Celli, M.; Lacomba, E. A.; Pérez-Chavela, E. On polygonal relative equilibria in the N-vortex problem. (English) Zbl 1272.76067 J. Math. Phys. 52, No. 10, 103101, 8 p. (2011). Summary: Helmholtz’s equations provide the motion of a system of \(N\) vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we first show that a relative equilibrium formed of a regular polygon and a possible vortex at the center, with more than three vertices on the polygon (two if there is a vortex at the center), requires equal vorticities on the polygon. We also provide an 8-vortex configuration, formed of two concentric squares making an angle of \(45^{\circ}\), with uniform vorticity on each square, which is in relative equilibrium for any value of the vorticities.{©2011 American Institute of Physics} Cited in 6 Documents MSC: 76B47 Vortex flows for incompressible inviscid fluids 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35Q35 PDEs in connection with fluid mechanics PDFBibTeX XMLCite \textit{M. Celli} et al., J. Math. Phys. 52, No. 10, 103101, 8 p. (2011; Zbl 1272.76067) Full Text: DOI arXiv References: [1] DOI: 10.1063/1.2425103 · Zbl 1144.81308 [2] DOI: 10.1016/S0065-2156(02)39001-X [3] DOI: 10.1063/1.1898143 · Zbl 1187.76024 [4] DOI: 10.1016/j.crma.2003.10.014 · Zbl 1079.70008 [5] Elmabsout B., Celest. Mech. Dyn. Astron. 41 pp 131– (1988) [6] DOI: 10.1090/S0002-9947-08-04685-0 · Zbl 1161.76011 [7] Havelock T. H., Philos. Mag. 11 (7) pp 617– (1931) [8] Helmholtz H., Philos. Mag. 33 pp 485– (1858) [9] DOI: 10.1175/1520-0469(2001)058<2196:MPFPAR>2.0.CO;2 [10] DOI: 10.1175/BAMS-85-2-151 [11] Morton W. B., Proc. R. Ir. Acad. A, Math. Phys. Sci. 41 pp 94– (1933) [12] DOI: 10.1175/BAMS-85-11-1663 [13] DOI: 10.1007/978-1-4684-9290-3 [14] DOI: 10.1016/j.physd.2007.07.015 · Zbl 1127.76016 [15] DOI: 10.1090/S0002-9939-1985-0784183-1 [16] DOI: 10.1103/PhysRevLett.43.214 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.