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Vortex filaments for Euler equations

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TURW02 - Rigorous analysis of incompressible fluid models and turbulence

We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a cruve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called “binormal curvature flow”. Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. In this talk I will discuss the construction of helical filaments, associated to a translating-rotating helix, and of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).

This talk is part of the Isaac Newton Institute Seminar Series series.

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