Nonlinear Stability of Columnar Euler Vortices

Definitions and mathematical statement

The incompressible three-dimensional Euler equations for velocity $u(t,x)\in{\mathbb{R}}^3$ and pressure $p(t,x)\in{\mathbb{R}}$ are \[ \partial_tu+(u\cdot\nabla)u=-\nabla p,\qquad \nabla\cdot u=0. \] Here the spatial domain $D$, boundary conditions, behavior at infinity, and solution regularity must be specified; the supplied record does not specify them. A steady solution $U$ satisfies \[ (U\cdot\nabla)U=-\nabla P,\qquad \nabla\cdot U=0 \] for a steady pressure $P$. Invariance along a column axis with unit direction $e$ means $(e\cdot\nabla)U=0$, wherever this expression is defined. The source does not specify the intended class of columnar vortex profiles, including whether axial flow is permitted.

Source question. What conditions, if any, ensure nonlinear stability of columnar vortices for the three-dimensional Euler equations?

A conditional mathematical template makes the missing stability choices explicit. Suppose an admissible metric space $(X,d_X)$ of divergence-free velocity fields, boundary conditions, and a solution class have been fixed, and suppose a global Euler evolution map $\Phi_t\colon X\to X$ is defined for every $t\geq0$. For a steady columnar vortex $U\in X$, nonlinear Lyapunov stability in $X$ would mean \[ \forall\varepsilon>0\ \exists\delta>0\ \forall u_0\in X:\quad d_X(u_0,U)<\delta \ \Longrightarrow\ \forall t\geq0,\quad d_X(\Phi_t(u_0),U)<\varepsilon. \] The intended problem could then ask for hypotheses on $U$, $X$, and the admissible perturbations ensuring this property. This is a conditional template, not a uniquely determined formulation of the source question. The record does not establish a global evolution map, choose the topology, distinguish stability of a fixed vortex from stability modulo symmetries, or specify whether perturbations may be arbitrary three-dimensional fields or must preserve a symmetry. If orbital stability is intended, the comparison with $U$ must instead be formulated using a specified symmetry orbit. Consequently, the supplied evidence supports the research question but does not determine a single fully quantified stability assertion.

Short English statement

Determine conditions under which columnar steady vortices of the three-dimensional incompressible Euler equations remain close to their initial equilibrium under small nonlinear perturbations. The domain, vortex profiles, perturbation class, notion of closeness, treatment of symmetries, and solution existence requirements remain unspecified. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry identified by the project instructions; supplied permanent UnsolvedMath ID $30004212$, source problem code catalogue entry, title Nonlinear Stability of Columnar Euler Vortices. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2019; no individual proposer is named.
• Supplied report citation: Mathematical Aspects of Hydrodynamics (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/38, pp. 2333--2393. https://doi.org/10.4171/owr/2019/38. The background describes this collection as drawn from workshop reports and their open-problem discussions. No exact problem page or speaker attribution is supplied.
• Supplied literature reference: Gallay and Smets, "stability of vortices." https://arxiv.org/abs/1901.02815. This is the supplied reference label, not an independently verified bibliographic title.
• Supplied literature reference: Miao, Ren, and Zhang, "asymptotic linear stability." https://arxiv.org/abs/2603.17268. This is the supplied reference label, not an independently verified bibliographic title.
• Supplied literature assessment, checked 2026-08-22: spectral and asymptotic linear stability results are available for important columnar vortices, but do not prove the requested nonlinear stability; the supplied assessment labels the problem open. This is supplied dated curation derived from the OWR collection, not independently verified here. The record explicitly calls for independent verification of time-sensitive status and cited claims.