# 二维玻尔兹曼方程的不可压缩极限Incompressible Limit of the Two Dimensional Boltzmann Equation

DOI: 10.12677/PM.2020.102019, PDF, HTML, XML, 下载: 177  浏览: 308

Abstract: In this paper, we study incompressible Navier-Stokes-Fourier limit of the two dimensional Boltzmann equation. The solution of the Boltzmann equation has no high order regularity in the bounded region, so we use a recent quantitative L2-L approach with a new L4 estimate for the hydrodynamic part, to obtain uniform upper estimation of the linear part of remainder equation, and then obtain the existence of the solution of remainder equation through iteration. Finally, we get existence of the solution of the Boltzmann equation and the convergence limit.

1. 引言

${\epsilon }^{-1}\stackrel{¯}{v}\cdot {\nabla }_{x}F+\epsilon \Phi \cdot {\nabla }_{v}F={\epsilon }^{-2}Q\left(F,F\right)$, $\left(x,v\right)\in \Omega ×{ℝ}^{3}$,

${F\left(x,v\right)|}_{n\left(x\right)\cdot \stackrel{¯}{v}<0}=\sqrt{\text{2}\pi }\mu {\int }_{n\left(x\right)\cdot \stackrel{¯}{u}>0}F\left(x,u\right)\left\{n\left(x\right)\cdot \stackrel{¯}{u}\right\}\text{d}u$, $x\in \partial \Omega$, (1)

$Q\left(F,H\right)\left(v\right)={\int }_{{ℝ}^{3}}{\int }_{{\mathbb{S}}^{2}}B\left(v-u,\omega \right)\left[F\left({v}^{\prime }\right)H\left({u}^{\prime }\right)-F\left(v\right)H\left(u\right)\right]\text{d}\omega \text{d}u={Q}_{+}\left(F,H\right)\left(v\right)-{Q}_{-}\left(F,H\right)\left(v\right)$.

$\stackrel{¯}{v}\in {ℝ}^{2},\stackrel{^}{v}\in ℝ$。定义 $\mu =\frac{1}{{\left(2\pi \right)}^{\frac{3}{2}}}{\text{e}}^{-\frac{|v{|}^{2}}{2}}$$B\left(V,\omega \right)=|V\cdot \omega |$$\Phi \left(x\right)=\left({\Phi }_{1},{\Phi }_{2},0\right)$ 表示给定的外力，F是稀薄

$\stackrel{¯}{v}\cdot {\nabla }_{x}f+{\epsilon }^{2}\frac{1}{\sqrt{\mu }}\Phi \cdot {\nabla }_{v}\left[\sqrt{\mu }f\right]+{\epsilon }^{-1}Lf=\Gamma \left(f,f\right)+\epsilon \Phi \cdot v\sqrt{\mu }$，在 $\Omega$ 上，

$f={P}_{\gamma }f$，在 ${\gamma }_{-}$ 上，

${‖f‖}_{2}+{‖Pf‖}_{4}+{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{\epsilon }^{-1/2}{|\left(1-{P}_{\gamma }\right)f|}_{2,+}+{\epsilon }^{1/2}{‖wf‖}_{\infty }\ll 1$,

$\stackrel{¯}{u}\cdot {\nabla }_{x}u+{\nabla }_{x}p=\sigma \Delta u+\Phi$${\nabla }_{x}\cdot u=0$，在 $\Omega$ 上，

$\stackrel{¯}{u}\cdot {\nabla }_{x}\theta =\kappa \Delta \theta$，在 $\Omega$ 上，

$u\left(x\right)=0,\theta \left(x\right)=0$，在 $\partial \Omega$ 上，其中 $\stackrel{¯}{u}=\left({u}_{1},{u}_{2}\right)$

2. 预备知识和主要引理

$X\underset{˜}{<}Y$ 等价于 $|X|\underset{˜}{<}CY$，C是与 $X,Y$ 无关的常数；定义 $Pf=a\sqrt{\mu }+v\cdot b\sqrt{\mu }+c\frac{{|v|}^{2}-3}{2}\sqrt{\mu }$

${\gamma }_{+}=\left\{\left(x,v\right)\in \partial \Omega ×{ℝ}^{3}:n\left(x\right)\cdot v>0\right\}$,

${\gamma }_{-}=\left\{\left(x,v\right)\in \partial \Omega ×{ℝ}^{3}:n\left(x\right)\cdot v<0\right\}$,

${\gamma }_{\text{0}}=\left\{\left(x,v\right)\in \partial \Omega ×{ℝ}^{3}:n\left(x\right)\cdot v=0\right\}$.

