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Determining nodes for the Kuramoto-Sivashinsky equation

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Abstract

We show that solutions of the 1D Kuramoto-Sivashinsky equation with periodic boundary conditions are asymptotically determined by their values at four points. That is, there existx 1,x 2,x 3, andx 4 in the (periodic) domainΩ such that if

$$\mathop {\lim }\limits_{t \to \infty } \left| {u_1 (x_j ,t) - u_2 (x_j ,t)} \right| = 0, j = 1,2,3,4$$

for two solutionsu 1 andu 2, then

$$\mathop {\lim }\limits_{t \to \infty } \left\| {u_1 ( \cdot ,t) - u_2 ( \cdot ,t)} \right\|_{L^2 (\Omega )} = 0$$

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, 47405, Bloomington, Indiana

    Ciprian Foias

  2. Department of Mathematics, University of Chicago, 60637, Chicago, Illinois

    Igor Kukavica

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  1. Ciprian Foias
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  2. Igor Kukavica
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Foias, C., Kukavica, I. Determining nodes for the Kuramoto-Sivashinsky equation. J Dyn Diff Equat 7, 365–373 (1995). https://doi.org/10.1007/BF02219361

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