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The Hele-Shaw flow and moduli of holomorphic discs

Part of: Incompressible viscous fluids Curves Differential operators in several variables Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  18 August 2015

Julius Ross  and
David Witt Nyström
Julius Ross
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK email j.ross@dpmms.cam.ac.uk
David Witt Nyström
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK email danspolitik@gmail.com
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Abstract

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We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

Keywords

Hele-Shaw flowLaplacian growthmoduli of holomorphic discshomogeneous complex Monge–Ampère equations

MSC classification

Primary: 76D27: Other free-boundary flows; Hele-Shaw flows 30E05: Moment problems, interpolation problems 14H15: Families, moduli (analytic) 32W20: Complex Monge-Ampère operators
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 12 , December 2015 , pp. 2301 - 2328
Copyright
© The Authors 2015 

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