  We prove that the volume preserving fractional mean curvature flow starting
from a convex set does not develop singularities along the flow. By the recent
result of Cesaroni-Novaga \cite{CN} this then implies that the flow converges
to a ball exponentially fast. In the proof we show that the apriori estimates
due to Cinti-Sinestrari-Valdinoci \cite{CSV2} imply the
$C^{1+\alpha}$-regularity of the flow and then provide a regularity argument
which improves this into $C^{2+\alpha}$-regularity of the flow. The regularity
step from $C^{1+\alpha}$ into $C^{2+\alpha}$ does not rely on convexity and can
probably be adopted to more general setting.
