$(\Psi_{k},\tilde{B}^{\intercal}\Phi)_{V^{1}_{per,0}(\Omega_{1})}=(\Psi_{k},\Phi_{0})_{V^{1}_{per,0}(\Omega_{1})}+\underbrace{(\Psi_{k},\Phi_{1})_{V^{1}_{per,0}(\Omega_{1})}}_{=0},$

where we used the equation of $\Phi_{1}$ and the fact that $\Psi_{k}$ has zero
boundary values in $\\{z=1\\}$. Hence we get by the properties of the basis
$\\{\Psi_{k}\\}_{k\text{ even}}$, that

$\lVert\sum_{k\leqslant N,k\text{
even}}(\Psi_{k},\tilde{B}^{\intercal}\Phi)_{V^{1}_{per,0}(\Omega_{1})}\Psi_{k}-\Phi_{0}\rVert_{H^{2}(\Omega_{1})}\to
0\text{ with }N\to\infty.$

  * •

Combining the arguments above we find that

$\lVert\mathcal{P}_{N}\Phi\rVert_{H^{2}(\Omega_{1})}\leqslant
C\lVert\Phi\rVert_{H^{2}(\Omega_{1})}+c\lVert\phi\rVert_{H^{2}(0,L)}.$

Moreover, $\mathcal{P}_{N}\Phi\to\Phi$ in $H^{2}(\Omega_{1})$ as $N\to\infty$.
Note that based on the similar but simpler analysis, the above estimate for
$\mathcal{P}_{N}\Phi$ holds also for the $H^{1}$ and $L^{2}$-norm.

We make the following ansatz, for every fixed $N\in{\mathbb{N}}$:

$\displaystyle\hat{u}_{N}(t,x,z):=\tilde{B}^{-\intercal}\sum_{k=1}^{N}\alpha^{k}_{N}(t)\Psi_{k}(x,z)\quad\text{
for all }t\in(0,T),$ (4.7) $\displaystyle
g_{N}(t,x):=\int_{0}^{t}\sum_{k=1}^{N}\alpha_{N}^{k}(\tau)\psi_{k}(x){\rm
d}\hbox{\hskip 0.5pt}\tau+\mathcal{P}_{N}h_{0}\quad\text{ for all }t\in(0,T).$

From the construction (4.7), we see that
$\hat{u}_{N}(t,x,1)={\partial}_{t}g_{N}{\hbox{\hskip 1.0pt}}{\rm e_{2}}$ for
$(t,x)\in(0,T)\times(0,L)$.

We construct now the solution $(\hat{u}_{N},g_{N})$ in four steps.

Step 1: Existence of ${\boldsymbol{\alpha}_{N}}$ for a given geometry. Assume
that $\tilde{h}\in C^{2}([0,T]\times[0,L];[h_{\min},h_{\max}])$ is the given
geometry and we still use the notation
$\tilde{B}=B_{\tilde{h}},\tilde{A}=A_{\tilde{h}}$ and
$\tilde{\chi}=\chi_{\tilde{h}}$.

In what follows we seek for the a couple of discrete solutions
$(\hat{u}_{N},g_{N})$ of the form (4.7) with time-dependent coefficients
${\boldsymbol{\alpha}_{N}}=(\alpha_{N}^{k})_{k=1}^{N}$, which solve the
following discrete equation, for $k=1,\cdots,N$:

$\displaystyle\begin{aligned}
&\left\langle\rho_{f}{\partial}_{t}\tilde{h}{\partial}_{t}\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\left\langle\rho_{f}\tilde{h}{\partial}_{t}^{2}\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\left\langle\rho_{f}\tilde{h}{\partial}_{t}\hat{u}_{N},{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle\\\
&+\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\hat{u}_{N}\left(\tilde{B}\nabla\right)\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left({\partial}_{t}\tilde{B}\nabla\right)\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle\\\
&+\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(\tilde{B}\nabla\right){\partial}_{t}\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(\tilde{B}\nabla\right)\hat{u}_{N},{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle\\\
&-\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\hat{u}_{N}\left(\tilde{B}\nabla\right)\left(\tilde{B}^{-\intercal}\Psi_{k}\right),\hat{u}_{N}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left({\partial}_{t}\tilde{B}\nabla\right)\left(\tilde{B}^{-\intercal}\Psi_{k}\right),\hat{u}_{N}\right\rangle\\\
&-\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(\tilde{B}\nabla\right)\left({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right),\hat{u}_{N}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(\tilde{B}\nabla\right)\left(\tilde{B}^{-\intercal}\Psi_{k}\right),{\partial}_{t}\hat{u}_{N}\right\rangle\\\
&+\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}^{2}\tilde{h}{\partial}_{t}g_{N},\psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\tilde{h}{\partial}_{t}^{2}g_{N},\psi_{k}\right\rangle\\\
&-\left\langle\rho_{f}{\partial}_{t}^{2}\tilde{\chi}(\tilde{B}\nabla)\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}({\partial}_{t}\tilde{B}\nabla)\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle\\\
&-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}(\tilde{B}\nabla){\partial}_{t}\hat{u}_{N},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}(\tilde{B}\nabla)\hat{u}_{N},{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle\\\
&+\left\langle\mu({\partial}_{t}\tilde{A}\nabla)\hat{u}_{N},\nabla(\tilde{B}^{-\intercal}\Psi_{k})\right\rangle+\left\langle\mu(\tilde{A}\nabla){\partial}_{t}\hat{u}_{N},\nabla(\tilde{B}^{-\intercal}\Psi_{k})\right\rangle\\\
&+\left\langle\mu(\tilde{A}\nabla)\hat{u}_{N},\nabla({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k})\right\rangle+\left\langle\rho_{s}{\partial}_{t}^{3}g_{N},\psi_{k}\right\rangle-\left\langle\beta{\partial}_{t}{\partial}_{x}^{2}g_{N},\psi_{k}\right\rangle\\\
&+\left\langle\alpha{\partial}_{t}{\partial}_{x}^{2}g_{N},{\partial}_{x}^{2}\psi_{k}\right\rangle=0,\end{aligned}$
(4.8)

where $(\Psi_{k},\psi_{k})$ are introduced in (4.5).

According to the definition of $\hat{u}_{N}$ and $g_{N}$ in (4.7), we have

$\displaystyle{\partial}_{t}\hat{u}_{N}=\sum_{j=1}^{N}{\partial}_{t}\tilde{B}^{-\intercal}{\alpha_{N}^{j}}(t)\Psi_{j}+\sum_{j=1}^{N}\tilde{B}^{-\intercal}{\alpha_{N}^{j}}^{\prime}(t)\Psi_{j},$
(4.9)
$\displaystyle{\partial}_{t}^{2}\hat{u}_{N}=\sum_{j=1}^{N}{\partial}_{t}^{2}\tilde{B}^{-\intercal}{\alpha_{N}^{j}}(t)\Psi_{j}+\sum_{j=1}^{N}2{\partial}_{t}\tilde{B}^{-\intercal}{\alpha_{N}^{j}}^{\prime}(t)\Psi_{j}+\sum_{j=1}^{N}\tilde{B}^{-\intercal}{\alpha_{N}^{j}}^{\prime\prime}(t)\Psi_{j},$
$\displaystyle\nabla\left({\partial}_{t}\hat{u}_{N}\right)=\sum_{j=1}^{N}{\alpha_{N}^{j}}^{\prime}(t)\nabla\left(\tilde{B}^{-\intercal}\Psi_{j}\right)+\sum_{j=1}^{N}{\alpha_{N}^{j}}(t)\nabla\left({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{j}\right),$
$\displaystyle{\partial}_{t}g_{N}=\sum_{j=1}^{N}{\alpha_{N}^{j}}(t)\psi_{j},\quad{\partial}_{t}^{2}g_{N}=\sum_{j=1}^{N}{\alpha_{N}^{j}}^{\prime}(t)\psi_{j},\quad{\partial}_{t}^{3}g_{N}=\sum_{j=1}^{N}{\alpha_{N}^{j}}^{\prime\prime}(t)\psi_{j}.$

We substitute (4.9) in the system (4.8) and obtain the second-order nonlinear
ODE system for ${\boldsymbol{\alpha}_{N}}=(\alpha_{N}^{k})_{k=1}^{N}$ as
follows:

$\mathcal{A}(t){\boldsymbol{\alpha}_{N}}^{\prime\prime}(t)+\mathcal{B}(t){\boldsymbol{\alpha}_{N}}^{\prime}(t)+\mathcal{C}(t){\boldsymbol{\alpha}_{N}}(t)+\left(\mathcal{D}(t)\cdot{\boldsymbol{\alpha}_{N}}^{\prime}(t)\right){\boldsymbol{\alpha}_{N}}(t)+\left(\mathcal{E}(t)\cdot{\boldsymbol{\alpha}_{N}}(t)\right){\boldsymbol{\alpha}_{N}}(t)=0,$
(4.10)

where the coefficients matrices $\mathcal{A}=(\mathcal{A}_{j,k})_{j,k=1}^{N}$,
$\mathcal{B}=(\mathcal{B}_{j,k})_{j,k=1}^{N}$,
$\mathcal{C}=(\mathcal{C}_{j,k})_{j,k=1}^{N}$,
$\mathcal{D}=(\mathcal{D}_{j,k}^{l})_{j,k,l=1}^{N}$ and
$\mathcal{E}=(\mathcal{E}_{j,k}^{l})_{j,k,l=1}^{N}$ are

$\mathcal{A}_{j,k}=\left\langle\rho_{f}\tilde{h}\tilde{B}^{-\intercal}\Psi_{j},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\left\langle\rho_{s}\psi_{j},\psi_{k}\right\rangle,$

$\displaystyle\mathcal{B}_{j,k}$
$\displaystyle=\left\langle\rho_{f}{\partial}_{t}\tilde{h}\tilde{B}^{-\intercal}\Psi_{j},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+2\left\langle\rho_{f}\tilde{h}{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{j},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad+\left\langle\rho_{f}\tilde{h}\tilde{B}^{-\intercal}\Psi_{j},{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad+\mu\left\langle\tilde{A}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\nabla(\tilde{B}^{-\intercal}\Psi_{k})\right\rangle+\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\tilde{h}\psi_{j},\psi_{k}\right\rangle,$

$\displaystyle\mathcal{C}_{j,k}$
$\displaystyle=\left\langle\rho_{f}{\partial}_{t}\tilde{h}{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{j},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\left\langle\rho_{f}\tilde{h}{\partial}_{t}^{2}\tilde{B}^{-\intercal}\Psi_{j},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad+\left\langle\rho_{f}\tilde{h}{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{j},{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}^{2}\tilde{\chi}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}{\partial}_{t}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}\tilde{B}\nabla({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad-\left\langle\rho_{f}{\partial}_{t}\tilde{\chi}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\left\langle\mu{\partial}_{t}\tilde{A}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\nabla(\tilde{B}^{-\intercal}\Psi_{k})\right\rangle$
$\displaystyle\quad+\left\langle\mu\tilde{A}\nabla({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{j}),\nabla(\tilde{B}^{-\intercal}\Psi_{k})\right\rangle+\left\langle\mu\tilde{A}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\nabla({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k})\right\rangle$
$\displaystyle\quad+\left\langle\beta{\partial}_{x}\psi_{j},{\partial}_{x}\psi_{k}\right\rangle+\left\langle\alpha{\partial}_{x}^{2}\psi_{j},{\partial}_{x}^{2}\psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}^{2}\tilde{h}\psi_{j},\psi_{k}\right\rangle,$

$\displaystyle D_{j,k}^{l}$
$\displaystyle=\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{l}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}{\hbox{\hskip
1.0pt}}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{l}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad-\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{l}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{k}),\tilde{B}^{-\intercal}\Psi_{l}\right\rangle,$

$\displaystyle\mathcal{E}_{j,k}^{l}$
$\displaystyle=\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{l}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}{\partial}_{t}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{l}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad+\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}\tilde{B}\nabla({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{l}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{l}),{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad-\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{l}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{j}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}{\partial}_{t}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{l}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle\quad-\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}\tilde{B}\nabla({\partial}_{t}\tilde{B}^{-\intercal}\Psi_{l}),\tilde{B}^{-\intercal}\Psi_{k}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\tilde{B}^{-\intercal}\Psi_{j}\tilde{B}\nabla(\tilde{B}^{-\intercal}\Psi_{k}),{\partial}_{t}\tilde{B}^{-\intercal}\Psi_{l}\right\rangle.$

We shall show in what follows that for given initial data
${\boldsymbol{\alpha}_{N}}(0)$ and ${\boldsymbol{\alpha}_{N}}^{\prime}(0)$,
the differential equation (4.10) allows a unique solution
${\boldsymbol{\alpha}_{N}}$. The choice of the initial value
${\boldsymbol{\alpha}_{N}}^{\prime}(0)$ will be discussed in Step 2 below. We
see that the matrix $\mathcal{A}$ is obviously symmetric and for every
$\xi\in{\mathbb{R}}^{N}\setminus\\{0\\}$, we have

$\displaystyle\mathcal{A}\xi\cdot\xi$
$\displaystyle=\rho_{f}\tilde{h}\sum_{j,k=1}^{N}\xi_{j}\xi_{k}\left\langle\tilde{B}^{-\intercal}\Psi_{j},\tilde{B}^{-\intercal}\Psi_{k}\right\rangle+\rho_{s}\sum_{j,k=1}^{N}\xi_{j}\xi_{k}\left\langle\psi_{j},\psi_{k}\right\rangle$
(4.11) $\displaystyle\geqslant
C\left\langle\tilde{B}^{-\intercal}\sum_{j=1}^{N}\xi_{j}\Psi_{j},\tilde{B}^{-\intercal}\sum_{k=1}^{N}\xi_{k}\Psi_{k}\right\rangle+c\sum_{k=1}^{N}\left\lVert\xi_{k}\psi_{k}\right\rVert^{2}$
$\displaystyle>0.$

Recalling that in the given geometry $\tilde{h}\in
C^{2}([0,T]\times[0,L];[h_{\min},h_{\max}])$, all the coefficients in (4.10)
are continuous in $t$ and all the non-linear quantites in (4.10) are locally
Lipschitz continuous in ${\boldsymbol{\alpha}_{N}}$ and
${\boldsymbol{\alpha}_{N}}^{\prime}$. According to the Picard-Lindelöf
theorem, there is a unique solution of (4.10) in short time. Therefore, we
obtain a solution $(\hat{u}_{N},g_{N})$ of (4.8) which is given in form of
(4.7).

Step 2: Choosing the initial values for the time-derivative. For the initial
data, we take

$\displaystyle\hat{u}_{N}(0,x,z)=\mathcal{P}_{N}\hat{u}_{0},\quad
g_{N}(0,x)=\mathcal{P}_{N}h_{0},\quad({\partial}_{t}g_{N})(0,x)=\mathcal{P}_{N}h_{1},$
(4.12)

where the projection $\mathcal{P}_{N}$ has been defined in (4.6). From the
definition of $\hat{u}_{N}$ in (4.7), we note that actually
${\boldsymbol{\alpha}_{N}}(0)$ is determined by $\hat{u}_{0}$ and $h_{0}$,
while ${\boldsymbol{\alpha}_{N}}^{\prime}(0)$ is free. With the assumption
(4.12) we choose the initial value ${\boldsymbol{\alpha}_{N}}^{\prime}(0)$ by
considering the following equality:

$\displaystyle\left\langle\rho_{f}h_{0}({\partial}_{t}\hat{u}_{N})(0),B^{-\intercal}_{h_{0}}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\mathcal{P}_{N}\hat{u}_{0}\left(B_{h_{0}}\nabla\right)\mathcal{P}_{N}\hat{u}_{0},B^{-\intercal}_{h_{0}}\Psi_{k}\right\rangle$
(4.13)
$\displaystyle-\frac{1}{2}\left\langle\rho_{f}\mathcal{P}_{N}\hat{u}_{0}\left(B_{h_{0}}\nabla\right)(B^{-\intercal}_{h_{0}}\Psi_{k}),\mathcal{P}_{N}\hat{u}_{0}\right\rangle+\frac{1}{2}\left\langle\rho_{f}h_{1}\mathcal{P}_{N}h_{1},\psi_{k}\right\rangle$
$\displaystyle-\left\langle\rho_{f}{\partial}_{t}\chi_{h_{0}}\left(B_{h_{0}}\nabla\right)\mathcal{P}_{N}\hat{u}_{0},B^{-\intercal}_{h_{0}}\Psi_{k}\right\rangle+\mu\left\langle\left(A_{h_{0}}\nabla\right)\mathcal{P}_{N}\hat{u}_{0},\nabla\left(B^{-\intercal}_{h_{0}}\Psi_{k}\right)\right\rangle$
$\displaystyle+\rho_{s}\left\langle({\partial}_{t}^{2}g_{N})(0),\psi_{k}\right\rangle-\beta\left\langle{\partial}_{x}^{2}\mathcal{P}_{N}h_{0},\psi_{k}\right\rangle+\alpha\left\langle{\partial}_{x}^{4}\mathcal{P}_{N}h_{0},\psi_{k}\right\rangle=0,$

where the projection operator $\mathcal{P}_{N}$ has been introduced in (4.6).
Using the expression in (4.9) for $({\partial}_{t}\hat{u}_{N})(0)$ and
$({\partial}_{t}^{2}g_{N})(0)$, we immediately obtain from (4.13) the equation
for ${\boldsymbol{\alpha}_{N}}^{\prime}(0)$:

$M{\boldsymbol{\alpha}_{N}}^{\prime}(0)=F({\boldsymbol{\alpha}_{N}}(0),\hat{u}_{0},h_{0},h_{1}),$
(4.14)

where the coefficient matrix $M=(M_{j,k})_{j,k=1}^{N}$ is given by

$M_{j,k}=\rho_{f}h_{0}\left\langle
B^{-\intercal}_{h_{0}}\Psi_{j},B^{-\intercal}_{h_{0}}\Psi_{k}\right\rangle+\rho_{s}\left\langle\psi_{j},\psi_{k}\right\rangle.$

By a similar analysis argument of $\mathcal{A}$ in (4.11), we know that the
matrix $M$ is invertible. Therefore, the initial value
${\boldsymbol{\alpha}_{N}}^{\prime}(0)$ is uniquely determined by the equality
(4.14).

Now we consider obtaining the initial value $({\partial}_{t}\hat{u}_{N})(0)$
and $({\partial}_{t}^{2}g_{N})(0)$ from the assumption of the regularity of
$\hat{u}_{0}$, $h_{0}$ and $h_{1}$. For this, we have the following
proposition.

######  Proposition 4.2.

With the assumption in (4.12), let the initial data $\hat{u}_{0}$, $h_{0}$ and
$h_{1}$ satisfy

$\hat{u}_{0}\in H^{2}({\Omega}_{1}),\quad h_{0}\in H^{4}(0,L),\quad h_{1}\in
H^{2}(0,L),$ (4.15)

then we have $({\partial}_{t}\hat{u}_{N})(0)\in L^{2}({\Omega}_{1})$ and
$({\partial}_{t}^{2}g_{N})(0)\in L^{2}(0,L)$.

###### Proof.

Still using the structure in (4.9) and recalling the projection
$\mathcal{P}_{N}$ defined in (4.6), we first take an integration by parts with
respect to space for the third and the sixth terms in (4.13):

$\displaystyle-\left\langle\mathcal{P}_{N}\hat{u}_{0}\left(B_{h_{0}}\nabla\right)(B^{-\intercal}_{h_{0}}\Psi_{k}),\mathcal{P}_{N}\hat{u}_{0}\right\rangle$
$\displaystyle=-\int_{{\Omega}_{1}}{\rm
div\,}\left((B^{\intercal}_{h_{0}}\mathcal{P}_{N}\hat{u}_{0})\otimes
B^{-\intercal}_{h_{0}}\Psi_{k}\right)\cdot\mathcal{P}_{N}\hat{u}_{0}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}$
$\displaystyle=-\int_{0}^{L}|\mathcal{P}_{N}h_{1}|^{2}\psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}x+\int_{{\Omega}_{1}}B^{-\intercal}_{h_{0}}\Psi_{k}\cdot\mathcal{P}_{N}\hat{u}_{0}(B_{h_{0}}\nabla)\mathcal{P}_{N}\hat{u}_{0}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z},$

and

$\displaystyle\left\langle\left(A_{h_{0}}\nabla\right)\hat{u}_{0}:\nabla\left(B^{-\intercal}_{h_{0}}\Psi_{k}\right)\right\rangle$
$\displaystyle=\int_{0}^{L}\left[A_{h_{0}}\nabla\hat{u}_{0}B^{-\intercal}_{h_{0}}\Psi_{k}\right](z=1)\cdot{\rm
e_{2}}{\rm d}\hbox{\hskip 0.5pt}x-\int_{{\Omega}_{1}}{\rm
div\,}(A_{h_{0}}\nabla\hat{u}_{0})\cdot
B^{-\intercal}_{h_{0}}\Psi_{k}{\hbox{\hskip 1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}.$

Multiply the equality (4.13) by $(\alpha_{N}^{k})^{\prime}(0)$ and sum over
$k=1,\cdots,N$. Note that we have

$\displaystyle\sum_{k=1}^{N}{\alpha_{N}^{k}}^{\prime}(0)\psi_{k}=({\partial}_{t}^{2}g_{N})(0),$
$\displaystyle\sum_{k=1}^{N}B_{h_{0}}^{-\intercal}{\alpha_{N}^{k}}^{\prime}(0)\Psi_{k}=B_{h_{0}}^{-\intercal}{\partial}_{t}\left(B_{h_{0}}^{\intercal}\hat{u}_{N}(0)\right)=B_{h_{0}}^{-\intercal}{\partial}_{t}B_{h_{0}}^{\intercal}\hat{u}_{N}(0)+({\partial}_{t}\hat{u}_{N})(0).$

Under the assumption (4.15), with (4.12) and (4.9), we derive that

$\lVert({\partial}_{t}\hat{u}_{N})(0)\rVert_{L^{2}({\Omega}_{1})}^{2}+\lVert({\partial}_{t}^{2}g_{N})(0)\rVert_{L^{2}(0,L)}^{2}\leqslant
c(C_{0})\left(\lVert\hat{u}_{0}\rVert_{H^{2}({\Omega}_{1})}^{2}+\lVert
h_{0}\rVert_{H^{4}(0,L)}^{2}+\lVert h_{1}\rVert_{H^{2}(0,L)}^{2}\right),$

which ends the proof. In the above estimate, we used the properties of the
projection discussed below (4.6) for initial data. ∎

Step 3: Energy estimate for the discrete decoupled solution. Now we derive the
energy estimate to extend the existence interval of
${\boldsymbol{\alpha}_{N}}$. Taking integration of (4.8) with respect to time
variable and recalling the equation (4.13), we thereby obtain

$\displaystyle\int_{{\Omega}_{1}}\rho_{f}\tilde{h}{\partial}_{t}\hat{u}_{N}\cdot\tilde{B}^{-\intercal}\Psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}+\frac{1}{2}\int_{{\Omega}_{1}}\rho_{f}\hat{u}_{N}(\tilde{B}\nabla)\hat{u}_{N}\cdot\tilde{B}^{-\intercal}\Psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}$ (4.16)
$\displaystyle-\frac{1}{2}\int_{{\Omega}_{1}}\rho_{f}\hat{u}_{N}(\tilde{B}\nabla)\left(\tilde{B}^{-\intercal}\Psi_{k}\right)\cdot\hat{u}_{N}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}+\frac{1}{2}\int_{0}^{L}\rho_{f}{\partial}_{t}\tilde{h}{\partial}_{t}g_{N}\psi_{k}{\rm
d}\hbox{\hskip 0.5pt}x$
$\displaystyle-\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\tilde{\chi}(\tilde{B}\nabla)\hat{u}_{N}\cdot\tilde{B}^{-\intercal}\Psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}+\mu\int_{{\Omega}_{1}}(\tilde{A}\nabla)\hat{u}_{N}:\nabla\left(\tilde{B}^{-\intercal}\Psi_{k}\right){\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}$
$\displaystyle+\rho_{s}\int_{0}^{L}{\partial}_{t}^{2}g_{N}\psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}x-\beta\int_{0}^{L}{\partial}_{x}^{2}g_{N}\psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}x+\alpha\int_{0}^{L}{\partial}_{x}^{2}g_{N}{\partial}_{x}^{2}\psi_{k}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}x=0.$

######  Proposition 4.3.

For the solution to the ODE we have the following energy estimates:

$\displaystyle
h_{\min}\rho_{f}\lVert\hat{u}_{N}\rVert^{2}_{L^{\infty}(0,T;L^{2}({\Omega}_{1}))}+\mu
c(\tilde{h})\lVert\nabla\hat{u}_{N}\rVert^{2}_{L^{2}(0,T;L^{2}({\Omega}_{1}))}$
(4.17)
$\displaystyle+\rho_{s}\lVert{\partial}_{t}g_{N}\rVert^{2}_{L^{\infty}(0,T;L^{2}(0,L))}+\beta\lVert{\partial}_{x}g_{N}\rVert^{2}_{L^{\infty}(0,T;L^{2}(0,L))}+\alpha\lVert{\partial}_{x}^{2}g_{N}\rVert^{2}_{L^{\infty}(0,T;L^{2}(0,L))}$
$\displaystyle\leqslant
C\left(\lVert\hat{u}_{0}\rVert_{L^{2}({\Omega}_{1})}+\lVert
h_{1}\rVert_{L^{2}(0,L)}^{2}+\lVert
h_{0}\rVert_{H^{2}(0,L)}^{2}\right)=:\tilde{C}.$

Please note that the constant $\tilde{C}$ does not depend on
$\tilde{h},h_{\min},h_{\max}$.

###### Proof.

Multiply (4.16) by $\alpha_{N}^{k}(t)$ and sum over $k=1,\cdots,N$. Note that

${\partial}_{t}\tilde{h}={\rm
div\,}(\tilde{B}^{\intercal}{\partial}_{t}\tilde{\chi}),$

by using integration by parts we have

$\displaystyle\frac{1}{2}\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\tilde{h}|\hat{u}_{N}|^{2}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}$
$\displaystyle=\frac{1}{2}\int_{{\Omega}_{1}}\rho_{f}{\rm
div\,}(\tilde{B}^{\intercal}{\partial}_{t}\tilde{\chi})|\hat{u}_{N}|^{2}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}$
$\displaystyle=\frac{1}{2}\int_{0}^{L}\rho_{f}{\partial}_{t}\tilde{h}|{\partial}_{t}g_{N}|^{2}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}x-\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\tilde{\chi}(\tilde{B}\nabla)\hat{u}_{N}\cdot\hat{u}_{N}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z},$

which further implies that

$\int_{{\Omega}_{1}}\rho_{f}\tilde{h}\hat{u}_{N}\cdot{\partial}_{t}\hat{u}_{N}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}+\frac{1}{2}\int_{0}^{L}\rho_{f}{\partial}_{t}\tilde{h}|{\partial}_{t}g_{N}|^{2}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}x-\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\tilde{\chi}(\tilde{B}\nabla)\hat{u}_{N}\cdot\hat{u}_{N}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}=\frac{1}{2}\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}\int_{{\Omega}_{1}}\rho_{f}\tilde{h}|\hat{u}_{N}|^{2}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}.$

Recalling the structure in (4.7) and integrating the equation with respect to
time, we immediately obtain the energy balance (4.17). ∎

Step 4: Performing the fixed-point theorem. We consider the following map for
a fixed point. Let $h_{\min},h_{\max}$ and $T$ be fixed by the initial
conditions (See Remark 2.3). Similar with ${\boldsymbol{\alpha}_{N}}$
introduced in (4.10), here we use the notation
${\boldsymbol{\beta}_{N}}:=\left(\beta_{N}^{k}\right)_{k=1}^{N}$. We take
$K_{N}$ as the set of all vector-valued functions ${\boldsymbol{\beta}_{N}}$
satisfying the following conditions:

$K_{N}:=\left\\{\begin{aligned} &{\boldsymbol{\beta}_{N}}\in
C^{1}([0,T];\mathbb{R}^{N})\left|{\hbox{\hskip
1.0pt}}h_{\beta}(t,x):=\int_{0}^{t}\sum_{k=1}^{N}\beta_{N}^{k}(\tau)\psi_{k}(x){\rm
d}\hbox{\hskip 0.5pt}\tau+\mathcal{P}_{N}h_{0}\right.\\\ &{\hbox{\hskip
1.0pt}}{\hbox{\hskip 1.0pt}}{\hbox{\hskip 1.0pt}}\text{such that}\quad
h_{\min}\leqslant h_{\beta}\leqslant
h_{\max},\quad\rho_{s}\lVert{\partial}_{t}h_{\beta}\rVert_{L^{\infty}(0,T;L^{2}(0,L))}^{2}\leqslant\tilde{C}.\end{aligned}\right\\},$

where the projection $\mathcal{P}_{N}$ has been introduced in (4.6). Taking
such a function $h_{\beta}$ as the geometry, for $N$ given we consider the
following map $F_{N}$:

$\displaystyle F_{N}:K_{N}$ $\displaystyle\to C^{2}([0,T];\mathbb{R}^{N})$
$\displaystyle{\boldsymbol{\beta}_{N}}$ $\displaystyle\mapsto
F_{N}({\boldsymbol{\beta}_{N}})={\boldsymbol{\alpha}_{N}},$

where ${\boldsymbol{\alpha}_{N}}$ is the unique solution to the ODE (4.10). We
shall use the Theorem of Schauder for fixed-point argument. For that we have
to check the following points:

  * •

The set $K_{N}$ is convex and closed, as can be seen from its definition.

  * •

The mapping is continuous and compact, as follows directly from classical ODE
theory as the system is continuous and solutions are in $C^{2}[0,T]$ which is
a compact subset of $C^{1}[0,T]$.

  * •

The mapping is onto $K_{N}$, because of Remark 2.3 and Proposition 4.3. Indeed
the energy estimate allows bounds on $\partial_{t}g_{N}$ and
$\partial_{x}^{2}g_{N}$ independent of $\tilde{h}$. This implies in particular
that for $T_{0}$ fixed by Remark 2.3 and according choices of $h_{\min}$ and
$h_{\max}$ that any function with bounded energy and with given initial values
stays in the interval $[h_{\min},h_{\max}]$.

Hence the map $F_{N}$ possesses a fixed point in the set $K_{N}$. This means
we find the existence of a coupled solution $(\hat{u}_{N},g_{N})$ to the
system:

$\displaystyle\left\langle\rho_{f}{\partial}_{t}g_{N}{\partial}_{t}\hat{u}_{N},B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle+\left\langle\rho_{f}{g_{N}}{\partial}_{t}^{2}\hat{u}_{N},B^{-\intercal}_{g_{N}}\Psi_{k}\right\rangle+\left\langle\rho_{f}{g_{N}}{\partial}_{t}\hat{u}_{N},{\partial}_{t}B^{-\intercal}_{g_{N}}\Psi_{k}\right\rangle$
(4.18)
$\displaystyle+\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\hat{u}_{N}\left(B_{g_{N}}\nabla\right)\hat{u}_{N},B^{-\intercal}_{g_{N}}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left({\partial}_{t}B_{g_{N}}\nabla\right)\hat{u}_{N},B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle+\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(B_{g_{N}}\nabla\right){\partial}_{t}\hat{u}_{N},B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle+\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(B_{g_{N}}\nabla\right)\hat{u}_{N},{\partial}_{t}B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle-\frac{1}{2}\left\langle\rho_{f}{\partial}_{t}\hat{u}_{N}\left(B_{g_{N}}\nabla\right)\left(B_{g_{N}}^{-\intercal}\Psi_{k}\right),\hat{u}_{N}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left({\partial}_{t}B_{g_{N}}\nabla\right)\left(B_{g_{N}}^{-\intercal}\Psi_{k}\right),\hat{u}_{N}\right\rangle$
$\displaystyle-\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(B_{g_{N}}\nabla\right)\left({\partial}_{t}B_{g_{N}}^{-\intercal}\Psi_{k}\right),\hat{u}_{N}\right\rangle-\frac{1}{2}\left\langle\rho_{f}\hat{u}_{N}\left(B_{g_{N}}\nabla\right)\left(B_{g_{N}}^{-\intercal}\Psi_{k}\right),{\partial}_{t}\hat{u}_{N}\right\rangle$
$\displaystyle+\left\langle\rho_{f}{\partial}_{t}^{2}g_{N}{\partial}_{t}g_{N},\psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}^{2}\chi_{g_{N}}(B_{g_{N}}\nabla)\hat{u}_{N},B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}\chi_{g_{N}}({\partial}_{t}B_{g_{N}}\nabla)\hat{u}_{N},B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle-\left\langle\rho_{f}{\partial}_{t}\chi_{g_{N}}(B_{g_{N}}\nabla){\partial}_{t}\hat{u}_{N},B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle-\left\langle\rho_{f}{\partial}_{t}\chi_{g_{N}}(B_{g_{N}}\nabla)\hat{u}_{N},{\partial}_{t}B_{g_{N}}^{-\intercal}\Psi_{k}\right\rangle$
$\displaystyle+\left\langle\mu({\partial}_{t}{A_{g_{N}}}\nabla)\hat{u}_{N},\nabla(B_{g_{N}}^{-\intercal}\Psi_{k})\right\rangle+\left\langle\mu(A_{g_{N}}\nabla){\partial}_{t}\hat{u}_{N},\nabla(B_{g_{N}}^{-\intercal}\Psi_{k})\right\rangle$
$\displaystyle+\left\langle\mu(A_{g_{N}}\nabla)\hat{u}_{N},\nabla({\partial}_{t}B_{g_{N}}^{-\intercal}\Psi_{k})\right\rangle+\left\langle\rho_{s}{\partial}_{t}^{3}g_{N},\psi_{k}\right\rangle-\left\langle\beta{\partial}_{t}{\partial}_{x}^{2}g_{N},\psi_{k}\right\rangle$
$\displaystyle+\left\langle\alpha{\partial}_{t}{\partial}_{x}^{2}g_{N},{\partial}_{x}^{2}\psi_{k}\right\rangle=0,$

where $\hat{u}_{N}(t,x,1)={\partial}_{t}g_{N}{\rm e_{2}}$ for all
$(t,x)\in(0,T)\times(0,L)$.

###  4.3 The higher order in time-estimate

It can be seen from the definition of the ODE, that multiplying (4.18) with
$\alpha_{N}^{\prime}$ is precisely testing the time-derivative equation with
the coupled test-function
$(\partial_{t}^{2}g_{N},\partial_{t}\hat{u}_{N}+G_{N})$ with $G_{N}$ similarly
defined as in (3.2), i.e.
$G_{N}=B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}$.
Hence the formal estimate of Section 3 can be performed rigorously on the
discrete level. Therefore, we have the following theorem. Please observe, that
together with Proposition 4.2 this implies uniform higher order estimates.

