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<p>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.</p>

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Ill-Posedness of the Hydrostatic Euler–Boussinesq Equations and Failure of Hydrostatic Limit

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Roberta Bianchini,
  • Michele Coti Zelati,
  • Lucas Ertzbischoff

摘要

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.