{"title":"利用凹性改进三层甲板及相关模型的适定性","authors":"David Gerard-Varet, Sameer Iyer, Yasunori Maekawa","doi":"10.1007/s00021-023-00809-4","DOIUrl":null,"url":null,"abstract":"<div><p>We establish linearized well-posedness of the Triple-Deck system in Gevrey-<span>\\(\\frac{3}{2}\\)</span> regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result (Dietert and Gerard-Varet in SIAM J Math Anal, 2021), one cannot expect a generic improvement of the result of Iyer and Vicol (Commun Pure Appl Math 74(8):1641–1684, 2021) to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier–Laplace side: (i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from Gerard-Varet et al. (Optimal Prandtl expansion around concave boundary layer, 2020. arXiv:2005.05022) (ii) smoothing properties of the Benjamin–Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier–Stokes equations: we show for this system a similar Gevrey-<span>\\(\\frac{3}{2}\\)</span> linear well-posedness result for concave data, improving at the linear level the recent work (Gérard-Varet et al. in Anal PDE 13(5):1417–1455, 2020).\n</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"25 3","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Improved Well-Posedness for the Triple-Deck and Related Models via Concavity\",\"authors\":\"David Gerard-Varet, Sameer Iyer, Yasunori Maekawa\",\"doi\":\"10.1007/s00021-023-00809-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We establish linearized well-posedness of the Triple-Deck system in Gevrey-<span>\\\\(\\\\frac{3}{2}\\\\)</span> regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result (Dietert and Gerard-Varet in SIAM J Math Anal, 2021), one cannot expect a generic improvement of the result of Iyer and Vicol (Commun Pure Appl Math 74(8):1641–1684, 2021) to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier–Laplace side: (i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from Gerard-Varet et al. (Optimal Prandtl expansion around concave boundary layer, 2020. arXiv:2005.05022) (ii) smoothing properties of the Benjamin–Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier–Stokes equations: we show for this system a similar Gevrey-<span>\\\\(\\\\frac{3}{2}\\\\)</span> linear well-posedness result for concave data, improving at the linear level the recent work (Gérard-Varet et al. in Anal PDE 13(5):1417–1455, 2020).\\n</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":\"25 3\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-07-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Fluid Mechanics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00021-023-00809-4\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-023-00809-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
摘要
在背景流的凹凸性假设下,建立了Gevrey- \(\frac{3}{2}\)三层系统的线性化适定性。由于最近的结果(Dietert和Gerard-Varet在SIAM J Math Anal, 2021),人们不能期望Iyer和Vicol (common Pure applied Math 74(8): 1641-1684, 2021)的结果一般改进到比实际分析性更弱的正则性类。通过对傅里叶-拉普拉斯侧时空模式的分析,我们的方法利用了两个成分:(i)涡度水平的稳定性估计,涉及凹性假设和改编自Gerard-Varet等人的微妙迭代方案(凹边界层周围的最优普朗特展开,2020)。(ii)无穷远处三层流所满足的类Benjamin-Ono方程的光滑性。有趣的是,我们对涡度方程的处理也适用于所谓的流体静力学Navier-Stokes方程:我们为该系统展示了一个类似的凹数据的Gevrey- \(\frac{3}{2}\)线性适定性结果,在线性水平上改进了最近的工作(g瓦德·瓦雷特等人在Anal PDE 13(5): 1417-1455, 2020)。
Improved Well-Posedness for the Triple-Deck and Related Models via Concavity
We establish linearized well-posedness of the Triple-Deck system in Gevrey-\(\frac{3}{2}\) regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result (Dietert and Gerard-Varet in SIAM J Math Anal, 2021), one cannot expect a generic improvement of the result of Iyer and Vicol (Commun Pure Appl Math 74(8):1641–1684, 2021) to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier–Laplace side: (i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from Gerard-Varet et al. (Optimal Prandtl expansion around concave boundary layer, 2020. arXiv:2005.05022) (ii) smoothing properties of the Benjamin–Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier–Stokes equations: we show for this system a similar Gevrey-\(\frac{3}{2}\) linear well-posedness result for concave data, improving at the linear level the recent work (Gérard-Varet et al. in Anal PDE 13(5):1417–1455, 2020).
期刊介绍:
The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.