= Solution
To first order in fluid <velocity> divided by the <speed of light>, the <Lorentz transformation of electromagnetic fields> gives $\mathbf E'=\mathbf E+\mathbf u\times\mathbf B$. Applying <Ohm's law> in the local fluid frame gives $\mathbf E+\mathbf u\times\mathbf B=\eta\mathbf J$. Neglecting the <displacement current> in <Ampère's law>, $\mathbf J=\nabla\times\mathbf B/\mu_0$. Substitution in <Faraday's law> gives
$$
\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)-\frac\eta{\mu_0}\nabla\times(\nabla\times\mathbf B)
=\nabla\times(\mathbf u\times\mathbf B)+\eta_m\nabla^2\mathbf B,\qquad\eta_m=\frac\eta{\mu_0}.
$$
Here $\eta_m$ is the <magnetic diffusivity>; the paper's $\eta$ is <electrical resistivity>, rather than <magnetic diffusivity>. Expanding the curl with the <solenoidal magnetic-field constraint> gives the <resistive induction equation> in its compressible form:
$$
\boxed{\partial_t\mathbf B+(\mathbf u\cdot\nabla)\mathbf B=(\mathbf B\cdot\nabla)\mathbf u-\mathbf B\,\nabla\cdot\mathbf u+\frac\eta{\mu_0}\nabla^2\mathbf B.}
$$
The left side is the <material derivative> of the <magnetic field>. The three right-hand terms respectively describe field-line stretching and reorientation, amplification or dilution by compression or expansion, and <magnetic diffusion>. In particular, compression has $\nabla\cdot\mathbf u<0$ and amplifies a field component perpendicular to the compression.
The assumptions are nonrelativistic bulk motion, the low-frequency approximation permitting omission of the <displacement current>, a conducting-fluid continuum, isotropic local <Ohm's law> with spatially constant scalar <electrical resistivity>, constant <magnetic permeability> $\mu_0$, and $\nabla\cdot\mathbf B=0$. Hall, electron-pressure and other non-Ohmic corrections to the generalized <Ohm's law> are excluded. <Incompressible flow> is not required: imposing it would remove the compression term.
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