  In this paper, we will prove the global existence of solutions to the three
dimensional axially symmetric Prandtl boundary layer equations with small
initial data, which lies in $H^1$ Sobolev space with respect to the normal
variable and is analytical with respect to the tangential variables. Proof of
the main result relies on the construction of a tangentially weighted analytic
energy functional, which acts on a specially designed good unknown. The
constructed energy functional can find its two dimensional parallel in
Ignatova-Vicol [2016ARMA] where no tangential weight is introduced and the
specially good unknown is set to control the lower bound of the analytical
radius, whose two dimensional similarity can be traced to Paicu-Zhang
[2021ARMA]. Our result is an improvement of that in Ignatova-Vicol [2016ARMA]
from the almost global existence to the global existence and an extension of
that in Paicu-Zhang [2021ARMA] from the two dimensional case to the three
dimensional axially symmetric case.
