Regularity for the Navier-Slip Thin-Film Equation in the perfect wetting regime

We investigate perturbations of traveling wave solutions to a thin-film equation with quadratic mobility and a zero contact angle condition at the triple junction between the three phases liquid, gas, and solid. This equation can be derived from the Navier Stokes system of a liquid droplet with a Navier-slip condition at the liquid-solid interface. Existence and uniqueness have been established joint with Giacomelli, Knüpfer, and Otto. As solutions are generically non-smooth, the approach relied on suitably subtracting the leading- order singular expansion at the free boundary. In this talk, we present a substantial improvement of this result by showing the regularizing effect of the degenerate-parabolic equation to arbitrary orders of the singular expansion. This result turns out to be natural in view of the properties of the source-type self-similar profile.