On the ill-posedness of the triple deck model

TL;DR

This paper investigates the stability of the triple deck model, revealing inherent instabilities that imply the local well-posedness results are optimal, especially for analytic data.

Contribution

It demonstrates the existence of unstable linearizations of the triple deck model, establishing fundamental limitations on its stability and well-posedness.

Findings

Unstable eigenmodes grow linearly with tangential frequency

The triple deck model exhibits inherent linear instabilities

Results support the optimality of recent well-posedness findings

Abstract

We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [2], based on complex analysis tools, and stability estimates inspired from [3], we exhibit unstable linearizations of the triple deck equation. The growth rates of the corresponding unstable eigenmodes scale linearly with the tangential frequency. This shows that the recent result of Iyer and Vicol [11] of local well-posedness for analytic data is essentially optimal.