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

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $φ$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $μ\nabla φ$, where $μ$ is the nonlocal chemical potential. In addition, the kinematic viscosity $η$ may depend on $φ$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $\mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $η$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $α$-Hölder continuous in space for $α\in (1/5,1)$.