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

Part of: Parabolic equations and systems Miscellaneous topics - Partial differential equations Partial differential equations

Published online by Cambridge University Press:  20 November 2024

David M. Ambrose Open the ORCID record for David M. Ambrose [Opens in a new window] ,
Milton C. Lopes Filho  and
Helena J. Nussenzveig Lopes
David M. Ambrose
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA ([email protected]) (corresponding author)
Milton C. Lopes Filho
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ 21941-909, Brazil ([email protected], [email protected])
Helena J. Nussenzveig Lopes
Affiliation:
Departamento de Matemática, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, Brazil ([email protected])
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Abstract

We prove the 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 that are in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data that are 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.

Keywords

AnalyticityexistenceKuramoto–Sivashinskypseudomeasuresingular data

MSC classification

Primary: 35K46: Initial value problems for higher-order parabolic systems
Secondary: 35R05: Partial differential equations with discontinuous coefficients or data 35B65: Smoothness and regularity of solutions 35A01: Existence problems 35A20: Analytic methods, singularities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 26
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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