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Department of Mathematics and Statistics
of University of Helsinki
Formula 1. CahnHilliardequation.
Formula 2. The parabolic equation for the perturbation.

The CahnHilliard equation (Formula 1) can be used to model phase separation
in mixtures of two substances A and B: the function
u has values
in the interval [1,1] with the endpoints describing the state
of pure phase of substance A, respectively, B. For a random
initial distribution h, numerical simulations show that
phase segregation evolves: domains of phase A and B start to
form. However, a rigorous mathematical analysis of this
phenomenon seems very difficult.
SOME RECENT ARTICLES

T.Korvola, A.Kupiainen, J.Taskinen:
Anomalous scaling for 3d CahnHilliard fronts
Comm. Pure Appl. Math.
LVIII, (2005), 10771115.
This work deals with the simple situation that the initial
data is a perturbation of the stationary kink solution
tanh (x/2) for the unbounded Euclidean space of at least
dimension 3. We prove the stability: the function u tends,
as expected, to the stationary solution, if the initial
perturbation is small enough.
The method is based on a detailed spectral analysis
of the linear elliptic fourth order operator L, see Formula 2
(using the Melnikov, or PegoWeinstein, method), and a fixed point argument
in some
properly chosen Banach spaces with weighted supnorms.

J.Bricmont, A.Kupiainen, J.Taskinen:
Stability of CahnHilliard fronts
Comm.Pure.Appl.Math. LII (1999), 839871.
This earlier work proves the stability of the stationary solution
in the case x ∈ R. The approach is based on the
renormalization group method.
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