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

2020/08/04 by Farhan Abedin, Abedin, Farhan, Russell W. Schwab +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.2008.01272

Modified the presentation of Theorem 1.2 to make more clear the dependence of the constants on various parameters and norms. Fixed various typos

arxiv created 2020/08/27 · arxiv updated 2020/08/28

Abstract

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,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.

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