**November 04, 2019 at 09:00 - November 08, 2019**
- BCAM & UPV/EHU

**Prof. Luis Vega (BCAM)**

**DATES:** 4 - 8 November 2019 (5 sessions)

**TIME:** 09:00 - 11:00 (a total of 10 hours)

**LOCATION:** BCAM Seminar room

**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.

** *Registration is free, but mandatory before October 31st.**
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