partial Differential Equations, S. HILDEBRANDT & H. KARCHER, Eds., Springer-Verlag, Berlin Heidelberg New York, 2003, pp. 637-648
21. DAL PASSO, R., GIACOMELLI, L., SHISHKOV, A.E.: The thin film equation with nonlinear diffusion. Comm. Partial Differential Equations 26, 1509-1557 (2001)
22. DE GENNES, P.G.: Wetting: Statics and dynamics. Rev. Mod. Phys. 57, 827-863 (1985)
23. DIBENEDETTO, E.: Degenerate parabolic equations. Springer-Verlag, New York, 1993
24. DUSSAN V., E.B., DAVIS, S.H.: On the motion of fluid-fluid interface along a solid surface. J. Fl. Mech. 65, 71-95 (1974)
25. GAGLIARDO, E.: Ulteriori proprietà di alcune classi di funzioni in piúvariabili. Ricerche di Mat. Napoli, 8, 24-51 (1959)
26. GIACOMELLI, L.: A fourth order degenerate parabolic equation describing thin viscous flows over an inclined plane. Appl. Math. Lett. 12, 107-111 (1999)
27. GIACOMELLI, L., OTTO, F.: Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Partial Differential Equations 13, 377-403 (2001)
28. GIACOMELLI, L., OTTO, F.: Droplet spreading: Intermediate scaling law by PDE methods. Comm. Pure Appl. Math. 55, 217-254 (2002)
29. GIACOMELLI, L., OTTO, R: Rigorous lubrication approximation. Interfaces Free Bound. 5, 483-529 (2003)
30. GRÜN, G.: Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening. Z. Anal. Anwendungen 14, 541-574 (1995)
31. GRÜN, G.: On free boundary problems arising in thin film flow. Habilitation thesis, University of Bonn, 2001
32. GRÜN, G.: On Bernis' interpolation inequalities in multiple space dimensions. Z. Anal. Anwendungen 20, 987-998 (2001)
33. GRÜN, G.: Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case. Interfaces Free Bound. 4, 309-323 (2002)
34. GRÜN, G.: Droplet spreading under weak slippage: the waiting time phenomenon. Ann. I.H.P., Analyse non lineaire, To appear.
35. HUH, C., SCRIVEN, L.E.: Hydrodynamic model of a steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85-101 (1971)
36. 1-norm. Adv. Differential Equations 3, 625-642 (1998)
37. KING, J.R.: Two generalisations of the thin film equation. Mathematical and Computer Modelling 34, 737-756 (2001)
38. KING, J.R.: Thin-film flows and higher-order degenerate parabolic equations. IUTAM Symposium on Free Surface Flows, A.C. KING & Y.D. SHIKHMURZAEV (eds), Kluwer Academic Publishers, 2001, pp. 7-18
39. KING, J.R.: The spreading of power-law fluids. IUTAM Symposium on Free Surface Flows, A.C. KING & Y.D. SHIKHMURZAEV (eds.), Kluwer Academic Publishers, 2001, pp.153-160
40. NIRENBERG, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115-162 (1959)
41. OPIC, B., KUFNER, A.: Hardy-type inequalities. Pitman Res. Notes 219, Longman, 1990
42. ORON, A., DAVIS, S.H., BANKOFF, G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931-980 (1997)
43. OTTO, F.: Lubrication approximation with prescribed non-zero contact angle. Comm. Partial Differential Equations 23, 2077-2164 (1998)
44. SHISHKOV, A.E.: Estimates of rate of propagation of perturbations in quasilinear divergent degenerate parabolic equations of high order. Ukrain. Math. J. 44, 1335-1340 (1992)
45. SHISHKOV, A.E., SHCHELKOV, A.G.: Dynamics of the supports of energy solutions of mixed problems for quasilinear parabolic equations of arbitrary order. Izv. Math. 62, 601-626 (1998)
46. P(0, T; B).Annali Mat. Pura Appl. 146, 65-97 (1987)
47. STAMPACCHIA, G.: Équations elliptiques du second ordre à coefficients discontinus. Les presses de l'Université de Montréal, Montréal, 1966
48. WEIDNER, D.E., SCHWARTZ, L.W.: Contact-line motion of shear-thinning liquids. Phys. Fluids 6. 3535-3538 (1994)