Well-posedness of the Hele-Shaw-Cahn-Hilliard system

Details Well-posedness of the Hele-Shaw-Cahn-Hilliard system Well-posedness of the Hele-Shaw-Cahn-Hilliard system

2010-12-14

arxiv.org/abs/1012.2944v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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