Well-posedness of a diffuse interface model for Hele-Shaw flows
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.
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