﻿ 一类相变模型的弱解存在性的研究

# 一类相变模型的弱解存在性的研究Existence of Weak Solutions for a Class of Phase Field Models

Abstract: We shall investigate a phase-field model with a non-conserved order parameter which is under Neumann boundary conditions and omitting the effect of elasticity. By introducing a parameter κ to construct a modified model, and then using Banach’s fixed point Theorem, Aubin-Lions lemma and a series of a-priori estimates, the existence of global weak solutions to the model is finally obtained.

1. 引言

$-di{v}_{x}T\left(t,x\right)=b\left(t,x\right)$ (1)

$T\left(t,x\right)=D\left(\epsilon \left({\nabla }_{x}u\left(t,x\right)\right)-\stackrel{¯}{\epsilon }S\left(t,x\right)\right)$ (2)

${S}_{t}\left(t,x\right)=-c\left({\psi }_{S}\left(\epsilon \left({\nabla }_{x}u\left(t,x\right)\right),S\left(t,x\right)\right)-\nu {\Delta }_{x}S\left(t,x\right)\right)|{\nabla }_{x}S\left(t,x\right)|$ (3)

$u\left(t,x\right)=\gamma \left(t,x\right),\text{\hspace{0.17em}}{S}_{x}\left(t,x\right)=0,\text{\hspace{0.17em}}\left(t,x\right)\in \left[0,\infty \right)×\partial \Omega ,$ (4)

$S\left(0,x\right)={S}_{0}\left(x\right),\text{\hspace{0.17em}}x\in \Omega .$ (5)

${\nabla }_{x}u\left(t,x\right)$ 是u的一阶微分， ${ℝ}^{3}$ 空间下，这是一个 $3×3$ 梯度矩阵， ${\left({\nabla }_{x}u\right)}^{\text{T}}$ 是其转置矩阵。应力张量： $\epsilon \left({\nabla }_{x}u\right)=\frac{1}{2}\left({\nabla }_{x}u+{\left({\nabla }_{x}u\right)}^{\text{T}}\right)$。错配应变 $\stackrel{¯}{\epsilon }\in {S}^{3}$ 是一个给定的矩阵。弹性张量： ${S}^{3}{S}^{3}$ 是一个线性的对称正定映射。总能量为

${\psi }^{*}\left(\epsilon ,S\right)\left(t,x\right)=\psi \left(\epsilon ,S\right)+\frac{\nu }{2}{|{\nabla }_{x}S\left(t,x\right)|}^{2}.$

$\begin{array}{l}{\psi }_{S}=-\frac{1}{2}\left(D\left(\stackrel{¯}{\epsilon }\right)\cdot \left(\epsilon -\stackrel{¯}{\epsilon }S\right)+D\left(\epsilon -\stackrel{¯}{\epsilon }S\right)\cdot \left(\stackrel{¯}{\epsilon }\right)\right)+\stackrel{^}{\psi }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-D\left(\epsilon -\stackrel{¯}{\epsilon }S\right)\cdot \left(\stackrel{¯}{\epsilon }\right)+{\stackrel{^}{\psi }}_{S}=-T\cdot \left(\stackrel{¯}{\epsilon }\right)+{\stackrel{^}{\psi }}_{S}={\stackrel{^}{\psi }}_{S},\end{array}$

$\psi \left(S\right)={S}^{2}{\left(1-S\right)}^{2}$ (6)

${\psi }^{*}\left(S,{S}_{x}\right)\left(t,x\right)=\psi \left(S\right)+\frac{\nu }{2}{|{S}_{x}\left(t,x\right)|}^{2}.$ (7)

${S}_{t}=-c\left({\psi }_{S}\left(S\right)-\nu {S}_{xx}\right)|{S}_{x}|,\text{\hspace{0.17em}}\left(t,x\right)\in \left(0,{T}_{e}\right)×\Omega$ (8)

${S}_{x}=0,\text{\hspace{0.17em}}\left(t,x\right)\in \left(0,{T}_{e}\right)×\partial \Omega ,$ (9)

$S\left(0,x\right)={S}_{0}\left(x\right),\text{\hspace{0.17em}}x\in \Omega ,$ (10)

$S\in {L}^{\infty }\left(0,{T}_{e};{H}^{1}\left(\Omega \right)\right)$ (11)

$-{\left(S,{\varphi }_{t}\right)}_{{Q}_{{T}_{e}}}=-c{\left({\psi }_{S}|{S}_{x}|,\varphi \right)}_{{Q}_{{T}_{e}}}-\frac{c\nu }{2}{\left(|{S}_{x}|{S}_{x},{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}+{\left({S}_{0},\varphi \left(0\right)\right)}_{\Omega }$ (12)

${S}_{xt}\in {L}^{\frac{4}{3}}\left(0,{T}_{e};{W}^{-1,\frac{4}{3}}\left(\Omega \right)\right),\text{\hspace{0.17em}}{S}_{t}\in {L}^{\frac{4}{3}}\left({Q}_{{T}_{e}}\right)$ (13)

${\left({S}_{x}|{S}_{x}|\right)}_{x}\in {L}^{\frac{4}{3}}\left({Q}_{{T}_{e}}\right)$${S}_{x}\in {L}^{\frac{8}{3}}\left(0,{T}_{e};{L}^{q}\left(\Omega \right)\right)$ 任意 $1 (14)

$S\in {L}^{\infty }\left(0,{T}_{e};{H}^{1}\left(\Omega \right)\right)$ (15)

$-{\left(S,{\varphi }_{t}\right)}_{{Q}_{{T}_{e}}}=-c{\left({\psi }_{S}{|{S}_{x}|}_{\kappa },\varphi \right)}_{{Q}_{{T}_{e}}}-c\nu {\left({\int }_{0}^{{S}_{x}}{|y|}_{\kappa }\text{d}y,{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}+{\left({S}_{0},\varphi \left(0\right)\right)}_{\Omega }$ (16)

${S}_{x}\in {L}^{\infty }\left(0,{T}_{e};{L}^{2}\left(\Omega \right)\right)$ (17)

${S}_{xx}\in {L}^{\infty }\left(0,{T}_{e};{L}^{2}\left(\Omega \right)\right),\text{\hspace{0.17em}}{S}_{t}\in {L}^{\frac{4}{3}}\left({Q}_{{T}_{e}}\right).$ (18)

2. 一类固固相变模型的弱解存在性

2.1. 修正模型的局部解存在性

${S}_{t}=-c\left({\psi }_{S}\left(S\right)-\nu {S}_{xx}\right){|{S}_{x}|}_{\kappa }$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\Omega ,$ (19)

${S}_{x}=0$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\partial \Omega ,$ (20)

$S\left(0,x\right)={S}_{0}\left(x\right)$$x\in \Omega .$ (21)

${S}_{t}-c\nu {S}_{xx}\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}=-c{\psi }_{S}\left(S\right)\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\Omega ,$ (22)

$X=\left\{S|S\in {L}^{\infty }\left(0,T;{H}^{1}\left(\Omega \right)\right)\cap {L}^{2}\left(0,T;{H}^{2}\left(\Omega \right)\right),{S}_{t}\in {L}^{\frac{4}{3}}\left(0,T;{L}^{\frac{4}{3}}\left(\Omega \right)\right)\right\}.$

${S}_{t}-F{S}_{xx}=-F{\psi }_{S}\left(S\right)$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\Omega ,$ (23)

${S}_{t}-F{S}_{xx}=-F{\psi }_{\stackrel{^}{S}}\left(\stackrel{^}{S}\right)$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\Omega .$ (24)

2.下面我们证明，当 $T>0$ 足够小时，映射A是紧压缩的。

Step1.根据上文中关于映射A的定义，当我们从X中选择两个函数 ${\stackrel{^}{S}}_{1}$${\stackrel{^}{S}}_{2}$ 时，就可以得到S1和S2，这两个新的函数即满足 ${S}_{1}=A\left[{\stackrel{^}{S}}_{1}\right]$${S}_{2}=A\left[{\stackrel{^}{S}}_{2}\right]$，也满足(20)-(21)和(24)。令 $W={S}_{1}-{S}_{2}$，得到了一个新的线性的偏微分方程组。

${W}_{t}-F{W}_{xx}=-F\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right)$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\Omega ,$ (25)

${W}_{x}=0$$\left(t,x\right)\in \left(0,{T}_{e}\right)×\partial \Omega ,$ (26)

$W\left(0,x\right)=0$$x\in \Omega .$ (27)

Step2.用W与(25)做内积可得

$\left({W}_{t},W\right)+\left(-F{W}_{xx},W\right)=\left(-F\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right),W\right),$

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖W‖}_{{L}^{2}\left(\Omega \right)}^{2}+\kappa {‖{W}_{x}‖}_{{L}^{2}\left(\Omega \right)}^{2}\le -C{\int }_{\Omega }\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right)W\text{d}x,$

$\begin{array}{l}{‖W‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+\kappa {\int }_{0}^{T}{‖{W}_{x}‖}_{{L}^{2}\left(\Omega \right)}^{2}\text{d}\tau \\ \le C{\int }_{{Q}_{{T}_{e}}}\left(-\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right)\right)W\text{d}\left(\tau ,x\right)\\ \le {C}_{\epsilon }{‖\left(-\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right)\right)‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}+\epsilon {‖W‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}\\ \le {C}_{\epsilon }{‖{\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}+\epsilon {‖W‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}{\int }_{0}^{T}\text{d}\tau \\ \le {C}_{\epsilon }{‖{\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}+\epsilon {‖W‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\end{array}$

$\begin{array}{l}{‖W‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+C{\int }_{0}^{T}{‖{W}_{x}‖}_{{L}^{2}\left(\Omega \right)}^{2}\text{d}\tau \\ \le {C}_{\epsilon }{‖{\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}\\ \le {C}_{\epsilon }{‖4{\stackrel{^}{S}}_{1}^{3}-6{\stackrel{^}{S}}_{1}^{2}+2{\stackrel{^}{S}}_{1}-4{\stackrel{^}{S}}_{2}^{3}+6{\stackrel{^}{S}}_{2}^{2}-2{\stackrel{^}{S}}_{2}‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}\\ \le {C}_{\epsilon }{‖\left({\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}\right)\left({\stackrel{^}{S}}_{1}^{2}+{\stackrel{^}{S}}_{1}{\stackrel{^}{S}}_{2}+{\stackrel{^}{S}}_{2}^{2}-6{\stackrel{^}{S}}_{1}-6{\stackrel{^}{S}}_{2}+2\right)‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}\\ \le C{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}{\int }_{0}^{T}\text{d}\tau \le CT{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\end{array}$

${‖W‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}+{‖{W}_{x}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}\le {\left(CT\right)}^{\frac{1}{2}}{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}.$ (28)

