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