Well-posedness of a diffuse interface model for Hele-Shaw flows
type: Item
instance of: scholarly article
title: Well-posedness of a diffuse interface model for Hele-Shaw flows
zbMATH Open document ID: 1435.35297
author: Andrea Giorgini
published in: Journal of Mathematical Fluid Mechanics
publication date: 2020-01-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1903.04457
review text: The purpose of this article is to show the uniqueness of strong or weak solutions for the Hele-Shaw-Cahn-Hilliard (HSCH) system or Cahn-Hilliard equation coupled with the Darcy's law in dimension two and three. First, some preliminary results are collected and the main assumptions are recalled. Then, uniqueness results of weak solutions in dimension two are proved first with logarithmic potential, then with regular potentials. Global strong solutions with unmatched viscosities and logarithmic potential in dimension two are carried out. Finally, the existence and uniqueness of strong solutions in dimension three is proved. The proofs use integration by parts, Leray projection operator, Young's inequality, Gronwall lemma, Brezis-Gallouet-Wainger inequality, Sobolev embedding, Hölder's inequality, generalized Gronwall lemma. Some extra proofs are made in an appendix.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35D35
Mathematics Subject Classification ID: 35K61
Mathematics Subject Classification ID: 76D27
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 76D45
zbMATH DE Number: 7159433
zbMATH Keywords: Darcy's law
zbMATH Keywords: Cahn-Hilliard equation
zbMATH Keywords: logarithmic potential
zbMATH Keywords: uniqueness
zbMATH Keywords: strong solutions
MaRDI profile type: Publication
OpenAlex ID: W2996961984
arXiv ID: 1903.04457
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Wikidata QID: Q126472001
DOI: 10.1007/S00021-019-0467-9
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description: scientific article