$\left({t}_{k+1},{x}_{k+1},{\stackrel{¯}{v}}_{k+1}\right)=\left({t}_{k}-{t}_{b}\left({x}_{k},{v}_{k}\right),{x}_{b}\left({x}_{k},{\stackrel{¯}{v}}_{k}\right),{\stackrel{¯}{v}}_{k+1}\right)$,

${X}_{cl}\left(s;t,x,\stackrel{¯}{v}\right)=\underset{k}{\sum }{1}_{\left[{t}_{k+1},{t}_{k}\right)}\left(s\right)\left\{{x}_{k}+\left(s-{t}_{k}\right){\stackrel{¯}{v}}_{k}\right\}$,

${\iint }_{\Omega ×{ℝ}^{3}}g\left(x,v\right)\sqrt{\mu }\text{d}x\text{d}v=0$, (2)

$\left[\lambda +\left(1-\tau \right){\epsilon }^{-1}\nu -\frac{1}{2}{\epsilon }^{2}\Phi \cdot v\right]f+\stackrel{¯}{v}\cdot {\nabla }_{x}f+{\epsilon }^{2}\Phi \cdot {\nabla }_{v}f+{\epsilon }^{-1}\tau Lf=g$，在 $\Omega ×{ℝ}^{\text{3}}$ 上，(3)

${f}_{-}={P}_{\gamma }f$，在 ${\gamma }_{-}$ 上，

${‖P\stackrel{\circ }{f}‖}_{\text{2}}\underset{˜}{<}{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }^{}+{|\left(1-{P}_{\gamma }\right)f|}_{2,+}^{}+{‖\frac{g}{\sqrt{\nu }}‖}_{2}^{}+o\left(1\right)|〈f〉|$,(4)

$\lambda |〈f〉|\underset{\sim }{<}\left(1-\tau \right){\epsilon }^{-1}{‖f‖}_{2}$(5)

${‖P\stackrel{\circ }{f}‖}_{4}\underset{˜}{<}{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{|\left(1-{P}_{\gamma }\right)f|}_{2,+}+{‖\frac{g}{\sqrt{\nu }}‖}_{2}+o\left(1\right)\left(|〈f〉|+{\epsilon }^{\frac{1}{2}}{‖wf‖}_{\infty }\right)$, (6)

${\stackrel{¯}{\omega }}_{\tau }=\lambda +\left(1-\tau \right){\epsilon }^{-1}\nu -\frac{1}{2}{\epsilon }^{2}\Phi \cdot v$，根据格林公式(见参考文献 [6])，得

${\iint }_{\Omega ×{ℝ}^{3}}\left\{\stackrel{¯}{v}\cdot {\nabla }_{x}f+{\epsilon }^{2}\Phi \cdot {\nabla }_{v}f\right\}\psi +\left\{\stackrel{¯}{v}\cdot {\nabla }_{x}\psi +{\epsilon }^{2}\Phi \cdot {\nabla }_{v}\psi \right\}f={\int }_{{\gamma }^{+}}f\psi -{\int }_{{\gamma }^{-}}f\psi$.

$\begin{array}{l}{\iint }_{\Omega ×{ℝ}^{3}}{\stackrel{¯}{\omega }}_{\tau }f\psi -\stackrel{¯}{v}\cdot f{\nabla }_{x}\psi -{\epsilon }^{2}f\Phi \cdot {\nabla }_{v}\psi +{\int }_{{\gamma }^{+}}f\psi -{\int }_{{\gamma }^{-}}f\psi \\ =-{\epsilon }^{-1}\tau {\iint }_{\Omega ×{ℝ}^{3}}\psi L\left(I-P\right)f+{\iint }_{\Omega ×{ℝ}^{3}}\psi g\end{array}$. (7)

$a-〈f〉=\stackrel{\circ }{a}$，则 $P\stackrel{\circ }{f}=\left(\stackrel{\circ }{a}\text{\hspace{0.17em}}+v\cdot b+c\frac{{|v|}^{2}-3}{2}\right)\sqrt{\mu }$，分以下三步证明。

${‖c‖}_{4}\underset{˜}{<}o\left(1\right){‖Pf‖}_{4}+{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{‖\left(1-P\right)f‖}_{4}+{|\left(1-{P}_{\gamma }\right)f|}_{2,+}+{‖\frac{g}{\sqrt{\nu }}‖}_{2}$. (8)

$N=2$ 时，我们想要 ${p}^{*}=2$，所以由 $\frac{1}{p}-\frac{1}{2}\le \frac{1}{2}$，得 $p\ge 1$，这里我们取 $p=\frac{4}{3}$，则对于任意的 $q\in \left[\frac{4}{3},2\right]$

${‖\nabla {\phi }_{c,4}‖}_{q}\underset{˜}{<}{‖\nabla {\phi }_{c,4}‖}_{{W}^{2,\frac{4}{3}}}$,

${‖\nabla {\phi }_{c,4}‖}_{\text{2}}\underset{˜}{<}{‖{\phi }_{c,4}‖}_{{W}^{2,\frac{4}{3}}}\underset{˜}{<}{‖{c}^{3}‖}_{\frac{4}{3}}={‖c‖}_{{}_{4}}^{3}$,(9)

(7)的右边 $\underset{˜}{<}{‖c‖}_{4}^{3}\left({\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{‖\frac{g}{\sqrt{\nu }}‖}_{2}\right)$(10)

$\psi ={\psi }_{c,\text{4}}=\left({|v|}^{2}-{\beta }_{c}\right)\sqrt{u}\stackrel{¯}{v}\cdot {\nabla }_{x}{\phi }_{c,4}$ 代入(7)，则(7)的左边可以写成如下形式：

${\iint }_{\partial \Omega ×{ℝ}^{3}}\left(n\left(x\right)\cdot \stackrel{¯}{v}\right)\left({|v|}^{2}-{\beta }_{c}\right)\sqrt{\mu }\underset{i=1}{\overset{2}{\sum }}{v}_{i}{\partial }_{i}{\phi }_{c,4}\left(x\right)\text{ }f\text{d}{S}_{x}\text{d}v$ (11)

$+{\int }_{\Omega ×{ℝ}^{3}}\left[\lambda +\left(1-\tau \right){\epsilon }^{-1}\nu \right]f\left({|v|}^{2}-{\beta }_{c}\right)\sqrt{\mu }\underset{i=1}{\overset{2}{\sum }}{v}_{i}\partial {\phi }_{c,4}\left(x\right)\text{ }\text{d}x\text{d}v$ (12)

$-{\iint }_{\Omega ×{ℝ}^{3}}\left({|v|}^{2}-{\beta }_{c}\right)\sqrt{\mu }\left\{\underset{i,j=1}{\overset{2}{\sum }}{v}_{i}{v}_{j}{\partial }_{ij}{\phi }_{c,4}\left(x\right)\right\}f\text{d}x\text{d}v$ (13)

$-{\epsilon }^{2}\underset{i,j}{\sum }{\iint }_{\Omega ×{ℝ}^{3}}\sqrt{\mu }{\Phi }_{i}\left\{{v}_{i}{v}_{j}\left({|v|}^{2}-{\beta }_{c}\right){\partial }_{j}{\phi }_{c,4}\left(x\right)+\left[{\delta }_{i,j}\left({|v|}^{2}-{\beta }_{c}\right)+2{v}_{i}{v}_{j}\right]{\partial }_{j}{\phi }_{c,4}\left(x\right)\right\}f$. (14)

$f=\left(a+v\cdot b+c\frac{{|v|}^{2}-3}{2}\right)\sqrt{\mu }+\left(I-P\right)f$，在 $\Omega ×{ℝ}^{3}$ 上， (15)

${f}_{\gamma }={P}_{\gamma }f+{1}_{{\gamma }^{+}}\left(1-{P}_{\gamma }\right)f$，在 $\gamma$ 上。 (16)

${\beta }_{c}=5$ 使得 ${\int }_{{ℝ}^{3}}\left({|v|}^{2}-{\beta }_{c}\right){v}_{i}^{2}u\left(v\right)\text{d}v=0,i=1,2$，则由奇函数性质，(13)表达式变成如下形式：

$\left(\text{13}\right)=-\underset{i=1}{\overset{2}{\sum }}{\int }_{{ℝ}^{3}}\left({|v|}^{2}-5\right){v}_{i}^{2}\frac{{|v|}^{2}-3}{2}\mu \left(v\right)\text{d}v{\int }_{\Omega }{\partial }_{ii}{\phi }_{c,4}\left(x\right)c\left(x\right)\text{d}x$ (17)

$-\underset{i=1}{\overset{2}{\sum }}{\iint }_{\Omega ×{ℝ}^{3}}\left({|v|}^{2}-5\right){v}_{i}\sqrt{\mu }\left(\stackrel{¯}{v}\cdot {\nabla }_{x}\right){\partial }_{i}{\phi }_{c,4}\left(I-P\right)f$. (18)

$\left(\text{17}\right)=-5{\int }_{\Omega }{\Delta }_{x}{\phi }_{c,4}c=5{‖c‖}_{4}^{4}$. (19)

$\left(\text{18}\right)\le {‖{\nabla }^{2}{\phi }_{c,2}‖}_{\frac{4}{3}}{‖\left(I-P\right)f‖}_{4}\le {‖c‖}_{4}^{3}{‖\left(I-P\right)f‖}_{4}$. (20)

${\beta }_{c}=5$ 时，由( ${v}_{i}{v}_{j}\left(i\ne j\right)$ 的奇性可得

${\iint }_{\partial \Omega ×{ℝ}^{3}}\left(n\left(x\right)\cdot \stackrel{¯}{v}\right)\left({|v|}^{2}-{\beta }_{c}\right)\sqrt{\mu }\underset{i=1}{\overset{2}{\sum }}{v}_{i}{\partial }_{i}{\phi }_{c,4}\left(x\right){P}_{\gamma }f\text{d}{S}_{x}\text{d}v=\text{0}$.

${\left({\int }_{\partial \Omega }\text{d}S\left(x\right){|u|}^{\frac{p\left(N-1\right)}{N-p}}\right)}^{\frac{N-p}{p\left(N-1\right)}}\le C\left(N,p\right){\left({\int }_{\Omega }\text{d}x{|u|}^{p}+{\int }_{\Omega }\text{d}x{|\nabla u|}^{p}\right)}^{\frac{1}{p}}$.

$p=\frac{4}{3}$$N=2$ 时， $\frac{p\left(N-1\right)}{N-p}=2$

$u=\nabla {\phi }_{c,4}$，由(9)得 ${‖{\nabla }_{x}{\phi }_{c,4}‖}_{{L}^{\frac{2}{\text{3}}}\left(\partial \Omega \right)}\underset{˜}{<}{‖{\nabla }_{x}{\phi }_{c,4}‖}_{{W}^{1,\frac{4}{3}}\left(\Omega \right)}\underset{˜}{<}{‖c‖}_{4}^{3}$，所以

$\left(\text{11}\right)\underset{˜}{<}{\mu }^{\frac{1}{2}}{|\left(1-{P}_{\gamma }\right)f|}_{2,+}{|{\nabla }_{x}{\phi }_{c,4}|}_{2,+}\underset{˜}{<}{|\left(1-{P}_{\gamma }\right)f|}_{2,+}{‖c‖}_{4}^{3}$. (21)

$\left(\text{12}\right)\underset{˜}{<}\left(\lambda +o\left(1\right)\right)×\left\{{‖Pf‖}_{4}+{‖\left(I-P\right)f‖}_{4}\right\}{‖c‖}_{4}^{3}$, (22)

$\left(\text{14}\right)\le {\epsilon }^{2}\left[{‖c‖}_{4}^{4}+{‖c‖}_{4}^{3}{‖\left(I-P\right)f‖}_{4}\right]{‖\Phi ‖}_{\infty }$. (23)

${‖c‖}_{4}\underset{˜}{<}o\left(1\right){‖Pf‖}_{4}+{|\left(I-{P}_{\gamma }\right)f|}_{2,+}+{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{‖\left(I-P\right)f‖}_{4}+{‖\frac{g}{\sqrt{\nu }}‖}_{2}$.

${‖b‖}_{4}\underset{˜}{<}o\left(1\right){‖Pf‖}_{4}+{|\left(I-{P}_{\gamma }\right)f|}_{2,+}+{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{‖\left(I-P\right)f‖}_{4}+{‖\frac{g}{\sqrt{\nu }}‖}_{2}$. (24)

$\psi ={\psi }_{b,\text{4}}^{i,j}\equiv \left({v}_{i}^{2}-{\beta }_{b}\right)\sqrt{\mu }{\partial }_{j}{\phi }_{b,\text{4}}^{j},\text{\hspace{0.17em}}i,j=1,2$, (25)

(7)右边 $\le {‖b‖}_{4}^{3}\left({\epsilon }^{-1}{‖\left(I-P\right)‖}_{\nu }+{‖\frac{g}{\sqrt{\nu }}‖}_{2}\right)$(26)

${\iint }_{\partial \Omega ×{ℝ}^{\text{3}}}\left(n\left(x\right)\cdot \stackrel{¯}{v}\right)\left({v}_{i}^{2}-{\beta }_{b}\right)\sqrt{\mu }{\partial }_{j}{\phi }_{b,4}^{j}\left[\left(1-{P}_{\gamma }\right)f\right]{1}_{{\gamma }_{+}}$ (27)

$+{\iint }_{\Omega ×{ℝ}^{\text{3}}}\left[\lambda +\left(1-\tau \right){\epsilon }^{-1}v\right]\text{ }f\left({v}_{i}^{2}-{\beta }_{b}\right)\sqrt{\mu }{\partial }_{j}{\phi }_{b,4}^{j}$ (28)

$+{\iint }_{\Omega ×{ℝ}^{\text{3}}}\left({v}_{i}^{2}-{\beta }_{b}\right)\sqrt{\mu }\left\{\underset{l}{\sum }{v}_{l}{\partial }_{lj}{\phi }_{b,4}^{j}\right\}f$ (29)

$-{\iint }_{\Omega ×{ℝ}^{3}}{\epsilon }^{2}\sqrt{\mu }f\left[\Phi \cdot v\left({v}_{i}^{2}-{\beta }_{b}\right){\partial }_{j}{\phi }_{b,4}^{j}+2{\Phi }_{i}{v}_{i}\right]{\partial }_{j}{\phi }_{b,4}^{j}$ (30)

$\left(\text{27}\right)\underset{˜}{<}{|\left(I-{P}_{\gamma }\right)f|}_{2,+}{|{\nabla }_{x}{\phi }_{b,4}|}_{2,+}\le {|\left(I-{P}_{\gamma }\right)f|}_{2,+}{‖b‖}_{4}^{3}$. (31)

$\left(\text{28}\right)\underset{˜}{<}\left(\lambda +o\left(1\right)\right)\left({‖Pf‖}_{4}+{‖\left(I-P\right)f‖}_{4}\right){‖b‖}_{4}^{3}$. (32)

$\left(\text{29}\right)=-\underset{i=1,2}{\sum }\left[{\iint }_{\Omega ×{ℝ}^{\text{3}}}\left({v}_{i}^{2}-{\beta }_{b}\right){v}_{l}^{2}\mu {\partial }_{lj}{\phi }_{b,4}^{j}\left(x\right){b}_{l}+{\iint }_{\Omega ×{ℝ}^{\text{3}}}\left({v}_{i}^{2}-{\beta }_{b}\right){v}_{l}\sqrt{\mu }{\partial }_{lj}{\phi }_{b,4}^{j}\left(x\right)\left(I-P\right)f\right]$. (33)

