Consider the vector valued system of PDE's for $\textbf{f} = [f_1,f_2,f_3]$ and $\textbf{g} = [g_1,g_2,g_3]$
\begin{equation} \dfrac{\partial}{\partial t}\textbf{f}- F\left(\textbf{f}, \nabla f_i, \Delta\textbf{f}\right) = 0 \tag{1} \end{equation}
\begin{equation} \dfrac{\partial}{\partial t}\textbf{g}- G\left(\textbf{g}, \nabla g_i, \Delta\textbf{g}\right) = 0 \tag{2} \end{equation}
with initial condition $g_i(x,0) = f_i(x,0)$. If $F_i\le G_i$ is it true that $|\textbf{f}|\le |\textbf{g}|$ and $f_i(x,t)\le g_i(x,t)$? Here $F$ and $G$ are both vector fields $(F = [F_1,F_2,F_3])$ not to be mistaken with $[f_1,f_2,f_3]$. For example, we can take $$F = \nu \Delta\mathbf{f}-\mathbf{f}\cdot \nabla f_i ,~~~ \nu>0$$ and attempt to find $G$ such that $F_i\le G_i.$ We can take $G = |F|$ because the norm $|F|$ is always larger than $F_i$ for each integer $i$, but this seem to further complicate the equation. A simple trick would be to find a $G$ so that we can solve Equation 2 is solvable or at least has an end behavior that we can determine (but does not blow up). In essence we want to bound $F$ with something we can solve.