  We derive functional flow equations for the two-particle vertex and the
self-energy in interacting fermion systems which capture the full frequency
dependence of both quantities. The equations are applied to the hole-doped
two-dimensional Hubbard model as a prototype system with entangled magnetic,
charge and pairing fluctuations. Each fluctuation channel acquires substantial
dependencies on all three Matsubara frequencies, such that the frequency
dependence of the vertex cannot be accurately represented by a channel sum with
only one frequency variable in each term. At the temperatures we are able to
access, the leading instabilities are mostly antiferromagnetic, with an
incommensurate wave vector. However, at large doping, a divergence in the
charge channel occurs at a finite frequency transfer, if the vertex flow is
computed without self-energy feedback. This enigmatic instability was already
observed in a calculation by Husemann et al. [Phys. Rev. B 85, 075121 (2012)],
who used an approximate separable ansatz for the frequency dependence of the
vertex. We identify a simple mechanism for this instability in terms of a
random phase approximation for the charge channel with a frequency dependent
effective magnetic interaction as input. In spite of the strong momentum and
frequency dependence of the vertex, the self-energy has a Fermi liquid form. At
the moderate interaction strength where our approach is applicable, we obtain a
moderate reduction of the quasi-particle weight and a sizable decay rate with a
pronounced momentum dependence. Nevertheless, the self-energy feedback into the
vertex flow turns out to be crucial, as it suppresses the unphysical finite
frequency charge instability.
