Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage

Archive for Rational Mechanics and Analysis

Volumn 211, Issue 3, 2014, Pages 771-818

  Fischer, Julian ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84895901236     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-013-0690-0     Document Type: Article

References (43)

1. Alikakos N. D.: On the pointwise behaviour of the solutions of the porous medium equation as t approaches zero or infinity. Nonlinear Anal. 9, 1085-1113 (1985).
2. Ansini L., Giacomelli L.: Doubly nonlinear thin-film equations in one space dimension. Arch. Rational Mech. Anal. 173, 89-131 (2004).
3. Aronson D., Caffarelli L.: The initial trace of a solution of the porous medium equations. Trans. Am. Math. Soc. 280, 351-366 (1983).
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.: Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differ. Equ. 1(3), 337-368 (1996).
6. Bernis, F.: Finite speed of propagation for thin viscous flows when 2 ≦ n < 3. C. R. Math. Acad. Sci. Paris322(12), 1169-1174 (1996).
7. Bernis F., Friedman A.: order nonlinear degenerate parabolic equations. J. Differ. Equ. 83, 179-206 (1990).
8. Bertozzi A., Pugh M.: Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J. 49(4), 1323-1366 (2000).
9. Bertozzi, A. L., Pugh, M. C.: The lubrication approximation for thin viscous films: the moving contact line with a 'porous media' cut-off of van der Waals interactions. Nonlinearity7, 1535-1564 (1994).
10. Bertozzi, A. L., Pugh, M. C.: Long-wave instabilities and saturation in thin film equations. Commun. Pure Appl. Math. 51, 0625-0661 (1998).
11. Bertsch M., Dal Passo R., Garcke H., Grün G.: The thin viscous flow equation in higher space dimensions. Adv. Differ. Equ. 3, 417- (1998).
12. Bertsch M., Giacomelli L., Karali G.: Thin-film equations with partial wetting energy: existence of weak solutions. Phys. D 209(1-4), 17-27 (2005).
13. Blowey J. F., King J. R., Langdon S.: Small- and waiting-time behaviour of the thin-film equation. SIAM J. Appl. Math. 67, 1776-1807 (2007).
14. Carrillo, J., Toscani, G.: Long-time asymptotics for strong solutions of the thin-film equation. Commun. Math. Phys. 225, 551-571 (2002).
15. Chipot M., Sideris T.: An upper bound for the waiting time for nonlinear degenerate parabolic equations. Trans. Am. Math. Soc. 288(1), 423-427 (1985).
16. Choi S., Kim I.: Waiting time phenomena for Hele-Shaw and Stefan problems. Indiana Univ. Math. J. 55(2), 525-552 (2006).
17. Dal Passo, R., Garcke, H.: Solutions of a fourth order degenerate parabolic equation with weak initial trace. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) tome 28(1), 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(2), 321-342 (1998).
19. Dal Passo R., Giacomelli L., Grün G. A waiting time phenomenon for thin film equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)tome 30(2): 437-463 (2001).
20. Djie, K. C.: An upper bound for the waiting time for doubly nonlinear parabolic equations. Interfaces Free Bound. 9, 95-105 (2007).
21. Fischer, J.: Infinite speed of propagation for the Derrida-Lebowitz-Speer-Spohn equation. NoDEA Nonlinear Differ. Equ. Appl. (2013). doi: 10. 1007/s00030-013-0235-0.
22. Fischer J.: Optimal lower bounds on asymptotic support propagation rates for the thin-film equation. J. Differ. Equ. 255(10), 3127-3149 (2013).
23. Giacomelli L., Grün G.: Lower bounds on waiting times for degenerate parabolic equations and systems. Interfaces Free Bound. 8, 111-129 (2006).
24. Giacomelli L., Knüpfer H.: A free boundary problem of fourth order: classical solutions in weighted Hölder spaces. Commun. Partial Differ. Equ. 35(11), 2059-2091 (2010).
25. Giacomelli L., Knüpfer H., Otto F.: Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ. 245, 1454-1506 (2008).
26. Giacomelli L., Otto F. (2001) Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Partial Differ. Equ. 13(3): 377-403.
27. Giacomelli L., Otto F.: Droplet spreading: Intermediate scaling law by PDE methods. Commun. Pure Appl. Math. 55(2), 217-254 (2002).
28. Giacomelli L., Otto F.: Rigorous lubrication approximation. Interfaces Free Bound. 5(4), 483-529 (2003).
29. Giacomelli L., Shishkov A.: Propagation of support in one-dimensional convected thin-film flow. Indiana Univ. Math. J. 54(4), 1181-1215 (2005).
30. Greenspan H.: On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125-143 (1978).
31. Grün G.: Degenerate parabolic partial differential equations of fourth order and a plasticity model with nonlocal hardening. Z. Anal. Anwend. 14, 541-573 (1995).
32. Grün G.: Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case. Interfaces Free Bound. 4(3), 309-323 (2002).
33. Grün, G.: Droplet spreading under weak slippage: existence for the Cauchy problem. Commun. Partial Differ. Equ. 29(11-12), 1697-1744 (2004).
34. 1-norm. Adv. Differ. Equ. 3(5), 625-642 (1998).
35. Jäger, W., Mikelic, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96-122 (2001).
36. Knüpfer H.: Well-posedness for the Navier slip thin-film equation in the case of partial wetting. Commun. Pure Appl. Math. 64(9), 1263-1296 (2011).
37. Novick-Cohen A., Shishkov A. E.: The thin film equation with backward second order diffusion. Interfaces Free Bound. 12, 463-496 (2010).
38. Oron A., Davis S. H., Bankoff S. G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 932-977 (1997).
39. Otto F.: Lubrication approximation with prescribed nonzero contact angle. Commun. Partial Differ. Equ. 23(11-12), 2077-2164 (1998).
40. Shishkov A. E.: Estimates for the rate of propagations in quasilinear degenerate higher-order parabolic equations in divergence form. Ukrain. Math. Zh. 44, 1335-1340 (1992).
41. Shishkov A. E.: Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Discrete Contin. Dyn. Syst., 2007(suppl), 938-945 (2007).
42. Shishkov A. E., Shchelkov A.: Dynamics of the supports of energy solutions of mixed problems for quasi-linear parabolic equations of arbitrary order. Izvestiya Math. 62(3), 601-626 (1998).
43. p(0, T; B). Ann. Mat. Pura Appl. 146(1), 65-96 (1986).