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

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.