Abstract

The Hele-Shaw Flow is a model for describing the propagation of fluid in a cell consisting of two parallel plates separated by a small gap. This model has been intensely studied for over a century, and is a paradigm for understanding more complicated systems such as the flow of water in porous media, melting of ice and models of tumor growth.

In this talk I will discuss how this flow fits into the more general framework of “inverse potential theory” through the idea of complex moments. I will then discuss joint work with David Witt Nystrom that connects to the moduli space of holomorphic discs with boundary in a totally real manifold. Using this we prove a number of short time existence/uniqueness results for the flow, including the case of the Hele Shaw flow with varying permeability starting from a smooth Jordan domain, and for the Hele Shaw flow starting from a single point.