Math @ Duke

Publications [#244230] of Thomas P. Witelski
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 Bernoff, AJ; Witelski, TP, Stability and dynamics of selfsimilarity in evolution equations,
Journal of Engineering Mathematics, vol. 66 no. 1
(March, 2010),
pp. 1131, ISSN 00220833 [s1066500993098], [doi]
(last updated on 2017/12/14)
Abstract: A methodology for studying the linear stability of selfsimilar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finitetime blowup, a secondorder semilinear heat equation with infinitetime spreading solutions, and the fourthorder Sivashinsky equation with finitetime selfsimilar blowup. These examples are used to show that selfsimilar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of selfsimilar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a selfsimilar profile. For blowup solutions it is demonstrated that the symmetries give rise to positive eigenvalues associated with the symmetries, and it is shown how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions. © Springer Science+Business Media B.V. 2009.


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