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Well-posedness of the Hele-Shaw-Cahn-Hilliard system. (English) Zbl 1291.35240
Summary: 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 \(H^s\), \(s>\frac{d}{2}+1\), the existence and uniqueness of solution in \(C([0,T];H^s)\cap L^2(0,T;H^{s+2})\) 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
76D27 Other free boundary flows; Hele-Shaw flows
76S05 Flows in porous media; filtration; seepage
42B25 Maximal functions, Littlewood-Paley theory
35B44 Blow-up in context of PDEs
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