该文研究的Canh-Hilliard-Brinkman系统如下

(1.1) $ \begin{equation} - \nu \Delta {{\bf u}} + \eta {{\bf u}} = - \nabla p - \gamma \varphi \nabla \mu , \end{equation} $

(1.2) $ \begin{equation} \nabla \cdot {{\bf u}} = 0, \end{equation} $

(1.3) $ \begin{equation} {\varphi _t} + {{\bf u}} \cdot \nabla \varphi - {\rm M}\Delta \mu = 0, \end{equation} $

(1.4) $ \begin{equation} \mu = - \varepsilon \Delta \varphi + \frac{1}{\varepsilon }f(\varphi ), \end{equation} $

其中$ (x, t) \in {\Omega _{{R^ + }}} = \Omega \times \left[ {0, + \infty } \right) $; $ \Omega $是一个具有光滑边界$ \partial \Omega $的三维有界区域; $ {\mathbb{R}^+ } = \left[ {0, + \infty } \right) $; $ \gamma > 0 $代表表面张力参数; $ \nu {\rm{ > 0}} $表示粘性系数; $ \eta > 0 $表示流体渗透性; $ p $是流体压力; $ {\rm M} > 0 $表示流动性; $ \varepsilon > 0 $与界面厚度有关; $ f $是非线性项. 在此系统中, $ {{\bf u}} $代表流体速度; $ \varphi $代表流体浓度差; $ \mu $代表所谓的化学势.

方程(1.1)–(1.4) 受以下初边值条件约束

(1.5) $ \begin{equation} {{\bf u}} = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}(x, t) \in \partial \Omega \times (0, T) , \end{equation} $

(1.6) $ \begin{equation} {\partial _{{\bf n}}}\varphi = {\partial _{{\bf n}}}\mu = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x, t) \in \partial \Omega \times (0, T), \end{equation} $

(1.7) $ \begin{equation} \varphi (x, 0) = {\varphi _0}(x), \end{equation} $

其中$ \partial \Omega $是$ \Omega $的边界; $ {\varphi _0}(x):\Omega \to \mathbb{R} $是一个已知的函数; $ {{\bf n}} $代表$ \partial \Omega $的外法向量.

Brinkman方程可以看作是油砂两种材料混合的模型. 在这个模型中, 沙子被视为坚硬的固体, 油的粘度和摩擦系数与压力有关. 因此, 当遇到在多孔介质中流动的压力差很大时, 这个模型是极其重要的, 通常会简化这个模型. Brinkman最初为了描述某一多孔介质中的流动, 通过改进Darcy方程

$ {{\bf u}} = - \nabla p, $

从而得到了Brinkman方程^[1]

$ - \nu \Delta {{\bf u}} + {{\bf u}} = - \nabla p. $

对流Cahn-Hilliard方程^[2]通过表面张力项$ \gamma \varphi \nabla \mu $与Brinkman方程耦合, 组成了系统(1.1)–(1.4). 因此, 系统(1.1)–(1.4) 被称作Cahn-Hilliard-Brinkman(CHB)系统, 该系统是描述多相流体行为的扩散界面模型. 在文献[3-5]中, 作者从数值和解析等角度分析了Cahn-Hilliard-Navier-Stokes系统. 当$ \nu = 0 $, 该系统变成Cahn-Hilliard-Hele-Shaw系统. 在文献[6] 中, Cahn-Hilliard-Hele-Shaw系统已经在数值计算和解的适定性方面得到了证明. 在文献[7] 中, 研究了Cahn-Hilliard-Darcy系统的适定性. 冯小兵等人研究了Cahn-Hilliard-Darcy系统^[8]的适定性与数值计算. 当$ v = \gamma = 0 $, 方程$ (1.1) $简化成了Darcy方程^[9]. 方程$ (1.2) $表明流体是不可压缩的. 当对流项消失时, 方程(1.3)–(1.4) 为Cahn-Hilliard方程^{[10, 11]}. 当$ v = 0 $时, 改变初边值条件, 系统可变成Cahn-Hilliard-Darcy系统.

