On the nonlocal CahnHilliardBrinkman and CahnHilliardHeleShaw systems
Francesco Della Porta  Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom (email) Abstract: The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the socalled Brinkman equation coupled with a convective CahnHilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the CahnHilliardBrinkman (CHB) system. In particular, the Brinkman equation is a Stokeslike equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the CahnHilliardHeleShaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study wellposedness for the CHB system, endowed with noslip and noflux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.
Keywords: Incompressible binary fluids, Brinkman equation, Darcy's law, diffuse interface models, CahnHilliard equation, weak solutions, existence, uniqueness, vanishing viscosity.
Received: September 2015; Revised: November 2015; Available Online: January 2016. 
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