Diffuse-interface phase-field problems: modeling, analysis and numerics.
In this short-course, we will explain the main ideas of diffuse-interface phase-field problems, appearing in many different interface problems, as for instance mixing of fluids, solidification process, liquid crystals, angiogenesis growth, etc. These models arise as an alternative to free-boundary models, introducing regular phase-field unknowns to determine where the different phases are.
Firstly, we state the two generic Partial Differential Equations related to these problems, Allen-Cahn and Cahn-Hilliard equations, and some characteristics to take into account; global conservation or non of the phases, the time dynamic is regulated by a dissipative energy law, maximum principle of the phase-field variable, etc..
Secondly, we introduce the corresponding variational formulations and some results about global in time weak solutions, with special attention to the behavior of solutions at infinite time.
Afterwards, we will study the problem of designing energy-stable and well-posed numerical schemes, comparing the numerical analysis with some numerical simulations.
Finally, some applications to two-fluids and liquid crystal models will be stated.