I am currently trying to fully understand the stationary Stokes equations of incompressible fluid. In the mixed form (homogeneous boundary data), for $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, a system like $$ \left\{ \begin{array}{rl} a(\mathbf{u},\mathbf{v}) + b(\mathbf{v},p) &= 0\\ b(\mathbf{u},q) &= 0 \end{array} \right. $$ for all $\mathbf v\in (H^1(\Omega))^d$ and $q\in L^2(\Omega)$ with $b(\mathbf v,p) = -(\nabla\cdot \mathbf v,p)_{L^2(\Omega)}$ arises. For proving existence and uniqueness of a solution one can follow the book of V. Girault and P.-A. Raviart.
However, they do not mention how to prove continuity of these two bilinear forms. For $a(\cdot,\cdot)$ I have managed to do the proof by using Korns inequality. Concerning the second bilinear form $b(\cdot,\cdot)$ I have found the estimate $$ \|\nabla\cdot \mathbf{v}\|_{L^2(\Omega)}\leq \sqrt{d}\|\nabla\mathbf{v}\|_{L^2(\Omega)}. $$ But it remains totally unclear to me, how to show this estimate. Help is very welcome and thanks in advance!
Update:
I have gotten so far, that $$ \|\nabla\mathbf{v}\|^2_{L^2(\Omega)} = \int_\Omega \sum_{i,j=1}^d (\frac{\partial v_i}{\partial x_j})^2 $$ and $$ \|\nabla\cdot\mathbf{v}\|^2_{L^2(\Omega)} = \int_\Omega (\sum_{i=1}^d \frac{\partial v_i}{\partial x_i})^2, $$ but how does the $\sqrt{d}$ come in? I can see that some terms of the Jacobian term appear in the divergence term, but what about the mixed terms?
First prove that if $a_i\in\mathbb{R}$, then $$\sum_{i=1}^d|a_i|\leq\sqrt{d}\sqrt{\sum_{i=1}^d|a_i|^2}\tag{1}$$
The last inequality says that if $x=(a_1,...,a_d)$, then $\|x\|_1\leq \sqrt{d}\|x\|_2$, i.e., these norms are equivalently in $\mathbb{R}^d$
Let $(a_{ij})$ be the coefficients of a matrix $n\times n$. Note that
\begin{eqnarray} \left|\sum_{i=1}^d a_{ii}\right| &\leq& \sum_{i=1}^d |a_{ii}| \nonumber \\ &=& \sqrt{d}\sqrt{\sum_{i=1}^da_{ii}^2} \nonumber \\ &\leq& \sqrt{d}\sqrt{\sum_{i,j=1}^d a_{ij}^2} \tag{2} \end{eqnarray}
where in the first inequality we have used $(1)$.
We conclude from $(1)$ that $$\left|\sum_{i=1}^d a_{ii}\right|^2\leq d\sum_{i,j=1}^d a_{ij}^2 \tag{3}$$
Now you can apply $(3)$ to the Jacobian matrix and the result follows.