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

We establish a correspondence between a parabolic complex Monge-Ampère equation and the $G_2$-Laplacian flow for initial data produced from a Kähler metric on a complex $2$- or $3$-fold. By applying estimate for the complex Monge-Ampère 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 Kähler Ricci-flat metric. Similar results are obtained for the $G_2$-Laplacian coflow, and in this case the coflow is related to the Kähler-Ricci flow.