Where can I find surveys on the mathematical aspects of the vortex filament equation?

In particular, I'm interested in the following topics:

  • physical motivation;
  • notion of solutions and wellposedness;
  • relationship with the Schrödinger equation;
  • relationship with Euler and Navier-Stokes equations;
  • formation of singularities.

Note 1. Vortex filament equation: $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$.

Note 2. Related questions have been asked at the following links:


The book by Majda and Bertozzi would seem to have some of what you're looking for https://www.cambridge.org/core/books/vorticity-and-incompressible-flow/393C35E544EDD0711CAA7F7AB05D7432

This paper by Jerrard and Seis also has some more recent work on weak solutions to the vortex filament equation https://arxiv.org/pdf/1603.00227.pdf

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  • $\begingroup$ Thank you. Do you know of references where the equation has been studied from the point of view and with the techniques of differential geometry? $\endgroup$ – Kei May 15 '19 at 21:12

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