# reference request – Relationship between the vortex filament equation and the transport equation

Consider the vortex filament equation
$$partial_t chi = partial_s chi wedge partial_ {ss} chi,$$
where $$chi (t, s)$$ it's a curve in $$mathbb R ^ 3$$.

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

Note 1. I have asked a more general question in the Vortex Strand Equation Survey.