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

TL;DR

This paper proves the well-posedness of the Hele-Shaw-Cahn-Hilliard system for binary fluid flow in porous media, establishing existence, uniqueness, and blow-up criteria using advanced mathematical tools.

Contribution

It provides the first rigorous analysis of well-posedness for the Hele-Shaw-Cahn-Hilliard system with arbitrary viscosity contrast.

Findings

Global existence in 2D for smooth initial data

Local existence in 3D with blow-up criteria

Use of Littlewood-Paley theory for key estimates

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.