Asymptotic properties of vortex-pair solutions for incompressible Euler equations in $\mathbb{R}^2$
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| Format: | Preprint |
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2023
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| _version_ | 1866914834067488768 |
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| author | Dávila, Juan del Pino, Manuel Musso, Monica Parmeshwar, Shrish |
| author_facet | Dávila, Juan del Pino, Manuel Musso, Monica Parmeshwar, Shrish |
| contents | A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ ω_t + \nabla^\perp Ψ\cdot \nabla ω= 0 , \quad Ψ= (-Δ)^{-1} ω, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $ω(x,t) = W(x_1-ct,x_2 )$ where $W(x)$ is compactly supported and odd in $x_2$. We revisit the problem of constructing solutions which are highly $\varepsilon$-concentrated around points $ (0, \pm q)$, more precisely with approximately radially symmetric, compactly supported bumps with radius $\varepsilon$ and masses $\pm m$. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters $q$ and $\varepsilon$ for the solution and its propagation speed $c$ are established. These results improve constructions through variational methods in [14] and in [5] for the case of a bounded domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12039 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic properties of vortex-pair solutions for incompressible Euler equations in $\mathbb{R}^2$ Dávila, Juan del Pino, Manuel Musso, Monica Parmeshwar, Shrish Analysis of PDEs A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ ω_t + \nabla^\perp Ψ\cdot \nabla ω= 0 , \quad Ψ= (-Δ)^{-1} ω, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $ω(x,t) = W(x_1-ct,x_2 )$ where $W(x)$ is compactly supported and odd in $x_2$. We revisit the problem of constructing solutions which are highly $\varepsilon$-concentrated around points $ (0, \pm q)$, more precisely with approximately radially symmetric, compactly supported bumps with radius $\varepsilon$ and masses $\pm m$. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters $q$ and $\varepsilon$ for the solution and its propagation speed $c$ are established. These results improve constructions through variational methods in [14] and in [5] for the case of a bounded domain. |
| title | Asymptotic properties of vortex-pair solutions for incompressible Euler equations in $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2311.12039 |