ABSTRACT

The Cahn-Hilliard system was introduced in (1) to describe at a macroscopic level the formation and evolution of microstructures during the phase separation in a binary alloys system below the critical temperature. Recently, Giacomin and Lebowitz have proposed in (2) and (3) a more realistic version of the Cahn-Hilliard system which takes into account long-range interactions between molecules. The resulting model is a conserved gradient flow associated with the free energy 𝜀(𝜑) = − 𝜖 2 ∫ ∫ 𝐽 (𝑥 − 𝑦)𝜑(𝑥)𝜑(𝑦)𝑑𝑥𝑑𝑦 + ∫ 1 𝜖 𝐹(𝜑)𝑑𝑥, 𝛺 𝛺 𝛺 (1) where 𝜑 represents the relative difference of the two phases, 𝐽 is an interaction kernel and 𝐹 is a potential. During the talk, we will discuss some recent analytical results on the nonlocal Cahn-Hilliard system with singular potential, namely we assume the physically relevant form for the potential 𝐹. In particular, we will focus on the validity of the separation property which plays a crucial role to understand the properties of weak solutions. We will then recover regularity results as well as the existence of finite dimensional attractors.