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Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

We establish that the $C^{1,γ}$ 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 $C^{1,γ}$ for a universal $γ$ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.