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.




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**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: 


* [Derivation of the vortex filament equation from Euler equation](https://mathoverflow.net/questions/331609/derivation-of-the-vortex-filament-equation-from-euler-equation),
* [Relationship between the vortex filament equation and the cubic Schrödinger equation](https://mathoverflow.net/questions/331614/relationship-between-the-vortex-filament-equation-and-the-cubic-schr%C3%B6dinger-equa),
* [Relationship between the vortex filament equation and the transport equation](https://mathoverflow.net/questions/331621/relationship-between-the-vortex-filament-equation-and-the-transport-equation).



Surveys/monographs on the vortex filament equation
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>


