David Witt Nyström: A criterion for the local existence of Hele-Shaw flow on a Riemann surface

The talk is based on joint work with Julius Ross.

Hele-Shaw flow models the propagation of a fluid confined between two parallell plates a small distance apart. In the classical setting one disregards surface tension, and one lets the continuous injection or suction of fluid at a fixed point be the driving force. Given some regularity of the boundary of the domain occupied by the fluid at time zero, the local existence of this Hele-Shaw flow was proved by Kufarev and Vinogradov in 1948. The modern proof using the Cauchy-Kowalevski theorem was given in 1993 by Reissig and von Wolfersdorf in 1993.

Hedenmalm and Shimorin studied the analog Hele-Shaw flow on a Riemann surface. This also goes under the name of elliptic growth of Beltrami type, and physically it corresponds to the fluid propagating through a medium with varying permeability. In their 2002 paper Hedenmalm-Shimorin proved the local existence of this flow on a real analytic Riemann surface.

We give a criterion for the local existence of the Hele-Shaw flow on a general Riemann surface. In particular we get a proof of the local existence in the classical setting not relying on the Cauchy Kowalevski theorem.