Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case

Abstract

We prove optimal estimates on the growth rate of the support of solutions to the thin-film equation ut + div(|u|n[nabla] [Dgr]u) = 0 in space dimensions N = 2 and N = 3 for parameters n [isin] [2, 3) which correspond to Navier's slip condition (n = 2) or certain variants modeling weaker slippage effects. Our approach relies on a new class of weighted energy estimates. It is inspired by the onedimensional technique of Hulshof and Shishkov Adv. Diff. Equations 3, (1998) 625-642, and it simplifies their method, mainly with respect to basic integral estimates to be used.