Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 61, pp. 1-51.

Title: Self-similar decay to the  marginally stable  ground state
       in a model for film flow over inclined wavy  bottoms

Authors: Tobias Hacker (Univ. Stuttgart,  Germany)
         Guido Schneider (Univ. Stuttgart,  Germany)
	 Hannes Uecker (Carl von Ossietzky Univ., Oldenburg, Germany)

Abstract: 
 The integral boundary layer system (IBL) with spatially periodic
 coefficients arises as a long wave approximation for the flow of a
 viscous incompressible fluid down a wavy inclined plane.
 The  Nusselt-like stationary solution of the IBL is linearly at best
 marginally stable; i.e., it has essential spectrum at least up to the
 imaginary axis. Nevertheless, in this stable  case we show that
 localized perturbations of the ground state decay  in a self-similar way.
 The proof uses the renormalization group method in Bloch variables
 and the fact that in the stable case the Burgers equation is the
 amplitude equation for long waves of small amplitude  in the IBL.
 It is the first time that such a proof is given for a quasilinear
 PDE with spatially periodic coefficients.

Submitted October 27, 2010. Published April 12, 2012.
Math Subject Classifications: 35Q35, 37E20, 35B35.
Key Words: Diffusive stability; renormalization; IBL system; periodic media.