The effect of linear stratification on the stability of a rest state in the 2D inviscid Boussinesq system
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915626439671808 |
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| author | Jurja, Catalina Ko, Haram |
| author_facet | Jurja, Catalina Ko, Haram |
| contents | We investigate and quantify the effect of stratification on the stability time of a stably stratified rest state for the 2D inviscid Boussinesq system on $\mathbb{R}^2$. As an important consequence, we obtain stability of the steady state starting from an $\varepsilon$-sized initial perturbation of Sobolev regularity $H^{3^+}$ on a timescale $\mathcal{O}(\varepsilon^{-4/3})$.
In our setting, stratification induces dispersion and at the core of our approach are inhomogeneous Strichartz estimates used to control nonlinear contributions. This allows to keep only $L^2-$based regularity assumptions on the initial perturbation, whereas previous works impose additional localizations to achieve this timescale.
We prove the analogous result for the related dispersive SQG equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The effect of linear stratification on the stability of a rest state in the 2D inviscid Boussinesq system Jurja, Catalina Ko, Haram Analysis of PDEs 35Q35, 35Q86, 35B35, 76B55, 76B15, 76B70, 76E20 We investigate and quantify the effect of stratification on the stability time of a stably stratified rest state for the 2D inviscid Boussinesq system on $\mathbb{R}^2$. As an important consequence, we obtain stability of the steady state starting from an $\varepsilon$-sized initial perturbation of Sobolev regularity $H^{3^+}$ on a timescale $\mathcal{O}(\varepsilon^{-4/3})$. In our setting, stratification induces dispersion and at the core of our approach are inhomogeneous Strichartz estimates used to control nonlinear contributions. This allows to keep only $L^2-$based regularity assumptions on the initial perturbation, whereas previous works impose additional localizations to achieve this timescale. We prove the analogous result for the related dispersive SQG equation. |
| title | The effect of linear stratification on the stability of a rest state in the 2D inviscid Boussinesq system |
| topic | Analysis of PDEs 35Q35, 35Q86, 35B35, 76B55, 76B15, 76B70, 76E20 |
| url | https://arxiv.org/abs/2508.04514 |