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 a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $mathbf{u}$ obeys a Darcys law depending on the so-called Korteweg force $mu abla varphi$, where $mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $eta$ may depend on $varphi$. 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 $eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $alpha$-H{o}lder continuous in space for $alphain (1/5,1)$.
تحميل البحث