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1 $\Downarrow$ $\Downarrow$ Cahn-Hilliard (Daisuke Furihata (Tomohiko Onda 1 (Masatake Mori Cahn-Hilliard Cahn-Hilliard ( $[1]^{1}$ reduce ( Cahn-Hilliard ( Cahn- Hilliard Cahn-Hilliard 2? $u(x, t_{0}$ ( 1

2 $\Downarrow$ $(p<0(q<0on\partial_{\omega}^{\omega}on_{<}^{in_{\partial_{r}^{\omega}}}$ $(2.\cdot.2(21(23(25(24$ 68 Cahn-Hilliard (CH eq $\{\begin{array}{l}u-\nabla^{2}\tilde{g}_{u}=0\nabla u\cdot n=0\nabla\tilde{g}_{u}\cdot n=0\tilde{g}(u=g-\frac{1}{2}q \nabla u ^{2}G(u=\frac{1}{2}pu^{2}+\frac{1}{4}ru^{4}\end{array}$ $G(u$ $\tilde{g}(u$ :Ginzburg-Landau ( : Ginzburg-Landau $- \frac{1}{2}q \nabla u ^{2}$ : ( Cahn-Hiliard $Eq.(22$ Eq.(2.3 $u_{t}=\nabla^{2}(pu+q\nabla^{2}u+ru^{3}$ in $\Omega(p, q<0<r$ (2.6 ( : Ginzburg-Landau $G(u$ double-well Eq.(2.5 ( $p,$ $q,$ $r$ 3 $\Uparrow$? $\perp_{t}\partial_{\partial}u>0$ ( ( 3 1. ( 2. ( 3. (

3 69 4. ( 5. ( 4 [1] ( ( ( ( ( 5? $\Uparrow$ ( : $<1$

4 $\Downarrow$ $\Downarrow$ $\Downarrow$ 70 ( 6 v(? $u$ $\{\begin{array}{l}v=\epsilon exp(ikx(\epsilon\ll 1(6.1u=u_{0}+v(u_{0}\cdot.Const.(6.2\end{array}$ Cahn-Hilliard Eq.(2.6 $\frac{\partial v}{\partial t}=\nabla^{2}\{(p+3ru_{0}^{2}v+q\nabla^{2}v\}$ in $\Omega$ (6.3 1;

5 $\lambda$ 71 ( $1$ $\lambda_{c}$ : $v_{c}$ $\lambda_{c}^{dcf}=\frac{v_{c}(t+\delta t}{v_{c}(t}$ $\in C$ (6.4 $\lambda$ : $v$ $\lambda-\frac{v(t+\delta t}{v(t}d\epsilon\lrcorner$ $\in C$ (6.5 $\Delta t$ 1:? 1 2 : Cahn-Hilliard $\lambda_{c}vs$ 2: $k(u_{0}=0$ $1$ ( $Eq.(10.1$

6 $\Downarrow$ $\Downarrow$ 72 $\lambda$ 3: vs $k(u_{0}=0$ ( $Eq.(10.3$ $1$ 7 $k$ ( ( $u_{0}$ ( 1. k \sim $k$ $\mathcal{k}$ 2. uo Cahn-Hilliard

7 $\mathcal{u}$ $\Downarrow$ 73 $\sim$ ( $u_{0}$ $\mathcal{u}$ ( 1 $\mathcal{k}=[-\frac{\pi}{\delta x}, \frac{\pi}{\delta x}]$ (7.1 $\mathcal{u}=[-\sqrt{\frac{-p}{r}}, \sqrt{\frac{-p}{r}}]$ (7.2 : 8 2( $I$ (8.1 $I(\lambda^{def}=\{ \lambda(k, u_{0} :\forall_{k}\in \mathcal{k}^{\forall}u_{0}\in \mathcal{u}\}$ $\approx I(\lambda\subseteq[0,1]$ 3(( 1 $I(\lambda\subseteq I(\lambda_{c}$ 4(( 1 $I(\lambda=I(\lambda_{c}$ lambda 2 ( 3. $\mathcal{k},$ $u_{0}$ ( :

