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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 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  We consider the KuramotoSivashinsky 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 u2 = 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  u2 = 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 KuramotoSivashinsky 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 
Type:  Text 
Language:  English 
Description:  126 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2006. 
URI:  http://hdl.handle.net/2142/86875 
Other Identifier(s):  (MiAaPQ)AAI3250245 
Date Available in IDEALS:  20150928 
Date Deposited:  2006 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois