Global Solutions to a Fourth-order Degenerate Model for Surface-Tension-Driven Convection

Abstract

This paper investigates the global existence and non-negativity of weak solutions to an initial-boundary value problem for a one-dimensional fourth-order nonlinear degenerate parabolic equation. This model governs the convection phenomena in thin films driven by surface tension. Our analytical approach begins with the formulation of a regularized problem and a corresponding Galerkin approximating scheme. We first establish the existence of solutions to the approximate problem. Subsequently, by constructing specialized energy and entropy functionals, we derive uniform a priori estimates for the approximating solutions. Leveraging the Aubin-Lions compactness lemma, we pass to the limit and establish the non-negativity of the limit function. Finally, we demonstrate that this limit is indeed a global weak solution to the original initial-boundary value problem.