  We investigate stability properties of the motion by curvature of planar
networks. We prove Lojasiewicz-Simon gradient inequalities for the length
functional of planar networks with triple junctions. In particular, such an
inequality holds for networks with junctions forming angles equal to
$\tfrac23\pi$ that are close in $H^2$-norm to minimal networks, i.e., networks
whose edges also have vanishing curvature. The latter inequality bounds a
concave power of the difference between length of a minimal network $\Gamma_*$
and length of a triple junctions network $\Gamma$ from above by the $L^2$-norm
of the curvature of the edges of $\Gamma$. We apply this result to prove the
stability of minimal networks in the sense that a motion by curvature starting
from a network sufficiently close in $H^2$-norm to a minimal one exists for all
times and smoothly converges. We further rigorously construct an example of a
motion by curvature having uniformly bounded curvature that smoothly converges
to a degenerate network in infinite time.