8 74 4: \mbox{\boldmath $\lambda$} vs $u_{0}$ 9 5( $u_{0}$ ( $\lambda$ $0$ 6( $u_{0}$ $ \lambda <1$ $\Re(\lambda>0$ $u_{0}$ 1. ( 2. (. 3. $\lambda_{c}$

9 $\Downarrow$ $(3r_{1}u_{0}^{0}3ru_{:^{2}\text{ ^{}}}^{2}$ $\Downarrow$ $\Downarrow$ $\Downarrow$ $\{\begin{array}{l} \end{array}$ 75 $\sim$ $\sim$ $\sim$ $I(\lambda_{c}$ CH-eq. Eq.(6.3 $veq.(6.1$ $\lambda_{c}=\exp[\{qk^{4}-(p+3ru_{0}^{2}k^{2}\}\triangle t]$ (10.1 $I( \lambda_{c}=[\exp\{\frac{\pi^{2}}{\triangle x^{2}}\delta t(2p+\frac{\pi^{2}}{\delta x^{2}}q\}, \exp(-\frac{p^{2}}{4q}\delta t]$ (10.2 $I(\lambda$ CH-eq. Eq.(6.3 $veq.(6.1$ $\lambda=\frac{1+\theta_{\zeta}^{1}\{(\cos k\delta x-1+\xi^{2}-\xi^{2}\}}{1-(1-\theta_{\zeta}^{1}\{(\cos k\triangle x-1+\xi^{2}-\xi^{2}\}}$ (10.3 ($p$ $==$ $\{\eta\xi\theta$ $\mathfrak{g}\backslash$ ]($p$,0:,other: $I( \lambda=[\frac{1+\theta\frac{4}{\delta x^{2}}\delta t(2p+\frac{4}{\delta x^{2}}q}{1-(1-\theta\frac{4}{\delta x^{2}}\triangle t(2p+\frac{4}{\delta x^{2}}q}\frac{1-\theta_{4}^{g}\frac{2}{q}\triangle t}{1+(1-\theta_{4q}^{l^{2}}\triangle t}]$ (10.6 $I(\lambda\subseteq I(\lambda_{c}$ 11 Cahn-Hilliard

10 $\theta$ 76 $CH- eq$ $\text{ _{ $\Delta x_{\theta:}$ }}$ $\}\Rightarrow^{\exists}$ $\not\in E?s.t.r$ $\text{ _{}D}\text{ _{ }}>^{\triangle t\theta_{x:^{*}}}$ ( $O$ : $x$ : ( 1 12 $\{\begin{array}{l}t\frac{\delta x^{l}}{4(2\theta-l(-2q-p\delta x^{2}}(\frac{1}{2}<\theta\leq 1\text{ }(0\leq\theta\leq\frac{1}{2}\end{array}$ (12.1 $\{\Delta t<\frac{\{\delta t<\triangle t<\frac{-4q}{\frac{(1-\thetap^{2}\delta x}{-1641-\theta}}(\triangle x^{2}\leq 8\Delta_{2}(8_{P}\Delta<\triangle x^{p}}{4(1-\theta(4q-p\delta x^{2}}$ (12.2 $(^{4}s<\triangle x^{2}p$ 1 (12.3 $\triangle t<\frac{\triangle x^{4}}{8\theta(-2q-p\triangle x^{2}}$ Eq.(12.2

11 77 $\triangle t$ ( } ( ( $p$ $=$ $-1.0$ $q$ $=$ $-0.001$ 5: $\lambda_{c}$ \mbox{\boldmath $\lambda$} ( ( $0$

12 78 $\triangle x$ $\Delta t$ $\theta$ 1 3 ( $\Delta t$ $\Delta t$ % Eq.(2.6 $t$ $\Delta t$ ( 6: (

