The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness

TL;DR

This paper establishes the existence and uniqueness of classical solutions for a 3D Navier-slip thin-film equation, advancing understanding of free boundary behavior in viscous fluid films with contact lines.

Contribution

It proves well-posedness and regularity for the 3D thin-film equation with Navier-slip, extending prior 1D results to more realistic three-dimensional settings.

Findings

Existence of classical solutions near a traveling-wave profile

Uniqueness of solutions under perturbations

Control over free boundary velocity

Abstract

We consider the thin-film equation $\partial_t h + \nabla \cdot \left(h^2 \nabla \Delta h\right) = 0$ in physical space dimensions (i.e., one dimension in time $t$ and two lateral dimensions with $h$ denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier-Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the $1+1$-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porous-medium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity.