On the regularity for the Navier-slip thin-film equation in the perfect wetting regime

TL;DR

This paper demonstrates that the degenerate-parabolic thin-film equation with Navier-slip boundary conditions exhibits a strong regularizing effect, improving the smoothness of solutions without extra initial data assumptions.

Contribution

It significantly advances understanding of solution regularity for the Navier-slip thin-film equation by removing previous compatibility constraints.

Findings

Solutions are regular to arbitrary orders of the singular expansion.

The regularizing effect holds without additional initial data assumptions.

The results align with properties of the source-type self-similar profile.

Abstract

We investigate perturbations of traveling wave solutions to a thin-film equation with quadratic mobility and a zero contact angle at the triple junction, where the three phases liquid, gas, and solid meet. This equation can be obtained in lubrication approximation from the Navier-Stokes system of a liquid droplet with a Navier-slip condition at the substrate. Existence and uniqueness have been established by the author together with Giacomelli, Kn\"upfer, and Otto in previous work. As solutions are generically non-smooth, the approach relied on suitably subtracting the leading-order singular expansion at the free boundary. In the present note, we substantially improve this result by showing the regularizing effect of the degenerate-parabolic equation to arbitrary orders of the singular expansion. In comparison to related previous work, our method does not require additional compatibility assumptions on the initial data. The result turns out to be natural in view of the properties of the source-type self-similar profile.