三维黏弹性相分离模型的局部强解

Local Strong Solution of 3D Viscoelastic Phase Separation Model

摘要: 本文给出了三维黏弹性相分离模型的局部强解，利用了Nirenberg不等式、Sobolev嵌入定理及能量估计等。

Abstract: In this paper, the local strong solution of the three-dimensional viscoelastic phase separation model is given by using Nirenberg inequality, Sobolev embedding theorem and energy estimation.

1. 引言

本文考虑黏弹性相分离模型

$\frac{\partial \varphi }{\partial t}+u\cdot \nabla \varphi =div\left(n^2\left(\varphi \right)\nabla \mu \right)-div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right),$ (1.1)

$\frac{\partial q}{\partial t}+u\cdot \nabla q=-\frac{1}{\tau \left(\varphi \right)}q+A\left(\varphi \right)\Delta \left(A\left(\varphi \right)q\right)-A\left(\varphi \right)div\left(n\left(\varphi \right)\nabla \mu \right)+{\epsilon }_1\Delta q,$ (1.2)

$\frac{\partial u}{\partial t}+\left(u\cdot \nabla \right)u=div\left(\eta \left(\varphi \right)Du\right)-\nabla p+\nabla \varphi \mu ,$ (1.3)

$\mu =-c_0\Delta \varphi +F^{\prime }\left(\varphi \right),$ (1.4)

$divu=0,$ (1.5)

方程组(1.1)~(1.5)是在 $\left(0,T\right)\times \Omega$ 上，其中 $\Omega \subset {ℝ}^3$ ，为有界区域且是充分光滑的。此模型具有初值和边界条件

$\left(\varphi ,q,u\right)|{}_{t=0}=\left({\varphi }_0,q_0,u_0\right),{\partial }_n\varphi |{}_{\partial \Omega }={\partial }_n\Delta \varphi |{}_{\partial \Omega }={\partial }_{n}q|{}_{\partial \Omega }=0,u|{}_{\partial \Omega }=0.$ (1.6)

其中， $\varphi$ 为聚合物分子的体积分数， $q{\rm I}$ 为聚合物相互作用产生的体应力，u为由溶剂速度和聚合物速度组成的体积平均速度。

简单二元流体的相分离过程是软物质物理学的核心现象，并且通常将混合物从高温状态淬火到低温状态来观察。对于牛顿流体，这一现象在20世纪70年代得到了很好的研究，通用的宏观模型是Hohenberg和Halperin的Cahn-Hilliard-Navier-Stokes系统 [1] 。如果用一个更大的组分取代混合物中的一个组分，例如聚合体，则产生的不对称性在动力学中起重要作用。Tanaka [2] 引入了一个数学模型并提出了“黏弹性相分离”一词正是指受外力影响的混合物，上述的动力不对称。不幸的是，这个模型与热力学第二定律不一致。在 [3] 中，Zhou，Zhang，E使用热力学一致性方法重新推导了一致模型 [4] [5] [6] 。上述模型的关键因素是附加条件与两种多相流体的速度差有关的压力，更多的相关信息详见 [7] 。这种压力，后来被命名为q，是由Tanaka引入的，考虑到聚合物–溶剂混合物的分子间相互作用，并被认为是一种黏弹性现象。

黏弹性相分离可以由相场演化的Cahn-Hilliard方程、流体流动的Navier-Stokes方程和黏弹性构象张量的时间演化组成的耦合系统来描述。在本文的模型中，为了更好地进行研究，没有考虑构象张量的影响。

黏弹性相分离模型是新推导出来的模型，目前已知的结论较少，其中包括三维整体弱解及其弱强唯一性 [8] ，还得到了二维整体弱解的存在性及其相关结果 [9] ，参考 [10] 中解的正则性的有关理论及更多相关研究 [11] [12] [13] ，得到了黏弹性相分离模型的三维局部强解。在一般的初值情况下，得不到此模型的三维整体强解，因此我们的三维局部强解是较优的结果。若对此模型进行更加深入的研究，可以在此基础上，考虑其正则性准则。

引理1.1 设 $u,a,b,k$ 为J中的非负连续函数， $J=\left[{\alpha }_1,{\beta }_1\right]$ ，设 $p>1$ 且为一个常数。假设 $a/b$ 在J中不减小，且

$u\left(t\right)\le a\left(t\right)+b\left(t\right){\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}k\left(s\right)u^p\left(s\right)\text{d}s}$ ， $t\in J$ (1.7)

就有

$u\left(t\right)\le a\left(t\right){\left\{1-\left(p-1\right){\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}k\left(s\right)b\left(s\right)a^{p-1}\left(s\right)\text{d}s}\right\}}^{\frac{1}{1-p}}$ ， $\alpha \le t\le {\beta }_p$ ， (1.8)

其中

${\beta }_p=\sup \left\{t\in J:\left(p-1\right){\displaystyle {\int }_{{\alpha }_1}^{t}k\left(s\right)b\left(s\right)a^{p-1}\left(s\right)\text{d}s}<1\right\}$ . (1.9)

利用引理1.1，我们给出了对于方程组(1.1)~(1.5)解的局部正则性结果，即定理1.2，本文的主要结论如下。

定理1.2 假设初始值 ${\varphi }_0\in H^2\left(\Omega \right)$ ， $q_0\in H^1\left(\Omega \right)$ ， $u_0\in H^1\left(\Omega \right)$ ，那么对于方程组(1.1)~(1.5)的光滑解 $\varphi$ ，q，u，存在正数T₀和常数C，使得对于每一个T，满足 $0<T<T_0$ ，有

$\underset{0<t<T}{\sup }\left[{\Vert \varphi \Vert }_{H^2}+{\Vert q\Vert }_{H^1}+{\Vert u\Vert }_{H^1}\right]\le C.$

其中T₀和C都只依赖于域 $\Omega$ ，参数 $\alpha$ ，初始值 ${\varphi }_0$ 在 $H^2\left(\Omega \right)$ 中的大小及初始值 $q_0$ ， $u_0$ 在 $H^1\left(\Omega \right)$ 中的大小。

2. 预备知识

以下将给出一些对定理及引理证明有用的定义、假设和引理。

定义2.1 给定初始值 $\left({\varphi }_0,q_0,u_0\right)\in \left[H^1\left(\Omega \right)\times L^2\left(\Omega \right)\times H\right]$ ，四元数组 $\left(\varphi ,q,\mu ,u\right)$ 为方程组(1.1)~(1.5)的弱解，如果有

$\varphi \in L^{\infty }\left(0,T;H^1\left(\Omega \right)\right)\cap L^2\left(0,T;H^2\left(\Omega \right)\right)\cap H^1\left(0,T;H^{-1}\left(\Omega \right)\right),$

$q\in L^{\infty }\left(0,T;L^2\left(\Omega \right)\right)\cap L^2\left(0,T;H^1\left(\Omega \right)\right)\cap W^{1,4/3}\left(0,T;H^{-1}\left(\Omega \right)\right),$

$u\in L^{\infty }\left(0,T;H\right)\cap L^2\left(0,T;V\right)\cap W^{1,4/3}\left(0,T;V^{\prime }\right),$

$A\left(\varphi \right)q,\mu \in L^{5/3}\left(0,T;W^{1,5/3}\left(\Omega \right)\right),$

$n\left(\varphi \right)\nabla \mu -\nabla \left(A\left(\varphi \right)q\right)\in L^2\left(0,T;L^2\left(\Omega \right)\right),$

和

${\displaystyle {\int }_{\Omega }{\partial }_t}\varphi \psi \text{d}x+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \varphi \right)\psi \text{d}x}+{\displaystyle {\int }_{\Omega }n\left(\varphi \right)}\left[n\left(\varphi \right)\nabla \mu -\nabla \left(A\left(\varphi \right)q\right)\right]\cdot \nabla \psi \text{d}x=0,$

$\begin{array}{l}{\displaystyle {\int }_{\Omega }{\partial }_t}q\zeta \text{d}x+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla q\right)\zeta \text{d}x}+{\displaystyle {\int }_{\Omega }\frac{q\zeta }{\tau \left(\varphi \right)}\text{d}x}+{\epsilon }_1{\displaystyle {\int }_{\Omega }\nabla q\cdot \nabla \zeta \text{d}x}\\ ={\displaystyle {\int }_{\Omega }\left[n\left(\varphi \right)\nabla \mu -\nabla \left(A\left(\varphi \right)q\right)\right]}\cdot \nabla \left(A\left(\varphi \right)\zeta \right)\text{d}x=0,\end{array}$

${\displaystyle {\int }_{\Omega }\mu \varsigma \text{d}x}-c_0{\displaystyle {\int }_{\Omega }\nabla \varphi \cdot \nabla }\varsigma \text{d}x-{\displaystyle {\int }_{\Omega }{F}^{\prime }\left(\varphi \right)\varsigma \text{d}x}=0,$

${\displaystyle {\int }_{\Omega }{\partial }_{t}u\cdot \nu \text{d}x}+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)u\cdot \nu \text{d}x}-{\displaystyle {\int }_{\Omega }\eta \left(\varphi \right)Du:D\nu \text{d}x}+{\displaystyle {\int }_{\Omega }\varphi \nabla \mu \cdot \nu \text{d}x}=0.$

上述方程组适用于任何检验函数 $\left(\psi ,\zeta ,\varsigma ,\nu \right)\in \left[H^1\left(\Omega \right)\times H^1\left(\Omega \right)\times H^1\left(\Omega \right)\times V\right]$ 和几乎每一个 $t\in \left(0,T\right)$ 。而且，它满足上面给出的初始条件，即 $\left(\varphi \left(0\right),q\left(0\right),u\left(0\right)\right)=\left({\varphi }_0,q_0,u_0\right)$ 。

假设2.2 1) 假设 $n\left(s\right)$ ， $\eta \left(s\right)$ ， $\tau \left(s\right)$ 都为 $C^1\left(ℝ\right)$ 函数，这三个函数以及它们的导数 $n^{\prime }\left(s\right)$ ， ${\eta }^{\prime }\left(s\right)$ ， ${\tau }^{\prime }\left(s\right)$ 都为正的、有上下界的函数。

2) 假设 $A\left(s\right)\in C^2\left(ℝ\right)$ ，它及其二阶导函数 $A^{″}\left(s\right)$ 都为非负的、有上下界的函数，同时有 ${\Vert A^{\prime }\Vert }_{L^{\infty }\left(ℝ\right)}\le c$ 。

3) 假设 $F\in C^3\left(ℝ\right)$ ，有常数 $c_{1,i},c_{2,i}>0$ ， $i=0,1,2,3$ ，同时有 $\left|F^{\left(i\right)}\left(x\right)\right|\le c_{1,i}{\left|x\right|}^{p-i}+c_{2,i}$ ，

$i=0,1,2,3$ 且 $2\le p\le 4$ 。

4) 为方便起见，假设常数 ${\epsilon }_1$ 充分大， $c_0$ 为1。

5) Einstein约定求和就是略去求和式中的求和号，在此规则中两个相同指标就表示求和，而不管指标是什么字母，有时亦称求和的指标为“哑指标”，

$a_i{b}_i={\displaystyle \underset{i=1}{\overset{3}{\sum }}{a}_i{b}_i=a_1}{b}_1+a_2{b}_2+a_3{b}_3.$

引理2.3 (Sobolev嵌入定理)

${\Vert \varphi \Vert }_{L^{\infty }}\le C{\Vert \varphi \Vert }_{H^2}.$

引理2.4 (Nirenberg不等式) 设 $\Omega \subset {ℝ}^N$ 为具光滑边界的有界区域， $u\in W^{m,r}\left(\Omega \right)$ ，则有

${\Vert D^{j}u\Vert }_{L^p}\le C_1{\Vert D^{m}u\Vert }_{L^r}^a{\Vert u\Vert }_{L^q}^{1-a}+C_2{\Vert u\Vert }_{L^q},$

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

进一步，若 $u\in W_0^{m,r}\left(\Omega \right)$ ，则有

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

3. 定理证明

在此部分，我们将给出引理1.1的证明，方程组(1.1)~(1.5)的解是光滑的，首先对(1.1)作用 $\Delta$ ，再乘 $\Delta \varphi$ ，结合(1.4)，在 $\Omega$ 上积分并利用分部积分，有

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \Delta \varphi \Vert }_2^2+{\displaystyle {\int }_{\Omega }{n}^2\left(\varphi \right)}{\left({\Delta }^2\varphi \right)}^2\text{d}x\\ =-{\displaystyle {\int }_{\Omega }\Delta \left(u\cdot \nabla \varphi \right)\Delta \varphi \text{d}x}+{\displaystyle {\int }_{\Omega }2n\left(\varphi \right)n^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu {\Delta }^2\varphi \text{d}x}\\ +{\displaystyle {\int }_{\Omega }{n}^2\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right){\Delta }^2\varphi \text{d}x}-{\displaystyle {\int }_{\Omega }\Delta div\left(n\left(\varphi \right)\nabla \left(A\left(\varphi \right)q\right)\right)\Delta \varphi \text{d}x}\\ =I_1+I_2+I_3+I_4.\end{array}$ (3.1)

对(1.2)先作用 $\nabla$ ，再乘 $\nabla q$ ，在 $\Omega$ 上积分并利用分部积分，得到

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla q\Vert }_2^2+{\epsilon }_1{\displaystyle {\int }_{\Omega }{\left|\Delta q\right|}^2\text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\nabla \left(u\cdot \nabla q\right)}\cdot \nabla q\text{d}x+{\displaystyle {\int }_{\Omega }\nabla \left(-\frac{1}{\tau \left(\varphi \right)}q\right)}\cdot \nabla q\text{d}x\\ +{\displaystyle {\int }_{\Omega }\nabla \left(A\left(\varphi \right)\Delta \left(A\left(\varphi \right)q\right)\right)}\cdot \nabla q\text{d}x-{\displaystyle {\int }_{\Omega }\nabla \left(A\left(\varphi \right)div\left(n\left(\varphi \right)\nabla \mu \right)\right)}\cdot \nabla q\text{d}x\\ =J_1+J_2+J_3+J_4.\end{array}$ (3.2)

对(1.3)作用 $-\Delta u$ ，在 $\Omega$ 上积分并利用分部积分，得到

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }_2^2+{\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\left(-\Delta u\right)\left(u\cdot \nabla u\right)\text{d}x}+{\displaystyle {\int }_{\Omega }\left(-\Delta u\right){\eta }^{\prime }\left(\varphi \right)}\nabla \varphi \cdot \nabla u\text{d}x\\ -{\displaystyle {\int }_{\Omega }\left(-\Delta u\right)\nabla p\text{d}x}+{\displaystyle {\int }_{\Omega }\left(-\Delta u\right)\nabla \varphi \mu \text{d}x}\\ =K_1+K_2+K_3+K_4.\end{array}$ (3.3)

接下来，将 $I_1$ 中的 $\Delta \left(u\cdot \nabla \varphi \right)$ 改写成分量形式，利用分部积分，结合(1.5)，可得到

$\begin{array}{c}{\partial }_k{\partial }_k\left(u_i{\partial }_i\varphi \right)={\partial }_k\left({\partial }_k{u}_i{\partial }_i\varphi +u_i{\partial }_i{\partial }_k\varphi \right)\\ ={\partial }_k^2{u}_i{\partial }_i\varphi +2{\partial }_k{u}_i{\partial }_i{\partial }_k\varphi +u_i{\partial }_i{\partial }_k^2\varphi ,\end{array}$

$\begin{array}{c}{I}_1=-{\displaystyle {\int }_{\Omega }\Delta u\nabla \varphi \Delta \varphi \text{d}x-2{\displaystyle {\int }_{\Omega }\left({\partial }_{k}u\cdot \nabla \right){\partial }_k\varphi \Delta \varphi \text{d}x}}-{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)\Delta \varphi \Delta \varphi \text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\Delta u\nabla \varphi \Delta \varphi \text{d}x}-2{\displaystyle {\int }_{\Omega }\left({\partial }_{k}u\cdot \nabla \right){\partial }_k\varphi \Delta \varphi \text{d}x}\\ =I_{11}+I_{12}.\end{array}$ (3.4)

继续将 $I_4$ 进行分部积分，整理之后可得到

$\begin{array}{c}{I}_4=-{\displaystyle {\int }_{\Omega }{n}^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \left(A\left(\varphi \right)q\right){\Delta }^2\varphi \text{d}x}-{\displaystyle {\int }_{\Omega }n\left(\varphi \right)\Delta \left(A\left(\varphi \right)q\right){\Delta }^2\varphi \text{d}x}\\ =-{\displaystyle {\int }_{\Omega }{n}^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \left(A\left(\varphi \right)q\right){\Delta }^2\varphi \text{d}x}-{\displaystyle {\int }_{\Omega }n\left(\varphi \right)A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q{\Delta }^2\varphi \text{d}x}\\ -{\displaystyle {\int }_{\Omega }n\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q{\Delta }^2\varphi \text{d}x}-{\displaystyle {\int }_{\Omega }2n\left(\varphi \right)A^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla q{\Delta }^2\varphi \text{d}x}\\ -{\displaystyle {\int }_{\Omega }n\left(\varphi \right)A\left(\varphi \right)\Delta q{\Delta }^2\varphi \text{d}x}\\ =I_{41}+I_{42}+I_{43}+I_{44}+I_{45}.\end{array}$ (3.5)

接下来对 $J_2$ 进行分部积分，可得

$\begin{array}{l}{J}_2={\displaystyle {\int }_{\Omega }\frac{1}{{\tau }^2\left(\varphi \right)}{\tau }^{\prime }\left(\varphi \right)\nabla \varphi q\nabla q\text{d}x}-{\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}{\left|\nabla q\right|}^2\text{d}x}\\ =J_{21}+J_{22}.\end{array}$ (3.6)

同样地，对 $J_3$ 进行分部积分，整理之后我们可得到

$\begin{array}{l}{J}_3=-{\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q\Delta q\text{d}x}-{\displaystyle {\int }_{\Omega }A\left(\varphi \right)A^{\prime }\left(\varphi \right)\Delta \varphi q\Delta q\text{d}x}\\ -{\displaystyle {\int }_{\Omega }2A\left(\varphi \right)A^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla q\Delta q\text{d}x}-{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}\\ =J_{31}+J_{32}+J_{33}+J_{34}.\end{array}$ (3.7)

对 $J_4$ 也进行分部积分，整理之后可得

$\begin{array}{l}{J}_4={\displaystyle {\int }_{\Omega }A\left(\varphi \right)\Delta q{n}^{\prime }\left(\varphi \right)\nabla \varphi \cdot \nabla \mu \text{d}x}-{\displaystyle {\int }_{\Omega }n\left(\varphi \right){\Delta }^2\varphi A\left(\varphi \right)\Delta q\text{d}x}\\ +{\displaystyle {\int }_{\Omega }A\left(\varphi \right)\Delta qn\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right)\text{d}x}\\ =J_{41}+J_{42}+J_{43}.\end{array}$ (3.8)

根据(1.5)可知， $K_3$ 为零，将 $K_4$ 结合(1.4)可得

$\begin{array}{l}{K}_4={\displaystyle {\int }_{\Omega }\Delta u\nabla \varphi \Delta \varphi \text{d}x}-{\displaystyle {\int }_{\Omega }\Delta u\nabla \varphi F^{\prime }\left(\varphi \right)\text{d}x}\\ =K_{41}+K_{42}.\end{array}$ (3.9)

接下来，我们开始对各项进行估计，结合Hölder不等式，Young’s不等式，定义2.1和引理2.4，对 $I_{12}$ 进行估计

$\begin{array}{l}{I}_{12}\le C{\Vert \nabla u\Vert }_2{\Vert \Delta \varphi \Vert }_4^2\\ \le C{\Vert \nabla u\Vert }_2^2+C{\Vert \Delta \varphi \Vert }_4^4\\ \le C{\Vert u\Vert }_2{\Vert \Delta u\Vert }_2+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}{\Vert \Delta \varphi \Vert }_2^{\frac{5}{2}}+C{\Vert \Delta \varphi \Vert }_2^4\\ \le C_{\epsilon }{\Vert u\Vert }_2^2+\epsilon {\Vert \Delta u\Vert }_2^2+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{10}+C{\Vert \Delta \varphi \Vert }_2^4\\ \le C+\epsilon {\Vert \Delta u\Vert }_2^2+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{10}+C{\Vert \Delta \varphi \Vert }_2^4\\ \le C\left(1+{\Vert \Delta \varphi \Vert }_2^{10}\right)+\epsilon {\Vert \Delta u\Vert }_2^2+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2.\end{array}$ (3.10)

结合(1.4)，定义2.1，假设2.2中 $n^{\prime }\left(s\right)$ 的有界性，引理2.3，Hölder不等式，Young’s不等式以及引理2.4，我们接下来对 $I_2$ 进行估计

$\begin{array}{l}{I}_2\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n^{\prime }\left(\varphi \right)\nabla \varphi \nabla \mu \right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi \right|}^4}{\left|\varphi \right|}^{2p-4}\text{d}x+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi \right|}^4\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \Delta \varphi \nabla \varphi \right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_4^4{\Vert \varphi \Vert }_{\infty }^{2p-4}+C{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \nabla \Delta \varphi \Vert }_4^2{\Vert \nabla \varphi \Vert }_4^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+\left(C{\Vert \Delta \varphi \Vert }_2^3{\Vert \nabla \varphi \Vert }_2+C{\Vert \nabla \varphi \Vert }_2^4\right){\Vert \Delta \varphi \Vert }_2^{2p-4}+C{\Vert \Delta \varphi \Vert }_2^3{\Vert \nabla \varphi \Vert }_2\\ +C{\Vert \nabla \varphi \Vert }_2^4+C\left({\Vert {\Delta }^2\varphi \Vert }_2^{\frac{7}{4}}{\Vert \Delta \varphi \Vert }_2^{\frac{1}{4}}+{\Vert \Delta \varphi \Vert }_2^2\right)\left({\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla \varphi \Vert }_2^{\frac{1}{2}}+{\Vert \nabla \varphi \Vert }_2^2\right)\end{array}$

$\begin{array}{l}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{2p-1}+C{\Vert \Delta \varphi \Vert }_2^{2p-4}+C{\Vert \Delta \varphi \Vert }_2^3+C+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2\\ +C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{14}+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^{\frac{7}{2}}+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^{14}\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2.\end{array}$ (3.11)

根据定义2.1，假设2.2，引理2.3，Hölder不等式，Young’s不等式和引理2.4，接下来对 $I_3$ 进行估计

$\begin{array}{l}{I}_3\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right)\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right)F^{‴}\left(\varphi \right){\left(\nabla \varphi \right)}^2\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right)F^{″}\left(\varphi \right)\Delta \varphi \right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \varphi \Vert }_{\infty }^{2p-6}{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \varphi \Vert }_{\infty }^{2p-4}{\Vert \Delta \varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{2p-6}{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \Delta \varphi \Vert }_2^{2p-4}{\Vert \Delta \varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^2\end{array}$

$\begin{array}{l}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{2p-6}\left({\Vert {\Delta }^2\varphi \Vert }_2{\Vert \nabla \varphi \Vert }_2^3+{\Vert \nabla \varphi \Vert }_2^4\right)+C{\Vert {\Delta }^2\varphi \Vert }_2{\Vert \nabla \varphi \Vert }_2^3\\ +C{\Vert \nabla \varphi \Vert }_2^4+C{\Vert \Delta \varphi \Vert }_2^{2p-2}+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{4p-12}+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^{2p-6}+C+C{\Vert \Delta \varphi \Vert }_2^{2p-2}+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^6\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2.\end{array}$ (3.12)

通过定义2.1，假设2.2中 $n\left(s\right)$ 、 $n^{\prime }\left(s\right)$ 的有界性， ${\Vert A^{\prime }\Vert }_{L^{\infty }\left(ℝ\right)}\le c$ ，Hölder不等式，Young’s不等式和引理2.4，接下来对 $I_4$ 中的第一项 $I_{41}$ 进行估计

$\begin{array}{l}{I}_{41}\le {\displaystyle {\int }_{\Omega }\left|n^{\prime }\left(\varphi \right)\nabla \varphi \nabla \left(A\left(\varphi \right)q\right){\Delta }^2\varphi \right|\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\frac{n^{\prime }\left(\varphi \right)}{n\left(\varphi \right)}\nabla \varphi \nabla \left(A\left(\varphi \right)q\right)\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\frac{n^{\prime }\left(\varphi \right)}{n\left(\varphi \right)}{A}^{\prime }\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q+\frac{n^{\prime }\left(\varphi \right)}{n\left(\varphi \right)}\nabla \varphi A\left(\varphi \right)\nabla q\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert A^{\prime }\left(\varphi \right)\Vert }_{\infty }^2{\Vert \nabla \varphi \Vert }_8^4{\Vert q\Vert }_4^2+C{\Vert \nabla \varphi \Vert }_4^2{\Vert \nabla q\Vert }_4^2\end{array}$

$\begin{array}{l}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left({\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla \varphi \Vert }_2^{\frac{5}{2}}+{\Vert \nabla \varphi \Vert }_2^4\right)\left({\Vert \nabla q\Vert }_2^{\frac{3}{2}}{\Vert q\Vert }_2^{\frac{1}{2}}+{\Vert q\Vert }_2^2\right)\\ +C\left({\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla \varphi \Vert }_2^{\frac{1}{2}}+{\Vert \nabla \varphi \Vert }_2^2\right)\left({\Vert \Delta q\Vert }_2^{\frac{7}{4}}{\Vert q\Vert }_2^{\frac{1}{4}}+{\Vert q\Vert }_2^2\right)\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \nabla q\Vert }_2^6+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C{\Vert \nabla q\Vert }_2^{\frac{3}{2}}\\ +C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{12}+\epsilon {\Vert \Delta q\Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}+\epsilon {\Vert \Delta q\Vert }_2^2+C\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left(1+{\Vert \nabla q\Vert }_2^6+{\Vert \Delta \varphi \Vert }_2^{12}\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+2\epsilon {\Vert \Delta q\Vert }_2^2,\end{array}$ (3.13)

根据定义2.1，假设2.2中 $A^{″}\left(s\right)$ 的有界性，Hölder不等式，Young’s不等式和引理2.4，接下来我们对 $I_4$ 中的第二项 $I_{42}$ 进行估计

$\begin{array}{l}{I}_{42}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_8^4{\Vert q\Vert }_4^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla q\Vert }_2^{\frac{3}{2}}+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}+C{\Vert \nabla q\Vert }_2^{\frac{3}{2}}+C\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \nabla q\Vert }_2^6+C{\Vert \nabla q\Vert }_2^{\frac{3}{2}}+C\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left(1+{\Vert \nabla q\Vert }_2^6\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2,\end{array}$ (3.14)

根据定义2.1，假设2.2中 ${\Vert A^{\prime }\Vert }_{L^{\infty }\left(ℝ\right)}\le c$ ，Hölder不等式，Young’s不等式和引理2.4，接下来我们对 $I_4$ 中的第三项 $I_{43}$ 进行估计

$\begin{array}{l}{I}_{43}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A^{\prime }\left(\varphi \right)\Delta \varphi q\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert A^{\prime }\left(\varphi \right)\Vert }_{\infty }^2{\Vert \Delta \varphi \Vert }_4^2{\Vert q\Vert }_4^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_4^2{\Vert q\Vert }_4^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left({\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{4}}{\Vert \Delta \varphi \Vert }_2^{\frac{5}{4}}+{\Vert \Delta \varphi \Vert }_2^2\right)\left({\Vert \nabla q\Vert }_2^{\frac{3}{2}}{\Vert q\Vert }_2^{\frac{1}{2}}+{\Vert q\Vert }_2^2\right)\end{array}$

$\begin{array}{l}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^2{\Vert \nabla q\Vert }_2^{\frac{12}{5}}+C{\Vert \Delta \varphi \Vert }_2^4+C{\Vert \nabla q\Vert }_2^3\\ +\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^4+C{\Vert \nabla q\Vert }_2^{\frac{24}{5}}+C{\Vert \nabla q\Vert }_2^3+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^4+{\Vert \nabla q\Vert }_2^{\frac{24}{5}}\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2,\end{array}$ (3.15)

结合定义2.1，假设2.2中 ${\Vert A^{\prime }\Vert }_{L^{\infty }\left(ℝ\right)}\le c$ ，Hölder不等式，Young’s不等式和引理2.4，接下来我们对 $I_4$ 中的第四项 $I_{44}$ 进行估计

$\begin{array}{l}{I}_{44}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A^{\prime }\left(\varphi \right)\nabla \varphi \nabla q\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert A^{\prime }\left(\varphi \right)\Vert }_{\infty }^2{\Vert \nabla \varphi \Vert }_4^2{\Vert \nabla q\Vert }_4^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_4^2{\Vert \nabla q\Vert }_4^2\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}{\Vert \Delta q\Vert }_2^{\frac{7}{4}}+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}+C{\Vert \Delta q\Vert }_2^{\frac{7}{4}}+C\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+2\epsilon {\Vert \Delta q\Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{12}+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}+C\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^{12}\right)+2\epsilon {\Vert \Delta q\Vert }_2^2,\end{array}$ (3.16)

结合假设2.2，Hölder不等式及Young’s不等式，接下来我们对 $I_4$ 中的第五项 $I_{45}$ 进行估计

$\begin{array}{l}{I}_{45}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta q\Vert }_2^2.\end{array}$ (3.17)

通过定义2.1，分部积分公式，Hölder不等式，Young’s不等式和引理2.4，接下来对 $J_1$ 进行估计

$\begin{array}{l}{J}_1=-{\displaystyle {\int }_{\Omega }{\partial }_k}\left(u_i{\partial }_{i}q\right){\partial }_{k}q\text{d}x\\ =-{\displaystyle {\int }_{\Omega }{\partial }_k{u}_i{\partial }_{i}q{\partial }_{k}q\text{d}x-{\displaystyle {\int }_{\Omega }{u}_i{\partial }_i}}{\partial }_{k}q{\partial }_{k}q\text{d}x\\ \le {\displaystyle {\int }_{\Omega }\left|{\partial }_k{u}_i{\partial }_{i}q{\partial }_{k}q\right|\text{d}x}\\ \le C{\Vert \nabla u\Vert }_2{\Vert \nabla q\Vert }_4^2\\ \le C{\Vert \nabla u\Vert }_2\left({\Vert \Delta q\Vert }_2^{\frac{3}{2}}{\Vert \nabla q\Vert }_2^{\frac{1}{2}}+{\Vert \nabla q\Vert }_2^2\right)\end{array}$

$\begin{array}{l}\le C{\Vert \nabla u\Vert }_2{\Vert \Delta q\Vert }_2^{\frac{3}{2}}{\Vert \nabla q\Vert }_2^{\frac{1}{2}}+C{\Vert \nabla u\Vert }_2{\Vert \nabla q\Vert }_2^2\\ \le \epsilon {\Vert \Delta q\Vert }_2^2+C_{\epsilon }{\Vert \nabla u\Vert }_2^4{\Vert \nabla q\Vert }_2^2+C{\Vert \nabla u\Vert }_2{\Vert \nabla q\Vert }_2^2\\ \le \epsilon {\Vert \Delta q\Vert }_2^2+C{\Vert \nabla u\Vert }_2^8+C{\Vert \nabla q\Vert }_2^4+C{\Vert \nabla u\Vert }_2^2\\ \le C\left(1+{\Vert \nabla u\Vert }_2^8+{\Vert \nabla q\Vert }_2^4\right)+\epsilon {\Vert \Delta q\Vert }_2^2.\end{array}$ (3.18)

通过定义2.1，假设2.2，Hölder不等式，Young’s不等式和引理2.4，接下来对 $J_{21}$ 进行估计

$\begin{array}{l}{J}_{21}\le \epsilon {\displaystyle {\int }_{\Omega }{\left|\sqrt{\frac{1}{\tau \left(\varphi \right)}}\nabla q\right|}^2\text{d}x}+C_{\epsilon }{\displaystyle {\int }_{\Omega }\frac{1}{{\tau }^3\left(\varphi \right)}{\left|{\tau }^{\prime }\left(\varphi \right)\right|}^2{\left|\nabla \varphi q\right|}^2\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}}{\left|\nabla q\right|}^2\text{d}x+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi q\right|}^2\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}}{\left|\nabla q\right|}^2\text{d}x+C{\Vert \nabla \varphi \Vert }_4^2{\Vert q\Vert }_4^2\\ \le \epsilon {\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}}{\left|\nabla q\right|}^2\text{d}x+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{1}{2}}{\Vert \Delta q\Vert }_2^{\frac{3}{4}}+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{1}{2}}+C{\Vert \Delta q\Vert }_2^{\frac{3}{4}}+C\\ \le \epsilon {\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}}{\left|\nabla q\right|}^2\text{d}x+C+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+2\epsilon {\Vert \Delta q\Vert }_2^2.\end{array}$ (3.19)

结合定义2.1，假设2.2中的 $A^{″}\left(s\right)$ 的有界性，Hölder不等式，Young’s不等式和引理2.4，接下来对 $J_3$ 中的第一项 $J_{31}$ 进行估计

$\begin{array}{l}{J}_{31}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A^{″}\left(\varphi \right){\left(\nabla \varphi \right)}^{2}q\right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_8^4{\Vert q\Vert }_4^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla q\Vert }_2^{\frac{3}{2}}+C{\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}+C{\Vert \nabla q\Vert }_2^{\frac{3}{2}}+C\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \nabla q\Vert }_2^6+C{\Vert \nabla q\Vert }_2^{\frac{3}{2}}+C\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C\left(1+{\Vert \nabla q\Vert }_2^6\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2,\end{array}$ (3.20)

根据定义2.1，假设2.2中的 ${\Vert A^{\prime }\Vert }_{L^{\infty }\left(ℝ\right)}\le c$ ，Hölder不等式，Young’s不等式和引理2.4，接下来对 $J_3$ 中的第二项 $J_{32}$ 进行估计

$\begin{array}{l}{J}_{32}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A^{\prime }\left(\varphi \right)\Delta \varphi q\right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert A^{\prime }\left(\varphi \right)\Vert }_{\infty }^2{\Vert \Delta \varphi \Vert }_4^2{\Vert q\Vert }_4^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_4^2{\Vert q\Vert }_4^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_4^4+C{\Vert q\Vert }_4^4\end{array}$

$\begin{array}{l}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C\left({\Vert {\Delta }^2\varphi \Vert }_2^{\frac{3}{2}}{\Vert \Delta \varphi \Vert }_2^{\frac{5}{2}}+{\Vert \Delta \varphi \Vert }_2^4\right)+C\left({\Vert \Delta q\Vert }_2^{\frac{3}{2}}{\Vert q\Vert }_2^{\frac{5}{2}}+{\Vert q\Vert }_2^4\right)\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{10}+C{\Vert \Delta \varphi \Vert }_2^4+\epsilon {\Vert \Delta q\Vert }_2^2+C\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^{10}\right)+\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+\epsilon {\Vert \Delta q\Vert }_2^2,\end{array}$ (3.21)

结合定义2.1，假设2.2中的 ${\Vert A^{\prime }\Vert }_{L^{\infty }\left(ℝ\right)}\le c$ ，Hölder不等式，Young’s不等式和引理2.4，接下来对 $J_3$ 中的第三项 $J_{33}$ 进行估计

$\begin{array}{l}{J}_{33}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A^{\prime }\left(\varphi \right)\nabla \varphi \nabla q\right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert A^{\prime }\left(\varphi \right)\Vert }_{\infty }^2{\Vert \nabla \varphi \Vert }_4^2{\Vert \nabla q\Vert }_4^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_4^2{\Vert \nabla q\Vert }_4^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}{\Vert \Delta q\Vert }_2^{\frac{7}{4}}+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}+C{\Vert \Delta q\Vert }_2^{\frac{7}{4}}+C\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{12}+2\epsilon {\Vert \Delta q\Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}+C\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^{12}\right)+2\epsilon {\Vert \Delta q\Vert }_2^2.\end{array}$ (3.22)

结合定义2.1，假设2.2中 $n^{\prime }\left(s\right)$ 的有界性，Hölder不等式，Young’s不等式，引理2.3和引理2.4，接下来对 $J_4$ 中的第一项 $J_{41}$ 进行估计

$\begin{array}{l}{J}_{41}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n^{\prime }\left(\varphi \right)\nabla \varphi \nabla \mu \right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi \right|}^4{\left|\varphi \right|}^{2p-4}\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi \right|}^4\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \Delta \varphi \nabla \varphi \right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_4^4{\Vert \varphi \Vert }_{\infty }^{2p-4}+C{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \nabla \Delta \varphi \Vert }_4^2{\Vert \nabla \varphi \Vert }_4^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+\left(C{\Vert \Delta \varphi \Vert }_2^3{\Vert \nabla \varphi \Vert }_2+C{\Vert \nabla \varphi \Vert }_2^4\right){\Vert \Delta \varphi \Vert }_2^{2p-4}+C{\Vert \Delta \varphi \Vert }_2^3{\Vert \nabla \varphi \Vert }_2\\ +C{\Vert \nabla \varphi \Vert }_2^4+C\left({\Vert {\Delta }^2\varphi \Vert }_2^{\frac{7}{4}}{\Vert \Delta \varphi \Vert }_2^{\frac{1}{4}}+{\Vert \Delta \varphi \Vert }_2^2\right)\left({\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla \varphi \Vert }_2^{\frac{1}{2}}+{\Vert \nabla \varphi \Vert }_2^2\right)\end{array}$

$\begin{array}{l}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{2p-1}+C{\Vert \Delta \varphi \Vert }_2^{2p-4}+C{\Vert \Delta \varphi \Vert }_2^3+C+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2\\ +C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{14}+C{\Vert \Delta \varphi \Vert }_2^{\frac{7}{2}}+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^{14}\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2,\end{array}$ (3.23)

结合假设2.2，Hölder不等式及Young’s不等式，接下来对 $J_4$ 中的第二项 $J_{42}$ 进行估计

$\begin{array}{l}{J}_{42}\le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}\\ \le \frac{1}{12}{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right){\Delta }^2\varphi \right|}^2\text{d}x}+C{\Vert \Delta q\Vert }_2^2,\end{array}$ (3.24)

结合定义2.1，假设2.2，Hölder不等式，Young’s不等式，引理2.3和引理2.4，接下来对 $J_4$ 中的第三项 $J_{43}$ 进行估计

$\begin{array}{l}{J}_{43}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right)\Delta F^{\prime }\left(\varphi \right)\right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right)F^{‴}\left(\varphi \right){\left(\nabla \varphi \right)}^2\right|}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|n\left(\varphi \right)F^{″}\left(\varphi \right)\Delta \varphi \right|}^2\text{d}x}\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \varphi \Vert }_{\infty }^{2p-6}{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \nabla \varphi \Vert }_4^4+C{\Vert \varphi \Vert }_{\infty }^{2p-4}{\Vert \Delta \varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^2\end{array}$

$\begin{array}{l}\le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{2p-6}\left({\Vert {\Delta }^2\varphi \Vert }_2{\Vert \nabla \varphi \Vert }_2^3+{\Vert \nabla \varphi \Vert }_2^4\right)+C{\Vert {\Delta }^2\varphi \Vert }_2{\Vert \nabla \varphi \Vert }_2^3\\ +C{\Vert \nabla \varphi \Vert }_2^4+C{\Vert \Delta \varphi \Vert }_2^{2p-2}+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{4p-12}+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2+C{\Vert \Delta \varphi \Vert }_2^{2p-6}+C+C{\Vert \Delta \varphi \Vert }_2^{2p-2}+C{\Vert \Delta \varphi \Vert }_2^2\\ \le \frac{1}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^6\right)+2\epsilon {\Vert {\Delta }^2\varphi \Vert }_2^2.\end{array}$ (3.25)

我们结合Hölder不等式，Young’s不等式和引理2.4，接下来对 $K_1$ 进行估计

$\begin{array}{l}{K}_1=-{\displaystyle {\int }_{\Omega }{\partial }_k}u{\partial }_k{u}_i{\partial }_{i}u\text{d}x\\ \le C{\Vert \nabla u\Vert }_3^3\\ \le C{\Vert \Delta u\Vert }_2^{\frac{3}{2}}{\Vert \nabla u\Vert }_2^{\frac{3}{2}}\\ \le C{\Vert \nabla u\Vert }_2^6+\epsilon {\Vert \Delta u\Vert }_2^2.\end{array}$ (3.26)

结合定义2.1，假设2.2，Hölder不等式，Young’s不等式和引理2.4，接下来对 $K_2$ 进行估计

$\begin{array}{l}{K}_2\le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C_{\epsilon }{\displaystyle {\int }_{\Omega }{\left|\frac{{\eta }^{\prime }\left(\varphi \right)}{\sqrt{\eta \left(\varphi \right)}}\nabla \varphi \nabla u\right|}^2\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_4^2{\Vert \nabla u\Vert }_4^2\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C\left({\Vert \Delta \varphi \Vert }_2^{\frac{3}{2}}{\Vert \nabla \varphi \Vert }_2^{\frac{1}{2}}+{\Vert \nabla \varphi \Vert }_2^2\right){\Vert \Delta u\Vert }_2^{\frac{7}{4}}{\Vert u\Vert }_2^{\frac{1}{4}}\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+2\epsilon {\Vert \Delta u\Vert }_2^2+C_{\epsilon }{\Vert \Delta \varphi \Vert }_2^{12}+C\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^{12}\right)+2\epsilon {\Vert \Delta u\Vert }_2^2.\end{array}$ (3.27)

结合定义2.1，假设2.2，Hölder不等式，Young’s不等式，引理2.3和引理2.4，接下来对 $K_4$ 中的第二项 $K_{42}$ 进行估计

$\begin{array}{l}{K}_{42}\le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C_{\epsilon }{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi F^{\prime }\left(\varphi \right)\right|}^2\frac{1}{\eta \left(\varphi \right)}\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi \right|}^2{\left|\varphi \right|}^{2p-2}\text{d}x}+C{\displaystyle {\int }_{\Omega }{\left|\nabla \varphi \right|}^2\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C{\Vert \nabla \varphi \Vert }_2^2{\Vert \varphi \Vert }_{\infty }^{2p-2}+C{\Vert \nabla \varphi \Vert }_2^2\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C{\Vert \Delta \varphi \Vert }_2^{2p-2}+C\\ \le \epsilon {\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}+C\left(1+{\Vert \Delta \varphi \Vert }_2^6\right).\end{array}$ (3.28)

将上述的估计(3.10)~(3.17)、(3.18)~(3.25)及(3.26)~(3.28)分别代入(3.1)、(3.2)及(3.3)中，再将三者相加，其中 $\epsilon$ 取充分小，有 $C_1,C_2,C_3,C_4$ 为正数，可得到

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \Delta \varphi \Vert }_2^2+\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla q\Vert }_2^2+\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }_2^2+C_1{\displaystyle {\int }_{\Omega }{n}^2\left(\varphi \right)}{\left({\Delta }^2\varphi \right)}^2\text{d}x+C_2{\displaystyle {\int }_{\Omega }{\left|\Delta q\right|}^2\text{d}x}\\ +C_3{\displaystyle {\int }_{\Omega }\frac{1}{\tau \left(\varphi \right)}}{\left|\nabla q\right|}^2\text{d}x+\frac{3}{8}{\displaystyle {\int }_{\Omega }{\left|A\left(\varphi \right)\Delta q\right|}^2\text{d}x}+C_4{\displaystyle {\int }_{\Omega }\eta \left(\varphi \right){\left(\Delta u\right)}^2\text{d}x}\\ \le C\left(1+{\Vert \Delta \varphi \Vert }_2^{14}+{\Vert \nabla q\Vert }_2^6+{\Vert \nabla u\Vert }_2^8\right),\end{array}$

最后，我们总结出

$\frac{\text{d}}{\text{d}t}{\Vert \Delta \varphi \Vert }_2^2+\frac{\text{d}}{\text{d}t}{\Vert \nabla q\Vert }_2^2+\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }_2^2\le C\left(1+{\Vert \Delta \varphi \Vert }_2^{14}+{\Vert \nabla q\Vert }_2^6+{\Vert \nabla u\Vert }_2^8\right).$ (3.29)

利用上式(3.29)，我们可以得到引理1.1，即证明完毕。