On the Motion of Rigid Bodies in Viscoelastic Fluids

Abstract

This paper is concerned with the two dimensional motion of a finite number of homogeneous rigid disks in a cavity filled with incompressible viscoelastic fluids which obey constitutive laws of differential type. We focus here on two types of differential constitutive laws which are commonly used to describe the flow of dilute solutions of polymeric liquids: the first one corresponds to the regularized Oldroyd model whereas the other one corresponds to the Oldroyd model. The movement of rigid bodies modifies the fluid domain and hence we are dealing with a free boundary problem. By using fixed point theorem, we prove the existence and uniqueness of local-in-time strong solutions of the considered moving-boundary problem for both models.

Appendix: Technical Details on the Change of Variables \(X\) and \(Y\)

In this appendix, we recall the transform $X$ and some easily verified properties of $X$ and its inverse mapping $Y$. To this end, we fix $k$ functions $h_{i}: t\mapsto h_{i}(t)$ such that for $i\in \{1,\ldots ,k\}$, we assume that $h_{i}\in {H^{2}(0,T;\mathbb{R}^{2})}$. Moreover, we define a family of regular cut-off function $\{\psi _{i}\}_{i=1}^{k}$ such that each has a compact support contained in $B(h_{i}(0),r_{i}+\frac{\gamma _{0}}{2})$ and equal 1 in a neighbourhood $V_{B_{i}}$ of $i$-th disk contained in $B(h_{i}(0),r_{i}+\frac{\gamma _{0}}{2})$, where $r_{i}$ denotes the radius of the $i$-th disk. Furthermore, we define the mapping $\Lambda : \mathbb{R}^{2}\times [0,T] \rightarrow \mathbb{R}^{2}$ by

$$ \Lambda (x,t)=\sum _{i=1}^{k}\nabla ^{\bot }(h'_{i}(t)\cdot x^{\bot }\psi _{i}(x)). $$

(A.1)

The mapping $X$ is defined as a solution of the following Cauchy problem:

$$ \textstyle\begin{cases} \displaystyle {\frac{\partial X}{\partial t}}(y,t)&=\Lambda (X(y,t),t), \:\:t\in ]0,T], \\ X(y,0)&=y \in \mathbb{R}^{2}. \end{cases} $$

(A.2)

For all $y\in \mathbb{R}^{2}$, the initial-value problem (A.2) admits a unique solution $X(y,.): [0,T]\rightarrow \mathbb{R}^{2}$, which is $\mathcal{C}^{1}$ on $[0,T]$. Moreover, the mapping $X(.,t)$ is a $\mathcal{C}^{\infty }$-diffeomorphism from $\mathcal{O}$ into itself and from $B_{i}$ onto $B_{i}(t)$ whenever $B_{i}(t)\subset V_{B_{i}}$. Furthermore, the inverse mapping $Y$ of $X$ satisfies

$$ \textstyle\begin{cases} \displaystyle {\frac{\partial Y}{\partial t}}(x,s)&=-\Lambda (Y(x,s),t-s), \:\:t\in ]0,T], \\ Y(x,0)&=x \in \mathbb{R}^{2}. \end{cases} $$

(A.3)

From the definition of $\Lambda$ in (A.1), one can check that for all $t$ such that $B_{i}(t)\subset V_{B_{i}}$ we have

$$\begin{aligned} Y(x,t)= x-h_{i}(t)+h_{i}(0), \hspace{0.4cm} &\text{if}\:x\in \partial B_{i}(t), \end{aligned}$$

(A.4)

$$\begin{aligned} Y(x,t)=0, \hspace{0.4cm} &\text{if}\:x\in \partial \mathcal{O}. \end{aligned}$$

(A.5)

Moreover, we have

$$\begin{aligned} \frac{d Y}{dt}(x,t)= -h'_{i}(t), \hspace{0.4cm} &\text{if}\:x\in \partial B_{i}(t), \end{aligned}$$

(A.6)

$$\begin{aligned} \frac{d Y}{dt}(x,t)=0, \hspace{0.4cm} &\text{if}\:x\in \partial \mathcal{O}. \end{aligned}$$

(A.7)

