Weak solutions for a Cahn-Hilliard equation with a non-local incompressibility constraint

The thesis focuses on a Cahn-Hilliard equation with a non-local incompressibility constraint, proposed by Felix Otto and Weinan E in 1997, which describes the dynamics of phase separation of an incompressible mixture. The key terms in the model are a degenerate mobility and the Flory-Huggins potential, as in the classical Cahn-Hilliard model, together with an additional non-local interaction involving the Leray projection operator. From a physical point of view, the importance of this new model is in the prediction of the dynamics of the interfacial regime and the dependence of the coarsening rate on the quench depth. The additional convective component leads to an extra mechanism in the energy dissipation that is not affected by the degeneracy of the mobility and does not weaken with the increasing quench depth. We demonstrate the existence of global-in-time weak solutions for the initial-boundary value problem associated to the Cahn-Hilliard equation with a non-local incompressibility constraint. We consider homogeneous Neumann boundary conditions for the concentration and the chemical potential. Following the method devised by Elliott and Garcke for the standard Cahn-Hilliard equation, the analysis is divided into two steps. In the first place, an existence result is proven for the model where the degenerate mobility and the logarithmic potential are approximated by a strictly positive mobility and a regular polynomial-like potential, respectively. This is achieved through a Galerkin approximation, taking advantage of compactness results in Lebesgue-Bochner spaces. In the second place, the existence of weak solutions for the original problem is obtained by a compactness method that relies on mass conservation and energy balance.