Step3.将(25)与 $-{W}_{xx}$ 做内积，有

$\left({W}_{t},-{W}_{xx}\right)+\left(-F{W}_{xx},-{W}_{xx}\right)=\left(-F\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right),-{W}_{xx}\right),$

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{‖{W}_{x}‖}_{{L}^{2}\left(\Omega \right)}^{2}+\kappa {‖{W}_{xx}‖}_{{L}^{2}\left(\Omega \right)}^{2}\\ \le C{\int }_{\Omega }\left({\psi }_{{\stackrel{^}{S}}_{1}}-{\psi }_{{\stackrel{^}{S}}_{2}}\right){W}_{xx}\text{d}x=C{\int }_{\Omega }{W}_{x}{\left({\psi }_{{\stackrel{^}{S}}_{2}}-{\psi }_{{\stackrel{^}{S}}_{1}}\right)}_{x}\text{d}x\\ =C{\int }_{\Omega }\left(\frac{\text{d}{\psi }_{{\stackrel{^}{S}}_{2}}}{\text{d}{\stackrel{^}{S}}_{2}}\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}-\frac{\text{d}{\psi }_{{\stackrel{^}{S}}_{1}}}{\text{d}{\stackrel{^}{S}}_{1}}\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right){W}_{x}\text{d}x\\ ={\int }_{\Omega }\left(\frac{\text{d}{\psi }_{{\stackrel{^}{S}}_{2}}}{\text{d}{\stackrel{^}{S}}_{2}}\left(\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right)+\left(\frac{\text{d}{\psi }_{{\stackrel{^}{S}}_{2}}}{\text{d}{\stackrel{^}{S}}_{2}}-\frac{\text{d}{\psi }_{{\stackrel{^}{S}}_{1}}}{\text{d}{\stackrel{^}{S}}_{1}}\right)\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right){W}_{x}\text{d}x\\ ={\int }_{\Omega }\frac{\text{d}{\psi }_{{\stackrel{^}{S}}_{2}}}{\text{d}{\stackrel{^}{S}}_{2}}\left(\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right){W}_{x}\text{d}x+{\int }_{\Omega }\left({\stackrel{^}{S}}_{2}-{\stackrel{^}{S}}_{1}\right)\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}{\psi }^{‴}\left(\xi \right){W}_{x}\text{d}x\\ \le {\int }_{\Omega }\left(\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right){W}_{x}\text{d}x+C{‖{\stackrel{^}{S}}_{2}-{\stackrel{^}{S}}_{1}‖}_{{L}^{2}\left(\Omega \right)}{‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}‖}_{{L}^{\infty }\left(\Omega \right)}{‖{W}_{x}‖}_{{L}^{2}\left( \Omega \right)}\end{array}$

$\begin{array}{l}{‖{W}_{x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+{‖{W}_{xx}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\\ \le {\int }_{{Q}_{T}}\left(\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right){W}_{x}\text{d}\left(x,\tau \right)+{\int }_{0}^{T}{‖{\stackrel{^}{S}}_{2}-{\stackrel{^}{S}}_{1}‖}_{{L}^{2}\left(\Omega \right)}{‖{W}_{x}‖}_{{L}^{2}\left(\Omega \right)}{‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}‖}_{{L}^{\infty }\left(\Omega \right)}\\ \le {\int }_{{Q}_{T}}\left(\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}\right){W}_{x}\text{d}\left(x,\tau \right)+{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{2}\left({Q}_{T}\right)}{‖{W}_{x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}{‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}‖}_{{L}^{2}\left(0,T;{L}^{\infty }\left(\Omega \right)\right)}\\ \le {‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}{‖{W}_{x}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}+{C}_{\epsilon }{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}+\epsilon {‖{W}_{x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\\ \le {C}_{\epsilon }{‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+{C}_{\epsilon }{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{2}\left({Q}_{T}\right)}^{2}+2\epsilon {‖{W}_{x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\end{array}$

${‖{W}_{x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+{‖{W}_{xx}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\le CT{‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+CT{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2},$

${‖{W}_{x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}+{‖{W}_{xx}‖}_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}\le {\left(CT\right)}^{\frac{1}{2}}\left({‖\frac{\partial {\stackrel{^}{S}}_{1}}{\partial x}-\frac{\partial {\stackrel{^}{S}}_{2}}{\partial x}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}+{‖{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{2}‖}_{{L}^{\infty }\left(0,T;{L}^{2}\left(\Omega \right)\right)}\right)$ (29)

${\int }_{\Omega }{\psi }^{*}\left(S,{S}_{x}\right)\left(t,x\right)\text{d}x\le C，$ (30)

${\int }_{{Q}_{{T}_{e}}}{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}\text{d}\left(\tau ,x\right)\le C$ (31)

$\begin{array}{c}\frac{\text{d}}{\text{d}t}{\int }_{\Omega }{\psi }^{\ast }\left(S,{S}_{x}\right)\left(t,x\right)\text{d}x=\frac{\text{d}}{\text{d}t}{\int }_{\Omega }\left(\psi \left(S\right)+\frac{\nu }{2}{S}_{x}^{2}\right)\text{d}x\\ ={\int }_{\Omega }\left(\psi {\left(S\right)}_{S}{S}_{t}+\nu {S}_{x}{S}_{tx}\right)\text{d}x\\ ={\int }_{\Omega }\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right){S}_{t}\text{d}x\\ =-c{\int }_{\Omega }{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}\text{d}x\end{array}$

