A thin film model for meniscus evolution

In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of fluid, i.e. a fluid contained between two solid surfaces and part of the fluid surface is in contact with the air. The fluid is governed by Navier-Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the fluid thickness (lubrication approximation) and the contact angle between the fluid-solid and the fluid-gas interfaces is close to $\pi$. This resulting model is a free boundary problem for the equation $h_t + (h^3h_{xxx})_x = 0$, for which we have $h>0$ at the contact point (different from the usual thin film equation with $h=0$ at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Also the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from no-slip condition. Furthermore, we show the global stability of steady state solutions in a periodic setting.