First, we recall that for $T>0$,

$$\begin{array}{c}\mathcal{K}=\{(W,Q,\mathcal{T},{(h_i,{\omega }_i)}_{i=1,\dots ,k})\in \mathcal{U}(0,T;{\mathrm{\Omega }}_F)\times L^2(0,T;{\dot{H}}^1({\mathrm{\Omega }}_F))\hfill \\ \times \mathfrak{T}(0,T;{\mathrm{\Omega }}_F)\times {(H^2(0,T;{\mathbb{R}}^2)\times H^1(0,T;\mathbb{R}))}^k:\\ \hfill {\parallel W\parallel }_{\mathcal{U}(0,T;{\mathrm{\Omega }}_F)}+{\parallel Q\parallel }_{L^2(0,T;{\dot{H}}^1({\mathrm{\Omega }}_F))}+\sum _{i=1}^k{\parallel h_i^{\prime \prime }\parallel }_{L^2(0,T;{\mathbb{R}}^2)}+{\parallel {\omega }_i^{\prime }\parallel }_{L^2(0,T;\mathbb{R})}+{\parallel \mathcal{T}\parallel }_{\mathfrak{T}(0,T;{\mathrm{\Omega }}_F)}\le R\}.\end{array}$$

The following lemma allows us to bound the coefficients of the operators in the source terms in the linearized problem corresponding to problem (1.14)-(1.24). We refer the reader to [25] for a similar proof.

Lemma A.1

Suppose that $(W,Q,\mathcal{T},(h_{i},\omega _{i})_{i=1,\ldots ,k})\in {\mathcal{K}}$, then there exists two constants $N_{K}$ and $N_{C}$ satisfying conditions (i) and (ii) respectively (see Sect. 2), such that

$$\begin{aligned} \displaystyle \Big\Vert \frac{\partial X}{\partial y}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K},\: \displaystyle \Big\Vert \frac{\partial ^{i+j} X}{\partial y_{1}^{i}\partial y_{2}^{j}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}}\leq N_{K}T,\: & 1< i+j \leq 3, \\ \displaystyle \Big\Vert \frac{\partial Y}{\partial x}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K},\: \displaystyle \Big\Vert \frac{\partial ^{i+j} Y}{\partial x_{1}^{i}\partial x_{2}^{j}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}}\leq N_{K}T, \:& 1< i+j \leq 3, \\ \Big\Vert \displaystyle \frac{\partial ^{2} X_{m}}{\partial t\partial y_{\ell }}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K},\: \Big\Vert \displaystyle \frac{\partial ^{2} Y_{m}}{\partial t\partial x_{\ell }} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K},\:&\ell , m, n\in \{1, 2\}, \\ \displaystyle \Big\Vert \frac{\partial X_{m}}{\partial y_{\ell }}- \delta _{m}^{\ell }\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T, \: \Big\Vert \displaystyle \frac{\partial Y_{m}}{\partial x_{\ell }}- \delta _{m}^{\ell }\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T, \:& \ell , m\in \{1, 2\}, \\ \Vert g^{m\ell }-\delta _{m}^{\ell }\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T,\: \Vert g_{m\ell }-\delta _{m}^{\ell }\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T,\:&\ell , m\in \{1, 2\}. \end{aligned}$$

By using Cauchy-Schwartz inequality and mean value theorem, one can easily check the following.

Lemma A.2

Suppose that $(W^{1},Q^{1},\mathcal{T}^{1},(h^{1}_{i},\omega ^{1}_{i})_{i=1, \ldots ,k})$ and $(W^{2},Q^{2},\mathcal{T}^{2},(h^{2}_{i},\omega ^{2}_{i})_{i=1, \ldots ,k})$ in $\mathcal{K}$, and let $Y^{i},X^{i}, \Gamma ^{ik}_{j,\ell }$, etc. the terms corresponding to $(W^{i},Q^{i},\mathcal{T}^{i},(h^{i}_{j},\omega ^{i}_{j})_{j=1, \ldots ,k})$. Then there exists a constant $N_{K}$ satisfying condition (i) (see Sect. 2), such that the functions $h_{i}=h_{i}^{1}-h_{i}^{2},\:X=X^{1}-X^{2}$, and $Y=Y^{1}-Y^{2}$ satisfy the following inequalities:

