  We consider the PDE flow associated to Riemann zeta and general Dirichlet
$L$-functions. These are models characterized by nonlinearities appearing in
classical number theory problems, and generalizing the classical holomorphic
Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet
$L$-function is an exact solution of the model. In this paper, we first show
local existence of bounded continuous solutions in the Duhamel sense to any
Dirichlet $L$-function flow with initial condition far from the pole (as long
as this exists). In a second result, we prove global existence in the case of
nonlinearities of the form Dirichlet $L$-functions and data initially on the
right of a possible pole. Additional global well-posedness and convergence
results are proved in the case of the defocusing Riemann zeta nonlinearity and
initial data located on the real line and close to the trivial zeros of the
zeta. The asymptotic stability of any stable zero is also proved. Finally, in
the Riemann zeta case, we consider the ``focusing'' model, and prove blow-up of
solutions near the pole $s=1$.
