  We develop foundational theory for the Laplacian flow for closed G_2
structures which will be essential for future study. (1). We prove Shi-type
derivative estimates for the Riemann curvature tensor Rm and torsion tensor T
along the flow, i.e. that a bound on $\Lambda(x,t)=\left(|\nabla
T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all
covariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blow
up at a finite-time singularity, so the flow will exist as long as
$\Lambda(x,t)$ remains bounded. (3). We give a new proof of forward uniqueness
and prove backward uniqueness of the flow, and give some applications. (4). We
prove a compactness theorem for the flow and use it to strengthen our long time
existence result from (2). (5). Finally, we study compact soliton solutions of
the Laplacian flow.