$$\begin{aligned} \Vert h'_{\ell }\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T^{1/2} \sum _{i=1}^{k}\Vert h_{i}\Vert _{L^{2}(0,T;\mathbb{R}^{2})}, \: 1 \leq \ell \leq k, \\ \displaystyle \Big\Vert \frac{\partial ^{m+n} X}{\partial y_{1}^{m}\partial y_{2}^{n}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}}\leq N_{K}T^{1/2} \sum _{i=1}^{k}\Vert h_{i}\Vert _{L^{2}(0,T;\mathbb{R}^{2})},\: 0 \leq m+n\leq 3, \\ \displaystyle \Big\Vert \frac{\partial ^{m+n} Y}{\partial x_{1}^{m}\partial x_{2}^{n}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}}\leq N_{K}T^{1/2} \sum _{i=1}^{k}\Vert h_{i}\Vert _{L^{2}(0,T;\mathbb{R}^{2})}, \: 0 \leq m+n\leq 3, \\ \Big\Vert \displaystyle \frac{\partial ^{2} X_{m}}{\partial t\partial y_{n}}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T^{1/2}\sum _{i=1}^{k} \Vert h_{i}\Vert _{L^{2}(0,T;\mathbb{R}^{2})}, \: m, n\in \{1, 2\} \\ \Big\Vert \displaystyle \frac{\partial ^{2} Y_{m}}{\partial t\partial x_{n}}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq N_{K}T^{1/2}\sum _{i=1}^{k} \Vert h_{i}\Vert _{L^{2}(0,T;\mathbb{R}^{2})},\: m, n\in \{1, 2\}. \end{aligned}$$

Next, we recall that

$$\begin{array}{c}\tilde{\mathcal{K}}=\{(W,Q,\mathcal{T},{(h_i,{\omega }_i)}_{i=1,\dots ,k})\in \tilde{\mathcal{S}}(0,T,{\mathrm{\Omega }}_F):{\parallel W\parallel }_{\tilde{\mathcal{U}}(0,T;{\mathrm{\Omega }}_F)}+{\parallel \mathcal{T}\parallel }_{L^{\mathrm{\infty }}(0,T;{\mathbf{H}}^2({\mathrm{\Omega }}_F))}\hfill \\ +{\parallel Q\parallel }_{L^2(0,T;{\mathbf{H}}^2({\mathrm{\Omega }}_F))}+{\parallel \mathrm{\nabla }Q\parallel }_{L^{\mathrm{\infty }}(0,T;{\mathbf{L}}^2({\mathrm{\Omega }}_F))}+{\parallel {\partial }_t\mathrm{\nabla }Q\parallel }_{L^2([0,T];H_{\mathrm{\Gamma }}^{-1}({\mathrm{\Omega }}_F))}+\sum _{i=1}^k{\parallel h_i^{\prime \prime }\parallel }_{L^{\mathrm{\infty }}([0,T]\times {\mathbb{R}}^2)}\\ \hfill +{\parallel {\omega }_i^{\prime }\parallel }_{L^{\mathrm{\infty }}([0,T]\times \mathbb{R})}\le R,{\parallel {\mathcal{T}}^{\prime }\parallel }_{L^{\mathrm{\infty }}(0,T;{\mathbf{H}}^1({\mathrm{\Omega }}_F))}\le R^{\prime }\}.\end{array}$$

The following lemma is essential to prove that the source term $F_{0}$ defined in (2.3) is in the good space to apply Proposition 3.1. We refer the reader again to [25] for a similar proof.

Lemma A.3

Suppose that $(W,Q,\mathcal{T},(h_{i},\omega _{i})_{i=1,\ldots ,k})\in \tilde{\mathcal{K}}$ and let $\Lambda , X$, and $Y$ be the terms corresponding to $(W,Q,\mathcal{T},(h_{i},\omega _{i})_{i=1,\ldots ,k})\in \tilde{\mathcal{K}}$. Then there exists a constant $K_{0}$ satisfying (i) and a constant $C_{0}$ satisfying (ii) (see Sect. 3) such that

$$\begin{aligned} \Vert h'_{\ell }\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq C_{0}+K_{0}T, \: \Big\Vert \displaystyle \frac{\partial ^{i+j}\Lambda }{\partial x_{1}^{i}\partial x_{2}^{j}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq C_{0}+K_{0}T,\: &0 \leq i+j\leq 4, \\ &1\leq \ell \leq k, \\ \displaystyle \Big\Vert \frac{\partial X}{\partial y}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\: \displaystyle \Big\Vert \frac{\partial ^{i+j} X}{\partial y_{1}^{i}\partial y_{2}^{j}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}}\leq K_{0}T,\: & 1< i+j \leq 4, \\ \displaystyle \Big\Vert \frac{\partial Y}{\partial x}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\: \displaystyle \Big\Vert \frac{\partial ^{i+j} Y}{\partial x_{1}^{i}\partial x_{2}^{j}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}}\leq K_{0}T, \:& 1< i+j \leq 4, \\ \Big\Vert \displaystyle \frac{\partial ^{2} X_{m}}{\partial t\partial y_{\ell }}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\: \Big\Vert \displaystyle \frac{\partial ^{3} X_{m}}{\partial t\partial y_{\ell }\partial y_{n}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\:&\ell , m, n\in \{1, 2\}, \\ \Big\Vert \displaystyle \frac{\partial ^{2} Y_{m}}{\partial t\partial x_{\ell }}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\: \Big\Vert \displaystyle \frac{\partial ^{3} Y_{m}}{\partial t\partial x_{\ell }\partial x_{n}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\:&\ell , m, n\in \{1, 2\}, \\ \Big\Vert \displaystyle \frac{\partial ^{2} X}{\partial t^{2}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\: \Big\Vert \displaystyle \frac{\partial ^{2} Y}{\partial t^{2}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\:& \\ \displaystyle \Big\Vert \frac{\partial X_{m}}{\partial y_{\ell }}- \delta _{m}^{\ell }\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}T, \: \Big\Vert \displaystyle \frac{\partial Y_{m}}{\partial x_{\ell }}- \delta _{m}^{\ell }\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}T, \:& \ell , m\in \{1, 2\}. \end{aligned}$$

