Fahran Abedin

Hele-Shaw Flow and Parabolic Integro-Differential Equations

I will present a regularization result for a special case of the two-phase Hele-Shaw free boundary problem (a.k.a. interfacial Darcy flow), which models the evolution of two immiscible fluids flowing in the narrow gap between two parallel plates and subject to an external pressure source. Assuming that the fluid interface is given by the graph of a function, recent work of Chang-Lara, Guillen, and Schwab establishes the equivalence between the Hele-Shaw free boundary problem and a first-order parabolic integro-differential equation. By exploiting this equivalence and using available regularity theory for nonlocal parabolic equations, we show that if the gradient of the graph of the fluid interface has a Dini modulus of continuity for all times, then the gradient must be Holder continuous. This is joint work with Russell Schwab (MSU).