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 phi is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity u obeys a Darcy's law depending on the so-called Korteweg force mu del phi, where mu is the nonlocal chemical potential. In addition, the kinematic viscosity eta may depend on phi. 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 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 eta is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is alpha-Hoelder continuous in space for alpha is an element of (1/5, 1).