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

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.