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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.
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Remarks on the Stabilization of Large-Scale Growth in the 2D Kuramoto–Sivashinsky Equation
In this article, some elementary observations are made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggest the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity con- dition for the two-dimensional Kuramoto–Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya–Prodi–Serrin-type cri- terion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solu- tions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers–Sivashinsky equation. The proofs are direct, elementary, and concise.
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- Award ID(s):
- 2206491
- PAR ID:
- 10546849
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Journal of Mathematical Fluid Mechanics
- Volume:
- 26
- Issue:
- 4
- ISSN:
- 1422-6928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00021-024-00890-3
