Łojasiewicz–Simon inequalities for minimal networks: stability and convergence

We investigate stability properties of the motion by curvature of planar networks. We prove Łojasiewicz–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 23π 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 Γ∗ and length of a triple junctions network Γ from above by the L^2-norm of the curvature of the edges of Γ. 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.