  In this paper, we address two main topics: steady propagation fields for
positive streamers in air and streamer deceleration in fields below the steady
propagation field. We generate constant-velocity positive streamers in air with
an axisymmetric fluid model, by initially adjusting the applied voltage based
on the streamer velocity. After an initial transient, we observe steady
propagation for velocities of $3\times10^4$ m/s to $1.2\times10^5$ m/s, during
which streamer properties and the background field do not change. This
propagation mode is not fully stable, in the sense that a small change in
streamer properties or background field eventually leads to acceleration or
deceleration. An important finding is that faster streamers are able to
propagate in significantly lower background fields than slower ones, indicating
that there is no unique stability field. We relate the streamer radius,
velocity, maximal electric field and background electric field to a
characteristic time scale for the loss of conductivity. This relation is
qualitatively confirmed by studying streamers in N2-O2 mixtures with less
oxygen than air. In such mixtures, steady streamers require lower background
fields, due to a reduction in the attachment and recombination rates. We also
study the deceleration of streamers, which is important to predict how far they
can propagate in a low field. Stagnating streamers are simulated by applying a
constant applied voltage. We show how the properties of these streamers relate
to the steady cases, and present a phenomenological model with fitted
coefficients that describes the evolution of the velocity and radius. Finally,
we compare the lengths of the stagnated streamers with predictions based on the
conventional stability field.
