Stability of torsion-free G_2 structures along the Laplacian flow

We prove that torsion-free G_2 structures are (weakly) dynamically stable along the Laplacian flow for closed G_2 structures. More precisely, given a torsion-free G_2 structure $\varphi$ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to $\varphi$ will converge to a torsion-free G_2 structure which is in the orbit of $\varphi$ under diffeomorphisms isotopic to the identity.