与许多扩散界面模型^{[12, 13]}一样, CHB系统也是耗散系统. 定义Cahn-Hilliard能量泛函

$ J(\varphi ) = \int_\Omega {\left[ {\frac{1}{2}{{\left| {\nabla \varphi } \right|}^2} + F(\varphi )} \right]{\kern 1pt} {\rm d}x}. $

一般来说, 对于耗散演化方程解的渐近性态, 可以用全局吸引子来描述. 动力学性质如解的全局渐近性和全局吸引子的存在性对于研究二元系统的相分离具有重要意义, 它们保证了相变的稳定性, 为相变动力学研究提供了数学基础. 因此, 在这个方面有许多的研究成果, 例如文献[14-16]. 在文献[17] 中, 研究了CHB系统在$ {H^1}(\Omega ) $中全局吸引子的存在性, 利用Lojasiewicz-Simon不等式建立了单个平衡的弱解的收敛性, 并给出了其收敛速度, 同时得到CHB系统的每个轨道$ \varphi (t) $将收敛到固定解$ \phi $, 该固定解满足

$ - \Delta \phi + f(\phi ) = \frac{1}{{\left| \Omega \right|}}\int_\Omega {f(\phi ){\rm d}x}. $

在文献[18] 中, 当$ f(s) = {s^3} - s $时, 证明了CHB系统在$ {H^s}(\Omega )(s = 1, 2, 3, 4) $中全局吸引子的存在性; 同时在半群可微的情况下, 通过计算Lyapunov指数, 用体积收缩的方法证明了有限分数维全局吸引子的存在性. 在文献[19] 中, 证明了二维Cahn-Hilliard-Navier-Stokes系统的全局吸引子的存在性.

本文对文献[18] 中的非线性条件进行了改进. 当$ {\rm M} $, $ \eta $以及$ \nu $为常数时, 得到了系统(1.1)–(1.7) 的一些结果, 主要研究了CHB系统弱解的适定性, 利用Galerkin方法得到了弱解的存在性, 并得到了弱解的唯一性; 最后得到了弱解的渐近估计, 并利用半群理论和紧嵌入定理, 得到了在$ {H^s}(\Omega ){\kern 1pt} {\kern 1pt} (s = 1, 2, 3, 4) $中全局吸引子的存在性.

令$ F(s) = \int_0^s {f(t){\rm d}t} $, 对非线性项进行如下假设

(1.8) $ \begin{equation} f(s)s \ge {c_1}F(s) - {c_2}, F(s) \ge - {c_3}, {c_1} > 0, {c_2}, {c_3} \ge 0, \end{equation} $

(1.9) $ \begin{equation} \left| {f(\ddot{s})} \right| \le c(1 + \left| s \right|), \end{equation} $

(1.10) $ \begin{equation} {f(\dot{s})} \ge - {c_0}, {c_0} \ge 0, \end{equation} $

(1.11) $ \begin{equation} \left| {f(s)} \right| \le {c_4}(1 + {\left| s \right|^3}). \end{equation} $

特别地, $ f(s) = {s^3} - s, F(s) = \frac{1}{4}{({s^2} - 1)^2} $满足以上假设条件, 因此该文进一步改进了文献[18] 的结果.

本文用$ \left\| \cdot \right\| $代表范数$ {\left\| \cdot \right\|_{{L^2}}} $, 令$ {I_0} = \int_\Omega {\varphi {\rm d}x} $. 特别地, 设定$ M = \varepsilon = v = \eta = \gamma = 1 $.

首先, 定义具有初边值条件(1.5)–(1.7) 的Cahn-Hilliard-Brinkman系统的弱解.

定义2.1  假设$ {\varphi _0} \in {H^1}(\Omega ) $, 若$ (\varphi , {{\bf u}}) $满足以下条件

$ \varphi \in {L^\infty }((0, T);{H^1}(\Omega )) \cap {L^2}((0, T);{H^3}(\Omega )), $

$ \mu \in {L^2}((0, T);{H^1}(\Omega )), $

$ {{\bf u}} \in {L^2}((0, T);{H^1}(\Omega )), $

$ {\varphi _t} \in {L^2}((0, T);{({W^{1, 3}}(\Omega ))^*}) \cap {L^{\frac{8}{5}}}((0, T);{H^1}{(\Omega )^*}), $

$ (\nabla {{\bf u}}, \nabla v) + ({{\bf u}}, v) = - (\nabla p, v) - (\varphi \nabla \mu , v), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall v \in H^1(\Omega ), $

$ \langle {\frac{\rm d}{{{\rm d}t}}\varphi (t), v} \rangle + (\varphi (t){{\bf u}}(t), \nabla v) = - (\nabla \mu , \nabla v), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall v \in {H^1}(\Omega ), $

则$ (\varphi , {{\bf u}}) $称为系统$ \rm (1.1)–(1.7) $的弱解.

定理2.1  假设$ {\varphi _0} \in {H^1}(\Omega ) $, 则Cahn-Hilliard-Brinkman系统存在定义2.1中的弱解$ (\varphi , {{\bf u}}) $; 对任意的$ T > 0 $, $ t \le T $, 有以下估计

$ {\left\| {\varphi (t)} \right\|_{{H^1}}} \le c, \; \; \; \int_0^t {\left\| {\nabla \mu (s)} \right\|} {\rm d}s \le c, \; \; \; \int_0^t {\left\| {{\bf u}} \right\|_{{H^1}}^2} {\rm d}s \le c, $

$ \int_0^t {\left\| \varphi \right\|_{{H^3}}^2 {\rm d}s \le c}, \; \; \; \; \int_0^t {{{\left\| {\Delta \varphi (s)} \right\|}^4}{\rm d}s \le c}. $

证  用Galerkin方法得到了全局弱解的存在性. 考虑满足Neumann边界的$ {\partial _n}w = 0 $特征值问题$ - \Delta w = \lambda w $, 存在特征值$ \left\{ {{\lambda _n}} \right\} $及其对应的特征函数$ \left\{ {{w_n}} \right\} $, 满足$ \left\{ {{\lambda _n}} \right\} $是非减序列, 且当$ n \to \infty $, 其极限趋于无穷. 序列$ \left\{ {{w_n}} \right\} $是$ {L^2}(\Omega ) $中的一列完备的标准正交基.

设$ {W_n} $是由$ \left\{ {{w_n}} \right\}_{n = 1}^k $所张成的有限维空间. 能找到$ {\varphi _n}(t, x) = \sum\limits_{i = 1}^n {{\varphi _{ni}}(t){w_i}(x)} $, $ {{{\bf u}}_n}(t, x) = \sum\limits_{i = 1}^n {{{{\bf u}}_{ni}}(t){w_i}(x)} $以及对应的$ {\mu _n} \in {W_n} $, $ {p_n} \in {W_n} $满足: 对于任意的$ v \in {H^1}(\Omega ) $, 有

$ (\nabla {{\bf u}}, \nabla v) + ({{{\bf u}}_n}, v) = - (\nabla {p_n}, v) - ({\varphi _n}\nabla {\mu _n}, v), $

$ \langle {\frac{\rm d}{{{\rm d}t}}{\varphi _n}(t), v} \rangle + ({\varphi _n}(t){{{\bf u}}_n}(t), \nabla v) = - (\nabla {\mu _n}, \nabla v), $

$ {\mu _n} = - \Delta {\varphi _n} + f({\varphi _n}), $

$ {\varphi _n}(0) = {\varphi _{n, 0}}. $

$ {{{\bf u}}_n} = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x, t) \in \partial \Omega \times (0, T), $

$ {\partial _{{\bf n}}}{\varphi _n} = {\partial _{{\bf n}}}{\mu _n} = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x, t) \in \partial \Omega \times (0, T). $

其中$ {\varphi _{n, 0}} $是$ {\varphi _0} $在$ {W_n} $上的投影.

在第一个方程中取$ v = {{{\bf u}}_n} $, 在第二个方程中取$ v = {\mu _n} $, 有

$ \frac{\rm d}{{{\rm d}t}}\int_\Omega {(\frac{1}{2}} {\left| {\nabla {\varphi _n}} \right|^2} + F({\varphi _n})){\rm d}x + {\left\| {\nabla {\mu _n}} \right\|^2} + \left\| {{{{\bf u}}_n}} \right\|_{{H^1}}^2= 0, $

进一步得到

$ \int_\Omega {(\frac{1}{2}{{\left| {\nabla {\varphi _n}} \right|}^2} + F({\varphi _n})){\rm d}x + \int_0^T {{{\left\| {\nabla {\mu _n}} \right\|}^2}{\rm d}t} + \int_0^T {\left\| {{\bf u}} \right\|} _{{H^1}}^2} {\rm d}t \le c. $

因此$ {\varphi _n} $在$ {L^\infty }((0, T);{H^1}(\Omega )) $中一致有界; $ {\mu _n} $在$ {L^2}((0, T);{H^1}(\Omega )) $中一致有界; $ {{{\bf u}}_n} $在$ {L^2}((0, T);{H^1}(\Omega )) $中一致有界.

