Ogasawara, Yosihito

Yanagiya, Akira

Kyoto University Research Information Repository

TitleOn the Stationary Solution of the Mathematical Model forGrain Boundary Grooving (Mathematical models and dynamicsof functional equations)Author(s)Yanagiya, Akira; Ogasawara, YosihitoCitation数理解析研究所講究録 (2004), 1372: 198-201Issue Date2004-04URL http://hdl.handle.net/2433/25513RightType Departmental Bulletin PaperTextversionpublisherKyoto University198On the Stationary Solution of the MathematicalModel for Grain Boundary GroovingAkira Yanagiya1and Yosihito Ogasawara2Advanced Institute for Complex Systems of Waseda University$\doteqdot f^{\backslash }\beta \mathrm{f}\mathrm{f}1\mathrm{x}’\frac{\backslash }{\mp}\not\in\Re_{f\mathrm{E}\Leftrightarrow\frac{\backslash \backslash }{\mp}}^{\tau_{\backslash }}|\overline{\mathrm{y}}ffl_{J\iota \mathrm{P}}^{*}T$3-31-1,Kamishakuzii,Nerima-ku,Tokyo, 177-0044,Japan.Te1 81-3-5991-4151,Fax 81-3-3928-4110,yanagiya@waseda.jpScience and Engeneering Department of Waseda University$\doteqdot\ovalbox{\tt\small REJECT}^{\backslash }\mathrm{f}\mathrm{f}1\lambda\doteqdot^{\backslash }\Phi \mathrm{I}^{\prime^{1^{\backslash }}}\neq^{4}\mathrm{f}\mathrm{f}1f\iota*\ovalbox{\tt\small REJECT}_{\underline{\backslash }}^{\backslash }\mathrm{t}\emptyset \mathrm{g}7\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{F}^{\frac{\backslash \backslash }{\neq}\ovalbox{\tt\small REJECT}_{-}^{\backslash }}\backslash \dagger$3-4-1,O0kub0,Sinjuku-ku,T0ky0,169-8555, Japanmugen@ruri.waseda.jp1. IntroductionIn this talk, we will present some stationary solution for nonlinear partial differ-ential equation called Mullins Equation which is occered in the theory of grainboundary grooving.$u_{t}=-C_{1}^{E}(u))(1+u_{x}^{2})^{1f2}exp(-C_{2}^{E}(u) \frac{u_{xx}}{(1+u_{x}^{2})^{3/2}})+C_{1}^{C}(u)(1+u_{x}^{2})^{1/2}-$ (1)The main tool, which we can use, is the admissibility property between weightedcontinuous function spaces for the integral $\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r},\mathrm{a}\mathrm{s}$ follows.$T_{\xi}x(t)=- \int_{f}^{\infty}.e^{\zeta_{1}(t-s)}F(x(s),y(s))$ds,$T_{\xi}y(t)= \xi e^{\zeta_{2}t}+\int_{0}^{t}e(2(t-s)F(x(s), y(s))ds$. (2)Prom this admissibility we can prove the existence theorem for the special si-multaneous differential equation. This existence theorem can be applied for thesecond order differential equation,$u’=f(u, u’)= \frac{kT(u)(1+u^{\prime 2})^{3\oint 2}}{v\gamma}ln(\frac{P_{0}(u)}{P_{c}})$ . (3)The solution of this equation is one of the stationary solution for Mullins Equa-tion.1372 2004 198-2011 se2. TheoremsOn the equation (1), we are interested in the stational solution. So we shallconsider the equation (3) which we can make by putting $ut=0$ for the equation(1). To prove the existence theorem for the stational solution, we use the nexttwo theorems.Theorem1For the second oreder differential equation,$u’=f(u, u’)$ , (4)suppose that the following hypotheses.$f$ (u, $p$) $\in C^{1}(R^{2})$ , $x>0,$ $\exists\lambda\in R^{1}$ $s.t$ . $f(\lambda, 0)=0,$ $f_{u}(\lambda, 0)>0$Then there exits the solution on $(0, \infty)$ and it satisfies that$\exists D>0$ $\mathrm{s}.\mathrm{t}$ . $|u(x)-\lambda|\leq Dexp(- \mathrm{r}x)$ ,where$0< \tau<|\frac{f_{p}(\lambda,0)-\sqrt{f_{p}(\lambda,0)^{2}+4f_{u}(\lambda,0)}}{2}|$ .TheOrem2On the differential equation,$\omega_{1}’=\zeta_{1}\omega_{1}+F(\omega_{1}, \omega_{2})$ , $\omega_{2}’=\zeta_{2}\omega_{2}+F(\omega_{1}, \omega_{2})$ , $x>0,$where,$f(\eta_{1}, \eta_{2})\in C^{1}(R^{2})$ , $F(0,0)=0,$ $F_{\eta_{1}}(0,0)=0,$ $\zeta_{1}>0$ , $S_{2}<0,$there exists some global nontrivial solution$\omega(x)=(\omega_{1}(x),\omega 2(X))$ , $x>0,$for every $\tau$, $0<\tau<|" 2$ $|$ , and the next inequality is satisfied.