  This paper contains a new proof of the short-time existence for the flow by
curvature of a network of curves in the plane. Appearing initially in
metallurgy and as a model for the evolution of grain boundaries, this flow was
later treated by Brakke \cite{Br} using varifold methods. There is good reason
to treat this problem by a direct PDE approach, but doing so requires one to
deal with the singular nature of the PDE at the vertices of the network. This
was handled in cases of increasing generality by Bronsard-Reitich \cite{BrRe},
Mantegazza-Novaga-Tortorelli \cite{MNT} and eventually, in the most general
case of irregular networks by Ilmanen-Neves-Schulze \cite{INS}. Although the
present paper proves a result similar to the one in \cite{INS}, the method here
provides substantially more detailed information about how an irregular network
`resolves' into a regular one. Either approach relies on the existence of
self-similar expanding solutions found in \cite{MS}. As a precursor to and
illustration of the main theorem, we also prove an unexpected regularity result
for the mixed Cauchy-Dirichlet boundary problem for the linear heat equation on
a manifold with boundary.
