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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2310.07238 |
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| _version_ | 1866918374332694528 |
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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 | We consider the problem of finding a solution to the incompressible Euler equations $$ ω_t + v\cdot \nabla ω= 0 \quad \hbox{ in } \mathbb{R}^2 \times (0,\infty), \quad v(x,t) = \frac 1{2π} \int_{{\mathbb R}^2} \frac {(y-x)^\perp}{|y-x|^2} ω(y,t)\, dy $$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$ ω(x,t) = ω_0(x-ct\, e ) - ω_0 ( x+ ct \, e) + o(1) \ \hbox{ as } t\to\infty $$ where $$ ω_0( x ) = \frac 1{\varepsilon^{2}} W \left ( \frac {x-q} \varepsilon \right ) - \frac 1{\varepsilon^{2}}W \left ( \frac {x+q} \varepsilon \right ) + o(1) \ \hbox{ as } \varepsilon \to 0 $$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global in Time Vortex Configurations for the $2$D Euler Equations Dávila, Juan del Pino, Manuel Musso, Monica Parmeshwar, Shrish Analysis of PDEs We consider the problem of finding a solution to the incompressible Euler equations $$ ω_t + v\cdot \nabla ω= 0 \quad \hbox{ in } \mathbb{R}^2 \times (0,\infty), \quad v(x,t) = \frac 1{2π} \int_{{\mathbb R}^2} \frac {(y-x)^\perp}{|y-x|^2} ω(y,t)\, dy $$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$ ω(x,t) = ω_0(x-ct\, e ) - ω_0 ( x+ ct \, e) + o(1) \ \hbox{ as } t\to\infty $$ where $$ ω_0( x ) = \frac 1{\varepsilon^{2}} W \left ( \frac {x-q} \varepsilon \right ) - \frac 1{\varepsilon^{2}}W \left ( \frac {x+q} \varepsilon \right ) + o(1) \ \hbox{ as } \varepsilon \to 0 $$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside. |
| title | Global in Time Vortex Configurations for the $2$D Euler Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.07238 |