{"title":"Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations","authors":"Denis Gaidashev, Alejandro Luque","doi":"10.1137/23m1551481","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4356-4374, August 2024. <br/> Abstract. The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier–Stokes equations. In [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950], Li and Sinai have proposed a renormalization approach to the problem of the existence of finite-time blow-up solutions of this equation. In this paper, we revisit the renormalization problem for the quasi-geostrophic blow-ups, prove the existence of a family of renormalization fixed points, and deduce the existence of real [math] solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1551481","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4356-4374, August 2024. Abstract. The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier–Stokes equations. In [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950], Li and Sinai have proposed a renormalization approach to the problem of the existence of finite-time blow-up solutions of this equation. In this paper, we revisit the renormalization problem for the quasi-geostrophic blow-ups, prove the existence of a family of renormalization fixed points, and deduce the existence of real [math] solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950].
SIAM 数学分析期刊》，第 56 卷第 4 期，第 4356-4374 页，2024 年 8 月。 摘要。一维准地转方程是著名的纳维-斯托克斯方程的一维傅里叶空间类似方程。在 [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp.在本文中，我们重新审视了准地转吹胀的重正化问题，证明了重正化定点族的存在，并推导出了与[D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp.
期刊介绍：
SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena.
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Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.
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