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arXiv · 2008.01272

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

Farhan Abedin·Russell W. Schwab
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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.

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Farhan Abedin, Russell W. Schwab. 2020-08-04. Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations. https://arxiv.org/abs/2008.01272

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