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

TL;DR

This paper proves that under certain smoothness assumptions, the free boundary in a specific two-phase Hele-Shaw flow becomes smoother instantly, using regularity theory for fractional parabolic equations.

Contribution

It introduces a regularity improvement result for free boundaries in two-phase Hele-Shaw flows based on fractional integro-differential equations and Dini continuity assumptions.

Findings

Free boundary becomes $C^{1, ext{ extgamma}}$ immediately.

Regularity results extend to one-phase problems.

Uses Krylov-Safonov estimates for fractional equations.

Abstract

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.