Global solutions to a fourth-order degenerate model for surface-tension-driven convection

This paper investigates the global existence and nonnegativity of weak solutions to an initial-boundary value problem for a one-dimensional fourth-order nonlinear degenerate parabolic equation. This model governs the convection in thin films driven by surface tension. Our analytical approach begins with the formulation of a regularized problem and an energy-stable mixed Galerkin approximation. 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 nonnegativity of the limit function. Finally, we demonstrate that this limit is indeed a global weak solution to the original initial-boundary value problem.