Jared C. Bronski, Tom Gambill
Published 2005-08-24, updated 2005-09-21Version 2
We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions \phi the exponent 1.5 is the best possible from this line of argument. Further, this result together with a result of Molinet gives an improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.
Comments: 17 pages, 1 figure; typos corrected, references added; figure modified
Existence and analyticity of solutions of the Kuramoto-Sivashinsky equation with singular data
Classical and Nonclassical symmetries of the (2+1)-dimensional Kuramoto-Sivashinsky equation
On a nonlocal analog of the Kuramoto-Sivashinsky equation
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