Electron. J. Diff. Equ., Vol. 2012 (2012), No. 61, pp. 1-51.

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

Tobias Häcker, Guido Schneider, Hannes Uecker

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

Show me the PDF file (710 KB), TEX file, and other files for this article.

Tobias Häcker
Institut für Analysis, Dynamik und Modellierung
Universität Stuttgart, Pfaffenwaldring 57
D-70569 Stuttgart, Germany
email: tobias.haecker@mathematik.uni-stuttgart.de
Guido Schneider
Institut für Analysis, Dynamik und Modellierung
Universität Stuttgart, Pfaffenwaldring 57
D-70569 Stuttgart, Germany
email: guido.schneider@mathematik.uni-stuttgart.de
Hannes Uecker
Institut für Mathematik
Carl von Ossietzky Universität Oldenburg
D-26111 Oldenburg, Germany
email: hannes.uecker@uni-oldenburg.de

Return to the EJDE web page