A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane
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.
scientific article; zbMATH DE number 493362