Justification of the linear long-wave approximation to viscous fluid flow down an inclined plane

The objective of this paper is to justify the linear long-wave approximation used in the derivation of approximate equations for long waves on the free surface of a two-dimensional viscous fluid flow down an inclined plane. To the first order of a small parameter, the approximate equation is a heat equation, which becomes ill-posed if a Reynolds number $R$ is greater than some critical value ${R_c}$. To overcome this difficulty we consider a higher-order approximate equation, which is well-posed even if $R > {R_c}$, and show that the solution of the higher-order equation is an approximation to the solution of the linearized Navier-Stokes equations. The justification is based upon a set of long-wave initial conditions, and the error bounds can also be expressed in terms of pointwise estimates.