No abstract

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the "complete wetting" regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide "minimal" conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.

We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourthorder parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x, x β ) (where x denotes the distance from the contact line) with β = √ 13−1 4 ≈ 0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second -order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L 2 -spaces for the linearized evolution, after suitable subtraction of a(t) + b(t)x β -terms.

We show that every L-periodic, mean-zero solution u of the Kuramoto-Sivashinsky equation is on average o(L) for L>>1, in the sense that for any T > 0 the space average of |u(t)| is bounded by LT for any t > T and any L sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this nonstandard perturbation of the Burgers equation is based on a “div-curl” argument

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 is an element of (3/2, 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 [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