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We address a physically-meaningful extension of the Prandtl system, alsoknown as hyperbolic Prandtl equations. We show that the linearised model arounda non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortlyin time, we generate solutions that experience a dispersion relation of orderk^(1/3) in the frequencies of the tangential direction, akin the pioneeringresult of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion wasof order k^(1/2)). We emphasise however that this growth rate does not implyill-posedness in Gevrey-class m, with m > 3 and we relate also these aspects tothe original Prandtl equations in Gevrey-class m, with m > 2. By relaxingcertain assumptions on the shear flow and on the solutions, namely allowing forunbounded flows in the vertical direction, we however determine a suitableshear flow for the mentioned ill-posedness in Gevrey spaces.

Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations