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Slow motion for the Cahn-Hilliard equation in one space dimension. (English) Zbl 0753.35042
The authors consider the Cahn-Hilliard equation in one space dimension $u_ t=(-\varepsilon^ 2 u_{xx}+W'(u))_{xx}\qquad \text{on}\qquad (0,1)\times(0,\infty),\tag{\text{CH}}$
$u_ x=u_{xx}=0,\quad x=0,1,\quad t>0;\qquad u(x,0)=u_ 0(x).$ Hypothesis: $$W(u)$$ is a $$C^ 4$$ function with exactly three critical points, $$\alpha<\beta<\gamma$$, with $$\alpha,\beta$$ local minima and $$\gamma$$ a local maximum and $$W\geq 0$$, $$W''(\alpha)$$, $$W''(\beta)$$, $$-W''(\gamma)>0$$.
The authors seek the invariant manifold $${\mathcal M}$$ for (CH) under the above hypothesis, i.e. seek the solution $$u(x,t)=\hat u(\xi(t),x)$$, where $$\hat u(\xi,x)$$ and $$\xi(t)$$ satisfy $$\dot\xi(t)=b(\xi)$$ and $$- \varepsilon^ 2\hat u_{xxxx}+(W'(\hat u_ x))_ x=b(\xi)\hat u$$.
They prove the Theorem: There exists $$\varepsilon_ 0>0$$ such that for $$\varepsilon<\varepsilon_ 0$$ there are functions $$\hat u(\xi,\cdot)\in H^{4\alpha}$$ and $$b(\xi)$$ defined in $$(-\ell+\delta,\ell-\delta)$$, with the following properties:
(i) $$\xi\to\hat u(\xi,\cdot)$$, $$\xi\to b(\xi)$$ are Lipschitz continuous, (ii) $${\mathcal M}=\{u=\hat u(\xi,\cdot);\;\xi\in(-\ell+\delta,\ell- \delta)\}$$ is an invariant manifold for the dynamical system defined by (CH) in $$H^{4\alpha}$$, (iii) The flow on $${\mathcal M}$$ is described by the ordinary differential equation $$\dot\xi=b(\xi)$$. (iv) The following estimates hold: $$\| \hat u-u(\xi,\cdot)\|_{H^ 2}=O(\exp(- \nu\delta_ \xi/\varepsilon))$$ and $$b(\xi)=O(\exp(-2\nu\delta_ \xi/\varepsilon))$$ and, $$b(0)=0$$, where $$u(\xi,x)=u(x-\xi)$$, $$\delta_ \xi=\ell-|\xi|$$, and $$u(x)$$ is the equilibrium (time independent solution) of (CH).

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 58J70 Invariance and symmetry properties for PDEs on manifolds 37C10 Dynamics induced by flows and semiflows
##### Keywords:
melted binary alloy; invariant manifold
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##### References:
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