Improved Well-Posedness for the Triple-Deck and Related Models via Concavity
Improved Well-Posedness for the Triple-Deck and Related Models via Concavity
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DOI：
10.1007/s00021-023-00809-4
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发表时间：
2022-05
影响因子：
1.3
通讯作者：
D. Gérard-Varet;Sameer Iyer;Yasunori Maekawa
中科院分区：
文献类型：
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作者：
D. Gérard-Varet;Sameer Iyer;Yasunori Maekawa
我们建立了三层系统在切向变量的Gevrey正则性下的线性化适定性。由于最近的结果（Dietert和Gerard-Varet in SIAM J Math Anal，2021），人们不能期望Iyer和Vicol（Commun Pure Appl Math 74（8）：1641-1684，2021）的结果一般改进为比真实的分析性更弱的正则性类。我们的方法利用两个成分，通过对Fourier-Laplace侧的时空模式的分析：（i）涡度水平的稳定性估计，涉及到假设和Gerard-Varet等人（凹边界层周围的最佳普朗特展开，2020年）改编的微妙迭代方案。arXiv：2005.05022）（ii）在无穷远处由三层流满足的Benjamin-Ono类方程的平滑性质。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：我们为这个系统展示了凹数据的类似Gevrey线性适定性结果，在线性水平上改进了最近的工作（Gérard-Varet et al. in Anal PDE 13（5）：1417-1455，2020）。
We establish linearized well-posedness of the Triple-Deck system in Gevrey-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-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).