$|e\tau x\mathrm{c}\mathrm{p}\mathrm{l}$ $(x)|+|e^{\tau x}\omega_{2}(x)|<\infty$ , $x>0.$At first we consider TheOrem2. By using the addmissibility of the integraloperator(2), we can establish the proof of TheOrem2. Let consider the integraloperator on the following function set $\mathrm{B}$ ,$B=\omega(x)=(\omega_{1}(x), \omega_{2}(x))\in C^{0}([0, \infty));||\mathrm{C}\mathrm{J}||\leq 2|\xi|$,$|| \omega||=\sup_{x\geq 0}(e^{\tau oe}\omega_{1}(x)+-e^{\tau x}\omega_{2}(x))$.200On this set the integral operator(2) satisfies the contraction princeple. Thenthe operator $T_{\xi}$ : $B$ $arrow B$ has the unique fixes point $\omega(x)$ $=$ $(\omega_{1}(x),\omega 2(x))$ .TheOrem3On the set $\mathrm{B}$ the integral operator(2) satisfies the contraction princeple. Thenthe integral equation which is made by the integral operator(2) has uniquesolution in the set B.The proof of this thre0rem3 is essentially depended the following inequalities.$|e\tau x7_{\mathrm{C}^{(\mathrm{J}_{1}}}$ $(x)|=|e \tau x\int_{x}^{\infty}e$”1 $(x-y)_{F(\omega_{1}(y),\omega_{2}(y))dy|}$$\leq\int_{x}^{\infty}e^{\zeta_{1}(x-y)}|e^{\tau x}(F(\omega_{1}(y), \omega_{2}(y))-F(0,0))|dy$$= \int_{l}^{\infty}e^{\zeta_{1}(x-y)}|$ $\mathrm{C}^{\mathrm{j}}’\eta 1$ $(\theta\omega_{1}(y), \theta\omega_{2}(y))\omega_{1}(y)e^{\tau x}$$+F_{\eta 2}(\theta\omega_{1}(y), \theta\omega_{2}(y))\omega_{2}(y)e^{\tau x}|dy$$\leq M\int_{x}$”$e^{\zeta_{1}(x-y)}(|e^{\tau x}\omega_{1}(y)|+|e^{\tau x}\omega_{2}(y)|)$dy,$|e\tau xT_{4^{\mathrm{p}}1}$ $(x)|=|e\tau x$ $( \xi e+\int_{0}^{x}e^{\zeta_{2}(x-y)}F(\omega_{1}(y),\omega_{2}(y))dy)$$\leq|\mathrm{c}|e^{(\tau_{2}):*}$ $+ \int_{0}^{x}e^{\tau x+\zeta_{2}x-\zeta_{2}y-\tau y}$$\mathrm{x}$ $|e^{\tau y}|F(\omega_{1}(y),\omega_{2}(y))|dy$$=|4|e^{(\tau+\mathrm{C}_{2}2\}$ $+ \int_{0}^{oe}e(\tau+\mathrm{C}_{2})(\mathrm{z}-y)$$\cross|e\tau y|F\mathrm{C}’ 1(y)$ , $\omega_{2}(y))|dy$$\leq|\xi|+\frac{M}{\tau_{2}}||\omega||$ ,where$0<\theta<1$ , $\tau+\zeta_{2}<0$ , $\tau+\zeta_{2}=-\tau_{i2}$ .Hence we can prove TheOrem2. Next we treat Theoreml, by using the results ofTheOrem2. Let define the function $F(\omega_{1},\omega_{2})$ in TheOrem2 by the next equation,$F( \eta_{1},\eta_{2})=f(\frac{\eta_{1}-\eta_{2}}{\zeta_{1}-\zeta_{2}}+ \lambda, \frac{\zeta_{1}\eta_{1}-\zeta_{2}\eta_{2}}{\zeta_{1}-\zeta_{2}})$ $- \frac{\eta_{1}-\eta_{2}}{\zeta_{1}-\zeta_{2}}f_{\mathrm{u}}(\lambda,0)-\frac{\zeta_{1}\eta_{1}-\zeta_{2}\eta_{2}}{\zeta_{1}-\zeta_{2}}f_{p}(\lambda, 0)$,where$\zeta_{1}=\frac{f_{p}(\lambda,0)+\sqrt{f_{p}(\lambda,0)^{2}+4f_{u}(\lambda,0)}}{2}>0$,201$\zeta_{2}=\frac{f_{p}(\lambda,0)-\sqrt f_{p}(\lambda,0)^{2}+4f_{u}(\lambda,0)}{2}<0,$where the function $f$ as in Theoreml. By the result of TheOrem2 there existsthe solution $\omega(x)=$ $(\omega_{1}(x), \omega 2(X))$ . Define$u(x)$ $= \frac{\omega_{1}(x)-\omega_{2}(x)}{(_{1}-(_{2}}+\lambda$, $x>0.$This function $u$ is the solution in Theoreml. At last, we can apply Theoremlfor the equation (3), we get the stational solution of (1).ReferencesA.Kitada,J.Math.Phys.(1986),(1987).P.Broadbridge,J.Math.Phys. (1989).

On the Stationary Solution of the Mathematical Model for Grain Boundary Grooving (Mathematical models and dynamics of functional equations)

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On the Stationary Solution of the Mathematical Model for Grain Boundary Grooving (Mathematical models and dynamics of functional equations)

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