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We establish a correspondence between a parabolic complex Monge-Amp\`ereequation and the $G_2$-Laplacian flow for initial data produced from a K\"ahlermetric on a complex $2$- or $3$-fold. By applying estimate for the complexMonge-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 areobtained for the $G_2$-Laplacian coflow, and in this case the coflow is relatedto the K\"ahler-Ricci flow.

Flows of $G_2$-Structures associated to Calabi-Yau Manifolds