On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation
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In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. This fact is mainly sustained with the calculation of the speed of the center of mass; in particular, we show that several conjectures made at the numerical level are in agreement with the theoretical expectations. Moreover, due to the spatial periodicity, the evolution of VFE at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner; and this interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M-corner case; and the numerical experiments carried out provide new arguments that support the multifractal character of the trajectory defined by one of the corners of the initial polygon.
KeywordsIntermittency Multifractality Talbot Effect Transfer of Energy Turbulence Vortex filament equation
Mathematics Subject Classification11L05 28A80 35Q55 65M70 65T50 76B47
We want to thank V. Banica and C. García-Cervera for very enlightening conversations concerning the last two sections of this paper. Part of this work was started while the second author was visiting MSRI, within the New Challenges in PDE 2015 program. We also want to thank the anonymous reviewers for their very valuable comments.