第三个方程乘以$ \nabla \Delta {\varphi _n} $, 有

$ \begin{eqnarray*} {\left\| {\nabla \Delta {\varphi _n}} \right\|^2}& \le& c{\left\| {\nabla {\mu _n}} \right\|^2} + \frac{1}{4}{\left\| {\nabla \Delta {\varphi _n}} \right\|^2} + {\left\| {f({\dot{\varphi} _n})} \right\|_{{L^3}}}{\left\| {\nabla {\varphi _n}} \right\|_{{L^6}}}\left\| {\nabla \Delta {\varphi _n}} \right\|\\ &\le& c{\left\| {\nabla {\mu _n}} \right\|^2} + \frac{1}{4}{\left\| {\nabla \Delta {\varphi _n}} \right\|^2} + C{\left\| {\nabla {\varphi _n}} \right\|^{\frac{1}{2}}}{\left\| {\nabla \Delta {\varphi _n}} \right\|^{\frac{3}{2}}}, \end{eqnarray*} $

得到$ {\varphi _n} \in {L^2}((0, T);{H^3}(\Omega )) $.

第二个方程乘以$ v \in {W^{1, 3}} $, 有

$ \begin{eqnarray*} \langle {{{({\varphi _n})}_t}, v} \rangle & =& \left( {{{{\bf u}}_n} \cdot {\varphi _n}, \nabla v} \right) - (\nabla {\mu _n}, \nabla v)\\ &\le &\left\| {{{{\bf u}}_n}} \right\|{\left\| {{\varphi _n}} \right\|_{{L^6}}}{\left\| {\nabla v} \right\|_{{L^3}}} + \left\| {\Delta {\mu _n}} \right\|\left\| {\nabla v} \right\|\\ &\le &c(\left\| {{{{\bf u}}_n}} \right\|\left\| {\nabla {\varphi _n}} \right\| + \left\| {\Delta {\mu _n}} \right\|){\left\| {\nabla v} \right\|_{{L^3}}}, \end{eqnarray*} $

可推出$ {\varphi _t} \in {L^2}((0, T);{({W^{1, 3}}(\Omega ))^*}) $.

对任意的$ \varphi \in {L^{\frac{8}{3}}}((0, T);{H^1}(\Omega )) $, 有

$ \left| {\int_0^T {\int_\Omega {{{{\bf u}}_n} \cdot \nabla {\varphi _n}\varphi {\rm d}x{\rm d}t} } } \right| \le {\bigg({\int_0^T {\left\| {{{{\bf u}}_n}} \right\|} ^2}{\rm d}t\bigg)^{\frac{1}{2}}} {\bigg(\int_0^T {\left\| {{\varphi _n}} \right\|} _{{L^\infty }}^8{\rm d}t\bigg)^{\frac{1}{8}}} {\bigg(\int_0^T {\left\| \varphi \right\|_{{H^1}}^{\frac{8}{3}}}{\rm d}t\bigg )^{\frac{3}{8}}}, $

可由插值不等式推出

$ \left| {\int_0^T {\int_\Omega {{{{\bf u}}_n} \cdot \nabla {\varphi _n}} } \varphi {\rm d}x{\rm d}t} \right| \le C. $

因此, 得到

$ {{{\bf u}}_n} \cdot \nabla {\varphi _n} \in {L^{\frac{8}{5}}}((0, T);{H^1}{(\Omega )^*}), \; \; \; \; {\varphi _t} \in {L^{\frac{8}{5}}}((0, T);{H^1}{(\Omega )^*}). $

