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.

does the vortex filament equation have any transport-like structure? $\endgroup$ – Kei May 15 at 21:07