Is it possible to show for $u:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ that $$\||\nabla(\nabla\cdot u)\||_2^2\leq C\||\Delta u\||_2^2?$$

Here $\||f\||_2$ is the norm in $(L^2(\Omega))^3$ and derivatives are taken in the sense of distributions.

Is it possible to show for $u:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ that $$\||\nabla(\nabla\cdot u)|\|_2^2\leq C\||\Delta u|\|_2^2?$$

Here $\||f|\|_2$ is the norm in $(L^2(\Omega))^3$ and derivatives are taken in the sense of distributions.