Comparison of polynomial matrix differential operators

We characterize matrix polynomials $P,Q$ such that the inequality $$\left\Vert Q(D)u\right\Vert _{L^{2}}\leq C\left\Vert P(D)u\right\Vert _{L^{2}}\quad\text{for all }u\in C_c^\infty(\Omega),$$ holds on bounded open sets $\Omega$. We also characterize the operators $P,Q$ for which the linear continuous embedding above is compact, i.e., if $u_n\in C_c^\infty(\Omega)$ are such that $(P(D)u_n)_{n\geq 1}$ is bounded in $L^2$, then $(Q(D)u_n)_{n\geq 1}$ is strongly compact in $L^2$.