identically zero in (\[E:eta-T-evolution\]) and (\[E:eta-norm-evolution\]).

\[ex:strip\] Consider the vertical strip $\Omega:=\{(x,y) \in  \mathbb{R}^2 : 0 \leq x \leq 1\}$ with boundary $\Sigma=\partial \Omega$ consists of two parallel vertical lines. Let $p_0=(0,-h/2)$ and $q_0=(1,h/2)$ for some $h>0$. It is easy to check that the solution to the chord shortening flow with initial chord $C_{p_0,q_0}$ is given by $p_t=(0,-h(t)/2)$, $q_t=(1,h(t)/2)$ where $h(t)$ is the unique solution to the ODE $$h'(t)=-\frac{2h(t)}{\sqrt{1+h^2(t)}}$$ with initial condition $h(0)=h$. From this we can see that the solution $h(t)$ exists for all $t \geq 0$. Moreover, $-h'(t) \leq 2h(t)$ implies $h(t) \leq h e^{-2t}$ and thus $h(t) \to 0$ exponentially as $t \to +\infty$. Therefore, the chord converges to a chord meeting $\partial \Omega$ orthogonally. In this case, we have $$-\eta^T(0)= \frac{1}{\sqrt{1+h^2(t)}} (0,h(t)) = \eta^T(1),$$ which satisfies the evolution equation (\[E:eta-T-evolution\]) and $\eta^T \to 0$ as $t \to +\infty$. See Figure \[F:strip\].

![A chord converging to a limit chord orthogonal to $\partial \Omega$.[]{data-label="F:strip"}](strip.png){height="6cm"}

Long time existence {#S:convergence}
===================

In this section, we prove our main convergence result which says that the only two possible convergence scenarios are given in Example \[ex:flat2\] and \[ex:strip\]. One should compare this convergence result with a similar result of Grayson [@Grayson89 Theorem 