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