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$$
森光太朗氏の修士論文の主結果
森 で得られた結果を概説する.
森 では, \(L>0\), \(d>0\), \(T>0\) として次の領域
\begin{align}
&\Omega \equiv \{(x,y)\in \mathbb{R}^2\mid 0 < x < L, \ -d < y < 0\},\\
&\Gamma_1 \equiv \{(x,y)\in \mathbb{R}^2\mid 0\le x\le L, \ y=0\},\\
&\Gamma_2 \equiv \{(x,y)\in \mathbb{R}^2\mid 0\le x\le L, \ y=-d\},
\end{align}
を定め, 以下の初期値境界値問題を考察する:
\[
\Phi(t;\cdot,y) \in C^2_\#(0,L), t\in[0,T],\ y\in[-d,0], \tag{1}
\]
\[
\Delta \Phi(t;x,y)=0, (x,y)\in \Omega,\ t \in (0,T), \tag{2}
\]
\[
\frac{\partial^2 \Phi}{\partial t^2}(t;x,y) + g \frac{\partial \Phi}{\partial y}(t;x,y) = 0, (x,y)\in \Gamma_1,\ t \in (0,T), \tag{3}
\]
\[
\frac{\partial \Phi}{\partial y}(t;x,y) + M\frac{\partial}{\partial x} \left( \frac{\partial \Phi}{\partial x}(t;x-\delta,y) \right)^m=0, (x,y)\in \Gamma_2,\ t \in (0,T), \tag{4}
\]
\[
\Phi(0;x,y)=f(x,y), (x,y)\in\Omega, \tag{5}
\]
\[
\frac{\partial \Phi}{\partial t}(0;x,y)=h(x,y), (x,y)\in\Omega. \tag{6}
\]
ただし, 初期値 \(f\), \(h\) は次の整合条件を満たすものとする: \(f,h\in C^2(\overline{\Omega})\) は \(\Omega\) 上の調和関数であり,
\begin{align*}
&f(\cdot,y), h(\cdot,y) \in C^{2}_{\#}(0,L),
& &y\in[-d,0],\\
&\frac{\partial f}{\partial y} + M \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x}(x-\delta,y) \right)^m =0,
& &(x,y)\in \Gamma_2,\\
&\frac{\partial h}{\partial y} + M\frac{\partial}{\partial x} \left( \frac{\partial h}{\partial x}(x-\delta,y) \right)^m =0,
& &(x,y)\in \Gamma_2
\end{align*}
を満たす.
境界条件 (4) が非線型でかつ位置 \(x\) について \(\delta\) だけずれが生じているため, 初期値境界値問題 (1)--(6) の直接的な解析は極めて困難である. そこで 森 では, \(m = 1\), \(\delta = 0\) の場合に限って解析を行い, 次の結果を得た: (1)--(6) を \(x\) 変数に Fourier 級数展開すると,
\[
\frac{\partial^2 \Phi_n}{\partial y^2}(t,y) = \mu_n^2 \Phi_n(t,y), t\in(0,T),\ y\in(-d,0), \tag{7}
\]
\[
\frac{\partial^2 \Phi_n}{\partial t^2}(t,y) + g \frac{\partial \Phi_n}{\partial y}(t,y)=0,t\in(0,T),\ y=0, \tag{8}
\]
\[
\frac{\partial \Phi_n}{\partial y}(t,y) - G\mu_n^2 \Phi_n(t,y)=0, t\in(0,T),\ y=-d, \tag{9}
\]
\[
\Phi_n(0,y) = f_n(y), y\in(-d,0), \tag{10}
\]
\[
\frac{\partial \Phi_n}{\partial t}(0,y) = h_n(y), y\in(-d,0) \tag{11}
\]
となる. ただし,
\begin{align*}
&X_n(x) = \frac{1}{\sqrt{L}}e^{i\mu_n x},\quad \mu_n = \frac{2n\pi}{L},\\
&\Phi_n(t,y) = \int_0^L \Phi(t;x,y)\overline{X_n(x)}\,dx,\\
&f_n(y) = \int_0^L f(x,y)\overline{X_n(x)}\,dx,\\
&h_n(y) = \int_0^L h(x,y)\overline{X_n(x)}\,dx
\end{align*}
とおいた. また \(f_n\), \(h_n\) の満たす整合条件は
\[
\frac{d^2 f_n}{dy^2}(y)=\mu_n^2 f_n(y),\, y\in(-d,0), \tag{12}
\]
\[
\frac{d^2 h_n}{dy^2}(y)=\mu_n^2 h_n(y),\, y\in(-d,0), \tag{13}
\]
\[
\frac{d f_n}{dy}(-d) - G\mu_n^2 f_n(-d)=0, \tag{14}
\]
\[
\frac{d h_n}{dy}(-d) - G\mu_n^2 h_n(-d)=0 \tag{15}
\]
と書き下される. 初期値境界値問題 (7)--(11) について, 次の定理が証明された.
\(n\in\mathbb{Z}\) に対して \(f_n(y),h_n(y)\) が整合条件 (12)--(15) を満たす時, 次が成り立つ.
- \(n = 0\) の時, \(f_0(y)\), \(h_0(y)\) に対してある定数 \(F_0\), \(H_0\) が存在して, \(f_0(y) = F_0\), \(h_0(y) = H_0\) と表すことができる. また, 初期値境界値問題 (7)--(11) を満たす関数は
$$
\Phi_0(t,y) = F_0 + H_0 t
$$
で与えられる.
- \(n \neq 0\) の時, \(f_n(y)\), \(h_n(y)\) に対してある定数 \(F_n\), \(H_n\) が存在して, \(f_n(y) = F_n Y_n(y)\), \(h_n(y) = H_n Y_n(y)\) と表すことができる. また, 初期値境界値問題 (7)--(11) を満たす関数は
$$
\Phi_n(t, y) = (F_n \cos (\omega_n t) + \frac{H_n}{\omega_n} \sin (\omega_n t)) Y_n(y)
$$
で与えられる. ただし,
\begin{align}
Y_n(y) =& \cosh(\mu_n y)+\theta_n\sinh(\mu_n y), \label{eq:Yn}\\
\theta_n =& \frac{\sinh(\mu_n d)+G\mu_n\cosh(\mu_n d)}{\cosh(\mu_n d)+G\mu_n\sinh(\mu_n d)}, \label{eq:thetan}\\
\omega_n =& \sqrt{g\mu_n \theta_n} \label{eq:omegan}
\end{align}
である.
一般に, \(m = 1\), \(\delta = 0\) の場合, (1)--(6) を満たす解は, 定理1. で求めた Fourier モード解の線形結合, すなわち
\[
\Phi(t; x, y) = (F_0 + H_0t) X_0(x) + \sum_{n \in \mathbb{Z}, n \neq 0} \left(F_n\cos(\omega_n t)+\frac{H_n}{\omega_n}\sin(\omega_n t)\right) Y_n(y) X_n(x) \tag{16}
\]
で表されると予想される. しかしこの無限和がどの位相で収束し, その極限が (1)--(6) を満たすかどうかを議論することは困難である.
参考文献
- Kennedy J. F., The mechanics of dunes and antidunes in erodible bed channels, J. Fluid Mech., Vol.16, pp.521--544, 1963.
- 森 光太朗, ポテンシャル流を用いた反砂堆現象の数理モデルとその解析, 京都大学大学院情報学研究科修士論文, 2021.