Logarithmic spirals in $2$d perfect fluids

Jeong, In-Jee; Said, Ayman R.

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 655-682

We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on $ℝ/\left(2\pi \mathbb{Z}\right)$. We show that this system is locally well-posed in $L^p$, $p\ge 1$, as well as for atomic measures, that is, logarithmic spiral vortex sheets. For logarithmic spiraling solutions, we make an observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time, which allows for a rather complete characterization of the long-time behavior. We prove global well-posedness for bounded logarithmic spirals as well as data that admit at most logarithmic singularities. We are then able to show a dichotomy in the long time behavior, solutions either blow up (either in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals should converge to constant steady states. For logarithmic spiral sheets, the dichotomy is shown to be even more drastic, where only finite time blow up or complete homogenization of the fluid can and does occur.

On étudie les spirales logarithmiques solutions des équations d’Euler incompressibles en deux dimensions d’espace qui résolvent un système de transport non linéaire sur $ℝ/\left(2\pi \mathbb{Z}\right)$. On montre que ce système est localement bien posé dans $L^p$, $p\ge 1$, ainsi que pour les nappes de tourbillon en spirale logarithmique. Pour ces spirales logarithmiques, nous observons que la circulation locale du tourbillon autour de l’origine est strictement monotone en temps, ce qui permet une caractérisation assez complète du comportement en temps long. On démontre le caractère bien posé global des spirales logarithmiques bornées ainsi que pour les données qui admettent au plus des singularités logarithmiques. Nous sommes alors en mesure de montrer une dichotomie dans le comportement en temps long : les solutions explosent (en temps fini ou infini) ou s’homogénéisent complètement. En particulier, les spirales logarithmiques bornées convergent vers des états stationnaires constants. Pour les nappes de tourbillon en spirale logarithmique, la dichotomie est encore plus radicale, où seule l’explosion en temps fini ou l’homogénéisation complète du fluide peuvent se produire et se produisent effectivement.

Classification : 76B47, 35Q35
Keywords: Logarithmic spirals, perfect fluids, longtime behavior, singularity formation
Mots-clés : Spirales logarithmiques, fluide parfait, comportement en temps long, formation de singularités

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