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

  • LORENZO GIACOMELLI (a1), MANUEL V. GNANN (a2) and FELIX OTTO (a2)

Abstract

In one space dimension, we consider source-type (self-similar) solutions to the thin-film equation with vanishing slope at the edge of their support (zero contact-angle condition) in the range of mobility exponents $n\in\left(\frac 3 2,3\right)$ . This range contains the physically relevant case n=2 (Navier slip). The existence and (up to a spatial scaling) uniqueness of these solutions has been established in [3] (Bernis, F., Peletier, L. A. & Williams, S. M. (1992) Source type solutions of a fourth-order nonlinear degenerate parabolic equation. Nonlinear Anal. 18, 217–234). It is also shown there that the leading-order expansion near the edge of the support coincides with that of a travelling-wave solution. In this paper we substantially sharpen this result, proving that the higher order correction is analytic with respect to two variables: the first one is just the spatial variable whereas the second one is a (generically irrational, in particular for n=2) power of it, which naturally emerges from a linearisation of the operator around the travelling-wave solution. This result shows that – as opposed to the case of n=1 (Darcy) or to the case of the porous medium equation (the second-order analogue of the thin-film equation) – in this range of mobility exponents, source-type solutions are not smooth at the edge of their support even when the behaviour of the travelling wave is factored off. We expect the same singular behaviour for a generic solution to the thin-film equation near its moving contact line. As a consequence, we expect a (short-time or small-data) well-posedness theory – of which this paper is a natural prerequisite – to be more involved than in the case n=1.

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[1]Angenent, S. (1988) Local existence and regularity for a class of degenerate parabolic equations. Math. Ann. 280 (3), 465482.
[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), 175200.
[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), 413439.
[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), 217234.
[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), 85123.
[6]Bowen, M. & King, J. R. (2001) Asymptotic behaviour of the thin film equation in bounded domains. Eur. J. Appl. Math. 12 (2), 135157.
[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. & Grün, G. (1998) On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29 (2), 321342 (electronic).
[10]Daskalopoulos, P. & Hamilton, R. (1998) Regularity of the free boundary for the porous medium equation. J. Am. Math. Soc. 11 (4), 899965.
[11]de Gennes, P. G. (Jul. 1985) Wetting: Statics and dynamics. Rev. Mod. Phys. 57, 827863.
[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), 273321.
[13]Ferreira, R. & Bernis, F. (1997) Source-type solutions to thin-film equations in higher dimensions. Eur. J. Appl. Math. 8 (5), 507524.
[14]Giacomelli, L. & Knüpfer, H. (2010) A free boundary problem of fourth order: Classical solutions in weighted Hölder spaces. Comm. Partial Differ. Equ. 35 (11), 20592091.
[15]Giacomelli, L., Knüpfer, H. & Otto, F. (2008) Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ. 245 (6), 14541506.
[16]Giacomelli, L. & Otto, F. (2003) Rigorous lubrication approximation. Interfaces Free Bound. 5 (4), 483529.
[17]Grün, G. (2004) Droplet spreading under weak slippage–-existence for the Cauchy problem. Comm. Partial Differ. Equ. 29 (11–12), 16971744.
[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. [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], with a foreword by Peter Bates).
[19]Knüpfer, H. (2011) Well-posedness for the Navier slip thin-film equation in the case of partial wetting. Comm. Pure Appl. Math. 64 (9), 12631296.
[20]Koch, H. (1999) Non-Euclidean Singular Intergrals and the Porous Medium Equation, Habilitation thesis, Universität 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, 931980.
[22]Smyth, N. F. & Hill, J. M. (1988) High-order nonlinear diffusion. IMA J. Appl. Math. 40 (2), 73–86.
[23]Vázquez, J. L. (2007) The Porous Medium Equation, Oxford Mathematical Monographs on Mathematical Theory. Oxford University Press (The Clarendon Press), Oxford, UK.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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