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 $\varphi$ is governed by convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. volume averaged fluid velocity $\mathbf{u}$ obeys Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, where $\mu$ chemical In addition, kinematic viscosity $\eta$ may depend $\varphi$. establish first existence global weak solution which satisfies energy identity. Then we prove strong solution. Further regularity results pressure are also obtained. Weak-strong uniqueness demonstrated in dimensional case. three-dimensional case, solutions holds if constant. Otherwise, weak-strong shown assuming that $\alpha$-H\"{o}lder continuous space $\alpha\in (1/5,1)$.