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
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Submission date: 11. Apr. 2012 (revised version: October 2012)
MSC-Numbers: 35C06, 35K65, 35B65, 34B16, 34C45
Keywords and phrases: self-similar solutions, degenerate parabolic equations, Smoothness and regularity of solutions, Singular nonlinear boundary value problems, Invariant manifolds
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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.