We recall that the functions $g^{ij},\:g_{i,j}$ and $\Gamma _{i,j}^{k}$ are defined as follows:

$$ g^{ij}=\sum _{k=1}^{2}\frac{\partial Y_{i}}{\partial x_{k}} \frac{\partial Y_{j}}{\partial x_{k}},\:\: g_{ij}=\sum _{k=1}^{2} \frac{\partial X_{k}}{\partial y_{i}} \frac{\partial X_{k}}{\partial y_{j}}, \:\: \Gamma ^{k}_{i,j}= \frac{1}{2}\sum _{\ell =1}^{2}g^{k\ell }\left \lbrace \frac{\partial g_{i\ell }}{\partial y_{j}}+ \frac{\partial g_{j\ell }}{\partial y_{i}}- \frac{\partial g_{ij}}{\partial y_{\ell }}\right \rbrace . $$

By noting that $g^{ij}(0)=g_{ij}(0)=\delta ^{i}_{j}$ and using mean-value theorem, we get

$$ \Vert g^{ij}-\delta ^{i}_{j}\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0}T,\:\Vert g_{ij}-\delta _{j}^{i}\Vert _{L^{\infty }([0,T] \times \mathcal{O})}\leq K_{0}T,\:\forall i,j\in \{1, 2\} $$

(A.8)

Moreover, we get the following as a direct consequence of Lemma A.3.

Corollary A.1

There exists a constant $K_{0}$ satisfying (i) (see Sect. 3) such that

$$\begin{aligned} \Vert g^{ij}\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}, \Vert \Gamma ^{i}_{j,k}\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0}T, \\ \Big\Vert \frac{\partial g^{ij}}{\partial y_{\ell }}\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}T, \Big\Vert \displaystyle \frac{\partial \Gamma ^{i}_{j,k}}{\partial y_{\ell }} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}T, \\ \Big\Vert \frac{\partial ^{2} g^{ij}}{\partial y_{\ell }\partial y_{m}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}T, \Big\Vert \displaystyle \frac{\partial ^{2}\Gamma ^{i}_{j,k}}{\partial y_{\ell }\partial y_{m}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}T. \\ \Big\Vert \frac{\partial g^{jk}}{\partial t}\Big\Vert _{L^{\infty }([0,T] \times \mathcal{O})}\leq K_{0},\: \Big\Vert \displaystyle \frac{\partial \Gamma ^{i}_{j,k}}{\partial t}\Big\Vert _{L^{\infty }([0,T] \times \mathcal{O})}\leq K_{0}, \\ \Big\Vert \frac{\partial ^{2} g^{jk}}{\partial t\partial y_{k}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0},\: \Big\Vert \displaystyle \frac{\partial ^{2}\Gamma ^{i}_{j,k}}{\partial t\partial y_{\ell }} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})}\leq K_{0}. \end{aligned}$$

We move now to derive some estimates which will be helpful in bounding the terms in the right hand side of (3.36) and (3.37) in terms of the terms in the left hand side of each one of them.

For $(W^{n},Q^{n},\mathcal{T}^{n},(h^{n}_{i},\omega ^{n}_{i})_{i=1, \ldots ,k})$ and $(W,Q,\mathcal{T},(h_{i},\omega _{i})_{i=1,\ldots ,k})\in \tilde{\mathcal{K}}$, we denote by $X^{n}, Y^{n},g^{ij,n}, \Gamma ^{k,n}_{i,j},\ldots$ the terms corresponding to $(W^{n},Q^{n},\mathcal{T}^{n},(h^{n}_{i},\omega ^{n}_{i})_{i=1, \ldots ,k})$ and by $X, Y, g^{ij}, \Gamma ^{k}_{i,j},\ldots$ the terms corresponding to $(W,Q,\mathcal{T},(h_{i},\omega _{i})_{i=1,\ldots ,k})$.

