  In the seminal paper (Alt, Caffarelli and Friedman, Trans. Amer. Math. Soc.,
282, (1984).), the regularity of the free boundary of two-phase fluid in two
dimensions via the so-called ACF energy functional was investigated. It was
shown the $C^1$ regularity of the free boundaries and asserted that the two
free boundaries coincide under some additional assumptions. Later on the
standard technique of Harnack inequality could be applied to improve the
regularity to $C^{1,\eta}$. A recent significant breakthrough in the regularity
of two-phase fluid is due to De Philippis, Spolaor and Velichkov, who
investigated the free boundary of the two-phase fluid with the two-phase
functional (De Philippis, Spolaor and Velichkov, Invent. Math., 225, (2021).),
and the $C^{1,\eta}$ regularity of the whole free boundaries was given in
dimension two. Moreover, the free boundaries of the two-phase fluids do not
coincide and the zero level set may process positive Lebesgue measure. In this
paper, we consider the free boundaries for the two-phase axisymmetric fluid and
show the free boundary is $C^{1,\eta}$ smooth. The Lebesgue measure of the zero
level set of may also be positive, and the main difference lies in the
degenerate elliptic operator and the free boundary conditions. More precisely,
we use partial boundary Harnack inequalities and establish a linearized
problem, whose regularity of the solutions implies the flatness decay of the
two-phase free boundaries. Then the iteration argument gives the smoothness of
the free boundaries.
