Abstract
We study the orbit structure of a vector fieldV defined on a three-dimensional Riemannian manifold which satisfiesV ^ curlV=0. Such a vector field represents the velocity of a stationary solution of Euler’s equation for a perfect fluid. In addition to several other results, we show that if the vector field admits a first integral, then each level set is toroidal and the induced flow on the level set is either periodic or conditionally periodic.
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Research supported by a grant from the Research Council of The Graduate School, University of Missouri.
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Chicone, C. The topology of stationary curl parallel solutions of Euler’s equations. Israel J. Math. 39, 161–166 (1981). https://doi.org/10.1007/BF02762862
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DOI: https://doi.org/10.1007/BF02762862