Geometric Hydrodynamics in Open Problems — Problem 23

Definitions and mathematical statement

Let $M$ denote one of the geometric settings listed in the source: a half-plane, a quarter-plane, a hemisphere, a quarter-sphere, a disk, a torus, a cylinder, or a half-cylinder. The source also allows other settings through the qualification "etc." Let $N$ denote the number of vortices under consideration; the source describes $N$ as small but supplies neither specific values nor an upper bound.

If vortices are intended to be represented by points, write their positions as \[ q(t)=(q_1(t),\ldots,q_N(t)),\qquad q_i(t)\in M. \] This notation is conditional: the supplied record does not explicitly specify a point-vortex model, a governing evolution equation, or an admissible configuration space.

Research problem. Study in detail the motion and bifurcations of a small number of vortices on the listed geometric settings, and potentially on further manifolds as indicated by the source.

Unresolved qualifications. The supplied statement is a research programme rather than a quantified conjecture. A precise dynamical formulation requires specifying the vortex model, the allowed values of $N$, vortex strengths and other parameters, the geometry of $M$, and boundary conditions where applicable. It also requires identifying which motions or invariant objects are to be studied and which parameter variations and notion of bifurcation are intended. Boundary inclusion, behavior at corners, and treatment of collisions are not specified. No particular choices of these data are imposed here.

Short English statement

Study the motion and bifurcations of a small number of vortices on half- and quarter-planes, hemispheres and quarter-spheres, disks, tori, cylinders, half-cylinders, and possibly other manifolds. The source leaves the dynamical model, boundary conditions, number of vortices, and intended bifurcations unspecified, so a precise theorem-level target requires further formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800824$, exact source problem code catalogue entry, titled Geometric Hydrodynamics in Open Problems --- Problem 23, in the AMR Open Problem Lists collection. The record attributes the problem to Khesin, Misiolek, and Modin and gives the proposed year as 2022.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 23. Supplied source URL: https://arxiv.org/abs/2205.01143.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; no independent source visit or verification is asserted here.
• The local record labels the problem open. This is a supplied registry classification, not an independently verified assessment of the literature or mathematical status.