We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White’s local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with sufficiently small Lipschitz constant.
"On short time existence for the planar network flow." J. Differential Geom. 111 (1) 39 - 89, January 2019. https://doi.org/10.4310/jdg/1547607687