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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, it is hard to treat the Navier--Stokesequations directly, so that a thin film approximation is often used. It is anapproximation obtained by the perturbation expansion with respect to the aspectratio $\delta$ of the film under the thin film regime $\delta\ll1$. Our purposeis to give a mathematically rigorous justification of the thin filmapproximation by establishing an error estimate between the solution of theNavier--Stokes equations and those of approximate equations. To this end, inthis paper we derive a uniform estimate for the solution of the Navier--Stokesequations with respect to $\delta$ under appropriate assumptions.

Uniform Estimates for the Flow of a Viscous Incompressible Fluid down an Inclined Plane in the Thin Film Regime