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Improved Well-Posedness for the Triple-Deck and Related Models via Concavity

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Abstract

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).

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Acknowledgements

SI acknowledges support from NSF Grant DMS-2306528 and NSF Grant DMS-1802940 when this project was initiated. D.G-V. acknowledges the support of SingFlows project, Grant ANR-18- CE40-0027 of the French National Research Agency (ANR) and of the Institut Universitaire de France. YM acknowledges the support of JSPS KAKENHI Grant Number 20K03698, 19H05597, 20H00118, 21H00991, 21H04433.

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Authors and Affiliations

  1. CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), Université Paris Cité and Sorbonne Université, 75013, Paris, France

    David Gerard-Varet

  2. Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

    Sameer Iyer

  3. Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan

    Yasunori Maekawa

Authors
  1. David Gerard-Varet
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  2. Sameer Iyer
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  3. Yasunori Maekawa
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Correspondence to David Gerard-Varet.

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Communicated by E. Feireisl

In memory of Antonín Novotný.

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Gerard-Varet, D., Iyer, S. & Maekawa, Y. Improved Well-Posedness for the Triple-Deck and Related Models via Concavity. J. Math. Fluid Mech. 25, 69 (2023). https://doi.org/10.1007/s00021-023-00809-4

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