2017/08/29 by David M. Ambrose, Anna L. Mazzucato, Ambrose, David M. +1 · 2 citations
Computer Science · Engineering · Mathematics · #35B10 #35B65 #35K25 #35K58 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations #math.AP #msc:35B10 #msc:35B65 #msc:35K25 #msc:35K58
paper · pdf · doi:10.48550/arxiv.1708.08752
26 pages
arxiv created 2017/08/29 · openalex publication_date 2017/08/29 · arxiv updated 2017/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.