$\begin{array}{l}{\int }_{\Omega }{\psi }^{\ast }\left(S,{S}_{x}\right)\left(t,x\right)\text{d}x+c{\int }_{{Q}_{{T}_{e}}}{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}\text{d}\left(\tau ,x\right)\\ \le {\int }_{\Omega }{\psi }^{\ast }\left(S,{S}_{x}\right)\left(0,x\right)\text{d}x\end{array}$ (32)

${\int }_{\Omega }{\psi }^{\ast }\left(S,{S}_{x}\right)\left(0,x\right)\text{d}x={\int }_{\Omega }\psi \left(S\left(0,x\right)\right)+\frac{\nu }{2}{S}_{x}{\left(0,x\right)}^{2}\text{d}x\le C，$

$S\in {L}^{\infty }\left(0,{T}_{e};{L}^{4}\left(\Omega \right)\right)，$ (33)

${S}_{x}\in {L}^{\infty }\left(0,{T}_{e};{L}^{2}\left(\Omega \right)\right)。$ (34)

${\int }_{\Omega }{S}^{2}-2{S}^{3}+{S}^{4}+\frac{\nu }{2}{|{S}_{x}|}^{2}\text{d}x={\int }_{\Omega }{\psi }^{\ast }\left(S,{S}_{x}\right)\left(t,x\right)\text{d}x\le C，$

${\int }_{\Omega }{S}^{2}+{S}^{4}+\frac{\nu }{2}{|{S}_{x}|}^{2}\text{d}x\le C+{\int }_{\Omega }2{S}^{3}\text{d}x\le C+2{\int }_{\Omega }\left(\epsilon {S}^{4}+{C}_{\epsilon }\right)\text{d}x\le C+2{\int }_{\Omega }\epsilon {S}^{4}\text{d}x，$

${\int }_{\Omega }{S}^{2}+\left(1-2\epsilon \right){S}^{4}+\frac{\nu }{2}{|{S}_{x}|}^{2}\text{d}x\le C，$

${\int }_{{Q}_{{T}_{e}}}{S}_{xx}^{2}\text{d}\left(\tau ,x\right)\le C$ (35)

${\left({\psi }_{S}-\nu {S}_{xx}-{\psi }_{S}\right)}^{2}<2\left({\left({\psi }_{S}-\nu {S}_{xx}\right)}^{2}+{\psi }_{S}^{2}\right)，$

$\begin{array}{l}{\int }_{{Q}_{{T}_{e}}}{\left({\psi }_{S}-\nu {S}_{xx}-{\psi }_{S}\right)}^{2}\text{d}\left(\tau ,x\right)\\ \le 2{\int }_{{Q}_{{T}_{e}}}{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}+{\psi }_{S}^{2}\text{d}\left(\tau ,x\right)\\ \le 2{\int }_{{Q}_{{T}_{e}}}{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}\text{d}\left(\tau ,x\right)+{\int }_{{Q}_{{T}_{e}}}{\psi }_{S}^{2}\text{d}\left(\tau ,x\right)\end{array}$

$\kappa {\int }_{{Q}_{{T}_{e}}}{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}\text{d}\left(\tau ,x\right)\le {\int }_{{Q}_{{T}_{e}}}{\left(\psi {\left(S\right)}_{S}-\nu {S}_{xx}\right)}^{2}\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}\text{d}\left(\tau ,x\right)\le C.$

${\int }_{{Q}_{{T}_{e}}}{|{S}_{t}|}^{\frac{4}{3}}\text{d}\left(\tau ,x\right)\le C$ (36)

$\begin{array}{c}{\int }_{{Q}_{{T}_{e}}}{|{S}_{t}|}^{r}\text{d}\left(\tau ,x\right)={\int }_{{Q}_{{T}_{e}}}{|-c\left({\psi }_{S}\left(S\right)-\nu {S}_{xx}\right)\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}|}^{r}\text{d}\left(\tau ,x\right)\\ ={\int }_{{Q}_{{T}_{e}}}{|-c\left({\psi }_{S}\left(S\right)-\nu {S}_{xx}\right)|}^{r}{\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}}^{\frac{r}{2}}{\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}}^{\frac{r}{2}}\text{d}\left(\tau ,x\right)\\ \le {\left({\int }_{{Q}_{{T}_{e}}}{|-c\left({\psi }_{S}\left(S\right)-\nu {S}_{xx}\right)|}^{qr}{\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}}^{\frac{rq}{2}}\text{d}\left(\tau ,x\right)\right)}^{\frac{1}{q}}{\left({\int }_{{Q}_{{T}_{e}}}{\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}}^{\frac{rp}{2}}\text{d}\left(\tau ,x\right)\right)}^{\frac{1}{p}}\\ \le C{\left({\int }_{{Q}_{{T}_{e}}}{\stackrel{˜}{{|{\left({S}_{x}\right)}_{\eta }|}_{\kappa }}}^{\frac{r}{2-r}}\text{d}\left(\tau ,x\right)\right)}^{\frac{2-r}{2}}\end{array}$

${\left({S}_{x}^{*}\right)}_{\eta }\to {S}_{x}$，强收敛于空间 ${L}^{2}\left({Q}_{{T}_{e}}\right)$ 内， (37)

$\stackrel{˜}{|{\left({S}_{x}^{*}\right)}_{\eta }|}\to |{S}_{x}|$，强收敛于空间内， (38)

${\stackrel{˜}{|{\left({S}_{x}^{*}\right)}_{\eta }|}}_{\kappa }\to {|{S}_{x}|}_{\kappa }$，强收敛于空间 ${L}^{2}\left({Q}_{{T}_{e}}\right)$ 内， (39)

${‖{\left({S}_{x}^{*}\right)}_{\eta }-{S}_{x}‖}_{{L}^{2}\left({Q}_{{T}_{e}}\right)}\to 0,$

${‖\stackrel{˜}{|{\left({S}_{x}^{*}\right)}_{\eta }|}-|{S}_{x}|‖}_{{L}^{2}\left({Q}_{{T}_{e}}\right)}\le {‖{\left({S}_{x}^{*}\right)}_{\eta }-|{S}_{x}|‖}_{{L}^{2}\left({Q}_{{T}_{e}}\right)}\to 0.$

${‖{\stackrel{˜}{|{\left({S}_{x}^{*}\right)}_{\eta }|}}_{\kappa }-{|{S}_{x}|}_{\kappa }‖}_{{L}^{2}\left({Q}_{{T}_{e}}\right)}\to 0.$

2.2. 与 无关的先验估计

${‖{S}_{0}^{\kappa }-{S}_{0}‖}_{{H}^{1}\left(\Omega \right)}\to 0$$\kappa \to 0$(40)

${\int }_{0}^{t}{\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x\text{d}\tau \le C,$ (41)

${\int }_{0}^{t}{\int }_{\Omega }{\left({|{S}_{x}^{\kappa }|}_{\kappa }|{S}_{xx}^{\kappa }|\right)}^{\frac{4}{3}}\text{d}x\text{d}\tau \le C$ (42)

$\begin{array}{l}\frac{\text{d}}{2\text{d}t}{‖{S}_{x}^{\kappa }‖}_{{L}^{2}\left(\Omega \right)}^{2}+{\int }_{\Omega }\nu {|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x\\ ={\int }_{\Omega }{\psi }_{S}{|{S}_{x}^{\kappa }|}_{\kappa }{S}_{xx}^{\kappa }\text{d}x\le C{\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }|{S}_{xx}^{\kappa }|\text{d}x\\ \le \epsilon {\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x+{C}_{\epsilon }{\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\end{array}$

$\frac{\text{d}}{2\text{d}t}{‖{S}_{x}^{\kappa }‖}_{{L}^{2}\left(\Omega \right)}^{2}+{\int }_{\Omega }\frac{\nu }{2}{|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x\le {C}_{\nu }{\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x.$

${\int }_{0}^{t}{\int }_{\Omega }{|{\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{x}|}^{\frac{4}{3}}\text{d}x\text{d}\tau \le C,$ (43)

${\int }_{0}^{t}{‖{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{\infty }}^{\frac{8}{3}}\text{d}\tau \le C,$ (44)

$|{\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{x}|=|{\left({|y|}_{\kappa }^{2}\right)}_{x}|\le 2|y||{y}_{x}|\le 2{|y|}_{\kappa }|{y}_{x}|=2{|{S}_{x}^{\kappa }|}_{\kappa }|{S}_{xx}^{\kappa }|,$ (45)

${\int }_{0}^{t}{\int }_{\Omega }{|{\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{x}|}^{\frac{4}{3}}\text{d}x\text{d}\tau \le {\int }_{0}^{t}{\int }_{\Omega }{|2{|{S}_{x}^{\kappa }|}_{\kappa }|{S}_{xx}^{\kappa }||}^{\frac{4}{3}}\text{d}x\text{d}\tau \le {2}^{\frac{4}{3}}{\int }_{0}^{t}{\int }_{\Omega }{|{|{S}_{x}^{\kappa }|}_{\kappa }|{S}_{xx}^{\kappa }||}^{\frac{4}{3}}\text{d}x\text{d}\tau \le C.$

$\begin{array}{c}{\int }_{0}^{t}{\int }_{\Omega }{|{|{S}_{x}^{\kappa }|}_{\kappa }^{2}|}^{\frac{4}{3}}\text{d}x\text{d}\tau ={\int }_{0}^{t}{\int }_{\Omega }{|{\left({S}_{x}^{\kappa }\right)}^{2}+{\kappa }^{2}|}^{\frac{4}{3}}\text{d}x\text{d}\tau \\ \le {\int }_{0}^{t}{\int }_{\Omega }{|{\left({S}_{x}^{\kappa }\right)}^{2}|}^{\frac{4}{3}}\text{d}x\text{d}\tau +{\int }_{0}^{t}{\int }_{\Omega }{|{\kappa }^{2}|}^{\frac{4}{3}}\text{d}x\text{d}\tau \\ \le C{\int }_{0}^{t}{\int }_{\Omega }{|{\left({\left({S}_{x}^{\kappa }\right)}^{2}\right)}_{x}|}^{\frac{4}{3}}\text{d}x\text{d}\tau +C\cdot 1\\ \le C{\int }_{0}^{t}{\int }_{\Omega }{|{\left({\left({S}_{x}^{\kappa }\right)}^{2}+{\kappa }^{2}\right)}_{x}|}^{\frac{4}{3}}\text{d}x\text{d}\tau +C\\ \le {\int }_{0}^{t}{\int }_{\Omega }{|{\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{x}|}^{\frac{4}{3}}\text{d}x\text{d}\tau +C\le C\end{array}$

${\int }_{0}^{t}{‖{|{S}_{x}^{\kappa }|}_{\kappa }^{2}‖}_{{L}^{\infty }\left(\Omega \right)}^{\frac{8}{3}}\text{d}\tau \le C{\int }_{0}^{t}{‖{|{S}_{x}^{\kappa }|}_{\kappa }^{2}‖}_{{W}^{1,\frac{4}{3}}\left(\Omega \right)}^{\frac{4}{3}}\text{d}\tau \le C,$

${\int }_{0}^{t}{‖{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{\infty }\left(\Omega \right)}^{\frac{8}{3}}\text{d}\tau \le {\int }_{0}^{t}{‖{|{S}_{x}^{\kappa }|}_{\kappa }^{2}‖}_{{L}^{\infty }\left(\Omega \right)}^{\frac{4}{3}}\text{d}\tau \le C.$

${‖{S}_{x}^{\kappa }{S}_{xt}^{\kappa }‖}_{{L}^{1}\left(0,{T}_{e};{H}^{-2}\left(\Omega \right)\right)}\le C,$ (46)

${‖{\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{t}‖}_{{L}^{1}\left(0,{T}_{e};{H}^{-2}\left(\Omega \right)\right)}\le C$ (47)

$|\left({S}_{x}^{\kappa }{S}_{xt}^{\kappa },\varphi \right)|\le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)},$

$\begin{array}{c}\left({S}_{x}^{\kappa }{S}_{xt}^{\kappa },\varphi \right)={\int }_{0}^{{T}_{e}}{\int }_{\Omega }-{S}_{xx}^{\kappa }{S}_{t}^{\kappa }\varphi \text{d}x\text{d}\tau -{\int }_{0}^{{T}_{e}}{\int }_{\Omega }{S}_{x}^{\kappa }{S}_{t}^{\kappa }{\varphi }_{x}\text{d}x\text{d}\tau \\ =c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }{S}_{xx}^{\kappa }\left({\psi }_{S}-\nu {S}_{xx}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }\varphi \text{d}x\text{d}\tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }{S}_{x}^{\kappa }\left({\psi }_{S}-\nu {S}_{xx}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }{\varphi }_{x}\text{d}x\text{d}\tau \\ =:{I}_{1}+{I}_{2}\end{array}$ (48)

$\begin{array}{c}|{I}_{1}|=|c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }{S}_{xx}^{\kappa }\varphi {\psi }_{S}{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau -c\nu {\int }_{0}^{{T}_{e}}{\int }_{\Omega }{\left({S}_{xx}^{\kappa }\right)}^{2}{|{S}_{x}^{\kappa }|}_{\kappa }\varphi \text{d}x\text{d}\tau |\\ \le c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }|\varphi {\psi }_{S}||{S}_{xx}^{\kappa }|{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau +c\nu {\int }_{0}^{{T}_{e}}{\int }_{\Omega }|\varphi |{\left({S}_{xx}^{\kappa }\right)}^{2}{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau .\end{array}$ (49)

$c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }|\varphi {\psi }_{S}||{S}_{xx}^{\kappa }|{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau \le C{‖\varphi ‖}_{{L}^{\infty }\left({Q}_{{T}_{e}}\right)}{‖{S}_{xx}^{\kappa }{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{1}\left({Q}_{{T}_{e}}\right)}\le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}.$ (50)

$c\nu {\int }_{0}^{{T}_{e}}{\int }_{\Omega }|\varphi |{\left({S}_{xx}^{\kappa }\right)}^{2}{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau \le C{‖\varphi ‖}_{{L}^{\infty }\left({Q}_{{T}_{e}}\right)}{‖{\left({S}_{xx}^{\kappa }\right)}^{2}{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{1}\left({Q}_{{T}_{e}}\right)}\le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}.$ (51)

$\begin{array}{c}|{I}_{2}|=|c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }{S}_{x}^{\kappa }{\psi }_{S}{|{S}_{x}^{\kappa }|}_{\kappa }{\varphi }_{x}\text{d}x\text{d}\tau -c\nu {\int }_{0}^{{T}_{e}}{\int }_{\Omega }{S}_{x}^{\kappa }{|{S}_{x}^{\kappa }|}_{\kappa }{S}_{xx}^{\kappa }{\varphi }_{x}\text{d}x\text{d}\tau |\\ \le c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }|{\varphi }_{x}{\psi }_{S}{S}_{x}^{\kappa }{|{S}_{x}^{\kappa }|}_{\kappa }|\text{d}x\text{d}\tau +c\nu {\int }_{0}^{{T}_{e}}{\int }_{\Omega }|{\varphi }_{x}{S}_{xx}^{\kappa }{S}_{x}^{\kappa }|{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau \\ =:|{I}_{2,1}|+|{I}_{2,2}\end{array}$ (52)

$|{I}_{2,1}|=c{\int }_{0}^{{T}_{e}}{\int }_{\Omega }|{\varphi }_{x}{\psi }_{S}{S}_{x}^{\kappa }{|{S}_{x}^{\kappa }|}_{\kappa }|\text{d}x\text{d}\tau \le C{\int }_{0}^{{T}_{e}}{\int }_{\Omega }|{\varphi }_{x}{|{S}_{x}^{\kappa }|}_{\kappa }^{2}|\text{d}x\text{d}\tau \le C{‖{\varphi }_{x}‖}_{{L}^{\infty }\left({Q}_{{T}_{e}}\right)}{‖{|{S}_{x}^{\kappa }|}_{\kappa }^{2}‖}_{{L}^{1}\left({Q}_{{T}_{e}}\right)}\le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}$ (53)

$\begin{array}{c}|{I}_{2,2}|=c\nu {\int }_{0}^{{T}_{e}}{\int }_{\Omega }|{S}_{x}^{\kappa }{S}_{xx}^{\kappa }{\varphi }_{x}|{|{S}_{x}^{\kappa }|}_{\kappa }\text{d}x\text{d}\tau \\ \le c\nu {\int }_{0}^{{T}_{e}}{‖{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{\infty }\left(\Omega \right)}^{\frac{1}{2}}{‖{\varphi }_{x}‖}_{{L}^{\infty }\left(\Omega \right)}{\int }_{\Omega }{\left({|{S}_{x}^{\kappa }|}_{\kappa }\right)}^{\frac{1}{2}}|{S}_{xx}^{\kappa }||{S}_{x}^{\kappa }|\text{d}x\text{d}\tau \\ \le c\nu {\int }_{0}^{{T}_{e}}{‖{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{\infty }\left(\Omega \right)}^{\frac{1}{2}}{‖{\varphi }_{x}‖}_{{L}^{\infty }\left(\Omega \right)}{\left({\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x\right)}^{\frac{1}{2}}{\left({\int }_{\Omega }{|{S}_{x}^{\kappa }|}^{2}\text{d}x\right)}^{\frac{1}{2}}\text{d}\tau \\ \le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}{\int }_{0}^{{T}_{e}}{‖{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{\infty }\left(\Omega \right)}^{\frac{1}{2}}{\left({\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x\right)}^{\frac{1}{2}}\text{d}\tau \\ \le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}{\left({\int }_{0}^{{T}_{e}}{‖{|{S}_{x}^{\kappa }|}_{\kappa }‖}_{{L}^{\infty }\left(\Omega \right)}\text{d}\tau \right)}^{\frac{1}{2}}{\left({\int }_{0}^{{T}_{e}}{\int }_{\Omega }{|{S}_{x}^{\kappa }|}_{\kappa }{|{S}_{xx}^{\kappa }|}^{2}\text{d}x\text{d}\tau \right)}^{\frac{1}{2}}\\ \le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}\end{array}$ (54)

${f}_{\kappa }:=-c\left(\psi {\left({S}^{\kappa }\right)}_{S}-\nu {S}_{xx}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }.$ (55)

$\varphi \in {L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)$，用 ${\left({S}_{x}^{\kappa }\varphi \right)}_{x}$$\left({S}_{t}^{\kappa }-{f}_{\kappa }\right)$ 作内积，并且关于 $\left(t,x\right)$ 做积分，利用(45)可得

$\begin{array}{c}0={\left(\left({S}_{t}^{\kappa }-{f}_{\kappa }\right),{\left({S}_{x}^{\kappa }\varphi \right)}_{x}\right)}_{{Q}_{{T}_{e}}}\\ ={\left({S}_{t}^{\kappa },{\left({S}_{x}^{\kappa }\varphi \right)}_{x}\right)}_{{Q}_{{T}_{e}}}-{\left({f}_{\kappa },{\left({S}_{x}^{\kappa }\varphi \right)}_{x}\right)}_{{Q}_{{T}_{e}}}\\ =-{\int }_{{Q}_{{T}_{e}}}{S}_{tx}^{\kappa }{S}_{x}^{\kappa }\varphi \text{d}\left(\tau ,x\right)-{\left({f}_{\kappa },{S}_{xx}^{\kappa }\varphi \right)}_{{Q}_{{T}_{e}}}-{\left({f}_{\kappa },{S}_{x}^{\kappa }{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}\\ =-\frac{1}{2}{\left({\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{t},\varphi \right)}_{{Q}_{{T}_{e}}}-{\left({f}_{\kappa },{S}_{xx}^{\kappa }\varphi \right)}_{{Q}_{{T}_{e}}}-{\left({f}_{\kappa },{S}_{x}^{\kappa }{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}\end{array}$

$|\frac{1}{2}{\left({\left({|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)}_{t},\varphi \right)}_{{Q}_{{T}_{e}}}|=|-{\left({f}_{\kappa },{S}_{xx}^{\kappa }\varphi \right)}_{{Q}_{{T}_{e}}}-{\left({f}_{\kappa },{S}_{x}^{\kappa }{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}|\le {\int }_{{Q}_{{T}_{e}}}|{f}_{\kappa }{S}_{xx}^{\kappa }\varphi |\text{d}\left(\tau ,x\right)+{\int }_{{Q}_{{T}_{e}}}|{f}_{\kappa }{S}_{x}^{\kappa }{\varphi }_{x}|\text{d}\left(\tau ,x\right)$ (56)

${\int }_{{Q}_{{T}_{e}}}|{f}_{\kappa }{S}_{xx}^{\kappa }\varphi |\text{d}\left(\tau ,x\right)={\int }_{{Q}_{{T}_{e}}}|{S}_{t}^{\kappa }{S}_{xx}^{\kappa }\varphi |\text{d}\left(\tau ,x\right)\le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}.$ (57)

${\int }_{{Q}_{{T}_{e}}}|{f}_{\kappa }{S}_{x}^{\kappa }{\varphi }_{x}|\text{d}\left(\tau ,x\right)={\int }_{{Q}_{{T}_{e}}}|{S}_{t}^{\kappa }{S}_{x}^{\kappa }{\varphi }_{x}|\text{d}\left(\tau ,x\right)\le C{‖\varphi ‖}_{{L}^{\infty }\left(0,{T}_{e};{H}^{2}\left(\Omega \right)\right)}.$ (58)

2.3. 一维Alber-Zhu模型弱解的存在性

${‖{S}^{\kappa }-S‖}_{{L}^{\frac{4}{3}}\left({Q}_{{T}_{e}}\right)}\to 0$${S}_{x}^{\kappa }⇀{S}_{x}$${S}_{t}^{\kappa }⇀{S}_{t}$(59)

