JUSTIFICATION OF THE LINEAR LONG-WAVE APPROXIMATION TO VISCOUS-FLUID PLOW 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.