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Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation

Jared C. Bronski·Tom Gambill
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Abstract

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 ϕthe 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.

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Jared C. Bronski, Tom Gambill. 2005-09-21. Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation. https://doi.org/10.1088/0951-7715%2F19%2F9%2F002

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