Let us consider the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$.

How is the Cauchy problem for the vortex filament equation related to the Cauchy problem for the linear transport equation $$ \partial_t u + \operatorname{div}(\boldsymbol{b} \,u) =0, $$ where $\boldsymbol b$ is a suitably chosen vector field?

In other words, the vortex filament equation comes from the Euler equation under suitable assumptions and can be transformed in the Schrödinger equation with a certain transformation, but does it also have any transport-like structure of the kind displayed above?

Note 1. I've asked a more general question at Surveys/monographs on the vortex filament equation.

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    $\begingroup$ apologies for interrupting, but wouldn't it make more sense to read some of the pointers to the literature that have been given in the previous questions you asked on this very same equation, in particular in the answer to your "survey" question, and then come back later when you have a more specific question on something that remains unclear after studying the literature? Firing off a series of closely related questions is not likely to be a productive way to engage with Mathoverflow. $\endgroup$ – Carlo Beenakker May 15 at 20:19
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    $\begingroup$ @CarloBeenakker Thank you for the suggestion, I'll be more careful. I chose so single out three questions from the most general one on surveys/monographs just because they are three specific issues that have come up (as opposed to a more general query) that are both related and different enough to be of interest for different communities. The present question particularly is related to the other two (on Schroedinger and Euler equations) because I wonder: other than coming from Euler and being transformed into Schroedinger, does the vortex filament equation have any transport-like structure? $\endgroup$ – Kei May 15 at 21:07

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