Founding bounds on a certain expression

Question

Suppose we have a bounded domain $\Omega$ with a boundary $\Gamma$. The space $L^2(\Omega)$ is equipped with the usual norm and inner product $|| \cdot||$ and $(\cdot , \cdot)$. I was working on a problem, and we want to find bounds on $L_E$ where: $$L_E = (f,\dot{u}) + (g,\dot{u})_{\Gamma} + (\dot{w}, \varphi) + (\dot{p}, \varphi)_{\Gamma} + (Q, \theta) - (q, \theta)_{\Gamma}$$ By using Holder's and Young's inequalities and trace theorem I found: $$L_E \leq C + \frac{C'}{2} \left[||\dot{u}||^2 + ||\nabla \dot{u}||^2 + ||\varphi||^2 + ||\nabla \varphi||^2 + ||\theta||^2 + ||\nabla \theta||^2 \right] $$ Where $C'$ is a positive constant (that comes from the trace inequality constants) and: $$ C = \frac{1}{2} \left[||f||^2 + ||g||^2_{\Gamma} + ||\dot{w}||^2 + ||\dot{p}||^2_{\Gamma} + ||Q||^2 + ||q||^2_{\Gamma} \right]$$ But in this article I am reading they found: $$L_E \leq C + \frac{1}{2} \left[||\dot{u}||^2 + ||\varphi||^2 + ||\theta||^2 \right] $$ Where did I go wrong? Or what am I missing?