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

Xiaoming Wang; Zhifei Zhang

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arXiv · 1012.2944

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

Xiaoming Wang·Zhifei Zhang

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Abstract

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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Xiaoming Wang, Zhifei Zhang. 2010-12-14. Well-posedness of the Hele-Shaw-Cahn-Hilliard system. https://arxiv.org/abs/1012.2944

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