Non-linear higher-order boundary value problems describing thin viscous flows near edges

Two boundary value problems for non-linear higher-order ordinary differential equations are analyzed, which have been recently proposed in the modeling of steady and quasi-steady thin viscous flows over a bounded solid substrate. The first problem concerns steady states and consists of a third-order ODE for the height of the liquid; the ODE contains an unknown parameter, the flux, and the boundary conditions relate, near the edges of the substrate, the height and its second derivative to the flux itself. For this problem, (non-)existence and non-uniqueness results are proved depending on the behavior, as the flux approaches zero, of the “height-function” (the function which relates the height to the flux near the edge out of which the liquid flows). The second problem concerns quasi-steady states and consists of a fourth-order ODE for the (suitably scaled) height of the liquid; non-linear boundary conditions relate the height to the flux near the edges of the substrate. For this problem, the existence of a solution is proved for a suitable class of height-functions.