  We study the fillability (or embeddability) of 3-dimensional $CR$ structures
under the geometric flows. Suppose we can solve a certain second order equation
for the geometric quantity associated to the flow. Then we prove that if the
initial $CR$ structure is fillable, then it keeps having the same property as
long as the flow has a solution. We discuss the situation for the torsion flow
and the Cartan flow. In the second part, we show that the above mentioned
second order operator is used to express a tangency condition for the space of
all fillable or embeddable $CR$ structures at one embedded in $\mathbb{C}^{2}.$
