Abstract
Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in \({\mathbb {C^n}}\) with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.
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Aleman, A., Constantin, A. Harmonic Maps and Ideal Fluid Flows. Arch Rational Mech Anal 204, 479–513 (2012). https://doi.org/10.1007/s00205-011-0483-2
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DOI: https://doi.org/10.1007/s00205-011-0483-2