On the viscous Allen-Cahn and Cahn-Hilliard systems with willmore regularization

We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation.