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Title:Application of Uncertainty Inequalities to Bound the Radius of the Attractor for the Kuramoto -Sivashinsky Equation
Author(s):Gambill, Thomas Naylor
Doctoral Committee Chair(s):Jared Bronski
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:We consider the Kuramoto-Sivashinsky equation in one 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. One attempt to improve the choice for &phis; for the Lyapunov function ||u( x,t) - &phis;(x)|| 22 using the calculus of variations gives an upper bound on the radius of the ball bounding the attractor of O(L 2.5). We apply Slepian's Uncertainty Theorem to a &phis; x that is a square well. However, the bound on the attractor, using the Slepian Theorem, only scales like O( L3.1). By means of numerical methods we obtain an optimal choice for &phis;. We then prove that for our choice of &phis; with ||&phis;||H2 = O( L1.5) that lim supt →infinity ||u||2 = O( L1.5) and that the set {u | || u - &phis;||2 < CL1.5} is invariant and exponentially attracting in forward time. This result is slightly weaker than that recently announced by Giacomelli and Otto, who showed that lim supt→infinity || u||2 = o(L1.5 ) by a very different method. We further show that for a class of functions &phis; the exponent 1.5 cannot be improved. Our results can be used with the destabilized Kuramoto-Sivashinsky equation to show that the optimal bound of the radius of the ball that contains the attractor of O( L1.5) is actually achieved.
Issue Date:2006
Description:126 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2006.
Other Identifier(s):(MiAaPQ)AAI3250245
Date Available in IDEALS:2015-09-28
Date Deposited:2006

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