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| We consider two-dimensional motion of liquid film of a viscous incompressible fluid down an inclined plane in the influence of the gravity and the surface tension. In order to investigate such a motion, a method of the thin film approximation is often used. It is the approximation by the perturbation expansion of the solution for the nondimensional parameter $\delta$ defined by ratio between the thickness of the liquid film and the typical wave length. In this study, we will give uniform estimates of the solution to the original Navier--Stokes equations in $\delta$ when the Reynolds number, the angle of inclination, and the initial date are sufficiently small. |
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