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MiS Preprint
23/2012

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

Lorenzo Giacomelli, Manuel Gnann and Felix Otto

Abstract

In one space dimension, we consider source-type (self-similar) solutions to the thin-film equation with vanishing slope at the edge of its support (zero contact-angle condition) in the range of mobility exponents n between 3/2 and 3. 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 [F. Bernis, L.A. Peletier & S. M. Williams, Nonlinear Anal. 18 (1992), 217-234]. There, it is also shown 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, even for n = 2) power of it, which naturally emerges from a linearization 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 behavior of the travelling wave is factored off. We expect the same singular behavior for a generic solution to the thin-film equation near its moving contact line. As a consequence, we expect a (short-time) well-posedness theory for classical solutions - of which this paper is a natural prerequisite - to be more involved than in the case n = 1.

Received:
Apr 11, 2012
Published:
Apr 12, 2012
MSC Codes:
35C06, 35K65, 35B65, 34B16, 34C45
Keywords:
self-similar solutions, degenerate parabolic equations, Smoothness and regularity of solutions, Singular nonlinear boundary value problems, Invariant manifolds

Related publications

inJournal
2013 Repository Open Access
Lorenzo Giacomelli, Manuel V. Gnann and Felix Otto

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

In: European journal of applied mathematics, 24 (2013) 5, pp. 735-760