  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.
