  We establish that the $C^{1,\gamma}$ regularity theory for translation
invariant fractional order parabolic integro-differential equations (via
Krylov-Safonov estimates) gives an improvement of regularity mechanism for
solutions to a special case of a two-phase free boundary flow related to
Hele-Shaw. The special case is due to both a graph assumption on the free
boundary of the flow and an assumption that the free boundary is
$C^{1,\text{Dini}}$ in space. The free boundary then must immediately become
$C^{1,\gamma}$ for a universal $\gamma$ depending upon the Dini modulus of the
gradient of the graph. These results also apply to one-phase problems of the
same type.
