Long-time stability of a stably stratified rest state in the inviscid 2D Boussinesq equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917404710273024 |
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| author | Jurja, Catalina Widmayer, Klaus |
| author_facet | Jurja, Catalina Widmayer, Klaus |
| contents | We establish the nonlinear stability on a timescale $O(\varepsilon^{-2})$ of a linearly, stably stratified rest state in the inviscid Boussinesq system on $\mathbb{R}^2$. Here $\varepsilon>0$ denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation.
At the core of this result is a dispersive effect due to anisotropic internal gravity waves. At the linearized level, this gives rise to amplitude decay at a rate of $t^{-1/2}$, as observed in [EW15]. We establish a refined version of this, and propagate nonlinear control via a detailed analysis of nonlinear interactions using the method of partial symmetries developed in [GPW23]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time stability of a stably stratified rest state in the inviscid 2D Boussinesq equation Jurja, Catalina Widmayer, Klaus Analysis of PDEs 35Q35, 35Q86, 35B35, 76B55, 76B15, 76E20 We establish the nonlinear stability on a timescale $O(\varepsilon^{-2})$ of a linearly, stably stratified rest state in the inviscid Boussinesq system on $\mathbb{R}^2$. Here $\varepsilon>0$ denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation. At the core of this result is a dispersive effect due to anisotropic internal gravity waves. At the linearized level, this gives rise to amplitude decay at a rate of $t^{-1/2}$, as observed in [EW15]. We establish a refined version of this, and propagate nonlinear control via a detailed analysis of nonlinear interactions using the method of partial symmetries developed in [GPW23]. |
| title | Long-time stability of a stably stratified rest state in the inviscid 2D Boussinesq equation |
| topic | Analysis of PDEs 35Q35, 35Q86, 35B35, 76B55, 76B15, 76E20 |
| url | https://arxiv.org/abs/2408.15154 |