Global-in-time vortex configurations for the $2$D Euler equations

Juan Dávila; Manuel del Pino; Monica Musso; Shrish Parmeshwar

doi:10.4171/jems/1776

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Global-in-time vortex configurations for the $2$ D Euler equations

Juan Dávila
University of Bath, UK
Manuel del Pino
University of Bath, UK
Monica Musso
University of Bath, UK
Shrish Parmeshwar
University of Bath, UK
- MR
- ORCID

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Abstract

We consider the problem of finding a solution to the incompressible Euler equations

ω_{t} + v \cdot \nabla ω = 0 in R^{2} \times (0, \infty), v (x, t) = \frac{1}{2 π} \int_{R^{2}} \frac{( y - x ) ^{⊥}}{∣ y - x ∣ ^{2}} ω (y, t) d y

that is close to a superposition of travelling vortices as $t \to \infty$ . We employ a constructive approach by gluing classical travelling waves: at main order two vortex-antivortex pairs. Each pair consists of two vortices a distance $2 q$ from each other, with wave speed $c$ , moving in the positive and negative $x_{1}$ directions respectively. More precisely, we find an initial condition that leads to a 4-vortex solution of the form

ω (x, t) = ω_{0} (x - c t e_{1}) - ω_{0} (x + c t e_{1}) + o (1) as t \to \infty,

where

ω_{0} (x) = \frac{1}{ε ^{2}} W (\frac{x - q}{ε}) - \frac{1}{ε ^{2}} W (\frac{x + q}{ε}) + o (1) as ε \to 0

and $W (y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disk and zero outside.

Cite this article

Juan Dávila, Manuel del Pino, Monica Musso, Shrish Parmeshwar, Global-in-time vortex configurations for the $2$ D Euler equations. J. Eur. Math. Soc. (2026), published online first

DOI 10.4171/JEMS/1776