######  Theorem 4.4.

Under the conditions (4.15) there exists a discrete solution to (4.18), that
satisfies the first-order equation (4.16) (for the coupled equation with
$\tilde{h}$ replaced by $g_{N}$), the energy inequality (4.17) and the
additional time-derivative estimate:

$\displaystyle\rho_{f}\lVert{\partial}_{t}\hat{u}_{N}\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{1}))}^{2}+\mu\lVert\nabla{\partial}_{t}\hat{u}_{N}\rVert^{2}_{L^{2}(0,T;L^{2}({\Omega}_{1}))}+\rho_{s}\lVert{\partial}_{t}^{2}g_{N}\rVert_{L^{\infty}(0,T;L^{2}(0,L))}$
(4.19)
$\displaystyle+\beta\lVert{\partial}_{t}{\partial}_{x}g_{N}\rVert_{L^{\infty}(0,T;L^{2}(0,L))}+\alpha\lVert{\partial}_{t}{\partial}_{x}^{2}g_{N}\rVert^{2}_{L^{\infty}(0,T;L^{2}(0,L))}$
$\displaystyle\leqslant
C(h_{\min},\lVert\hat{u}_{0}\rVert_{L^{2}({\Omega}_{1})},\lVert
h_{0}\rVert_{H^{2}(0,L)}),\lVert
h_{1}\rVert_{L^{2}(0,L)})\Big{(}\lVert({\partial}_{t}\hat{u})(0)\rVert_{L^{2}({\Omega}_{1})}^{2}+\lVert({\partial}_{t}^{2}h)(0)\rVert_{L^{2}(0,L)}^{2}+\lVert
h_{1}\rVert_{H^{2}(0,L)}^{2}\Big{)},$

which holds in particular independent of $N$.

###### Proof.

The existence follows by the fixed point performed in the last subsection. We
are left to show the time-derivative estimate of $\hat{u}_{N}$, i.e. (4.19).
To do this, we first derive the following identities:

$\displaystyle\sum_{k=1}^{N}{\alpha_{N}^{k}}^{\prime}(t)\psi_{k}={\partial}_{t}^{2}g_{N},$
$\displaystyle\sum_{k=1}^{N}B_{g_{N}}^{-\intercal}{\alpha_{N}^{k}}^{\prime}(t)\Psi_{k}=B_{g_{N}}^{-\intercal}{\partial}_{t}\left(B_{g_{N}}^{\intercal}\hat{u}_{N}\right)=B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}+{\partial}_{t}\hat{u}_{N},$
$\displaystyle\sum_{k=1}^{N}{\partial}_{t}B_{g_{N}}^{-\intercal}{\alpha_{N}^{k}}^{\prime}(t)\Psi_{k}={\partial}_{t}B_{g_{N}}^{-\intercal}{\partial}_{t}\left(B_{g_{N}}^{\intercal}\hat{u}_{N}\right)={\partial}_{t}B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}+{\partial}_{t}B_{g_{N}}^{-\intercal}B_{g_{N}}^{\intercal}{\partial}_{t}\hat{u}_{N}.$

By using the fact that $\nabla(Av)=(\nabla A)v+A(\nabla v)$, where $A$ is a
$2\times 2$ matrix and $v$ is a $2\times 1$ matrix, we also derive that

$\displaystyle\sum_{k=1}^{N}{\alpha_{N}^{k}}^{\prime}(t)\nabla\left(B_{g_{N}}^{-\intercal}\Psi_{k}\right)=\nabla\left(B_{g_{N}}^{-\intercal}{\partial}_{t}\left(B_{g_{N}}^{\intercal}\hat{u}_{N}\right)\right)$
$\displaystyle=\nabla
B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}+\nabla
B_{g_{N}}^{-\intercal}B_{g_{N}}^{\intercal}{\partial}_{t}\hat{u}_{N}+B_{g_{N}}^{-\intercal}\nabla{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}$
$\displaystyle\quad+B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\nabla\hat{u}_{N}+B_{g_{N}}^{-\intercal}\nabla
B_{g_{N}}^{\intercal}{\partial}_{t}\hat{u}_{N}+\nabla{\partial}_{t}\hat{u}_{N}$
and
$\displaystyle\sum_{k=1}^{N}{\alpha_{N}^{k}}^{\prime}(t)\nabla\left({\partial}_{t}B_{g_{N}}^{-\intercal}\Psi_{k}\right)=\nabla\left({\partial}_{t}B_{g_{N}}^{-\intercal}{\partial}_{t}\left(B_{g_{N}}^{\intercal}\hat{u}_{N}\right)\right)$
$\displaystyle=\nabla{\partial}_{t}B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}+\nabla{\partial}_{t}B_{g_{N}}^{-\intercal}B_{g_{N}}^{\intercal}{\partial}_{t}\hat{u}_{N}+{\partial}_{t}B_{g_{N}}^{-\intercal}\nabla{\partial}_{t}B_{g_{N}}^{\intercal}\hat{u}_{N}$
$\displaystyle\quad+{\partial}_{t}B_{g_{N}}^{-\intercal}{\partial}_{t}B_{g_{N}}^{\intercal}\nabla\hat{u}_{N}+{\partial}_{t}B_{g_{N}}^{-\intercal}\nabla
B_{g_{N}}^{\intercal}{\partial}_{t}\hat{u}_{N}+{\partial}_{t}B_{g_{N}}^{-\intercal}B_{g_{N}}^{\intercal}\nabla{\partial}_{t}\hat{u}_{N}.$

Now multiply (4.18) by ${\alpha_{N}^{k}}^{\prime}(t)$, sum over $k=1,\cdots,N$
and substitute the above sum expressions in the resulting equation, which
allows to reconstruct (3.1). First taking the integration with time on $(0,t)$
for $t\in(0,T)$, we note that the terms kept on the left side are

$\displaystyle\rho_{f}\lVert{\partial}_{t}\hat{u}_{N}\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{1}))}^{2}+\mu\lVert\nabla{\partial}_{t}\hat{u}_{N}\rVert^{2}_{L^{2}(0,T;L^{2}({\Omega}_{1}))}+\rho_{s}\lVert{\partial}_{t}^{2}g_{N}\rVert_{L^{\infty}(0,T;L^{2}(0,L))}$
$\displaystyle+\beta\lVert{\partial}_{t}{\partial}_{x}g_{N}\rVert_{L^{\infty}(0,T;L^{2}(0,L))}+\alpha\lVert{\partial}_{t}{\partial}_{x}^{2}g_{N}\rVert^{2}_{L^{\infty}(0,T;L^{2}({\Omega}_{1}))}.$

which precisely corresponds to the left hand side of (3.1). We show in what
follows that, multiplying (4.18) by ${\alpha_{N}^{k}}^{\prime}(t)$, the
resulting discrete equation also has the same right hand side as (3.1) in the
continuous level. For that we first note from the above series structure that
$\sum_{k=1}^{N}B_{g_{N}}^{-\intercal}{\alpha_{N}^{k}}^{\prime}(t)\Psi_{k}$
represents ${\partial}_{t}\hat{u}_{N}+G_{N}$. With the definition of $\varphi$
in (3.3), we derive from the formula of $G$ in (3.2) that

$\varphi=B_{h}^{-\intercal}{\partial}_{t}B_{h}^{\intercal}({\partial}_{t}\hat{u}+B_{h}^{-\intercal}{\partial}_{t}B_{h}^{\intercal}\hat{u})=-{\partial}_{t}B_{h}^{-\intercal}{\partial}_{t}B_{h}^{\intercal}\hat{u}-{\partial}_{t}B_{h}^{-\intercal}B_{h}^{\intercal}{\partial}_{t}\hat{u},$

which corresponds to
$\sum_{k=1}^{N}{\partial}_{t}B_{g_{N}}^{-\intercal}{\alpha_{N}^{k}}^{\prime}(t)\Psi_{k}$
in the discrete level. We used in the above the fact that
$B_{h}^{-\intercal}{\partial}_{t}B_{h}^{\intercal}=-{\partial}_{t}B_{h}^{-\intercal}B_{h}^{\intercal}$.
Hence, the series
$\sum_{k=1}^{N}{\alpha_{N}^{k}}^{\prime}(t)\nabla\left({\partial}_{t}B_{g_{N}}^{-\intercal}\Psi_{k}\right)$
thereby corresponds to $\nabla\varphi$. Form here, we realized that the terms
depending on $\varphi$ on the right hand side in (3.1) appear here due to the
time derivative of $\partial_{t}B_{g_{N}}^{-\intercal}$, which is precisely
related to the pressure.

Comparing the right hand sides of the continuous equation (3.1) with the
discrete equation, we realize that, after doing the integration by parts for
the solid part, only the convective terms need to be clarified. For that we
rewrite the conventive term as it appears in the continuous equation in (3.1),
such that it connects to (4.18) multiplied by ${\alpha_{N}^{k}}^{\prime}(t)$.
Using the structure (4.2) we derive that

$\displaystyle-\frac{1}{2}\int_{0}^{t}\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\hat{u}(B_{h}\nabla)\hat{u}\cdot({\partial}_{t}\hat{u}+G){\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}=\frac{1}{2}\int_{0}^{t}\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\hat{u}(B_{h}\nabla)({\partial}_{t}\hat{u}+G)\cdot\hat{u}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}-\frac{1}{2}\int_{0}^{t}\rho_{f}|{\partial}_{t}^{2}h|^{2}{\partial}_{t}h{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}x,$

and

$-\frac{1}{2}\int_{0}^{t}\int_{{\Omega}_{1}}\rho_{f}\hat{u}(B_{h}\nabla){\partial}_{t}\hat{u}\cdot({\partial}_{t}\hat{u}+G){\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}=\frac{1}{2}\int_{0}^{t}\int_{{\Omega}_{1}}\rho_{f}\hat{u}(B_{h}\nabla)({\partial}_{t}\hat{u}+G)\cdot{\partial}_{t}\hat{u}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}-\frac{1}{2}\int_{0}^{t}\rho_{f}|{\partial}_{t}^{2}h|^{2}{\partial}_{t}h{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}x,$
$-\frac{1}{2}\int_{0}^{t}\int_{{\Omega}_{1}}\rho_{f}\hat{u}({\partial}_{t}B_{h}\nabla)\hat{u}\cdot({\partial}_{t}\hat{u}+G){\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}=\frac{1}{2}\int_{0}^{t}\int_{{\Omega}_{1}}\rho_{f}\hat{u}({\partial}_{t}B_{h}\nabla)({\partial}_{t}\hat{u}+G)\cdot\hat{u}{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z},$

where we observe that no boundary term in the last equality since
${\partial}_{t}B_{h}^{\intercal}\hat{u}\otimes\hat{u}({\partial}_{t}\hat{u}+G)$
vanishes at $z=1$.

Based on the above analysis, we obtain that the resulting equation by
multipying (4.18) by ${\alpha_{N}^{k}}^{\prime}(t)$ is exactly corresponds to
the structure (3.1) in the continuous sense. Moreover, $(\hat{u}_{N},g_{N})$
preserves the energy estimate (4.17). Therefore, the proof follows line by
line using the estimates of (3.1). This finishes the proof. ∎

###  4.4 Existence of a strong solution

By Theorem 4.4 together with Proposition 4.2 we got the time-derivative
estimate on the discrete level. This allows us to reconstruct a weak solution,
with additional regularity properties. Together this implies the existence of
a strong solution. With the help of (4.17) and (4.19), we are able to deal
with the limit procedure.

Limit passage for $\hat{u}$ and $h$.

