I'm studying regularity for linear elasticity problem where I encountered
$\nabla (\nabla \cdot u) = \nabla \times (\nabla \times u) + \Delta u$
I am hoping to find a bound of $\|\nabla (\nabla \cdot u)\|$ with respect to $\|u\|_2$ or any norm of $u$. something like
$\|\nabla (\nabla \cdot u)\|_{L^2} \leq \|\nabla \times (\nabla \times u)\|_{L^2} + \|\Delta u\|_{L^2} \leq f(\|u\|_{H^2}, \|u\|_{H1}, ...)$
But I could not find one. I only manage to show $\|\nabla \times u\|_{L^2} \leq C\|u\|_{H^1}$, I would like to ask is there an inequality for $\|\nabla(\nabla\cdot u)\|_{L^2}$ and $\|\nabla \times (\nabla \times u) \|_{L^2}$ with respect to $\|u\|_{H^2}$.
One way to look at this is to recall that $$ \nabla \times (\nabla \times u) = \det \begin{pmatrix} e_1 & e_2 & e_3\\ \partial_1 & \partial_2 & \partial_3\\ v_1 & v_2 & v_3 \end{pmatrix} , $$ where $v=\nabla \times u$, and $e_i$ are the canonical vectors in $\mathbb{R}^3$. Put another way we have $$ \nabla \times(\nabla \times u)= \left( \partial_2 v_3 - \partial_3 v_2, \partial_3 v_1 - \partial_1 v_3, \partial_1 v_2- \partial_2 v_1 \right). $$ Now, for instance, lets focus on the first term: We know $$ v= \left( \partial_2 u_3- \partial_3 u_2, \partial_3 u_1 - \partial_1 u_3, \partial_1 u_2 - \partial_2 u_1\right), $$ and so $$ \partial_2 v_3 - \partial_3 v_2 = \partial_2(\partial_1 u_2 - \partial_2 u_1) - \partial_3(\partial_3 u_1- \partial_1 u_3), $$ and it's clear from the triangle inequality that the $L^2$ norm of this quantity is controlled by the $H^2$ norm of $u$.