We study the Schrödinger equation in a periodic domain
\begin{equation} \left\{ \begin{array}{ll} i\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} = 0 \qquad & x \in [0, \ell(t)] \\ u(0, t) = 0, u(\ell(t), t) = 0 \qquad & t \ge 0\\ u(x,0) = u_0 \qquad & x \in [0,1]\\ \end{array} \right. \end{equation}
Here, $t \in [0,T]$, $\ell(t) = 1+\varepsilon \sin(\omega t)$ where $\varepsilon \in (0,1)$ and $\omega \in (0,\frac{4}{\varepsilon(1+\varepsilon)})$.
Due to my research, after some transformation, we need the following condition:
\begin{equation} \sup_{t \in [0,T]}|\ell(t )u_x(\ell(t),t)|^2 \leq c_T^2 < \frac{1-\varepsilon}{\varepsilon}\int_{0}^{1}|u_x(x,0)|^2 dx \end{equation}
where $c_T$ is some constant depending on $T$.
Can anyone help me to explain the physical meaning behind this formula. Thank you very much.