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 u obeys a Darcy’s law depending on the so-called Korteweg force μ∇ 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 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∈ (1 / 5 , 1).