On evolution equations for moving domains
type: Item
instance of: scholarly article
title: On evolution equations for moving domains
zbMATH Open document ID: 0920.35174
DOI: 10.4171/ZAA/870
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-04-26T00:00:00Z
full work available at URL: https://eudml.org/doc/48217
review text: Summary: We introduce a general framework for the description of the autonomous motion of closed surfaces which are diffeomorphic images of spheres. The governing surface motion laws are in general nonlocal and lead therefore to nonlocal evolution equations for a perturbation function on a fixed reference domain. Although this evolution equation is fully nonlinear, the invariance of the problem with respect to a certain class of reparametrizations and the corresponding chain rule allow a quasilinearization of the evolution equation. Hence, as far as short-time existence and uniqueness of the solution and stability of equilibria are concerned, the analysis of the problems is reduced to the study of their linearizations and the application of known techniques for quasilinear Cauchy problems. Using a priori estimates and Galerkin approximations in Sobolev spaces, both parabolic and first-order hyperbolic equations can be treated. In the case of parabolic problems, the smoothing property of the evolution can be proved. This general approach can be applied to a broad class of moving boundary problems. We will briefly discuss Hele-Shaw flow and Stokes flow driven by surface tension as well as classical Hele-Shaw flow with advancing liquid boundary as examples for parabolic evolutions.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 76D07
zbMATH DE Number: 1278887
zbMATH Keywords: autonomous motion of closed surfaces
zbMATH Keywords: surface motion laws
zbMATH Keywords: nonlocal evolution equations
zbMATH Keywords: quasilinearization
zbMATH Keywords: short-time existence
zbMATH Keywords: uniqueness
zbMATH Keywords: stability
zbMATH Keywords: a priori estimates
zbMATH Keywords: Galerkin approximations
zbMATH Keywords: Hele-Shaw flow
zbMATH Keywords: Stokes flow
zbMATH Keywords: surface tension
author: Georg Prokert
MaRDI profile type: Publication
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OpenAlex ID: W2075102227
description: scientific article; zbMATH DE number 1278887