Existence and analyticity of solutions of the Kuramoto-Sivashinsky equation with singular data

2023/08/16 by David M. Ambrose, Ambrose, David M., Milton C. Lopes Filho +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.08078

openalex publication_date 2023/08/16 · openalex created_date 2023/08/18 · openalex updated_date 2026/07/28

Abstract

We prove existence of solutions to the Kuramoto-Sivashinsky equation with low-regularity data, in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data which is in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data which is in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.

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