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We develop foundational theory for the Laplacian flow for closed G_2structures which will be essential for future study. (1). We prove Shi-typederivative estimates for the Riemann curvature tensor Rm and torsion tensor Talong the flow, i.e. that a bound on $\Lambda(x,t)=\left(|\nablaT(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on allcovariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blowup 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 uniquenessand prove backward uniqueness of the flow, and give some applications. (4). Weprove a compactness theorem for the flow and use it to strengthen our long timeexistence result from (2). (5). Finally, we study compact soliton solutions ofthe Laplacian flow.

Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness