粘弹性相分离模型在二维空间中强解的整体存在性

The Overall Existence of Strong Solution for Viscoelastic Phase Separation Model in the Two Dimensional Space

摘要: 本文给出了粘弹性相分离模型在二维空间下强解的整体存在性，使用Gagliardo-Nirenberg不等式、Sobolev不等式和Gronwall不等式进行证明，还使用了先验估计的证明方法。

Abstract: We present the overall existence of strong solutions for the viscoelastic phase separation model in two dimensions, using Gagliardo-Nirenberg inequality, Sobolev inequality and Gronwall inequality, and also the proof method of prior estimation.

1. 介绍

本文讨论在二维空间下粘弹性相分离模型强解的存在性

$\frac{\partial \varphi }{\partial t}+u\cdot \nabla \varphi =div\left(m\left(\varphi \right)\nabla \mu \right)-\kappa div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right),$ (1.1)

$\frac{\partial q}{\partial t}+u\cdot \nabla q=-\frac{1}{\tau \left(\varphi \right)}q+A\left(\varphi \right)\Delta \left(A\left(\varphi \right)q\right)-\kappa A\left(\varphi \right)div\left(n\left(\varphi \right)\nabla \mu \right)+\epsilon \Delta q,$ (1.2)

$\frac{\partial u}{\partial t}+\left(u\cdot \nabla \right)u=div\left(\eta \left(\varphi \right)\left(\nabla u+{\left(\nabla u\right)}^{\text{T}}\right)\right)-\nabla P+\nabla \varphi \mu ,$ (1.3)

$div\left(u\right)=0,$ (1.4)

$\mu =-\Delta \varphi +f\left(\varphi \right),$ (1.5)

其中 $\varphi$ 为聚合物分子的体积分数，q为聚合物相互作用产生的体应力，u为由溶剂与聚合物速度组成的体积平均速度， $m\left(\varphi \right)$ 和 $n\left(\varphi \right)$ 代表移动性函数， $\tau \left(\varphi \right)$ 代表广义弛豫时间， $A\left(\varphi \right)$ 代表体模量， $\eta \left(\varphi \right)$ 为粘度， $\epsilon$ 和 $\kappa$ 是正常数， $f\left(\varphi \right)=F^{\prime }\left(\varphi \right)$ 而 $F\left(\varphi \right)=a{\varphi }^2{\left(\varphi -1\right)}^2$ 且满足 $\left|F\left(s\right)\right|\le c_1{\left|s\right|}^p+c_2$ ， $\left|F^{\prime }\left(\varphi \right)\right|\le c_3{\left|\varphi \right|}^{p-1}+c_4$ ， $\left|F^{″}\left(\varphi \right)\right|\le c_5{\left|s\right|}^{p-2}+c_6$ ，对于 $p\ge 2$ ，其中， $F\in C^2\left(R\right)$ 并且 $c_i>0$ ，对于 $i\in \left\{1,\cdots ,6\right\}$ 。

这里 $\Omega \in R^2$ 是具有利普希茨边界条件的有界域，该模型具有以下初始条件和边界条件

${\left(\varphi ,q,u\right)|}_{t=0}=\left({\varphi }_0,q_0,u_0\right),{{\partial }_n\varphi |}_{\partial \Omega }={{\partial }_n\mu |}_{\partial \Omega }={{\partial }_{n}q|}_{\partial \Omega }=0,{u|}_{\partial \Omega }=0.$

以下是论文中涉及到的符号假设。在后续内容中，将假设所有共函数都是连续正有界的， $0<{\tau }_1\le \tau \left(s\right)\le {\tau }_2$ ， $0<A_1\le A\left(s\right)\le A_2$ ， $0<{\eta }_1\le \eta \left(s\right)\le {\eta }_2$ 对于 $s\in R$ ，其中 ${\Vert {\tau }^{\prime }\Vert }_{\infty }\le {{\tau }^{\prime }}_2$ ， ${\Vert A^{\prime }\Vert }_{\infty }\le {A^{\prime }}_2$ ， ${\Vert A^{″}\Vert }_{\infty }\le {A^{″}}_2$ ， ${\Vert {\eta }^{\prime }\Vert }_{\infty }\le {{\eta }^{\prime }}_2$ 。还需假设函数对于 $m,n$ 是连续正有界的，表示如下 $0<m_1\le m\left(s\right)\le m_2$ ， $0<n_1\le n\left(s\right)\le n_2$ 对于 $s\in R$ ，其中 ${\Vert m^{\prime }\Vert }_{\infty }\le {m^{\prime }}_2$ ， ${\Vert n^{\prime }\Vert }_{\infty }\le {n^{\prime }}_2$ 。

二元流体的相分离是软物质物理学中的一个基本过程。对于牛顿流体来说，这种现象是很普遍的。粘弹性相分离可以用由相位演化的Cahn-Hilliard方程、低流体演化的Navier-Stokes方程和粘弹性构象张量的时间演化组成的耦合系统来描述。其中 $\varphi$ 的相位变量主要受自由能泛函的低梯度控制，并与所涉及过程的热力学密切相关。对于该模型已经有人得出三维空间整体弱解的存在性，而更加复杂的模型如 [1] 中，也已经得出模型二维空间弱解的存在性以及弱解强解唯一性的结论，本文主要是对 [1] 中的模型简化处理后的方程，讨论研究其在二维空间的整体强解的存在性问题，采用对方程进行更高阶的处理方式，故而得到比 [1] 中的解阶数更高的整体强解，这是本文的重要创新点，这样得到的强解识别性更强，但是进行更高阶处理的同时，对后续估计工作增加的难度，这也是本文的主要难点。此外，本文对原方程进行简单的处理，并且增加了控制项，使得后续的计算证明变得更加简便明确。

本文使用的模型类似于 [1] 、 [2] 和 [3] 中的粘弹性相分离模型，主要研究粘弹性考虑了弹性构象张量的时间演化的模糊彼得林模型，而不是经典Oldroyd-B模型。彼得林模型 [2] 可以看作是经典Oldroyd-B模型的非线性推广。因此，本文整体强解的存在性结果也适用于 [3] 和 [4] 中的粘弹性相分离模型。文献 [5] 中研究了强解的良好性，文中使用的迁移率函数和融合共聚物的模型可以在 [6] 和 [7] 中得到详细解释。更多的关于粘弹性相分离模型的研究，请读者参考 [8] [9] [10] 。

主要结论为以下定理

定理1 假设 $\left({\varphi }_0,q_0,u_0\right)\in H^2\left(\Omega \right)\times H^1\left(\Omega \right)\times H^1\left(\Omega \right)$ ，这里 $\Omega$ 是 $R^2$ 上的光滑区域，对于任意有限时间 $T>0$ ，方程(1.1)~(1.5)在区域 $\Omega$ 上存在强解 $\left(\varphi ,q,u\right)$ 。

下面将介绍一些定理证明所用到的定义和引理。

2. 准备工作

以下定义和引理参考 [1] [2]

定义2.1 假设 $\left({\varphi }_0,q_0,u_0\right)\in H^2\left(\Omega \right)\times H^1\left(\Omega \right)\times H^1\left(\Omega \right)$ ，这里 $\Omega$ 是 $R^2$ 上光滑区域，若对 $\left(\varphi ,q,u\right)$ 有 $\varphi \in L^{\infty }\left(0,T;H^2\left(\Omega \right)\right)$ ， $q\in L^{\infty }\left(0,T;H^1\left(\Omega \right)\right)$ ， $u\in L^{\infty }\left(0,T;H^1\left(\Omega \right)\right)$ 。 $\left(\varphi ,q,u\right)$ 是方程(1.1)~(1.5)的弱解。

定义2.2 假设 $\left({\varphi }_0,q_0,u_0\right)\in H^2\left(\Omega \right)\times H^1\left(\Omega \right)\times H^1\left(\Omega \right)$ ，这里 $\Omega$ 是 $R^2$ 上的光滑区域，对于任意有限时间 $T>0$ ，若 $\left(\varphi ,q,u\right)$ 满足 $\varphi \in L^{\infty }\left(0,T;H^2\left(\Omega \right)\right)$ ， $q\in L^{\infty }\left(0,T;H^1\left(\Omega \right)\right)$ ， $u\in L^{\infty }\left(0,T;H^1\left(\Omega \right)\right)$ 。 $\left(\varphi ,q,u\right)$ 是方程(1.1)~(1.5)的强解。

引理2.1 (Sobolev不等式)设p满足 $1\le p<n$ ，对于一切 $u\in C_0^{\infty }\left(R^n\right)$ ，

${\Vert u\Vert }_{L^q}\le C{\Vert Du\Vert }_{L^p},$

其中q满足 $\frac{1}{q}=\frac{1}{p}-\frac{1}{n}$ 。

引理2.2 (Gagliardo-Nirenberg不等式)对于 $0<q,r\le \infty$ ， $0<\alpha <1$ ， $u\in W^{m,r}\left(\Omega \right)$ ，这里 $\Omega$ 是边界光滑的区域，有

${\Vert D^{j}u\Vert }_{L^p}\le C{\Vert D^{m}u\Vert }_{L^r}^{\alpha }{\Vert u\Vert }_{L^q}^{1-\alpha },$

其中 $\frac{j}{m}\le \alpha <1$ ， $\frac{1}{p}=\frac{j}{n}+\alpha \left(\frac{1}{r}-\frac{m}{n}\right)+\left(1-\alpha \right)\frac{1}{q}$ 。

引理2.2 (Gronwall不等式)设 $\eta$ 是 $\left[0,T\right]$ 到 $\left[0,\infty \right)$ 的连续函数， $\eta \left(t\right)\ge 0$ 且满足

${\eta }^{\prime }\left(t\right)\le \varphi \left(t\right)\eta \left(t\right)+\psi \left(t\right)$ ，有 $\eta \left(t\right)\le {\text{e}}^{{\displaystyle {\int }_0^t\varphi \left(s\right)}\text{d}s}\left[\eta \left(0\right)+{\displaystyle {\int }_0^t\psi \left(s\right)\text{d}s}\right]$ 。

3. 二维空间的强解存在性证明

本节中将给出本文中的定理1的证明。

证明：若要证明定理1，则要假设解 $\left(\varphi ,q,u\right)$ 是光滑的，对(1.1)作用 $\Delta$ ，右乘 $\Delta \varphi$ ，对(1.2)作用 $\nabla$ ，右乘 $\nabla q$ ，对(1.3)右乘 $-\Delta u$ ，得

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{\left|\Delta \varphi \right|}^2\text{d}x}=-{\displaystyle {\int }_{\Omega }\Delta \left(u\cdot \nabla \varphi \right)\Delta \varphi \text{d}x}+{\displaystyle {\int }_{\Omega }\Delta \left(div\left(m\left(\varphi \right)\nabla \mu \right)\right)\Delta \varphi }-{\displaystyle {\int }_{\Omega }\Delta \left(\kappa div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right)\right)\Delta \varphi \text{d}x}\\ =-{\displaystyle {\int }_{\Omega }m\left(\varphi \right)}{\left|{\Delta }^2\varphi \right|}^2\text{d}x-{\displaystyle {\int }_{\Omega }\Delta \left(u\cdot \nabla \varphi \right)\Delta \varphi \text{d}x}+{\displaystyle {\int }_{\Omega }\Delta \left(m^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu +m\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right)\right)\Delta \varphi \text{d}x}\\ -{\displaystyle {\int }_{\Omega }\kappa div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right){\Delta }^2\varphi \text{d}x},\end{array}$

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{\left|\nabla q\right|}^2\text{d}x}=-{\displaystyle {\int }_{\Omega }\nabla \left(u\cdot \nabla q\right)\nabla q\text{d}x}-{\displaystyle {\int }_{\Omega }\nabla \left(\frac{1}{\tau \left(\varphi \right)}q\right)\nabla q\text{d}x}+{\displaystyle {\int }_{\Omega }\nabla \left(A\left(\varphi \right)\Delta \left(A\left(\varphi \right)q\right)\right)\nabla q\text{d}x}\\ -{\displaystyle {\int }_{\Omega }\nabla \left(\kappa A\left(\varphi \right)div\left(n\left(\varphi \right)\nabla \mu \right)\right)\nabla q\text{d}x}+{\displaystyle {\int }_{\Omega }\nabla \left(\epsilon \Delta q\right)\nabla q\text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\epsilon {\left|\Delta q\right|}^2\text{d}x}-{\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}{\left|\nabla q\right|}^2\text{d}x}-{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}-{\displaystyle {\int }_{\Omega }\nabla \left(u\cdot \nabla q\right)\nabla q\text{d}x}\\ -{\displaystyle {\int }_{\Omega }\nabla \left(\frac{1}{\tau \left(\varphi \right)}q\right)\nabla q\text{d}x}-{\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q\Delta q\text{d}x}-{\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q\Delta q\text{d}x}\\ -{\displaystyle {\int }_{\Omega }2A\left(\varphi \right)A^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla q\Delta q\text{d}x}-{\displaystyle {\int }_{\Omega }\nabla \left(\kappa A\left(\varphi \right)div\left(n\left(\varphi \right)\nabla \mu \right)\right)\nabla q\text{d}x},\end{array}$

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^2\text{d}x}={\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)u\left(-\Delta u\right)\text{d}x}+{\displaystyle {\int }_{\Omega }div\left(\eta \left(\varphi \right)Du\right)\left(-\Delta u\right)\text{d}x}-{\displaystyle {\int }_{\Omega }\nabla P\left(-\Delta u\right)\text{d}x}+{\displaystyle {\int }_{\Omega }\nabla \varphi }\mu \left(-\Delta u\right)\text{d}x\\ =-{\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left|\Delta u\right|}^2\text{d}x}+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)u\left(-\Delta u\right)\text{d}x}-{\displaystyle {\int }_{\Omega }{\eta }^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla u\Delta u\text{d}x}-{\displaystyle {\int }_{\Omega }\nabla \varphi \mu \Delta u\text{d}x},\end{array}$

其中 $I_1=-{\displaystyle {\int }_{\Omega }\Delta \left(u\cdot \nabla \varphi \right)\Delta \varphi \text{d}x}$ 结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，得到

$\begin{array}{c}{I}_1=2{\displaystyle {\int }_{\Omega }\left({\partial }_{k}u\cdot \nabla \right){\partial }_k\varphi \Delta \varphi \text{d}x}+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)\Delta \varphi \Delta \varphi \text{d}x}\\ \le C{\Vert \nabla u\Vert }_{L^2}{\Vert \Delta \varphi \Vert }_{L^4}^2\\ \le C{\Vert \nabla u\Vert }_{L^2}{\Vert \nabla \varphi \Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le \epsilon {\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+C\left(\epsilon \right){\Vert \nabla u\Vert }_{L^2}^2.\end{array}$

再令 $I_2={\displaystyle {\int }_{\Omega }\Delta \left(m^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu +m\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right)\right)\Delta \varphi \text{d}x}$

$\begin{array}{c}{I}_2={\displaystyle {\int }_{\Omega }\Delta \left(m^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu +m\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right)\right)\Delta \varphi \text{d}x}\\ ={\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu {\Delta }^2\varphi \text{d}x}+{\displaystyle {\int }_{\Omega }m\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right){\Delta }^2\varphi \text{d}x}\\ =I_{21}+I_{22},\end{array}$

其中

$\begin{array}{c}{I}_{21}={\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu {\Delta }^2\varphi \text{d}x}\\ ={\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \nabla \left(-\Delta \varphi +{\left|\varphi \right|}^{p-1}+C\right){\Delta }^2\varphi \text{d}x}\\ \le {\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \left(\nabla \Delta \varphi \right){\Delta }^2\varphi \text{d}x}+{\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \nabla \left({\left|\varphi \right|}^3\right){\Delta }^2\varphi \text{d}x}+C{\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi {\Delta }^2\varphi \text{d}x},\end{array}$

结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，对 $I_{21}$ 的第一部分进行估计得到

$\begin{array}{l}{\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \left(\nabla \Delta \varphi \right){\Delta }^2\varphi \text{d}x}\\ \le c{\Vert {\Delta }^2\varphi \Vert }_{L^2}{\Vert \nabla \Delta \varphi \Vert }_{L^4}{\Vert {\Delta }^2\varphi \Vert }_{L^4}\\ \le c{\Vert {\Delta }^2\varphi \Vert }_{L^2}{\Vert \nabla \varphi \Vert }_{L^2}^{\frac{1}{2}}{\Vert \Delta \varphi \Vert }_{L^2}^{\frac{1}{2}}{\Vert {\Delta }^2\varphi \Vert }_{L^2}^{\frac{1}{2}}{\Vert \nabla \Delta \varphi \Vert }_{L^2}^{\frac{1}{2}}\\ \le c{\Vert {\Delta }^2\varphi \Vert }_{L^2}^{\frac{3}{2}}{\Vert \Delta \varphi \Vert }_{L^2}^{\frac{1}{2}}{\Vert \nabla \Delta \varphi \Vert }_{L^2}^{\frac{1}{2}}\\ \le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left({\epsilon }_1\right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2,\end{array}$

同理，对 $I_{21}$ 的第二部分进行估计得到

$\begin{array}{l}{\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi \nabla \left({\left|\varphi \right|}^3\right){\Delta }^2\varphi \text{d}x}\\ \le c{\Vert \nabla \varphi \Vert }_{L^4}^2{\Vert \varphi \Vert }_{L^{\infty }}^2{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le c{\Vert \nabla \varphi \Vert }_{L^2}^{\frac{3}{2}}{\Vert \nabla \Delta \varphi \Vert }_{L^2}^{\frac{1}{2}}{\Vert \varphi \Vert }_{L^2}^{\frac{3}{2}}{\Vert {\Delta }^2\varphi \Vert }_{L^2}^{\frac{3}{2}}\\ \le {\epsilon }_2{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left({\epsilon }_2\right){\Vert \nabla \Delta \varphi \Vert }_{L^2}^2,\end{array}$

再对 $I_{21}$ 的第三部分进行估计得到

$\begin{array}{l}C{\displaystyle {\int }_{\Omega }{m}^{\prime }\left(\varphi \right)\nabla \varphi {\Delta }^2\varphi \text{d}x}\\ \le c{\Vert \nabla \varphi \Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le c{\Vert \Delta \varphi \Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_3{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left({\epsilon }_3\right){\Vert \Delta \varphi \Vert }_{L^2}^2,\end{array}$

将上述不等式相加可以得到

$I_{21}\le \epsilon {\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left(\epsilon \right)\left({\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+{\Vert \Delta \varphi \Vert }_{L^2}^2\right).$

对于

$\begin{array}{c}{I}_{22}={\displaystyle {\int }_{\Omega }m\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right){\Delta }^2\varphi \text{d}x}\\ \le {\displaystyle {\int }_{\Omega }m\left(\varphi \right)\left(\left|\varphi \right|{\left(\nabla \varphi \right)}^2+{\left|\varphi \right|}^{p-1}\Delta \varphi +C\right){\Delta }^2\varphi \text{d}x}\\ \le {\displaystyle {\int }_{\Omega }m\left(\varphi \right)\left|\varphi \right|{\left(\nabla \varphi \right)}^2{\Delta }^2\varphi \text{d}x}+{\displaystyle {\int }_{\Omega }m\left(\varphi \right){\left|\varphi \right|}^2\Delta \varphi {\Delta }^2\varphi \text{d}x}+C{\displaystyle {\int }_{\Omega }m\left(\varphi \right){\Delta }^2\varphi \text{d}x},\end{array}$

结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，对 $I_{22}$ 的第一部分进行估计得到

$\begin{array}{l}{\displaystyle {\int }_{\Omega }m\left(\varphi \right)\left|\varphi \right|{\left(\nabla \varphi \right)}^2{\Delta }^2\varphi \text{d}x}\\ \le C{\Vert \varphi \Vert }_{L^{\infty }}{\Vert \nabla \varphi \Vert }_{L^4}^2{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left({\epsilon }_1\right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2,\end{array}$

同理，对 $I_{22}$ 的第二部分进行估计得到

$\begin{array}{l}{\displaystyle {\int }_{\Omega }m\left(\varphi \right){\left|\varphi \right|}^2\Delta \varphi {\Delta }^2\varphi \text{d}x}\\ \le c{\Vert \varphi \Vert }_{L^{\infty }}^2{\Vert \Delta \varphi \Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_2{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left({\epsilon }_2\right){\Vert \Delta \varphi \Vert }_{L^2}^4,\end{array}$

将上述不等式相加可以得到

$I_{22}\le \epsilon {\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left(\epsilon \right)\left({\Vert \Delta \varphi \Vert }_{L^2}{\Vert \nabla \Delta \varphi \Vert }_{L^2}+{\Vert \Delta \varphi \Vert }_{L^2}^4\right).$

令 $I_3=-{\displaystyle {\int }_{\Omega }\kappa div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right){\Delta }^2\varphi \text{d}x}$ 则

$\begin{array}{c}{I}_3={\displaystyle {\int }_{\Omega }\kappa div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right){\Delta }^2\varphi \text{d}x}\\ ={\displaystyle {\int }_{\Omega }\kappa n^{\prime }\left(\varphi \right)\nabla \varphi \nabla \left(A\left(\varphi \right)q\right){\Delta }^2\varphi \text{d}x}+{\displaystyle {\int }_{\Omega }\kappa n\left(\varphi \right)A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q{\Delta }^2\varphi \text{d}x}+{\displaystyle {\int }_{\Omega }\kappa n\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q{\Delta }^2\varphi \text{d}x}\\ +{\displaystyle {\int }_{\Omega }2\kappa n\left(\varphi \right)A^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla q{\Delta }^2\varphi \text{d}x}+{\displaystyle {\int }_{\Omega }\kappa n\left(\varphi \right)A\left(\varphi \right)\Delta q{\Delta }^2\varphi \text{d}x}\\ =I_{31}+I_{32}+I_{33}+I_{34}+I_{35},\end{array}$

对于

$\begin{array}{c}{I}_{31}={\displaystyle {\int }_{\Omega }\kappa n^{\prime }\left(\varphi \right)\nabla \varphi \nabla \left(A\left(\varphi \right)q\right){\Delta }^2\varphi \text{d}x}\\ =\kappa {\displaystyle {\int }_{\Omega }{n}^{\prime }\left(\varphi \right)A\left(\varphi \right)\nabla \varphi \cdot \nabla q{\Delta }^2\varphi \text{d}x}+\kappa {\displaystyle {\int }_{\Omega }{n}^{\prime }\left(\varphi \right)A^{\prime }\left(\varphi \right)}{\left(\nabla \varphi \right)}^{2}q{\Delta }^2\varphi \text{d}x,\end{array}$

同理，对 $I_{31}$ 的第一部分进行估计得到

$\begin{array}{l}\kappa {\displaystyle {\int }_{\Omega }{n}^{\prime }\left(\varphi \right)A\left(\varphi \right)\nabla \varphi \cdot \nabla q{\Delta }^2\varphi \text{d}x}\\ \le \kappa c{\Vert \nabla \varphi \Vert }_{L^4}{\Vert \nabla q\Vert }_{L^4}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_2{\Vert \Delta q\Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1,{\epsilon }_2\right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2,\end{array}$

同理，对 $I_{31}$ 的第二部分进行估计得到

$\kappa {\displaystyle {\int }_{\Omega }{n}^{\prime }\left(\varphi \right)A^{\prime }\left(\varphi \right)}{\left(\nabla \varphi \right)}^{2}q{\Delta }^2\varphi \text{d}x\le \kappa c{\Vert q\Vert }_{L^4}{\Vert {\left(\nabla \varphi \right)}^2\Vert }_{L^4}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1\right){\Vert \Delta \varphi \Vert }_{L^2}^2$

将上述不等式相加可以得到

$I_{31}\le 2{\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_2{\Vert \Delta q\Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1,{\epsilon }_2\right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1\right){\Vert \Delta \varphi \Vert }_{L^2}^2,$

由于 $I_{32}$ 和 $I_{31}$ 的第一部分估计方式相同，而 $I_{34}$ 和 $I_{31}$ 的第二部分估计方式相同，所以

$\begin{array}{l}{I}_{32}\le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_2{\Vert \Delta q\Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1,{\epsilon }_2\right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2\\ I_{34}\le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1\right){\Vert \Delta \varphi \Vert }_{L^2}^2,\end{array}$

结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，对 $I_{33}$ 进行估计得到

$\begin{array}{c}{I}_{33}=\kappa {\displaystyle {\int }_{\Omega }n\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q{\Delta }^2\varphi \text{d}x}\\ \le \kappa c{\Vert q\Vert }_{L^4}{\Vert \Delta \varphi \Vert }_{L^4}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1,{\epsilon }_2\right){\Vert \Delta \varphi \Vert }_{L^2}^2,\end{array}$

同理，对 $I_{35}$ 进行估计得到

$\begin{array}{c}{I}_{35}={\displaystyle {\int }_{\Omega }\kappa n\left(\varphi \right)A\left(\varphi \right)\Delta q{\Delta }^2\varphi \text{d}x}\\ \le \kappa c{\Vert \Delta q\Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_2{\Vert \Delta q\Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1,{\epsilon }_2\right){\Vert \nabla q\Vert }_{L^2}^2\end{array}$

将上述不等式相加得到 $I_3$ 的估计为

$I_3\le \epsilon {\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_1{\Vert \Delta q\Vert }_{L^2}^2+{\epsilon }_2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+c\left(\kappa ,\epsilon ,{\epsilon }_1,{\epsilon }_2\right)\left({\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla q\Vert }_{L^2}^2\right).$

令 $I_4=-{\displaystyle {\int }_{\Omega }\nabla \left(u\cdot \nabla q\right)\nabla q\text{d}x}$ ，结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，得到

$\begin{array}{c}{I}_4={\displaystyle {\int }_{\Omega }\nabla \left(u\cdot \nabla q\right)\nabla q\text{d}x}\\ \le C{\Vert u\Vert }_{L^4}{\Vert \nabla q\Vert }_{L^4}{\Vert \Delta q\Vert }_{L^2}\\ \le \epsilon {\Vert \Delta q\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert u\Vert }_{L^2}^2{\Vert \nabla u\Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2.\end{array}$

令 $I_5=-{\displaystyle {\int }_{\Omega }\nabla \left(\frac{1}{\tau \left(\varphi \right)}q\right)\nabla q\text{d}x}$ ，同上可以得到

$I_5={\displaystyle {\int }_{\Omega }\nabla \left(\frac{1}{\tau \left(\varphi \right)}q\right)\nabla q\text{d}x}\le \epsilon {\Vert \nabla q\Vert }_{L^2}^2+c\left(\tau \right){\Vert \nabla \varphi \Vert }_{L^2}^2{\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert q\Vert }_{L^2}^2.$

而对 $I_6$ 的估计可以写为

$\begin{array}{c}{I}_6={\displaystyle {\int }_{\Omega }\nabla \left(A\left(\varphi \right)\Delta \left(A\left(\varphi \right)q\right)\right)\nabla q\text{d}x}\\ ={\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q\Delta q\text{d}x}+{\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q\Delta q\text{d}x}\\ +{\displaystyle {\int }_{\Omega }2A\left(\varphi \right)A^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla q\Delta q\text{d}x}+{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}\\ =I_{61}+I_{62}+I_{63}+I_{64},\end{array}$

对 $I_{61},I_{62},I_{63}$ 分别进行估计可以得到

$\begin{array}{l}{I}_{61}={\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q\Delta q\text{d}x}\le \epsilon {\Vert A\left(\varphi \right)\Delta q\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \nabla \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2,\\ I_{62}={\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q\Delta q\text{d}x}\le \epsilon {\Vert \Delta q\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \nabla \Delta \varphi \Vert }_{L^2}^2{\Vert q\Vert }_{L^2}^2,\\ I_{63}={\displaystyle {\int }_{\Omega }2A\left(\varphi \right)A^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla q\Delta q\text{d}x}\le \epsilon {\Vert \Delta q\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2,\end{array}$

将上述不等式相加可以得到

$I_6\le {\epsilon }_1{\Vert A\left(\varphi \right)\Delta q\Vert }_{L^2}^2+{\epsilon }_2{\Vert \Delta q\Vert }_{L^2}^2+c\left({\epsilon }_1,{\epsilon }_2\right)\left({\Vert \nabla \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2{\Vert q\Vert }_{L^2}^2+{\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla q\Vert }_{L^2}^2\right)$

令 $I_7=-{\displaystyle {\int }_{\Omega }\nabla \left(\kappa A\left(\varphi \right)div\left(n\left(\varphi \right)\nabla \mu \right)\right)\nabla q\text{d}x}$ ，

$\begin{array}{c}{I}_7=\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n^{\prime }\left(\varphi \right)\Delta q\nabla \varphi \cdot \nabla \mu \text{d}x}+\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q{\Delta }^2\varphi \text{d}x}+\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q\Delta F^{\prime }\left(\varphi \right)\text{d}x}\\ =I_{71}+I_{72}+I_{73},\end{array}$

其中 $I_{71}$ 可以写成下式

$\begin{array}{c}{I}_{71}=\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n^{\prime }\left(\varphi \right)\Delta q\nabla \varphi \cdot \nabla \mu \text{d}x}\\ =\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n^{\prime }\left(\varphi \right)\Delta q\nabla \varphi \cdot \nabla \Delta \varphi \text{d}x}+\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n^{\prime }\left(\varphi \right)\Delta q{\left(\nabla \varphi \right)}^2{\left|\varphi \right|}^2\text{d}x},\end{array}$

对 $I_{71}$ 的第一部分进行估计可以得到

$\begin{array}{l}\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n^{\prime }\left(\varphi \right)\Delta q\nabla \varphi \cdot \nabla \Delta \varphi \text{d}x}\\ \le \kappa c{\Vert \Delta q\Vert }_{L^2}{\Vert \nabla \varphi \Vert }_{L^4}{\Vert \nabla \Delta \varphi \Vert }_{L^4}\\ \le {\epsilon }_1{\Vert \Delta q\Vert }_{L^2}^2+{\epsilon }_2{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left({\epsilon }_1,{\epsilon }_2,\kappa \right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2,\end{array}$

对 $I_{71}$ 的第二部分进行估计可以得到

$\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n^{\prime }\left(\varphi \right)\Delta q{\left(\nabla \varphi \right)}^2{\left|\varphi \right|}^2\text{d}x}\le c\kappa {\Vert \varphi \Vert }_{L^{\infty }}^2{\Vert \Delta q\Vert }_{L^2}{\Vert \nabla \varphi \Vert }_{L^4}^2\le \epsilon {\Vert \Delta q\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \Delta \varphi \Vert }_{L^2}^4,$

由于 $I_{72}$ 和 $I_{35}$ 的估计方式相同，所以

$\begin{array}{c}{I}_{72}\le {\displaystyle {\int }_{\Omega }\kappa n\left(\varphi \right)A\left(\varphi \right)\Delta q{\Delta }^2\varphi \text{d}x}\\ \le \kappa c{\Vert \Delta q\Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\\ \le {\epsilon }_1{\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+{\epsilon }_2{\Vert \Delta q\Vert }_{L^2}^2+c\left(\kappa ,{\epsilon }_1,{\epsilon }_2\right){\Vert \nabla q\Vert }_{L^2}^2,\end{array}$

同理 $I_{73}$ 可以也写成下式

$\begin{array}{c}{I}_{73}=\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q\Delta F^{\prime }\left(\varphi \right)\text{d}x}\\ =\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q\left|\varphi \right|{\left(\nabla \varphi \right)}^2\text{d}x}+\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q{\left|\varphi \right|}^2\Delta \varphi \text{d}x},\end{array}$

对 $I_{73}$ 的第一部分进行估计可以得到

$\begin{array}{l}\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q\left|\varphi \right|{\left(\nabla \varphi \right)}^2\text{d}x}\\ \le \kappa c{\Vert \Delta q\Vert }_{L^2}{\Vert \varphi \Vert }_{L^{\infty }}{\Vert \nabla \varphi \Vert }_{L^4}^2\\ \le {\epsilon }_1{\Vert \Delta q\Vert }_{L^2}^2+{\epsilon }_2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+c\left({\epsilon }_1,{\epsilon }_2,\kappa \right){\Vert \Delta \varphi \Vert }_{L^2}^2,\end{array}$

对 $I_{73}$ 的第二部分进行估计可以得到

$\kappa {\displaystyle {\int }_{\Omega }A\left(\varphi \right)n\left(\varphi \right)\Delta q{\left|\varphi \right|}^2\Delta \varphi \text{d}x}\le \kappa c{\Vert \varphi \Vert }_{L^{\infty }}^2{\Vert \Delta \varphi \Vert }_{L^2}{\Vert {\Delta }^2\varphi \Vert }_{L^2}\le \epsilon {\Vert {\Delta }^2\varphi \Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \Delta \varphi \Vert }_{L^2}^4,$

令 $I_8={\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)u\left(-\Delta u\right)\text{d}x}$ ，结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，得到

$I_8={\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)u\cdot \left(\Delta u\right)\text{d}x}\le c{\Vert u\Vert }_{L^4}{\Vert \nabla u\Vert }_{L^4}{\Vert \Delta u\Vert }_{L^2}\le \epsilon {\Vert \Delta u\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert u\Vert }_{L^2}^2{\Vert \nabla u\Vert }_{L^2}^4.$

令 $I_9=-{\displaystyle {\int }_{\Omega }{\eta }^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla u\Delta u\text{d}x}$ ，同理可以得到

$I_9\le c{\Vert \nabla \varphi \Vert }_{L^4}{\Vert \nabla u\Vert }_{L^4}{\Vert \Delta u\Vert }_{L^2}\le \epsilon {\Vert \Delta u\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \Delta \varphi \Vert }_{L^2}^2{\Vert \nabla u\Vert }_{L^2}^2.$

令 $I_{10}=-{\displaystyle {\int }_{\Omega }\nabla \varphi \mu \Delta u\text{d}x}$ ，同理可以得到

$I_{10}={\displaystyle {\int }_{\Omega }\nabla \varphi F^{\prime }}\left(\varphi \right)\Delta u\text{d}x\le c{\Vert \varphi \Vert }_{L^{\infty }}^3{\Vert \nabla \varphi \Vert }_{L^2}{\Vert \Delta u\Vert }_{L^2}\le \epsilon {\Vert \Delta u\Vert }_{L^2}^2+c\left(\epsilon \right){\Vert \nabla \varphi \Vert }_{L^2}^4{\Vert \nabla u\Vert }_{L^2}^4.$

再综合对 $I_1~I_{10}$ 的估计可以得到

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }\left({\left|\Delta \varphi \right|}^2+{\left|\nabla q\right|}^2+{\left|\nabla u\right|}^2\right)\text{d}x}\\ \le C_1\left({\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+{\Vert \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla \varphi \Vert }_{L^2}^2+{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \nabla u\Vert }_{L^2}^2+1\right)\left({\Vert \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \nabla u\Vert }_{L^2}^2\right)+C_2{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2,\end{array}$

由定理2.1的假设条件可知，对任意有限时间T

${\displaystyle {\int }_0^T\left({\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+{\Vert \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla \varphi \Vert }_{L^2}^2+{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \nabla u\Vert }_{L^2}^2+1\right)\text{d}x}\le C,$

${\displaystyle {\int }_0^T{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2\text{d}x}\le C,$

又因为(1.1)~(1.5)对任意有限时间T存在弱解，所以有

${\displaystyle {\int }_0^T{\Vert \nabla \varphi \Vert }_{L^2}^2\text{d}x}\le C,{\displaystyle {\int }_0^T{\Vert \nabla q\Vert }_{L^2}^2\text{d}x}\le C,{\displaystyle {\int }_0^T{\Vert \nabla u\Vert }_{L^2}^2\text{d}x}\le C,$

设 $A\left(t\right)={\Vert \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \nabla u\Vert }_{L^2}^2,t\in \left(0,T\right)$ 且 $A\left(0\right)={\Vert \Delta {\varphi }_0\Vert }_{L^2}^2+{\Vert \nabla q_0\Vert }_{L^2}^2+{\Vert \nabla u_0\Vert }_{L^2}^2$ ，

利用引理2.2 (Gronwall不等式)，带入对 $I_1~I_{10}$ 的估计可以得到

$\begin{array}{l}\sup A\left(t\right)\\ \le \left(A\left(0\right)+{\displaystyle {\int }_0^T{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2\text{d}x}\right)\exp \left\{T+{\displaystyle {\int }_0^T{\Vert \nabla \Delta \varphi \Vert }_{L^2}^2+{\Vert \Delta \varphi \Vert }_{L^2}^2+{\Vert \nabla \varphi \Vert }_{L^2}^2+{\Vert \nabla q\Vert }_{L^2}^2+{\Vert \nabla u\Vert }_{L^2}^2\text{d}x}\right\}\\ <\infty \end{array}$

综上所述，可以证明定理2.1成立，那么由弱解的存在性，可以得到粘弹性相分离模型在二维空间中整体强解的存在性。