Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage

We derive upper bounds on the waiting time of solutions to the thin-film equation in the regime of weak slippage ${n\in 2,\frac{32}{11})}$ . In particular, we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary. For ${n\in (2,\frac{32}{11})}$ , our estimates are sharp, for n = 2, they are sharp up to a logarithmic correction term. Note that the case n = 2 corresponds—with a grain of salt—to the assumption of the Navier slip condition at the fluid-solid interface. We also obtain results in the regime of strong slippage ${n \in (1,2)}$ ; however, in this regime we expect them not to be optimal. Our method is based on weighted backward entropy estimates, Hardy’s inequality and singular weight functions; we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long.