(33)中第一项 $=2{\int }_{\Omega }\left({\partial }_{i}{\partial }_{j}{\Delta }^{-1}{b}_{j}^{3}\left(x\right)\right){b}_{i}$(34)

(33)中第二项 $\underset{˜}{<}{‖{\nabla }^{2}{\phi }_{b,4}‖}_{\frac{4}{3}}\cdot {‖\left(I-P\right)f‖}_{4}\le {‖b‖}_{4}^{3}\cdot {‖\left(I-P\right)f‖}_{4}$(35)

$\left(\text{3}0\right)\underset{˜}{<}{\epsilon }^{2}{‖\Phi ‖}_{\infty }\left({‖b‖}_{4}^{4}+{‖b‖}_{4}^{3}\cdot {‖\left(I-P\right)f‖}_{4}^{4}\right)$. (36)

$|{\int }_{\Omega }\left({\partial }_{i}{\partial }_{j}{\Delta }^{-1}{b}_{j}^{3}\right){b}_{i}|\underset{˜}{<}{‖b‖}_{4}^{3}\left({\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{‖\frac{g}{\sqrt{\nu }}‖}_{2}+{|\left(I-{P}_{\gamma }\right)f|}_{2,+}+o\left(1\right){‖Pf‖}_{4}\right)$. (37)

(7)右边 $\underset{˜}{<}{‖b‖}_{4}^{3}\left({\epsilon }^{-1}{‖\left(I-P\right)f‖}_{\nu }+{‖\frac{g}{\sqrt{\nu }}‖}_{2}\right)$

$\begin{array}{c}-{\iint }_{\Omega ×{ℝ}^{3}}\stackrel{¯}{v}\cdot {\nabla }_{x}\psi f=-{\iint }_{\Omega ×{ℝ}^{3}}{|v|}^{2}{v}_{i}{v}_{j}\sqrt{\mu }\left\{\underset{l=1}{\overset{2}{\sum }}{v}_{l}{\partial }_{lj}{\phi }_{b,4}^{i}\right\}f\\ =-{\iint }_{\Omega ×{ℝ}^{3}}{|v|}^{2}{v}_{i}^{2}{v}_{j}^{2}\mu \left[{\partial }_{ij}{\phi }_{b,4}^{i}{b}_{j}+{\partial }_{jj}{\phi }_{b,4}^{i}{b}_{j}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\underset{l=1}{\overset{2}{\sum }}{\iint }_{\Omega ×{ℝ}^{3}}{|v|}^{2}{v}_{i}{v}_{j}{v}_{l}\sqrt{\mu }{\partial }_{lj}{\phi }_{b,4}^{i}\left(x\right)\left(I-P\right)f\end{array}$. (38)

(38)中第一项 $=-4\left[{\int }_{\Omega }\left({\partial }_{i}{\partial }_{j}{\Delta }^{-1}{b}_{i}^{\text{3}}\right){b}_{j}+\left({\partial }_{j}{\partial }_{j}{\Delta }^{-1}{b}_{i}^{\text{3}}\right){b}_{i}\right]$

$\begin{array}{c}{‖\stackrel{\circ }{a}‖}_{\text{4}}\underset{˜}{<}o\left(1\right){‖Pf‖}_{\text{4}}+{‖\left(I-P\right)f‖}_{\text{4}}+{\epsilon }^{-1}{‖\left(I-P\right)f‖}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{|\left(I-{P}_{\gamma }\right)f|}_{2,+}^{2}+{‖\frac{g}{\sqrt{\nu }}‖}_{2}+o\left(1\right)|〈f〉|\end{array}$. (39)

(7)右边

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,

,

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. (40)

. (41)

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,

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，在上，

,

, (42)

. (43)

3. 定理1.1的证明

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,

,

,

.(44)

. (45)

, (46)

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