Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914729769828352 |
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| author | Bianchini, Roberta Zelati, Michele Coti Ertzbischoff, Lucas |
| author_facet | Bianchini, Roberta Zelati, Michele Coti Ertzbischoff, Lucas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17857 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit Bianchini, Roberta Zelati, Michele Coti Ertzbischoff, Lucas Analysis of PDEs 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. |
| title | Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.17857 |