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Article

Convergence of the one-dimensional Cahn-Hilliard equation

SIAM Journal on Mathematical Analysis. 2012. Vol. 44. No. 5. P. 3458-3480.
Bellettini G., Bertini L., Mariani M., Novaga M.

We consider the Cahn-Hilliard equation in one space dimension with scaling parameter epsilon, i.e., u(t) = (W'(u) - epsilon(2)u(xx))(xx), where W is a nonconvex potential. In the limit epsilon down arrow 0, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation.