A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane
type: Item
instance of: scholarly article
title: A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane
zbMATH Open document ID: 0802.35168
DOI: 10.1007/BF02559501
author: Lothar von Wolfersdorf
author: Michael Reissig
published in: Arkiv för Matematik
publication date: 1994-12-11T00:00:00Z
review text: A Hele-Shaw free boundary problem in a plane \(x\), \(y\) admits the following formulation in terms of the complex variable \(z = x + iy\): given \(f_ 0(z)\) analytic and univalent in a neighborhood of \(| z | \leq 1\), \(f_ 0 (0) = 0\), a function \(f(z,t)\) is sought such that for each \(t>0\), \(f\) has the same properties as \(f_ 0\), \(f_ t\) is continuous in some interval \((0,T)\), and \(\text{Re} ({1 \over z} {\partial f \over \partial t} {\partial f \over \partial z}) = 1\), for \(| z | = 1\); \(f(z,0) = f_ 0(z)\), for \(| z | \leq 1\); \(f(0,t) = 0\). The aim of this paper is to study several generalizations of the problem above and to give existence and uniqueness of the solutions. Even in the case of the classical Hele-Shaw problem, this proof (which is based on the application of a nonlinear abstract version of the Cauchy-Kovalevski theorem) simplifies the classical proofs in a relevant way.
reviewed by: Mario Primicerio
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 35A10
zbMATH DE Number: 493362
zbMATH Keywords: univalent function
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: Hele-Shaw problem
MaRDI profile type: Publication
cites work: On a differential equation arising in a Hele Shaw flow moving boundary problem
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description: scientific article; zbMATH DE number 493362