Based on the uniform estimates of Proposotion 4.3 and Theorem 4.4, sending
$N\to\infty$, we obtain the following convergence, up to a subsequence, for
every $T>0$,

$\displaystyle\hat{u}_{N}$
$\displaystyle\rightharpoonup\hat{u}\quad\text{weak-$\ast$ in}\quad
L^{\infty}(0,T;L^{2}({\Omega}_{1}))\cap L^{2}(0,T;H^{1}({\Omega}_{1})),$
(4.20) $\displaystyle{\partial}_{t}\hat{u}_{N}$
$\displaystyle\rightharpoonup{\partial}_{t}\hat{u}\quad\text{weak-$\ast$
in}\quad L^{\infty}(0,T;L^{2}({\Omega}_{1}))\cap
L^{2}(0,T;H^{1}({\Omega}_{1})),$ $\displaystyle g_{N}$
$\displaystyle\rightharpoonup h\quad\text{weak-$\ast$ in}\quad
L^{\infty}(0,T;H^{2}(0,L)),$ $\displaystyle{\partial}_{t}g_{N}$
$\displaystyle\rightharpoonup{\partial}_{t}h\quad\text{weak-$\ast$ in}\quad
L^{\infty}(0,T;H^{2}(0,L)),$ $\displaystyle{\partial}_{t}^{2}g_{N}$
$\displaystyle\rightharpoonup{\partial}_{t}^{2}h\quad\text{weak-$\ast$
in}\quad L^{\infty}(0,T;L^{2}(0,L)).$

This implies by compactness that up to a subsequence, we have using the
analysis (2.9) that

$\displaystyle\hat{u}_{N}$ $\displaystyle\to\hat{u}\quad\text{ in }\quad
L^{2}((0,T)\times{\Omega}_{1}),$ $\displaystyle{\partial}_{t}g_{N}$
$\displaystyle\to{\partial}_{t}h\quad\text{ in }\quad
C^{0,\alpha}([0,T]\times[0,L]),$ $\displaystyle g_{N}$ $\displaystyle\to
h\quad\text{ in }\quad C^{1}([0,T]\times[0,L]),$

for $0<\alpha<\frac{2}{3}$. This allows us to pass to the limit with (4.16)
(the version by replacing $\tilde{h}$ by $g_{N}$ due to the fixed point
procedure) and we thereby obtain

$\displaystyle\int_{{\Omega}_{1}}\rho_{f}h{\partial}_{t}\hat{u}\cdot
B_{h}^{-\intercal}\Psi{\hbox{\hskip 1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}+\int_{{\Omega}_{1}}\rho_{f}\hat{u}(B_{h}\nabla)\hat{u}\cdot
B_{h}^{-\intercal}\Psi{\hbox{\hskip 1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}$
$\displaystyle-\int_{{\Omega}_{1}}\rho_{f}{\partial}_{t}\chi_{h}(B_{h}\nabla)\hat{u}\cdot
B_{h}^{-\intercal}\Psi{\hbox{\hskip 1.0pt}}{\rm d}\hbox{\hskip
0.5pt}\mathbf{z}+\mu\int_{{\Omega}_{1}}(A_{h}\nabla)\hat{u}:\nabla\left(B_{h}^{-\intercal}\Psi\right){\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}\mathbf{z}$
$\displaystyle+\rho_{s}\int_{0}^{L}{\partial}_{t}^{2}h\psi{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip
0.5pt}x-\beta\int_{0}^{L}{\partial}_{x}^{2}h\psi{\hbox{\hskip 1.0pt}}{\rm
d}\hbox{\hskip
0.5pt}x+\alpha\int_{0}^{L}{\partial}_{x}^{2}h{\partial}_{x}^{2}\psi{\hbox{\hskip
1.0pt}}{\rm d}\hbox{\hskip 0.5pt}x=0,$

for all sufficiently smooth $\psi(t,x)=\Psi(t,x,1)$ with ${\rm div\,}\Psi=0$.

Regularity of $u$, $p$ and $h$.

Here we rely on the properties of the steady Stokes equation. Let us consider
the steady Stokes system

$\left\\{\begin{aligned} &\Delta u-\nabla\pi=f,\\\ &{\rm div\,}u=0,\\\
&u|_{\partial{{\Omega}_{h}}}=u_{\partial},\end{aligned}\right.$ (4.21)

in a domain ${{\Omega}_{h}}\subset{\mathbb{R}}^{2}$ with unit normal ${\bf
n}$. The result given in the following theorem is a maximal regularity
estimate for the solution of (4.21) in terms of the right-hand side. We quote
here the Stokes estimate from [11, Theorem 3.1] and [12, Theorem 2.8] for the
system (4.21).

######  Theorem 4.5.

Let $q\in(1,\infty)$, $s\geqslant 1+\frac{1}{q}$ and

$\displaystyle\varrho\geqslant q\quad\text{if}\quad q(s-1)\geqslant
3;\quad\varrho\geqslant\tfrac{2q}{q(s-1)-1}\quad\text{if}\quad q(s-1)<3,$

such that $3\big{(}\frac{1}{q}-\frac{1}{2}\big{)}+1\leqslant s$. Suppose that
${{\Omega}_{h}}$ is a $B^{\theta}_{\varrho,q}$-domain333For bounded domain
$\mathcal{O}$, the Besov spaces are defined as
$B^{\theta}_{\varrho,q}(\mathcal{O}):=\\{f|_{\mathcal{O}}:f\in
B^{\theta}_{\varrho,q}({\mathbb{R}}^{n})\\}$ with the norm $\lVert
g\rVert_{B^{\theta}_{\varrho,q}(\mathcal{O})}=\inf\\{\lVert
f\rVert_{B^{\theta}_{\varrho,q}({\mathbb{R}}^{n})}:f|_{\mathcal{O}}=g\\}$. and
$h\in L^{\infty}(\bar{I};B^{\theta}_{\varrho,q}\cap C^{1}[0,L])$ for some
$\theta>s-1/q$ with locally small Lipschitz constant, $f\in
W^{s-2,q}({{\Omega}_{h}})$ and $u_{\partial}\in
W^{s-1/q,q}(\partial{{\Omega}_{h}})$ with
$\int_{\partial{{\Omega}_{h}}}u_{\partial}\cdot{\bf n}{\rm d}\hbox{\hskip
0.5pt}S=0$. Then there is a unique solution to (4.21) with
$\int_{\Omega_{h}}\pi\mathrm{d}x=0$ that satisfies

$\displaystyle\|u\|_{W^{s,q}({{\Omega}_{h}})}+\|\pi\|_{W^{s-1,q}({{\Omega}_{h}})}\lesssim\|f\|_{W^{s-2,q}({{\Omega}_{h}})}+\|u_{\partial}\|_{W^{s-1/q,q}(\partial{{\Omega}_{h}})}.$
(4.22)

######  Remark 4.6.

We remark here that the above result in [11] and [12] is stated in a general
$B^{\theta}_{\varrho,q}$-domain $\mathcal{O}$. It is also available for the
moving domain ${\Omega}_{h}$ with $h\in
L^{\infty}(\bar{I};B^{\theta}_{\varrho,q}\cap C^{1}[0,L])$, and it suffices to
verify that ${\partial}{\Omega}_{h}\in B^{\theta}_{\varrho,q}$. This has been
explained in [11, Remark 3.4] and [12, Remark 2.9] in a detailed way.

Based on the weakly lower semi-continuity of the weak convergence in (4.20),
we note that $(\hat{u},h)$ also satisfy the energy balance (4.17) and the
time-derivative estimate (4.19). We will explain in the following that this,
together with Lemma 4.5, implies more regularity for $\hat{u}$. Moreover, with
the help of Proposition 2.1, we could derive more regularity directly for $u$.

######  Theorem 4.7.

Assume that the initial data $(u_{0},h_{0},h_{1})$ satisfy $h_{0}\in
H^{4}(0,L)$, $h_{1}\in H^{2}(0,L)$, $u_{0}\in H^{2}({\Omega}_{h_{0}})$ and
$\min_{x\in(0,L)}h_{0}=\delta>0$, then the system (1.1)–(1.7) admits a unique
solution $(u,p,h)$, for $T>0$, satisfying

$\displaystyle h\in W^{1,\infty}(0,T;H^{2}(0,L))\cap
W^{2,\infty}(0,T;L^{2}(0,L))\cap L^{\infty}(0,T;H^{4}(0,L)),$ (4.23)
$\displaystyle u\in L^{\infty}(0,T;H^{2}({\Omega}_{h}))\cap
L^{4}(0,T;H^{\frac{5}{2}}({\Omega}_{h})),$ $\displaystyle p\in
L^{\infty}(0,T;H^{1}({\Omega}_{h}))\cap
L^{4}(0,T;H^{\frac{3}{2}}({\Omega}_{h})).$

This solution exists until the self-intersection of ${\Omega}_{h}$.

###### Proof.

The regularity of $h$ can be obtained directly from the estimate (4.17) and
(4.19) that

$h\in W^{1,\infty}(0,T;H^{2}(0,L))\cap W^{2,\infty}(0,T;L^{2}(0,L)).$

For the initial data, recalling that
$\hat{u}_{0}(x,z)=u_{0}(x,h_{0}z)=u_{0}(x,y)$, we notice from Proposition 2.1
that $\lVert\hat{u}_{0}\rVert_{H^{2}(\Omega_{1})}$ is equivalent to $\lVert
u_{0}\rVert_{H^{2}({\Omega}_{h_{0}})}$. This implies that the initial
conditions (4.15) is satisfied. The pressure is constructed in two parts. For
that we decompose it into

$\displaystyle p(t,x)=p_{0}(t,x)+p_{1}(t)\text{ such that
}\int_{\Omega_{h(t)}}p_{0}(t,x)\,\mathrm{d}x=0.$ (4.24)

Now $p_{0}$ is directly constructed via Stokes equation, while $p_{1}$ is
constant in space.

To present the proof clearly, we divide it in the following steps.

Step 1: We show that $u\in L^{\infty}(0,T;W^{2,r}({\Omega}_{h}))$ for every
$1<r<2$. Note that we have the regularity for $\hat{u}$:

$\hat{u}\in W^{1,\infty}(0,T;L^{2}({\Omega}_{1}))\cap
H^{1}(0,T;H^{1}({\Omega}_{1}))$

from the weak convergence (4.20). This gives us that $\nabla\hat{u}\in
L^{\infty}(0,T;L^{2}({\Omega}_{1}))$. From Proposition 2.1 we obtain that
$\nabla u\in L^{\infty}(0,T;L^{2}({\Omega}_{h}))$. Using Hölder’s inequality
and Sobolev embedding $H^{1}({\Omega}_{h})\hookrightarrow
L^{\frac{2r}{2-r}}({\Omega}_{h})$, for every $1<r<2$ we have

$\lVert u\cdot\nabla u\rVert_{L^{r}({\Omega}_{h})}\lesssim\lVert
u\rVert_{L^{\frac{2r}{2-r}}({\Omega}_{h})}\lVert\nabla
u\rVert_{L^{2}({\Omega}_{h})}\lesssim\lVert\nabla
u\rVert_{L^{2}({\Omega}_{h})}^{2}.$

Hence we obtain that $u\cdot\nabla u\in L^{\infty}(0,T;L^{r}({\Omega}_{h}))$
for every $1<r<2$.

Now we apply the Stokes estimate (4.22) for $s=2$, $q=r\in(1,2)$,
$f={\partial}_{t}u+u\cdot\nabla u$ and $u_{{\partial}}={\partial}_{t}h$.
Thereby we have

$\displaystyle\lVert u\rVert_{W^{2,r}({\Omega}_{h})}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{W^{1,r}({\Omega}_{h})}$
$\displaystyle\lesssim\lVert{\partial}_{t}u+u\cdot\nabla
u\rVert_{L^{r}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{W^{2-\frac{1}{r},r}(0,L)}$
(4.25)
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{L^{r}({\Omega}_{h})}+\lVert
u\cdot\nabla
u\rVert_{L^{r}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{W^{2-\frac{1}{r},r}(0,L)}.$

Note that ${\partial}_{t}\hat{u}\in L^{\infty}(0,T;L^{2}({\Omega}_{1}))$, then
through Proposition 2.1 we get that ${\partial}_{t}u\in
L^{\infty}(0,T;L^{2}({\Omega}_{h}))$. Moreover, recall that
${\partial}_{t}h\in L^{\infty}(0,T;H^{2}(0,L))$, using the continuous
embedding $H^{2}(0,L)\hookrightarrow W^{2-\frac{1}{r},r}(0,L)$, we derive from
(4.25) that $u\in L^{\infty}(0,T;W^{2,r}({\Omega}_{h}))$ for $1<r<2$.

