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We consider a two-dimensional motion of a thin film flowing down an inclinedplane under the influence of the gravity and the surface tension. In order toinvestigate the stability of such flow, we often use a thin film approximation,which is an approximation obtained by the perturbation expansion with respectto the aspect ratio of the film. The famous example of the approximateequations are the Burgers equation, Kuramoto--Sivashinsky equation,KdV--Burgers equation, KdV--Kuramoto--Sivashinsky equation, and so on. In thispaper, we give a mathematically rigorous justification of a thin filmapproximation by establishing an error estimate between the solution of theNavier--Stokes equations and those of approximate equations.

A Mathematical Justification of a Thin Film Approximation for the Flow down an Inclined Plane