Analysis of Diffuse Interface Models (M. Conti, M. Grasselli)

The diffuse interface method gives a rather satisfactory description of phenomena characterized by interfaces separating different phases through nonlinear partial differential equations. This approach, also known as phase field modeling, has proved to be extremely powerful in the description and numerical simulation of the development of microstructure without having to track the evolution of individual interfaces, as is the case with sharp interface problems.

Phase field systems now play a basic role in several applications of technological as well as social impact. Concerning the former, we recall the Cahn-Hilliard equation which is a cornerstone of mathematical modeling in Materials Science. Indeed it governs phase separation phenomena in binary alloys subject to deep quenching, a crucial issue to understand their microstructural properties. For instance, Cahn-Hilliard type equations are employed to model the self-assembly of polymers which are used to create nanopatterned systems or ultra-light and very strong materials. Besides, the phase field approach is also applied in image reconstruction (e.g., inpainting) and in the mathematical description of liquid crystal behavior. As far as the social impact is concerned, there is an increasing international interest in the application of the diffuse interface methodology to model the evolution of different kind of tumors (for instance, vascular tumor growth). Typically they are based on equations of Cahn-Hilliard type coupled with suitable reaction-diffusion equations and the Navier-Stokes system or the Darcy law. Thanks to these models, large scale simulations of tumor growth with complex morphologies can be performed.

A deeper mathematical knowledge of more refined diffuse interface models seem necessary to improve the understanding of complex phenomena like the ones just briefly described. The main issues are the existence, regularity and longtime behavior of solutions to suitable initial and boundary value problems. A further area of interest is the optimal control. Collaborations or interactions with experts in modeling and/or in numerical simulation are encouraged.

Politecnico di Milano - Dipartimento di Matematica