Step 2: We prove qualitatively that $u\in L^{\infty}(0,T;H^{2}({\Omega}_{h}))$
and $p\in L^{\infty}(0,T;H^{1}({\Omega}_{h}))$. Using the regularity $u\in
L^{\infty}(0,T;W^{2,r}({\Omega}_{h}))$ and the continuous embedding
$W^{2,r}({\Omega}_{h})\hookrightarrow L^{\infty}({\Omega}_{h})$ for $1<r<2$,
we have $u\in L^{\infty}((0,T)\times{\Omega}_{h})$. Moreover, we note that

$\lVert u\cdot\nabla u\rVert_{L^{2}({\Omega}_{h})}\lesssim\lVert
u\rVert_{L^{\infty}({\Omega}_{h})}\lVert\nabla u\rVert_{L^{2}({\Omega}_{h})}.$

This, together with $\nabla u\in L^{\infty}(0,T;L^{2}({\Omega}_{h}))$, gives
us that $u\cdot\nabla u\in L^{\infty}(0,T;L^{2}({\Omega}_{h}))$. At this
stage, we apply the Stokes estimate (4.22) again for $s=2=q$ and obtain

$\displaystyle\lVert u\rVert_{H^{2}({\Omega}_{h})}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{H^{1}({\Omega}_{h})}$
$\displaystyle\lesssim\lVert{\partial}_{t}u+u\cdot\nabla
u\rVert_{L^{2}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{H^{\frac{3}{2}}(0,L)}$
(4.26)
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{L^{2}({\Omega}_{h})}+\lVert
u\cdot\nabla
u\rVert_{L^{2}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{H^{\frac{3}{2}}(0,L)}.$

Since we already notice that ${\partial}_{t}u\in
L^{\infty}(0,T;L^{2}({\Omega}_{h}))$, then (4.26) shows that $u\in
L^{\infty}(0,T;H^{2}({\Omega}_{h}))$ and $p_{0}\in
L^{\infty}(0,T;H^{1}({\Omega}_{h}))$.

Based on the decomposition (4.24), now we define $p_{1}$ in such a way that
(1.2) is satisfied. Recall that we have the zero mean condition (2.1) for the
source term $\phi(u,p,h)$ defined in (1.4). This, together with (4.24), gives
us that $\sigma(u,p)=\sigma(u,p_{0})-p_{1}\mathbb{I}_{2\times 2}$ and thereby
we have

$Lp_{1}(t)=\int_{0}^{L}{\rm
e_{2}}\cdot\sigma(u,p_{0})(t,x,h(t,x))(-{\partial}_{x}h{\hbox{\hskip
1.0pt}}{\rm e_{1}}+{\rm e_{2}})\mathrm{d}x\in L^{\infty}(0,T).$

Hence, with the current regularity of $u$ and $p$, we derive that the Cauchy
stress tensor $\sigma(u,p)\in L^{\infty}(0,T;H^{1}({\Omega}_{h}))$. Thereby
the source term of the structure $\phi(u,p,h)$ belongs to
$L^{\infty}(0,T;H^{\frac{1}{2}}(0,L))$. Based on the beam equation (1.2) and
the regularity of $h$, we finally have ${\partial}_{x}^{4}h\in
L^{\infty}(0,T;L^{2}(0,L))$. Therefore, the strong solution $(u,p,h)$ of the
system (1.1)–(1.7) is established in both space and time.

Step 3: We have further regularity: $u\in
L^{4}(0,T;H^{\frac{5}{2}}({\Omega}_{h}))$ and $p\in
L^{4}(0,T;H^{\frac{3}{2}}({\Omega}_{h}))$. Recalling the regularity of
$\hat{u}$, we have ${\partial}_{t}\hat{u}\in
L^{\infty}(0,T;L^{2}({\Omega}_{1}))\cap L^{2}(0,T;H^{1}({\Omega}_{1}))$. With
the result in Step 2, we know that $u\in L^{\infty}(0,T;H^{2}({\Omega}_{h}))$
and then by Proposition 2.1 we get ${\partial}_{t}u\in
L^{\infty}(0,T;L^{2}({\Omega}_{h}))\cap L^{2}(0,T;H^{1}({\Omega}_{h}))$ as
well. By the interpolation inequality:

$\lVert{\partial}_{t}u\rVert_{L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))}\lesssim\lVert{\partial}_{t}u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{\frac{1}{2}}\lVert{\partial}_{t}u\rVert_{L^{2}(0,T;H^{1}({\Omega}_{h}))}^{\frac{1}{2}},$
(4.27)

we obtain further that ${\partial}_{t}u\in
L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))$. Now we use the Stokes estimate
(4.22) for $s=\frac{5}{2}$ and $q=2$ once more and have

$\displaystyle\lVert
u\rVert_{H^{\frac{5}{2}}({\Omega}_{h})}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{H^{\frac{3}{2}}({\Omega}_{h})}$
$\displaystyle\lesssim\lVert{\partial}_{t}u+u\cdot\nabla
u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{H^{2}(0,L)}$
(4.28)
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}+\lVert
u\cdot\nabla
u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{H^{2}(0,L)}$
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}+\lVert
u\rVert_{L^{\infty}({\Omega}_{h})}\lVert\nabla
u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}+\lVert{\partial}_{t}h\rVert_{H^{2}(0,L)}.$

Combining with the above analysis, we obtain from (4.28) that

$\displaystyle\lVert
u\rVert_{L^{4}(0,T;H^{\frac{5}{2}}({\Omega}_{h}))}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{L^{4}(0,T;H^{\frac{3}{2}}({\Omega}_{h}))}$
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))}+C(T)\left(\lVert
u\rVert_{L^{\infty}(0,T;H^{2}({\Omega}_{h}))}+\lVert{\partial}_{t}h\rVert_{L^{\infty}(0,T;H^{2}(0,L))}\right).$

Using the decomposition argument again, we obtain the regularity of $p$.

Putting the above results together, we conclude that the regularity in (4.23)
holds, which ends the proof. ∎

###  4.5 Proof of Theorem 1.1

Now we are in a position to finish the proof of Theorem 1.1. The existence of
a strong solution was obtained in Theorem 4.7. Hence we are left to show
quantitatively that the estimate is linear with respect to the higher order
norms of the initial conditions.

Beginning from Step 2 in the proof of Theorem 4.7, we derive the following
estimate of $u$ in the spaces $L^{\infty}(0,T;H^{2}({\Omega}_{h}))$ and
$L^{4}(0,T;H^{\frac{5}{2}}({\Omega}_{h}))$, which is linear with respect to
the higher order norms of the initial conditions. Based on the inequality
(4.26), we have

$\displaystyle\lVert
u\rVert_{L^{\infty}(0,T;H^{2}({\Omega}_{h}))}^{2}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{L^{\infty}(0,T;H^{1}({\Omega}_{h}))}^{2}$
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}+\lVert
u\cdot\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}+\lVert{\partial}_{t}h\rVert_{L^{\infty}(0,T;H^{\frac{3}{2}}(0,L))}^{2}.$

Note that for the convective term above, we estimate by using Ladyzhenskaya
inequality that

$\displaystyle\lVert u\cdot\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}$
$\displaystyle\leqslant\lVert
u\rVert_{L^{\infty}(0,T;L^{4}({\Omega}_{h}))}^{2}\lVert\nabla
u\rVert_{L^{\infty}(0,T;L^{4}({\Omega}_{h}))}^{2}$
$\displaystyle\lesssim\lVert
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}\lVert\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}\lVert\nabla^{2}u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}$
$\displaystyle\leqslant{\varepsilon}\lVert\nabla^{2}u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}+C({\varepsilon})\lVert
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}\lVert\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{4}.$

According to Proposition 2.1, we know that $\lVert\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}\sim\lVert\nabla\hat{u}\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{1}))}$.
Then we consider the following interpolation inequality:

$\lVert\nabla\hat{u}\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{1}))}\lesssim\lVert\nabla\hat{u}\rVert_{L^{2}(0,T;L^{2}({\Omega}_{1}))}^{\frac{1}{2}}\lVert\nabla\hat{u}\rVert_{H^{1}(0,T;L^{2}({\Omega}_{1}))}^{\frac{1}{2}}\lesssim
C_{0}^{\frac{1}{4}}\lVert\nabla\hat{u}\rVert_{H^{1}(0,T;L^{2}({\Omega}_{1}))}^{\frac{1}{2}},$

This gives us that

$\displaystyle\lVert u\cdot\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}$
$\displaystyle\leqslant{\varepsilon}\lVert\nabla^{2}u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}+C({\varepsilon})C_{0}^{2}\lVert\nabla\hat{u}\rVert_{H^{1}(0,T;L^{2}({\Omega}_{1}))}^{2},$
(4.29)

where we used the energy balance (2.8). Combining with the above estimate, we
derive from Proposition 4.2 and Theorem 4.4 that

$\displaystyle\lVert
u\rVert_{L^{\infty}(0,T;H^{2}({\Omega}_{h}))}^{2}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{L^{\infty}(0,T;H^{1}({\Omega}_{h}))}^{2}$
$\displaystyle\leqslant
c(C_{0})\left(\lVert\hat{u}_{0}\rVert_{H^{2}({\Omega}_{1})}^{2}+\lVert
h_{0}\rVert_{H^{4}(0,L)}^{2}+\lVert h_{1}\rVert_{H^{2}(0,L)}^{2}+1\right).$

Similarly, we can derive a ’linear estimate’ from (4.28). We consider

$\displaystyle\lVert
u\rVert_{L^{4}(0,T;H^{\frac{5}{2}}({\Omega}_{h}))}^{4}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{L^{4}(0,T;H^{\frac{3}{2}}({\Omega}_{h}))}^{4}$
$\displaystyle\lesssim\lVert{\partial}_{t}u\rVert_{L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))}^{4}+\lVert
u\cdot\nabla
u\rVert_{L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))}^{4}+\lVert{\partial}_{t}h\rVert_{L^{2}(0,T;H^{2}(0,L))}^{4}.$
(4.30)

By using the interpolation (4.27), we deal with the convective term again as
follows:

$\displaystyle\lVert u\cdot\nabla
u\rVert_{L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))}^{4}$
$\displaystyle\lesssim\lVert u\cdot\nabla
u\rVert_{L^{\infty}(0,T;L^{2}({\Omega}_{h}))}^{2}\lVert u\cdot\nabla
u\rVert_{L^{2}(0,T;H^{1}({\Omega}_{h}))}^{2}.$ (4.31)

Observe that

$\lVert u\cdot\nabla
u\rVert_{L^{2}(0,T;H^{1}({\Omega}_{h}))}^{2}=\int_{0}^{T}\left(\lVert\nabla
u\rVert_{L^{2}({\Omega}_{h})}^{4}+\lVert
u\cdot\nabla^{2}u\rVert_{L^{2}({\Omega}_{h})}^{2}\right){\rm d}\hbox{\hskip
0.5pt}t:={\rm I+II}.$

