Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit

Roberta Bianchini; Michele Coti Zelati; Lucas Ertzbischoff

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arXiv · 2403.17857

Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit

Roberta Bianchini·Michele Coti Zelati·Lucas Ertzbischoff

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Abstract

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler-Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler-Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces.

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Roberta Bianchini, Michele Coti Zelati, Lucas Ertzbischoff. 2024-03-26. Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit. https://arxiv.org/abs/2403.17857

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