Kuramoto-Sivashinsky weak turbulence, in the symmetry unrestricted space
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Go beyond reproducing the Kuramoto-Sivashinsky results for the subspace of
antisymmetric solutions - discussed in the Webbook) by looking at the full
space of solutions (with zero mean velocity).
This is related to Yueheng Lan's thesis, and both Roman Grigoriev and I
are interested in it as an open research problem, so you can get some good
interactions with several people as you work on it. Later in the semester
we will also have a visitor, C Dettmann from Bristol UK, working with us
on this kind of problem.
For starters, I recommend reading Chapter 7 of
P. Holmes, J.L. Lumley and G. Berkooz,
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
(Cambridge U. Press, Cambridge 1996).
While I do not think that the Proper Orthogonal Decomposition is the right
tool (I am hoping that the recurrent patterns will do the job), the
physics and the motivation is described very nicely in this book.
Predrag Cvitanovic, Sep 2 2003