Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3

European Journal of Applied Mathematics

Volumn 24, Issue 5, 2013, Pages 735-760

  Giacomelli, Lorenzo ^a   Gnann, Manuel V ^b   Otto, Felix ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

Degenerate parabolic equations; Invariant manifolds; Key words: Self similar solutions; Singular nonlinear boundary value problems; Smoothness and regularity of solutions

Indexed keywords

DEGENERATE PARABOLIC EQUATION; INVARIANT MANIFOLDS; NONLINEAR BOUNDARY VALUE PROBLEMS; REGULARITY OF SOLUTIONS; SELF-SIMILAR SOLUTION;

POROUS MATERIALS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 84883187117     PISSN: 09567925     EISSN: 14694425     Source Type: Journal    
DOI: 10.1017/S0956792513000156     Document Type: Article

References (23)

1. Angenent, S. (1988) Local existence and regularity for a class of degenerate parabolic equations. Math. Ann. 280(3), 465-482.
2. Beretta, E., Bertsch, M. & Dal Passo, R. (1995) Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Anal. 129(2), 175-200.
3. Bernis, F., Hulshof, J. & King, John R. (2000) Dipoles and similarity solutions of the thin film equation in the half-line. Nonlinearity 13(2), 413-439.
4. Bernis, F., Peletier, L. A. & Williams, S. M. (1992) Source-type solutions of a fourth-order nonlinear degenerate parabolic equation. Nonlinear Anal. 18(3), 217-234.
5. Bertozzi, A. L. & Pugh, M. (1996) The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions. Comm. Pure Appl. Math. 49(2), 85-123.
6. Bowen, M. & King, J. R. (2001) Asymptotic behaviour of the thin film equation in bounded domains. Eur. J. Appl. Math. 12(2), 135-157.
7. Chiricotto, M. & Giacomelli, L. (2011) Droplets spreading with contact-line friction: Lubrication approximation and traveling wave solutions. Commun. Appl. Ind. Math. 2(2), e-388 (1-16).
8. Coddington, Earl A. & Levinson, N. (1955) Theory of Ordinary Differential Equations, McGraw-Hill, New York.
9. Dal Passo, R., Garcke, H. & Gr?un. G. (1998) On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29(2), 321-342 (electronic).
10. Daskalopoulos, P. & Hamilton, R. (1998) Regularity of the free boundary for the porous medium equation. J. Am. Math. Soc. 11(4), 899-965.
11. de Gennes, P. G. (Jul. 1985) Wetting: Statics and dynamics. Rev. Mod. Phys. 57, 827-863.
12. Evans, J.D., Galaktionov, V. A. & King, J.R. (2007) Source-type solutions of the fourth-order unstable thin film equation. Eur. J. Appl. Math. 18(3), 273-321.
13. Ferreira, R. & Bernis, F. (1997) Source-type solutions to thin-film equations in higher dimensions. Eur. J. Appl. Math. 8(5), 507-524.
14. Giacomelli, L. & Kn?upfer, H. (2010) A free boundary problem of fourth order: Classical solutions in weighted H?older spaces. Comm. Partial Differ. Equ. 35(11), 2059-2091.
15. Giacomelli, L., Kn?upfer, H. & Otto, F. (2008) Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ. 245(6), 1454-1506.
16. Giacomelli, L. & Otto, F. (2003) Rigorous lubrication approximation. Interfaces Free Bound. 5(4), 483-529.
17. Gr?un, G. (2004) Droplet spreading under weak slippage - existence for the Cauchy problem. Comm. Partial Differ. Equ. 29(11-12), 1697-1744.
18. Hartman, P. (2002) Ordinary Differential Equations, Vol. 38, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. (Corrected reprint of the second (1982) ed. [Birkh?auser, Boston, MA; MR0658490 (83e:34002)], with a foreword by Peter Bates).
19. Kn?upfer, H. (2011) Well-posedness for the Navier slip thin-film equation in the case of partial wetting. Comm. Pure Appl. Math. 64(9), 1263-1296.
20. Koch, H. (1999) Non-Euclidean Singular Intergrals and the Porous Medium Equation, Habilitation thesis, Universit?at Heidelberg, Heidelberg, Germany.
21. Oron, A., Davis, Stephen H. & Bankoff, S. G. (Jul. 1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931-980.
22. Smyth, N. F. & Hill, J. M. (1988) High-order nonlinear diffusion. IMA J. Appl. Math. 40(2), 73-86.
23. Vá zquez, J. L. (2007) The Porous Medium Equation, Oxford Mathematical Monographs on Mathematical Theory. Oxford University Press (The Clarendon Press), Oxford, UK.