Behaviour of free boundaries in thin-film flow: The regime of strong slippage and the regime of very weak slippage

Abstract

We analyze the behaviour of free boundaries in thin-film flow in the regime of strong slippage $n\in [1,2)$ and in the regime of very weak slippage $n\in [\frac{32}{11},3)$ qualitatively and quantitatively. In the regime of strong slippage, we construct initial data which are bounded from above by the steady state but for which nevertheless instantaneous forward motion of the free boundary occurs. This shows that the initial behaviour of the free boundary is not determined just by the growth of the initial data at the free boundary. Note that this is a new phenomenon for degenerate parabolic equations which is specific for higher-order equations. Furthermore, this result resolves a controversy in the literature over optimality of sufficient conditions for the occurrence of a waiting time phenomenon. In contrast, in the regime of very weak slippage we derive lower bounds on free boundary propagation which are optimal in the sense that they coincide up to a constant factor with the known upper bounds. In particular, in this regime the growth of the initial data at the free boundary fully determines the initial behaviour of the interface.