Peralta-Salas, Daniel; Rechtman, Ana; Torres De Lizaur, Francisco A characterization of 3D steady Euler flows using commuting zero-flux homologies. (English) Zbl 1468.35122 Ergodic Theory Dyn. Syst. 41, No. 7, 2166-2181 (2021). Summary: We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved. Cited in 10 Documents MSC: 35Q31 Euler equations 76B99 Incompressible inviscid fluids 57R25 Vector fields, frame fields in differential topology 57R30 Foliations in differential topology; geometric theory Keywords:smooth dynamics; symplectic dynamics; Euler equations; volume-preserving flows × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Arnold, V. I. and Khesin, B. A.. Topological Methods in Hydrodynamics. Springer, New York, 1998. · Zbl 0902.76001 [2] Cardona, R., Miranda, E. and Peralta-Salas, D.. Euler flows and singular geometric structures. Phil. Trans. R. Soc. A377 (2019), 0034. · Zbl 1462.76048 [3] Cieliebak, K. and Volkov, E.. First steps in stable Hamiltonian topology. J. Eur. Math. Soc. (JEMS)17 (2015), 321-404. · Zbl 1315.53097 [4] Cieliebak, K. and Volkov, E.. A note on the stationary Euler equations of hydrodynamics. Ergod. Th. & Dynam. Sys.37 (2017), 454-480. · Zbl 1362.35218 [5] Etnyre, J. and Ghrist, R.. Contact topology and hydrodynamics I. Nonlinearity13 (2000), 441-458. · Zbl 0982.76021 [6] Etnyre, J. and Ghrist, R.. Contact topology and hydrodynamics II. Ergod. Th. & Dynam. Sys.22 (2002), 819-833. · Zbl 1098.76011 [7] Ginzburg, V. and Khesin, B.. Steady fluid flows and symplectic geometry. J. Geom. Phys.14 (1994), 195-210. · Zbl 0805.58023 [8] Izosimov, A. and Khesin, B.. Characterization of steady solutions to the 2D Euler equation. Int. Math. Res. Not. IMRN2017 (2017), 7459-7503. · Zbl 1405.35151 [9] Kuperberg, G.. A volume-preserving counterexample to the Seifert conjecture. Comment. Math. Helv.71 (1996), 70-97. · Zbl 0859.57017 [10] Morgan, F.. Geometric Measure Theory. Academic Press, Boston, 1988. · Zbl 0671.49043 [11] Peralta-Salas, D.. Selected topics on the topology of ideal fluid flows. Int. J. Geom. Methods Mod. Phys.13 (2016), 1630012 (1-23). · Zbl 1469.76011 [12] Rechtman, A.. Use and disuse of plugs in foliations. PhD Thesis, ENS Lyon, 2009. [13] Rechtman, A.. Existence of periodic orbits for geodesible vector fields on closed 3-manifolds. Ergod. Th. & Dynam. Sys.30 (2010), 1817-1841. · Zbl 1214.37015 [14] Simon, B.. Convexity: An Analytic Viewpoint. Cambridge University Press, Cambridge, 2011. · Zbl 1229.26003 [15] Sullivan, D.. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math.36 (1976), 225-255. · Zbl 0335.57015 [16] Sullivan, D.. A foliation of geodesics is characterized by having no tangent homologies. J. Pure Appl. Algebra13 (1978), 101-104. · Zbl 0402.57015 [17] Tao, T.. On the universality of potential well dynamics. Dyn. PDE14 (2017), 219-238. · Zbl 1383.37016 [18] Tischler, D.. On fibering certain foliated manifolds over 𝕊^1. Topology9 (1970), 153-154. · Zbl 0177.52103 [19] Wilson, F. W.. On the minimal sets of non-singular vector fields. Ann. of Math. (2)84 (1966), 529-536. · Zbl 0156.43803 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.