Approximation of the Integral Boundary Layer Equation by the Kuramoto--Sivashinsky Equation

In suitable parameter regimes the Integral Boundary Layer equation (IBLe) can be formally derived as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane. For very long waves with small amplitude, the IBLe can be further reduced to the Kuramoto–Sivashinsky equation (KSe). Here we justify this reduction of the IBL to the KSe. Using energy estimates we show that solutions of the KSe approximate solutions of the IBLe over sufficiently long time scales. This is a step towards understanding the approximation properties of the KSe for the full Navier–Stokes system describing the inclined film flow.