Communications in Mathematical Sciences

Volume 13 (2015)

Number 5

Dynamic transitions and pattern formations for a Cahn–Hilliard model with long-range repulsive interactions

Pages: 1289 – 1315



Honghu Liu (Department of Atmospheric & Oceanic Sciences, University of California at Los Angeles)

Taylan Sengul (Department of Mathematics, Yeditepe University, Istanbul, Turkey)

Shouhong Wang (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)

Pingwen Zhang (Department of Mathematics, Beijing University, Beijing, China)


The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on a Cahn–Hilliard model with a nonlocal term in the corresponding energy functional, representing certain long-range repulsive interaction. We show that as soon as the trivial steady state loses its linear stability, the system always undergoes a dynamic transition of one of three types—continuous, catastrophic and random—forming different patterns/structures, such as lamellae, hexagonally packed cylinders, rectangles, and spheres. The types of transitions are dictated by a non-dimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. In particular, the hexagonal pattern is unique to this long-range interaction, and it is captured by the corresponding two-dimensional reduced equations on the center manifold, which involve (degenerate) quadratic terms and non-degenerate cubic terms. Explicit information on the metastability and basins of attraction of different ordered states, corresponding to different patterns, are derived as well.


phase transition, pattern formation, long-range interaction, a Cahn–Hilliard model, center manifold reduction, hexagonal pattern

2010 Mathematics Subject Classification

35B32, 35B36, 37L05, 37L10

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