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Variety of unsymmetric multibranched logarithmic vortex spirals

Published online by Cambridge University Press:  22 December 2017

V. ELLING
Affiliation:
Department of Mathematics, Academia Sinica, 6F Astronomy-Mathematics Bldg., No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan email: velling@math.sinica.edu.tw
M. V. GNANN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching near Munich, Germany email: gnann@ma.tum.de

Abstract

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

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Papers
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

The authors' research was partially supported by the National Science Foundation under Grant NSF DMS-1054115 and by Taiwan MOST grant 105-2115-M-001-007-MY3.

References

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