  Dynamical density functional theory (DDFT) has been successfully derived and
applied to describe on the one hand passive colloidal suspensions, including
hydrodynamic interactions between individual particles. On the other hand,
active "dry" crowds of self-propelled particles have been characterized using
DDFT. Here we go one essential step further and combine these two approaches.
We establish a DDFT for active microswimmer suspensions. For this purpose,
simple minimal model microswimmers are introduced. These microswimmers
self-propel by setting the surrounding fluid into motion. They hydrodynamically
interact with each other through their actively self-induced fluid flows and
via the common "passive" hydrodynamic interactions. An effective soft steric
repulsion is also taken into account. We derive the DDFT starting from common
statistical approaches. Our DDFT is then tested and applied by characterizing a
suspension of microswimmers the motion of which is restricted to a plane within
a three-dimensional bulk fluid. Moreover, the swimmers are confined by a
radially symmetric trapping potential. In certain parameter ranges, we find
rotational symmetry breaking in combination with the formation of a
"hydrodynamic pumping state", which has previously been observed in the
literature as a result of particle-based simulations. An additional instability
of this pumping state is revealed.
