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We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for anincompressible mixture of two fluids. The relative concentration difference$\varphi$ is governed by a convective nonlocal Cahn-Hilliard equation withdegenerate mobility and logarithmic potential. The volume averaged fluidvelocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Kortewegforce $\mu\nabla \varphi$, where $\mu$ is the nonlocal chemical potential. Inaddition, the kinematic viscosity $\eta$ may depend on $\varphi$. We establishfirst the existence of a global weak solution which satisfies the energyidentity. Then we prove the existence of a strong solution. Further regularityresults on the pressure and on $\mathbf{u}$ are also obtained. Weak-stronguniqueness is demonstrated in the two dimensional case. In thethree-dimensional case, uniqueness of weak solutions holds if $\eta$ isconstant. Otherwise, weak-strong uniqueness is shown by assuming that thepressure of the strong solution is $\alpha$-H\"{o}lder continuous in space for$\alpha\in (1/5,1)$.

Nonlocal Cahn-Hilliard-Hele-Shaw systems with singular potential and degenerate mobility