${‖{g}_{n}‖}_{{L}^{q}\left(\left(0,{T}_{e}\right)×\Omega \right)}\le C$${g}_{n}\to g$，a.e.在 $\left(0,{T}_{e}\right)×\Omega$ 内，

${S}_{x}^{\kappa }\to {S}_{x}$，a.e. in ${Q}_{{T}_{e}}$(60)

$\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }\to {S}_{x}$${|{S}_{x}^{\kappa }|}_{\kappa }\to |{S}_{x}|$，a.e. in ${Q}_{{T}_{e}}$(61)

$\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }⇀{S}_{x}$${|{S}_{x}^{\kappa }|}_{\kappa }⇀|{S}_{x}|$，弱收敛于空间 ${L}^{\frac{4}{3}}\left({Q}_{{T}_{e}}\right)$(62)

$\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }^{2}\to {S}_{x}|{S}_{x}|$，强收敛于空间 ${L}^{\frac{4}{3}}\left(0,{T}_{e};{L}^{2}\left(\Omega \right)\right)$(63)

${B}_{0}={W}^{1,\frac{4}{3}}\left(\Omega \right)$$B={L}^{2}\left(\Omega \right)$${B}_{1}={H}^{-2}\left(\Omega \right)$

$0={\left(\left({S}_{t}^{\kappa }-{f}_{\kappa }\right),-{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}={\int }_{{Q}_{{T}_{e}}}{S}_{tx}^{\kappa }\varphi \text{d}\left(\tau ,x\right)+{\left({f}_{\kappa },{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}},$

${\int }_{{Q}_{{T}_{e}}}{S}_{tx}^{\kappa }\varphi \text{d}\left(\tau ,x\right)\le {‖{f}_{\kappa }‖}_{{L}^{\frac{4}{3}}\left({Q}_{{T}_{e}}\right)}{‖{\varphi }_{x}‖}_{{L}^{4}\left({Q}_{{T}_{e}}\right)}\le C{‖\varphi ‖}_{{L}^{4}\left(0,{T}_{e};{W}^{1,4}\left(\Omega \right)\right)},$

${‖{S}_{xt}^{\kappa }‖}_{{L}^{\frac{4}{3}}\left(0,{T}_{e};{W}^{-1,\frac{4}{3}}\left(\Omega \right)\right)}\le C$。联结此式和(59)，第一项立即可得。

${\left({S}_{0}^{\kappa },\varphi \left(0\right)\right)}_{\Omega }\to {\left({S}_{0},\varphi \left(0\right)\right)}_{\Omega },$ (64)

${\left({S}^{\kappa },{\varphi }_{t}\right)}_{{Q}_{{T}_{e}}}\to {\left(S,{\varphi }_{t}\right)}_{{Q}_{{T}_{e}}},$ (65)

${\left({\int }_{0}^{{S}_{x}^{\kappa }}{|y|}_{\kappa }\text{d}y,{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}}\to {\left(\frac{1}{2}|{S}_{x}|{S}_{x},{\varphi }_{x}\right)}_{{Q}_{{T}_{e}}},$ (66)

${\left({\psi }_{S}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa },\varphi \right)}_{{Q}_{{T}_{e}}}\to {\left({\psi }_{S}|{S}_{x}|,\varphi \right)}_{{Q}_{{T}_{e}}},$ (67)

$\kappa \to 0$。(64)和(65)由(40)和(59)可得。为了证明(66)

${\int }_{0}^{{S}_{x}^{\kappa }}{|y|}_{\kappa }\text{d}y-\frac{1}{2}|{S}_{x}|{S}_{x}=\left({\int }_{0}^{{S}_{x}^{\kappa }}{|y|}_{\kappa }\text{d}y-\frac{1}{2}\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)+\frac{1}{2}\left(\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }^{2}-|{S}_{x}|{S}_{x}\right),$

${‖\frac{1}{2}\left(\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }^{2}-|{S}_{x}|{S}_{x}\right)‖}_{{L}^{\frac{4}{3}}\left(0,{T}_{e};{L}^{2}\left(\Omega \right)\right)}\to 0$

$\kappa \to 0$。对于第一项我们有

$\begin{array}{l}|\left({\int }_{0}^{{S}_{x}^{\kappa }}{|y|}_{\kappa }\text{d}y-\frac{1}{2}\mathrm{sgn}\left({S}_{x}^{\kappa }\right){|{S}_{x}^{\kappa }|}_{\kappa }^{2}\right)|\\ =|\left({\int }_{0}^{{S}_{x}^{\kappa }}{|y|}_{\kappa }\text{d}y-\frac{1}{2}\mathrm{sgn}\left({S}_{x}^{\kappa }\right)\left({\left({S}_{x}^{\kappa }\right)}^{2}+{\kappa }^{2}\right)\right)|\\ \le |\left({\int }_{0}^{{S}_{x}^{\kappa }}{|y|}_{\kappa }\text{d}y-\frac{1}{2}\mathrm{sgn}\left({S}_{x}^{\kappa }\right){\left({S}_{x}^{\kappa }\right)}^{2}\right)|+|\frac{1}{2}\mathrm{sgn}\left({S}_{x}^{\kappa }\right){\kappa }^{2}|\\ \le {\int }_{0}^{|{S}_{x}^{\kappa }|}|{|y|}_{\kappa }-|y||\text{d}y+{C}_{\kappa }\le \kappa |{S}_{x}^{\kappa }|+{C}_{\kappa }\end{array}$

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