It is important to note that the transforms $X$ and $X^{n}$ satisfy the estimates in Lemma A.3 independent of $n$. We denote by $\bar{X}^{n}=X-X^{n}$ and $\bar{Y}^{n}=Y-Y^{n}$. Then using arguments identical to that given in [25] shows that $\bar{X}^{n}$ and $\bar{Y}^{n}$ satisfy the following.

Lemma A.4

Assume that $(W,Q,\mathcal{T},(h_{i},\omega _{i})_{i=1,\ldots ,k})$ and $(W^{n},Q^{n},\mathcal{T}^{n},(h_{i}^{n},\omega _{i}^{n})_{i=1, \ldots ,k})\in \tilde{\mathcal{K}}$, $\forall n\geq 1$.

Then there exists a constant $K_{0}$ satisfying (i) and a positive constant $C$ such that

$$\begin{aligned} \displaystyle \Big\Vert \frac{\partial ^{\ell +m} \bar{X}^{n}}{\partial y_{1}^{\ell }\partial y_{2}^{m}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}} \leq K_{0}T \displaystyle \sum _{i=1}^{k}\Vert h'_{i}-(h^{n}_{i})'\Vert _{L^{\infty }([0,T]\times \mathcal{O})},\: & 0\leq \ell +m\leq 3, \\ \displaystyle \Big\Vert \frac{\partial ^{\ell +m} \bar{Y}^{n}}{\partial x_{1}^{\ell }\partial x_{2}^{m}} \Big\Vert _{{L^{\infty }([0,T]\times \mathcal{O})}} \leq K_{0}T \displaystyle \sum _{i=1}^{k}\Vert h'_{i}-(h^{n}_{i})'\Vert _{L^{\infty }([0,T]\times \mathcal{O})}, \:& 0\leq \ell +m\leq 3, \\ \Big\Vert \displaystyle \frac{\partial ^{1+\ell +m} \bar{X}^{n}}{\partial t\partial y_{1}^{\ell }\partial y_{2}^{m}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0} \displaystyle \sum _{i=1}^{k}\Vert h'_{i}-(h^{n}_{i})'\Vert _{L^{\infty }([0,T]\times \mathcal{O})},\:&1\leq \ell + m\leq 3, \\ \Big\Vert \displaystyle \frac{\partial ^{1+\ell +m} \bar{Y}^{n}}{\partial t\partial x_{1}^{\ell }\partial x_{2}^{m}} \Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0} \displaystyle \sum _{i=1}^{k}\Vert h'_{i}-(h^{n}_{i})'\Vert _{L^{\infty }([0,T]\times \mathcal{O})},\:&1\leq \ell + m\leq 3. \end{aligned}$$

Finally, the following is a direct consequence of Lemma A.4.

Lemma A.5

There exists a positive constant $K_{0}$ satisfying $(i)$ such that

$$\begin{aligned} \Vert g^{ij}(X)-g^{ij,n}(X^{n})\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0}T\displaystyle \sum _{\ell =1}^{k}\Vert h'_{\ell }-(h^{n}_{\ell })'\Vert _{L^{\infty }([0,T]\times \mathcal{O})}, \\ \Big\Vert \frac{\partial }{\partial y_{k}}\Big(g^{ij}(X)-g^{ij,n}(X^{n}) \Big)\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0}T \displaystyle \sum _{\ell =1}^{k}\Vert h'_{\ell }-(h^{n}_{\ell })'\Vert _{L^{\infty }([0,T]\times \mathcal{O})}, \\ \big\Vert \Gamma ^{i}_{j,k}(X)-\Gamma ^{i,n}_{j,k}(X^{n})\big\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0}T\displaystyle \sum _{ \ell =1}^{k}\Vert h'_{\ell }-(h^{n}_{\ell })'\Vert _{L^{\infty }([0,T] \times \mathcal{O})}, \\ \Big\Vert \frac{\partial }{\partial y_{k}} \Big(\Gamma ^{i}_{j,\ell }(X)- \Gamma ^{i,n}_{j,\ell }(X^{n})\Big)\Big\Vert _{L^{\infty }([0,T]\times \mathcal{O})} \leq K_{0}T\displaystyle \sum _{m=1}^{k}\Vert h'_{m}-(h^{n}_{m})' \Vert _{L^{\infty }([0,T]\times \mathcal{O})}. \end{aligned}$$