A mathematical justification of a thin film approximation for the flow down an inclined plane

We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous examples of the approximate equations are the Burgers equation, Kuramoto-Sivashinsky equation, KdV-Burgers equation, KdV-Kuramoto-Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier-Stokes equations and those of approximate equations.