Now we estimate ${\rm I}$ and ${\rm II}$, respectively. We first have

$\displaystyle{\rm I}$ $\displaystyle\lesssim\int_{0}^{T}\lVert
u\rVert_{L^{2}({\Omega}_{h})}^{\frac{12}{5}}\lVert\nabla
u\rVert_{H^{\frac{3}{2}}({\Omega}_{h})}^{\frac{8}{5}}{\rm d}\hbox{\hskip
0.5pt}t$ $\displaystyle\lesssim\left(\int_{0}^{T}\lVert
u\rVert_{L^{2}({\Omega}_{h})}^{4}{\rm d}\hbox{\hskip
0.5pt}t\right)^{\frac{3}{5}}\left(\int_{0}^{T}\lVert\nabla
u\rVert_{H^{\frac{3}{2}}({\Omega}_{h})}^{4}\right)^{\frac{2}{5}},$

where we used Hölder’s inequality and the interpolation inequality:

$\lVert\nabla u\rVert_{L^{2}({\Omega}_{h})}\lesssim\lVert\nabla
u\rVert_{H^{-1}({\Omega}_{h})}^{\frac{3}{5}}\lVert\nabla
u\rVert_{H^{\frac{3}{2}}({\Omega}_{h})}^{\frac{2}{5}}.$

For ${\rm II}$, we also have

$\displaystyle{\rm II}$ $\displaystyle\leqslant\int_{0}^{T}\lVert
u\rVert_{L^{4}({\Omega}_{h})}^{2}\lVert\nabla^{2}u\rVert_{L^{4}({\Omega}_{h})}^{2}{\rm
d}\hbox{\hskip 0.5pt}t$ $\displaystyle\lesssim\int_{0}^{T}\lVert
u\rVert_{L^{2}({\Omega}_{h})}\lVert\nabla
u\rVert_{L^{2}({\Omega}_{h})}\lVert\nabla^{2}u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}^{2}{\rm
d}\hbox{\hskip 0.5pt}t$
$\displaystyle\leqslant\left(\int_{0}^{T}\lVert\nabla^{2}u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}^{4}{\rm
d}\hbox{\hskip 0.5pt}t\right)^{\frac{1}{2}}\left(\int_{0}^{T}\lVert\nabla
u\rVert_{L^{2}({\Omega}_{h})}^{2}{\rm d}\hbox{\hskip
0.5pt}t\right)^{\frac{1}{2}}.$

This implies using (4.29) and (4.31) that

$\displaystyle\lVert u\cdot\nabla
u\rVert_{L^{4}(0,T;H^{\frac{1}{2}}({\Omega}_{h}))}^{4}$
$\displaystyle\lesssim\Bigg{(}\left(\int_{0}^{T}\lVert\nabla^{2}u\rVert_{H^{\frac{1}{2}}({\Omega}_{h})}^{4}{\rm
d}\hbox{\hskip 0.5pt}t\right)^{\frac{1}{2}}+\left(\int_{0}^{T}\lVert\nabla
u\rVert_{H^{\frac{3}{2}}({\Omega}_{h})}^{4}\right)^{\frac{2}{5}}\Bigg{)}$
$\displaystyle\times\lVert\nabla^{2}u\rVert_{L^{\infty}(0,T;L^{2}(\Omega_{h}))}\lVert\nabla\hat{u}\rVert_{H^{1}(0,T;L^{2}(\Omega_{1}))}$
$\displaystyle\leqslant{\varepsilon}\int_{0}^{T}\lVert
u\rVert_{H^{\frac{5}{2}}({\Omega}_{h})}^{4}{\rm d}\hbox{\hskip
0.5pt}t+\left(\lVert u_{0}\rVert_{H^{2}({\Omega}_{1})}+\lVert
h_{0}\rVert_{H^{4}(0,L)}+\lVert h_{1}\rVert_{H^{2}(0,L)}+1\right)^{4}.$

Hence, we conclude from (4.5) and the above that

$\displaystyle\lVert
u\rVert_{L^{4}(0,T;H^{\frac{5}{2}}({\Omega}_{h}))}^{4}+\lVert{\hbox{\hskip
1.0pt}}p_{0}\rVert_{L^{4}(0,T;H^{\frac{3}{2}}({\Omega}_{h}))}^{4}$
$\displaystyle\leqslant
c(C_{0})\left(\lVert\hat{u}_{0}\rVert_{H^{2}({\Omega}_{1})}^{4}+\lVert
h_{0}\rVert_{H^{4}(0,L)}^{4}+\lVert h_{1}\rVert_{H^{2}(0,L)}^{4}+1\right).$

This finishes the proof of Theorem 1.1.

## Funding

S. Schwarzacher and P. Su are partially supported by the ERC-CZ Grant CONTACT
LL2105 funded by the Ministry of Education, Youth and Sport of the Czech
Republic and the University Centre UNCE/SCI/023 of Charles University. S.
Schwarzacher also acknowledges the support of the VR Grant 2022-03862 of the
Swedish Science Foundation.

## Compliance with Ethical Standards

Conflict of Interest. The authors declare that they have no conflict of
interest.

Data Availability. Data sharing is not applicable to this article as no
datasets were generated or analyzed during the current study.

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Inputs      16       3.000000     3      3
Extracted Context Information Store      47    6.941176      2      11

Location Proofs Store      38     9.441176      2         20

Verified Information Store      25     6.764706      0         18

Accept / Reject Location Proof Store     7     1.705882      0         5

The CPN model describes the complex network in great detail, explains the way in which it works, and details the various ways information flows through it. Moreover, the CPN models are designed to ensure the integrity of and to demonstrate the presence of deadlocks in a system. From the above CPNs models and the result analysis of the marking size of places in these models, it is clear that all of the model's states are accessible. Each model transition is enabled/fired during the simulation. Additionally, these models have no bottlenecks, indicating that they are entirely free of deadlocks. There may be a range of tokens for each state, as the execution pattern or the number of times a transition is fired will influence this number.

§ CONCLUSION

This paper has studied, examined, and analysed our existing proposed clone node attack detection scheme for IoT networks. We modelled the proposed scheme using HLPNs, which have the advantage of giving a stable mathematical description and analysing its functional and structural characteristics. We specified several properties defining the proposed scheme's behaviour using the Z specification language, and if the models satisfy particular properties, they are considered correct. Further, we validated our models using a model checking technique, utilising the SMT-Lib and Z3 solver. Verification results demonstrated in this work indicate that the models used in the proposed framework are adequate for behavioural aspects. We extended our work by modelling the proposed scheme using CPNs, which extend Petri Nets with high-level programming language-like properties, making them more effective and ideal for modelling complex systems. Finally, we verified the logical validity and performance of the proposed approach using both untimed and timed models of CPNs.

§ DECLARATION OF COMPETING INTEREST

The authors of this paper declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

§ FUNDING

This research did not receive any specific grant from funding agencies in the public, commercial, or
not-for-profit sectors

§ PIPE+ ILLUSTRATIONS OF HLPN MODELS

This appendix includes Pipe+ illustrations for the following HLPN models: ECDSA* key generation, ECDSA* signature generation, ECDSA* signature verification, calculate location, generate location proof, and verify location proof.

A Pipe+ View of HLPN Model of an ECDSA* Key Generation

Fig. <ref> depicts the HLPN model of an ECDSA* Key Generation, which consists of three transitions and three places. All transitions in this HLPN are timed.

A Pipe+ View of HLPN Model of an ECDSA* Signature Generation

Fig. <ref> shows the HLPN model for ECDSA* Signature Generation. This HLPN has four locations and five transitions. Only "Generate Signature Pair 2" is utilised as an immediate transition.

A Pipe+ View of HLPN Model of an ECDSA* Signature Verification

Fig. <ref> shows the HLPN model of an ECDSA* Signature Verification. This HLPN has six places and six transitions. Two of the six transitions are immediate: "Compute Coordinates" and "Verify Signatures".

A Pipe+ View of HLPN Model of the Calculate Location in LPS

Fig. <ref> depicts the HLPN of Calculate Location in LPS, which has three places and three transitions.  All transitions in this HLPN are timed.

A Pipe+ View of HLPN Model of Generate Location Proof in LPS

Fig. <ref> illustrates the HLPN model for Generate Location Proof in LPS, which has five places and five transitions. A "Generate Location Proof" is the only immediate transition.

Fig. <ref> illustrates the HLPN model of the verify location proof in LPS, which comprises five places and five transitions. From the five transitions, two are used as immediate transitions: "Verify Context Information", "Verify Location Proof".

A Pipe+ View of HLPN Model of the Verify Location Proof in LPS

§ INCIDENCE MARKING

This appendix presents the results of the incidence markings of HLPN models, which comprise ECDSA* signature generation, ECDSA* signature verification, calculate location, generate location proof and verify location proof. The results of the incidence markings are determined using several matrices, including forwards, backwards, combined, and inhibition.

High Level Petri Net Incidences - ECDSA* Signature Generation

6|l|Forwards Incidence Matrix $I^{+}$
Start      Compute
Coordinates       Compute
Hash        Generate
Signature  Pair 1     Generate
Signature Pair 2
$\varphi$ $(Inputs)$     1       0        0          0   0
$\varphi$ $\makecell{(Coordinates \\ Store)}$      0       1       0       0    0
$\varphi$ $\makecell{(Hash Integer\\ Store)}$         0    0          1    0     0
$\varphi$ $\makecell{(Signature \\ Store)}$     0       0       0       1     1
6|l|Backwards Incidence Matrix $I^{-}$
Start      Compute
Coordinates       Compute
Hash        Generate
Signature  Pair 1     Generate
Signature Pair 2
$\varphi$ $(Inputs)$      0         1     1     0      1
$\varphi$ $\makecell{(Coordinates \\ Store)}$       0       0       0       1     0
$\varphi$ $\makecell{(Hash Integer\\ Store)}$     0       0       0       0     1
$\varphi$ $\makecell{(Signature \\ Store)}$      0      0        0          0   0

6|l|Combined Incidence Matrix I
Start      Compute
Coordinates       Compute
Hash        Generate
Signature  Pair 1     Generate
Signature Pair 2
$\varphi$ $(Inputs)$      1          -1     -1      0     -1
$\varphi$ $\makecell{(Coordinates \\ Store)}$          0    1          0    -1     0
$\varphi$ $\makecell{(Hash Integer\\ Store)}$      0       0      1       0     -1
$\varphi$ $\makecell{(Signature \\ Store)}$        0          0     0     1     1

6|l|Inhibition Matrix H
Start      Compute
Coordinates       Compute
Hash        Generate
Signature  Pair 1     Generate
Signature Pair 2
$\varphi$ $(Inputs)$       0     0       0       0     0
$\varphi$ $\makecell{(Coordinates \\ Store)}$         0    0          0    0     0
$\varphi$ $\makecell{(Hash Integer\\ Store)}$      0       0       0       0     0
$\varphi$ $\makecell{(Signature \\ Store)}$       0          0     0         0    0

Table <ref> presents the results of forwards, backwards, combined, and inhibition incidence matrices for an ECDSA* Signature Generation Petri net.

The forwards, backwards, combined, and inhibition matrices for the ECDSA* Signature Verification Petri Net are shown in Table <ref>.

