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arXiv · 2305.04664

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

Francesco De Anna·Joshua Kortum·Stefano Scrobogna
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Abstract

We address a physically-meaningful extension of the Prandtl system, also known as hyperbolic Prandtl equations. We show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order k^(1/3) in the frequencies of the tangential direction, akin the pioneering result of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion was of order k^(1/2)). We emphasise however that this growth rate does not imply ill-posedness in Gevrey-class m, with m > 3 and we relate also these aspects to the original Prandtl equations in Gevrey-class m, with m > 2. By relaxing certain assumptions on the shear flow and on the solutions, namely allowing for unbounded flows in the vertical direction, we however determine a suitable shear flow for the mentioned ill-posedness in Gevrey spaces.

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Francesco De Anna, Joshua Kortum, Stefano Scrobogna. 2023-05-08. Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations. https://arxiv.org/abs/2305.04664

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