Estimates in Hölder spaces

Question

Let $u,v\in C^{2,\alpha}\left(\overline{\Omega}\right)$. Proof that there exists a constant $C>0$ so that \begin{equation} \|\Delta v\left(|\nabla v|^2-|\nabla u|^2\right)\|_0\leq C\left(\|u\|_2+\|v\|_2\right)^2\|v-u\|_2. \end{equation} I've tried to solve this problem but the best I could do is to estimate this quantity by \begin{equation} \|\Delta v\left(|\nabla v|^2-|\nabla u|^2\right)\|_0\leq\|v\|_2 \cdot\left(\||\nabla u|\|_0+\||\nabla v|\|_0\right)\||\nabla(v-u)|\|_0,\end{equation} although I don't know if it's correct. If this estimate was correct, I just have to prove that if $u\in C^{2,\alpha}\left(\overline{\Omega}\right)$ then \begin{equation} \||\nabla u|\|_0\leq\|u\|_2, \end{equation} but I don't know how to do it...