  We establish a correspondence between a parabolic complex Monge-Amp\`ere
equation and the $G_2$-Laplacian flow for initial data produced from a K\"ahler
metric on a complex $2$- or $3$-fold. By applying estimate for the complex
Monge-Amp\`ere equation, we show that for this class of initial data the
$G_2$-Laplacian flow exists for all time and converges to a torsion-free
$G_2$-structure induced by a K\"ahler Ricci-flat metric. Similar results are
obtained for the $G_2$-Laplacian coflow, and in this case the coflow is related
to the K\"ahler-Ricci flow.
