Improved well-posedness for the Triple-Deck and related models via concavity

TL;DR

This paper proves linearized well-posedness of the Triple-Deck system and hydrostatic Navier-Stokes equations in Gevrey-3/2 regularity under concavity assumptions, using stability estimates and smoothing properties.

Contribution

It establishes the first well-posedness results in Gevrey-3/2 regularity for these models under concavity, extending previous analyticity results.

Findings

Linearized well-posedness in Gevrey-3/2 regularity for Triple-Deck system

Extension of Gevrey-3/2 well-posedness to hydrostatic Navier-Stokes equations

Use of stability estimates and smoothing properties in the analysis

Abstract

We establish linearized well-posedness of the Triple-Deck system in Gevrey-$\frac32$ regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result \cite{DietertGV}, one cannot expect a generic improvement of the result of \cite{IyerVicol} 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 \cite{GVMM} 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-$\frac32$ linear well-posedness result for concave data, improving at the linear level the recent work \cite{MR4149066}.