I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$
\begin{align*} \mathrm{d} \omega_R+\kappa_R\left(\omega_R\right) \mathcal{L}_{v_R} \omega_R \mathrm{~d} t+\sum_{k=1}^{\infty} \mathcal{L}_{\xi_k} \omega_R \mathrm{~d} B_t^k=\frac{1}{2} \sum_{k=1}^{\infty} \mathcal{L}_{\xi_k}^2 \omega_R \mathrm{~d} t,\left.\quad \omega_R\right|_{t=0}=\omega_0 \end{align*}
where $\mathcal{L}_v\omega$ denotes the Lie derivative defined as $(v.\nabla)\omega -(\omega.\nabla)v$, $\omega_R= \text{curl}~v_R$, the processes $B^k$ with $k \in \mathbb{N}$ are scalar independent Borwnian motions, $v, \xi_k$ are some vector fields in $\mathbb{R}^3$, $\kappa_R(\omega):=f_R\left(\|\nabla v\|_{\infty}\right)$, where $f_R$ is a smooth function, equal to 1 on $[0, R]$, equal to 0 on $[R+1, \infty)$ and decreasing in $[R, R+1]$. They consider a stopping time $\tau_R = \inf \{ t \ge 0 : \|\omega\|_{H^2} \ge \frac{R}{C}\}$ where $C$ is the constant from Sobolev embedding $\| \nabla v\|_\infty \le C \| \omega \|_{H^2}$. The proof starts with the possibility of two solutions $\omega^{(1)}_R, \omega^{(2)}_R$ of the equation under consideration and claims that the term $$\Phi:= \kappa_R(\omega^{(1)}) \mathcal{L}_{v^{(1)}} \omega^{(1)}- \kappa_R(\omega^{(2)}) \mathcal{L}_{v^{(2)}} \omega^{(2)}$$ is identically zero on the set $\tau^{(2)} \le \tau^{(1)}$ if $\| w^{(1)}\|_{H^2} \ge \frac{R}{C}$. I could not figure out why this is true.
I think the operator $\mathcal{L}_v\omega$ playes no role in this claim and somehow the coefficients $\kappa_R(\omega^{(1)})$ and $\kappa_R(\omega^{(2)})$ vanish on this set. For this we require $\|\nabla v^{(1)}\|_\infty \ge R+1$ and $\|\nabla v^{(2)}\|_\infty \ge R+1$. Now, from the definition of stopping time $\|\omega^{(1)}\|_{H^2} \ge R/C$ but this does not imply $\| \nabla v^{(1)}\|_\infty \ge R+1$. So how shall I use the definition of stopping time?