High Level Petri Net Incidences - ECDSA* Signature Verification

7|l|Forwards Incidence Matrix $I^{+}$
Start      Get
Integers       Compute
Hash        Calculate
Point     Compute
Coordinates     Verify
$\varphi$ $(Inputs)$     1       0        0          0   0   0
$\varphi$ $\makecell{(Signatures \\ Store)}$      0       1       0       0    0     0
$\varphi$ $\makecell{(Hash \\ Integer\\ Store)}$         0    0          1    0     0     0
$\varphi$ $\makecell{(Point \\ Store)}$     0       0       0       1     0    0
$\varphi$ $\makecell{(Coordinates \\ Store)}$     0       0       0       0     1     0
$\varphi$ $\makecell{(Accept \\ Reject)}$     0       0       0       0     0   1
7|l|Backwards Incidence Matrix $I^{-}$
Start      Get
Integers       Compute
Hash        Calculate
Point     Compute
Coordinates     Verify

$\varphi$ $(Inputs)$     0       1        1      0       0   1
$\varphi$ $\makecell{(Signatures \\ Store)}$      0       0      0       1    1   0
$\varphi$ $\makecell{(Hash \\ Integer\\ Store)}$         0    0          0    0     1    0
$\varphi$ $\makecell{(Point \\ Store)}$     0       0       0       0     1    0
$\varphi$ $\makecell{(Coordinates \\ Store)}$     0       0       0       0     0    1
$\varphi$ $\makecell{(Accept \\ Reject)}$     0       0       0       0    0   0
7|l|Combined Incidence Matrix I

Start      Get
Integers       Compute
Hash        Calculate
Point     Compute
Coordinates     Verify
$\varphi$ $(Inputs)$     1       -1        -1          0   0   -1
$\varphi$ $\makecell{(Signatures \\ Store)}$      0       1       0       -1    -1   0
$\varphi$ $\makecell{(Hash \\ Integer\\ Store)}$         0    0          1    0     -1   0
$\varphi$ $\makecell{(Point \\ Store)}$     0       0       0       1     -1   0
$\varphi$ $\makecell{(Coordinates \\ Store)}$     0       0       0       0    1   -1
$\varphi$ $\makecell{(Accept \\ Reject)}$     0       0       0       0    0   1

7|l|Inhibition Matrix H
Start      Get
Integers       Compute
Hash        Calculate
Point     Compute
Coordinates     Verify

$\varphi$ $(Inputs)$     0   0       0        0          0   0
$\varphi$ $\makecell{(Signatures \\ Store)}$      0       0   0       0       0    0
$\varphi$ $\makecell{(Hash \\ Integer\\ Store)}$         0    0             0    0     0
$\varphi$ $\makecell{(Point \\ Store)}$     0       0       0       0     0   0
$\varphi$ $\makecell{(Coordinates \\ Store)}$     0       0       0       0    0   0
$\varphi$ $\makecell{(Accept \\ Reject)}$     0       0       0       0    0   0

Table <ref> shows the results of forward, backward, combined and inhibition matrices for the scenario of Calculate Location in the proposed clone node detection mechanism.

High Level Petri Net Incidences - Calculate Location

4|l|Forwards Incidence Matrix $I^{+}$     4l|Backwards Incidence Matrix $I^{-}$
Start        Determine
Space         Calculate
Distance                Start        Determine
Space         Calculate
$\varphi$ $(Inputs)$           1      0                   0      $\varphi$ $(Inputs)$      0        1     0
$\varphi$ $\makecell{(Points\\Store)}$        0      1       0         $\varphi$ $\makecell{(Points \\Store)}$           0        0        1
$\varphi$ $\makecell{(Provers' \\ Distance \\ Store)}$       0          0    1         $\varphi$ $\makecell{(Provers' \\ Distance\\ Store)}$           0        0       0
4|l|Combined Incidence Matrix I     4l|Inhibition Matrix H

Start        Determine
Space         Calculate
Distance                Start        Determine
Space         Calculate
$\varphi$ $(Inputs)$          1       -1                   0      $\varphi$ $(Inputs)$     0          0     0
$\varphi$ $\makecell{(Points\\Store)}$        0      1      -1          $\varphi$ $\makecell{(Points\\Store)}$           0        0     0
$\varphi$ $\makecell{(Provers' \\ Distance \\ Store)}$        0         0     1        $\varphi$ $\makecell{(Provers' \\ Distance \\ Store)}$          0        0     0

High Level Petri Net Incidences - Generate Location Proof

6|l|Forwards Incidence Matrix $I^{+}$
Start      Sense
Information       Stored
Information    Request
Proof     Generate
Location Proof
$\varphi$ $(Inputs)$     1       0        0          0   0
$\varphi$ $\makecell{(Context \\ Information \\ Store)}$      0       1       0       0    0
$\varphi$ $\makecell{(LBS\\ Information \\ Store)}$         0    0          1    0     0
$\varphi$ $\makecell{(Location \\ Proof \\Store)}$     0       0       0       1     0
$\varphi$ $\makecell{(Signed \\ Location \\ Proofs \\ Store)}$     0       0       0       0     1
6|l|Backwards Incidence Matrix $I^{-}$
Start      Sense
Information       Stored
Information    Request
Proof     Generate
Location  Proof
$\varphi$ $(Inputs)$      0         1     0     0      1
$\varphi$ $\makecell{(Context \\ Information \\ Store)}$      0       0       1       0     1
$\varphi$ $\makecell{(LBS\\ Information \\ Store)}$     0       0       0       1     0
$\varphi$ $\makecell{(Location \\ Proof \\Store)}$      0      0        0          0   1
$\varphi$ $\makecell{(Signed \\ Location \\ Proofs \\Store)}$     0       0       0       0    0
6|l|Combined Incidence Matrix I
Start      Sense
Information       Stored
Information    Request
Proof     Generate
Location Proof
$\varphi$ $(Inputs)$      1          -1     0     0     -1
$\varphi$ $\makecell{(Context \\ Information \\ Store)}$          0    1          -1    0     -1
$\varphi$ $\makecell{(LBS\\ Information \\ Store)}$     0        0     1       -1     0
$\varphi$ $\makecell{(Location \\ Proof \\Store)}$        0          0     0     1     -1
$\varphi$ $\makecell{(Signed \\ Location \\ Proofs \\Store)}$     0       0       0       0     1

6|l|Inhibition Matrix H
Start      Sense
Information       Stored
Information    Request
Proof     Generate
Location Proof
$\varphi$ $(Inputs)$       0     0       0       0     0
$\varphi$ $\makecell{(Context \\ Information \\ Store)}$         0    0          0    0     0
$\varphi$ $\makecell{(LBS\\ Information \\ Store)}$      0       0       0       0     0
$\varphi$ $\makecell{(Location \\ Proof \\Store)}$       0          0     0         0    0
$\varphi$ $\makecell{(Signed \\ Location \\ Proofs \\Store)}$     0       0       0       0     0

Table <ref> presents the results of forwards, backwards, combined, and inhibition incidence matrices for the Generate Location Proof Petri net in the proposed clone node attack detection technique.

High Level Petri Net Incidences - Verify Location Proof

6|l|Forwards Incidence Matrix $I^{+}$
Start      Extract
Information       Accept
Request    Verify
Information     Verify
$\varphi$ $(Inputs)$     1       0        0          0   0
$\varphi$ $\makecell{(Extracted \\ Context  Information \\ Store)}$      0       1       0       0    0
$\varphi$ $\makecell{(Location\\ Proofs \\ Store)}$         0    0          1    0     0
$\varphi$ $\makecell{(Verified \\ Information \\Store)}$     0       0       0       1     0
$\varphi$ $\makecell{(Accept / \\ Reject \\ Location \\Proof)}$     0       0       0       0     1
6|l|Backwards Incidence Matrix $I^{-}$
Start      Extract
Information       Accept
Request    Verify
Information     Verify
$\varphi$ $(Inputs)$      0         1     0     0      1
$\varphi$ $\makecell{(Extracted \\ Context  Information \\ Store)}$     0       0       1       1    0
$\varphi$ $\makecell{(Location\\ Proofs \\ Store)}$     0       0       0       1     0
$\varphi$ $\makecell{(Verified \\ Information \\Store)}$      0      0        0          0   1
$\varphi$ $\makecell{(Accept / \\ Reject \\ Location \\Proof)}$     0       0       0       0     0
6|l|Combined Incidence Matrix I
Start      Extract
Information       Accept
Request    Verify
Information     Verify
$\varphi$ $(Inputs)$      1          -1     0     0     -1
$\varphi$ $\makecell{(Extracted \\ Context  Information \\ Store)}$         0    1          -1    -1     0
$\varphi$ $\makecell{(Location\\ Proofs \\ Store)}$     0        0     1       -1     0
$\varphi$ $\makecell{(Verified \\ Information \\Store)}$        0          0     0     1     -1
$\varphi$ $\makecell{(Accept / \\ Reject \\ Location \\Proof)}$     0       0       0       0     1

6|l|Inhibition Matrix H
Start      Extract
Information       Accept
Request    Verify
Information     Verify

$\varphi$ $(Inputs)$      0         0     0     0     0

$\varphi$ $\makecell{(Extracted \\ Context  Information \\ Store)}$    0       0       0       0     0

$\varphi$ $\makecell{(Location\\ Proofs \\ Store)}$      0       0       0       0     0
$\varphi$ $\makecell{(Verified \\ Information \\Store)}$     0       0       0       0     0
$\varphi$ $\makecell{(Accept / \\ Reject \\ Location \\Proof)}$     0       0       0       0     0

The forwards, backwards, combined, and inhibition matrices for the Verify Location Proof Petri net of the proposed clone node attack detection scheme are shown in Table <ref>.

§ CONFIDENCE INTERVALS

This appendix contains the confidence intervals for several HLPNs in the average number of tokens with minimum and maximum threshold values. These HLPNs include ECDSA* signature generation, ECDSA* signature verification, calculate location, generate location proof, and verify location proof.

Table <ref> shows the results of ECDSA* signature generation HLPN in the form of the minimum and maximum threshold values obtained by throwing an average number of tokens at each place.

Average Number of Token with Minimum and Maximum Threshold Values - ECDSA* Signature Generation

Places     Average Number
of Tokens     Minimum
Values     Average Number
of Tokens     Maximum

Inputs          0.86139             0.356              1.02997            0.196

Store         0.26733           0.4366              0.2684              0.57542
Store       0.23762             0.0734           0.028                     0.21279
Store        13.67327               0.5872           2.612                  144.92008

Table <ref> shows the results of ECDSA* signature verification HLPN in the form of the minimum and maximum threshold values obtained by throwing an average number of tokens at each place.

Average Number of Token with Minimum and Maximum Threshold Values - ECDSA* Signature Verification

Places     Average Number
of Tokens     Minimum
Values     Average Number
of Tokens     Maximum

Inputs       1.0297               0.3697               1.17183             0.27562

Store         0.68317             0.35744            0.46254               0.13784
Store        1.9604             4.48262              45.97802              17.2552
Store        0.45545              0.29094                0.32068           0.16362
Store      0.17822                0.0359            0.12987                 0.01708
Reject        4.08911               0.84402               40.39461           3.425

Table <ref> shows the results of calculate location HLPN in the form of the minimum and maximum threshold values obtained by throwing an average number of tokens at each place.

Average Number of Token with Minimum and Maximum Threshold Values - Calculate Location

Places     Average Number
of Tokens     Minimum
Values     Average Number
of Tokens     Maximum
Inputs         2.65347              5.153333                33.81019            15.71
Store          1.60396                3.087666         9.436666                13.28272
Store      14.71287                 2.4348          146.54146                 8.12

Table <ref> shows the results of generate location proof HLPN in the form of the minimum and maximum threshold values obtained by throwing an average number of tokens at each place.

Average Number of Token with Minimum and Maximum Threshold Values - Generate Location Proof

Places     Average Number
of Tokens     Minimum
Values     Average Number
of Tokens     Maximum

Inputs       1.33663                  1.2186            2.27473             1.074

Store          0.71287             0.512                1.02897          0.8534
Store      1.0396               0.3306                    1.3037      0.3034
Store        1.62376            0.617                  0.86513            0.361
Store       4.70297             0.3852               61.03696             0.2952

Table <ref> shows the results of verify location proof HLPN in the form of the minimum and maximum threshold values obtained by throwing an average number of tokens at each place.

Average Number of Token with Minimum and Maximum Threshold Values - Verify Location Proof

Places     Average Number
of Tokens     Minimum
Values     Average Number
of Tokens     Maximum

Inputs           1.11881              2.968            4.7033              1.456

Store         0.78218               0.57056            0.91409               0.30618
Store      0.24752                 0.0806             0.26174               0.033476
Store       0.19802              0.082582                   0.14785         0.02452
Accept / Reject
Store         5.67327            0.4008              61.47453                 0.199048