The Lubrication Approximation For Thin Viscous Films: The Moving Contact Line With A `porous Media' Cut Off Of Van Der Waals Interactions

. We consider the effect of a second order `porous media' [25] term on the evolution of weak solutions of the fourth order degenerate diffusion equation (1) h t = \Gammar \Delta (h n r\Deltah \Gamma rh m ) in one space dimension. The equation without the second order term is derived from a `lubrication approximation' and models surface tension dominated motion of thin viscous films and spreading droplets [18]. Here h(x; t) is the thickness of the film, and the physical problem corresponds to n = 3. For simplicity we consider periodic boundary conditions which has the physical interpretation of modeling a periodic array of droplets. In a previous work [6] we studied the above equation without the second order `porous media' term. In particular we showed the existence of nonnegative weak solutions with increasing support for 0 ! n ! 3 but the techniques failed for n 3. This is consistent with the fact that, in this case, nonnegative self-similar source-type solutions do not exist fo..