  In this paper, we prove that on a Fano manifold $M$ which admits a
K\"ahler-Ricci soliton $(\om,X)$, if the initial K\"ahler metric
$\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak
K\"ahler-Ricci flow exists globally and converges in Cheeger-Gromov sense.
Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified
K\"ahler-Ricci flow converges exponentially to a unique K\"ahler-Ricci soliton
nearby. Especially, if the Futaki invariant vanishes, we may delete the
$K_X$-invariant assumption. The methods based on the metric geometry of the
space of the K\"ahler metrics are potentially applicable to other stability
problem of geometric flow near a critical metric.
