  Given a $7$-dimensional compact Riemannian manifold $\left( M,g\right) $ that
admits $G_{2}$-structure, all the $G_{2}$-structures that are compatible with
the metric $g$ are parametrized by unit sections of an octonion bundle over
$M$. We define a natural energy functional on unit octonion sections and
consider its associated heat flow. The critical points of this functional and
flow precisely correspond to $G_{2}$-structures with divergence-free torsion.
In this paper, we first derive estimates for derivatives of $V\left( t\right) $
along the flow and prove that the flow exists as long as the torsion remains
bounded. We also prove a monotonicity formula and and an $\varepsilon
$-regularity result for this flow. Finally, we show that within a metric class
of $G_{2}$-structures that contains a torsion-free $G_{2}$-structure, under
certain conditions, the flow will converge to a torsion-free $G_{2}$-structure.
