Regularized shock solutions in coating flows with small surface tension

We study the dynamics of thin liquid films on the surface of a rotating horizontal cylinder in the presence of gravity in the small surface tension limit. Using dynamical system methods, we show that the continuum of shock solutions increasing across the jump point persists in the small surface tension limit, whereas the continuum of shock solutions decreasing across the jump point terminates in the limit. Using delicate numerical computations, we show that the number of steady states with equal mass increases as the surface tension parameter goes to zero. This corresponds to an increase in the number of loops on the mass-flux bifurcation diagram. If n is the number of loops in the mass-flux diagram with 2n + 1 solution branches, we show that n + 1 solution branches are stable with respect to small perturbations in the time evolution of the liquid film.