We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. for initial data in Hs, s>d2+1, the existence and uniqueness of solution in C([0,T];Hs) ∪L2(0,T;Hs+2) that is global in time in the two-dimensional case (d=2) and local in time in the three-dimensional case (d=3) are established. Several blow-up criterions in the three-dimensional case are provided as well. One of the tools that we utilized is the Littlewood-Paley theory in order to establish certain key commutator estimates. © 2012 Elsevier Masson SAS.
X. Wang and Z. Zhang, "Well-Posedness of the Hele-Shaw-Cahn-Hilliard System," Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, vol. 30, no. 3, pp. 367 - 384, EMS Press, Jan 2013.
The definitive version is available at https://doi.org/10.1016/j.anihpc.2012.06.003
Mathematics and Statistics
Keywords and Phrases
Blow-Up Criterion; Hele-Shaw-Cahn-Hilliard; Well-Posedness
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 2013
National Science Foundation, Grant 1008852