  We define functionals generalising the Seiberg-Witten functional on closed
$spin^c$ manifolds, involving higher order derivatives of the curvature form
and spinor field. We then consider their associated gradient flows and, using a
gauge fixing technique, are able to prove short time existence for the flows.
We then prove energy estimates along the flow, and establish local
$L^2$-derivative estimates. These are then used to show long time existence of
the flow in sub-critical dimensions. In the critical dimension, we are able to
show that long time existence is obstructed by an $L^{k+2}$ curvature
concentration phenomenon.
