The vortex filament equation, the Talbot effect and non-circular jets

DATES: 4 - 8 November 2019 (5 sessions)
TIME: 09:40 - 11:40 (a total of 10 hours)
LOCATION: Campus UPV/EHU Leioa, Faculty of Science and Technology
CLASSROOM:
- Monday: Classroom 0.22
- Tuesday: Classroom 0.26
- Wednesday: Classroom 0.22
- Thursday: Classroom 2.5
- Friday: Classroom 0.22

PROGRAMME:
1. Vortex Filament Equation: the localized induction approximation of Biot-Savart Integral and Hasimoto transformation.
2. Self-similar solutions: characterization and their connection with the case of regular polygons. The turbulence of non-circular jets. Non-linear Talbot effect
3. Skew polygonal lines: self-similar solutions have finite energy. Formation of singularities and continuation of the solutions beyond the blow-up time.

ABSTRACT:
We propose the vortex filament equation as a possible toy model for turbulence, in particular because of its striking similarity to the dynamics of non-circular jets. This similarity implies the existence of some type of Talbot effect due to the interaction of non-linear waves that propagate along the filament. Another consequence of this interaction is the existence of a new class of multi-fractal sets that can be seen as a generalization of the graph of Riemann´s non-differentiable function. Theoretical and numerical arguments about the transfer of energy will be also given. This a joint work with V. Banica and F. de la Hoz.

BIBLIOGRAPHY:
[1] V. Banica and L. Vega, The initial value problem for the binormal flow with rough data, Ann. Sci. E ́c. Norm. Sup ́er. 48 (2015), 1421-1453.
[2] V. Banica and L. Vega, Singularity formation for the 1-D cubic NLS and the Schro ̈dinger map on S2, Comm. Pure Appl. Anal. 17 (2018), 1317-1329.
[3] V. Banica and L. Vega, Evolution of polygonal lines by the binormal flow, ArXiv 1807.06948.
[4] H. Hashimoto, A soliton on a vortex filament. J. Fluid Mech. 51(3), 477-485 (1972)
[5] L.S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo 22 (1906), 117-135.
[6] F. de la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity 27 (2014), 3031-3057.
[7] F. de la Hoz and L. Vega, On the relationship between the one-corner problem and the M-corner problem for the vortex filament equation, J. Nonlinear Sci., 28 (2018), 2275-2327.
[8] S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Commun. PDE 28 (2003) 927-968.
[9] H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech. 51 (1972), 477-485.

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• BCAM_UPV_Course_2019_11_04