Global Existence and Analyticity for the 2D Kuramoto–Sivashinsky Equation

David M. Ambrose, Anna L. Mazzucato

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Abstract

There is little analytical theory for the behavior of solutions of the Kuramoto–Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the linearized behavior depends on the size of the torus—in particular, for different sizes of the domain, there are different numbers of linearly growing modes. We prove that small solutions exist for all time if there are no linearly growing modes, proving also in this case that the radius of analyticity of solutions grows linearly in time. In the general case (i.e., in the presence of a finite number of growing modes), we make estimates for how the radius of analyticity of solutions changes in time.

Original languageEnglish (US)
Pages (from-to)1525-1547
Number of pages23
JournalJournal of Dynamics and Differential Equations
Volume31
Issue number3
DOIs
StatePublished - Sep 1 2019

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All Science Journal Classification (ASJC) codes

  • Analysis

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