Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators

We show that $C_0$-semigroups generated by first order partial differential operators on $L^p(\Omega,\mu)$ and $C_{0,\rho}(\Omega)$, respectively, are hypercyclic if and only if they are weakly mixing, where $\Omega\subset\mathbb{R}^d$ is open. In the case of $d=1$ we give an easy to check characterization of when this happens. Furthermore, we give an example of a hypercyclic evolution family such that not all of the operators of the family are hypercyclic themselves. This stands in complete contrast to hypercyclic $C_0$-semigroups.