  In this paper we consider the steepest descent $H^{-1}$-gradient flow of the
length functional for immersed plane curves, known as the curve diffusion flow.
It is known that under this flow there exist both initially immersed curves
which develop at least one singularity in finite time and initially embedded
curves which self-intersect in finite time. We prove that under the flow closed
curves with initial data close to a round circle in the sense of normalised
$L^2$ oscillation of curvature exist for all time and converge exponentially
fast to a round circle. This implies that for a sufficiently large `waiting
time' the evolving curves are strictly convex. We provide an optimal estimate
for this waiting time, which gives a quantified feeling for the magnitude to
which the maximum principle fails. We are also able to control the maximum of
the multiplicity of the curve along the evolution. A corollary of this estimate
is that initially embedded curves satisfying the hypotheses of the global
existence theorem remain embedded. Finally, as an application we obtain a
rigidity statement for closed planar curves with winding number one.
