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.