Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

Abstract

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation.

References in corpus (6)

• Laplacian Growth and Whitham Equations of Soliton Theory
• Planar elliptic growth
• Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
• The Variable Coefficient Hele-Shaw Problem, Integrability and Quadrature Identities
• Quadrature domains and kernel function zipping
• Self-similarity in Laplacian Growth

Cited by in corpus (3)

• Laplacian Growth II: Saffman - Taylor Problem Without Surface Tension in Filtration Combustion: Formation of One Finger with Half of the Channel Width
• Non-algebraic quadrature domains
• On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth