Vortices for lake equations (review with questions and speculations)
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915129517408256 |
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| author | Koiller, Jair |
| author_facet | Koiller, Jair |
| contents | The `lake equation' on a planar domain D with bathymetry b(x,y) is given by $ \partial_t u + (u \cdot {\rm grad}) u= -{\rm grad}\, p \,, \,\,{\rm div} (b u) = 0 \,,\, \text{with}\,\, u \parallel \partial D.$ % \, \,\, \,\,\, \text{),}$$ We focus on Geometric Mechanics aspects, glossing over hard analysis issues. % related to the desingularization. Motivating example is a `rip current' produced by vortex pairs near a beach shore. For uniform slope beach there is a perfect analogy with \ Thomson's vortex rings. The stream function produced by a vortex is defined as the Green function of the operator $- {\rm div} ( {\rm grad} ψ/b)$ with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudoanalytical functions and quasiconformal mappings. Uniformly elliptic equations on closed Riemann surfaces could be called `planet equations'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vortices for lake equations (review with questions and speculations) Koiller, Jair Mathematical Physics Analysis of PDEs 76B47, 76M60, 34C23, 37E35 The `lake equation' on a planar domain D with bathymetry b(x,y) is given by $ \partial_t u + (u \cdot {\rm grad}) u= -{\rm grad}\, p \,, \,\,{\rm div} (b u) = 0 \,,\, \text{with}\,\, u \parallel \partial D.$ % \, \,\, \,\,\, \text{),}$$ We focus on Geometric Mechanics aspects, glossing over hard analysis issues. % related to the desingularization. Motivating example is a `rip current' produced by vortex pairs near a beach shore. For uniform slope beach there is a perfect analogy with \ Thomson's vortex rings. The stream function produced by a vortex is defined as the Green function of the operator $- {\rm div} ( {\rm grad} ψ/b)$ with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudoanalytical functions and quasiconformal mappings. Uniformly elliptic equations on closed Riemann surfaces could be called `planet equations'. |
| title | Vortices for lake equations (review with questions and speculations) |
| topic | Mathematical Physics Analysis of PDEs 76B47, 76M60, 34C23, 37E35 |
| url | https://arxiv.org/abs/2501.10433 |