13 79 7: 1 $\Delta x^{--}1/30$. $\Delta t=1/19765$ ( $\cross 1.02$ 8: 2

14 80 14 Cahn-Hilliard $\Delta x=1/30,$ $\Delta t=1/10080$ ( $\Delta x=1/30$. $\Delta t=1/10080$ ( 10: : :

15 81 11; : : 12: : :

16 82 13: : : $\Delta x=1/30,$ $\Delta t=1/124.5$. $\theta= $ ( 14: : :

17 83 15: : : $\Delta x=1/30,$ $\Delta t=1/124.5,$ $\theta= $ ( 16: : :

18 84 $\Delta x=1/30,$ $\Delta t=1/10123$ ( 17: : : $\Delta x=1/30,$ $\Delta t=1/10123,$ $\theta= $ ( 18; : :

19 $\{\begin{array}{l}l\backslash K\overline{\pi} \ovalbox{\tt\small REJECT}_{\prime}M\not\equiv\neq^{\backslash }EM9\cross\neq-\text{ }\neq-a2\backslash \ \overline{\pi}\text{ }\#j\lessgtr 9\text{ }+-A\end{array}$ $CH-eq$. Fig.19 1 Cahn-Hilliard $p=-1.0,$ $q=-0.03,$ $r= $ 6 $1/30$ 1/ Fig 20 1 Cahn-Hilhard $p=-1.0,$ $q=-0.001,$ $r=1.0$ 1 1/50 1/ Fig Cahn-Hilliard $p=-1.0,$ $q=-0.001,$ $r=1.0$ 1 $1/30$ 1/66584

20 86 Explicit(Large COM 19: : : :1

21 21: :2 87

22 88 22: :2

23 23: :2 89

24 90 24; :2

25 25: :2 91

26 $1t$. $064$ 26: :2

27 93 16 Cahn-Hilliard ( [1] Charles M.Elliott. Numerical studies of the Cahn-Hilliard equation for phase separation. $IMA$ J.Appl.Math., Vol.$38:pp $, Cahn-Hilliard. [2] J.W.Cahn and J.E.Hilliard. Free energy of a non-uniform system.i.interfacial free energy. J. Chem.Phys., Vol.$28;pp $, [3] J.W.Cahn. Phase separation by spinodal decomposition in isotropic systems. $1$ J. Chem.Phys., Vol.42(No. $:pp.93-99$, Jan [4] J.S.Langer, M.Bar-on, and Harold D.Miller. New computational method in the theory of spinodal decomposition. Phys. Rev. $A$, Vol.ll (No.4:pp , Apr (i(ii. [5]. $-$ $162$.1978., Vol.13(No. $1,3$ $:pp.1-10,155-$ [6] Y.Oono and S.Puri. Study of phase-separation dynamics by use of cell dynamical systems.i.modeling. Phys.Rev. $A,$ $Vol.38(No.1:pp.434\triangleleft 53$, Jul [7].., Vol.6(No.8:pp , 1985.

42 1 ( ) 7 ( ) $\mathrm{s}17$ $-\supset$ 2 $(1610?\sim 1624)$ 8 (1622) (3 ), 4 (1627?) 5 (1628) ( ) 6 (1629) ( ) 8 (1631) (2 ) $\text{ }$ ( ) $\text{

$42 1 ( ) 7 ( ) $\mathrm{s}17$ $-\supset$ 2 $(1610?\sim 1624)$ 8 (1622) (3 ), 4 (1627?) 5 (1628) ( ) 6 (1629) ( ) 8 (1631) (2 ) $\text{ }$ ( ) $\text{$ 26 [\copyright 0 $\perp$ $\perp$ 1064 1998 41-62 41 REJECT}$ $=\underline{\not\equiv!}\xi*$ $\iota_{arrow}^{-}\approx 1,$ $\ovalbox{\tt\small ffl $\mathrm{y}