# Well-posedness of a diffuse interface model for Hele-Shaw flows

## TL;DR

This paper establishes the well-posedness, including existence and uniqueness, of solutions for a diffuse interface model describing two viscous fluids with different viscosities in a Hele-Shaw cell, extending previous matched viscosity results.

## Contribution

It proves existence and uniqueness of solutions for the model with logarithmic Helmholtz potential in both two and three dimensions, including the unmatched viscosities case.

## Key findings

- Uniqueness of weak solutions in 2D under regularity conditions
- Existence and uniqueness of global strong solutions in 2D
- Local and global strong solutions in 3D depending on data size

## Abstract

We study a diffuse interface model describing the motion of two viscous fluids driven by the surface tension in a Hele-Shaw cell. The full system consists of the Cahn-Hilliard equation coupled with the Darcy's law. We address the physically relevant case in which the two fluids have different viscosities (unmatched viscosities case) and the free energy density is the logarithmic Helmholtz potential. In dimension two we prove the uniqueness of weak solutions under a regularity criterion, and the existence and uniqueness of global strong solutions. In dimension three we show the existence and uniqueness of strong solutions, which are local in time for large data or global in time for appropriate small data. These results extend the analysis obtained in the matched viscosities case by Giorgini, Grasselli and Wu (Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire 35 (2018), 318-360). Furthermore, we prove the uniqueness of weak solutions in dimension two by taking the well-known polynomial approximation of the logarithmic potential.