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.