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Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation | | Jared C. Bronski
; Tom Gambill
; | | Date: |
24 Aug 2005 | | Subject: | Analysis of PDEs; ; Mathematical Physics MSC-class: 35G25 (primary); 35P15 (secondary) | math.AP math-ph math.MP | | 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 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. | | Source: | arXiv, math.AP/0508481 | | Services: | Forum | Review | PDF | Favorites | |
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