Well-Posedness and Optimal Control of Phase-Field Separations in Multi-Components Systems Described by Cahn-Hilliard and Allen-Cahn Equations

We consider a moving interface problem given by the phase-field model of the coupled Cahn-Hilliard Navier-Stokes system. The main objective is to analyze this model, and study the existence, regularity as well as the continuity of solutions with respect to the boundary control of the fluid velocity field. We also formulate the optimal control problem for the motion of the phase separation of the two fluids by the boundary control. We use the Lagrange Calculus in the analysis of this highly non-linear and non-smooth control problem. Also, some theoretical properties of the proposed discretized system will be discussed. We study the energy functional for solutions of Cahn-Hilliard Navier-Stokes system, and prove that it satisfies a certain evolution equation. The coupled Cahn-Hilliard Navier-Stokes system describes the separation of two fluids in a binary fluid mixture by the boundary control of the fluid velocity field. The Navier-Stokes equations are coupled to the Cahn-Hilliard system in the following way. The velocity field introduces the transport term of the concentrations in the Cahn-Hilliard equations, the fluid structure interaction force is added into the incompressible Navier-Stokes equations as an interaction force, and our control enters as a Dirichlet boundary condition for the velocity field. We will study the well-posedness of the Cahn-Hilliard Navier-Stokes system, i.e. we introduce the weak form of equations and then show the existence of weak solutions as well as proving the Lipschitz continuity of solutions with respect to control and initial conditions. Then we extend our study to Allen-Cahn equations and compare the two systems to apply similar techniques to both models. Lastly, we develop numerical computations to show our results numerically.