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The Cahn-Hilliard equation: Mathematical and modeling perspectives. (English) Zbl 0917.35044

It is discussed the prototype modeling role of the Cahn-Hilliard equation \[ \begin{alignedat}{2} u_t=&\nabla \cdot M\nabla (-\varepsilon^2 \Delta u-u+u^3),\quad&& (x,t)\in\Omega\times {\mathbb R}^+,\\ u {\mathbf n}\cdot \nabla u=&{\mathbf n}\cdot \nabla\Delta u=0,\quad&&(x,t)\in\partial\Omega\times {\mathbb R}^+\end{alignedat} \] as progressively more involved models are proposed. The mobility \(M\) depends on the concentration \(u(x,t)\). \(\Omega\) is a bounded domain in \(\mathbb R^n\), \(n=1,2,3\). The author presents the mathematical progress which has been made towards analyzing the dynamics of the Cahn-Hilliard equation and some open questions which would be interesting to address.

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
35K55 Nonlinear parabolic equations
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