Remarks on the stabilization of large-scale growth in the 2D Kuramoto-Sivashinsky equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911979860393984 |
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| author | Larios, Adam Martinez, Vincent R. |
| author_facet | Larios, Adam Martinez, Vincent R. |
| contents | 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 condition 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 criterion 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 solutions, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04888 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on the stabilization of large-scale growth in the 2D Kuramoto-Sivashinsky equation Larios, Adam Martinez, Vincent R. Analysis of PDEs 35Q35, 35A01, 35K25, 35K52, 35K58, 35B65, 35B10, 65M70 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 condition 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 criterion 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 solutions, 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. |
| title | Remarks on the stabilization of large-scale growth in the 2D Kuramoto-Sivashinsky equation |
| topic | Analysis of PDEs 35Q35, 35A01, 35K25, 35K52, 35K58, 35B65, 35B10, 65M70 |
| url | https://arxiv.org/abs/2401.04888 |