对任意的$ v \in {L^{\frac{8}{3}}}((0, T);{L^2}(\Omega ) $, 有

$ \begin{eqnarray*} &&\left| {\int_0^T {({\mu _n}\nabla {\varphi _n}) \cdot v{\rm d}t} } \right|\\ & \le& c{\bigg(\int_0^T {\left\| {{\mu _n}} \right\|_{{H^1}}^2}{\rm d}t\bigg )^{\frac{1}{2}}} {\bigg(\int_0^T {\left\| {\nabla {\varphi _n}} \right\|_{{L^3}}^8} {\rm d}t\bigg)^{\frac{1}{8}}} {\bigg(\int_0^T {{{\left\| v \right\|}^{\frac{8}{3}}}} {\rm d}t\bigg)^{\frac{3}{8}}}\\ & \le& c{\bigg(\int_0^T {\left\| {{\mu _n}} \right\|_{{H^1}}^2}{\rm d}t\bigg )^{\frac{1}{2}}} {\bigg(C\int_0^T {({{\left\| {\nabla {\varphi _n}} \right\|}^{\frac{3}{4}}}{{\left\| {\nabla \Delta {\varphi _n}} \right\|}^{\frac{1}{4}}}} + \left\| {\nabla {\varphi _n}} \right\|)^8}{\rm d}t{\bigg)^{\frac{1}{8}}} {\bigg(\int_0^T {{{\left\| v \right\|}^{\frac{8}{3}}}} {\rm d}t\bigg)^{\frac{3}{8}}} \\ & \le& C. \end{eqnarray*} $

因此, 可推出

$ {\mu _n}\nabla {\varphi _n} \in {L^{\frac{8}{5}}}((0, T);{L^2}(\Omega )). $

通过上述估计, 得到

$ {在 {L^2}((0, T);{H^3}(\Omega )) 中, {\varphi _n} 弱收敛到 \varphi ;} \\ {在 {L^\infty }((0, T);{H^1}(\Omega )) 中, {\varphi _n} 强收敛到 \varphi ;} \\ {在 {L^2 }((0, T);{H^1}(\Omega )) 中, {\mu _n} 弱收敛到 \mu ;} \\ {在 {L^2 }((0, T);{H^1}(\Omega )) 中, {{{\bf u}}_n} 弱收敛到 {{\bf u}} ;} \\ {在 {L^{\frac{8}{5}}}((0, T);{H^1}{(\Omega )^*}) 中, {\rm{ }}{({\varphi _n})_t} 弱收敛到 {\varphi _t} .} $

对于$ f({\varphi _n}) $的收敛性: 对任意的$ v \in {H^1}(\Omega ) $, 有

$ \int_\Omega {f({\varphi _n})v{\rm d}x} \le \int_\Omega {(1 + {{\left| {{\varphi _n}} \right|}^3}} )v{\rm d}x \le c\left\| {{\varphi _n}} \right\|_{{L^6}}^3\left\| v \right\| \le C\left\| v \right\|. $

通过Lebesgue控制收敛定理, 可得到$ f({\varphi _n}) $的收敛性.

对于弱解形式的非线性项的收敛性

$ \int_0^T {({\varphi _n}\nabla {\mu _n}, v){\rm d}t}, $

以及

$ \int_0^T {({\varphi _n}{{{\bf u}}_n}, \nabla v){\rm d}t}. $

首先有

$ ({\varphi _n}\nabla {\mu _n}, v) - (\varphi \nabla \mu , v) = (({\varphi _n} - \varphi )\nabla {\mu _n}, v) + (\varphi (\nabla {\mu _n} - \nabla \mu ), v), $

对方程右边第一项

$ \left| {\int_0^T {(({\varphi _n} - \varphi )\nabla {\mu _n}, v)} {\rm d}t} \right| \le {\left\| {{\varphi _n} - \varphi } \right\|_{{L^2}(0, T;{L^3})}}{\left\| {{\mu _n}} \right\|_{{L^2}(0, T;{H^1})}}{\left\| v \right\|_{{H^1}}} \to 0, $

由于$ {\varphi _n} \in {L^2}((0, T);{H^3}(\Omega )) \subset {L^2}((0, T);{L^\infty }(\Omega )) $以及$ {\mu _n} \in {L^2}((0, T);{H^1}(\Omega )) $, 可推得

$ \int_0^T {(\varphi (\nabla {\mu _n} - \nabla \mu ), v){\rm d}t \to 0}. $

因此

$ \int_0^T {({\varphi _n}\nabla {\mu _n} - \varphi \nabla \mu , v){\rm d}t \to 0}. $

接下来探讨另一非线性项的收敛性

$ ({\varphi _n}{{{\bf u}}_n} - \varphi {{\bf u}}, \nabla v) = (({\varphi _n} - \varphi ){{{\bf u}}_n}, \nabla v) + (\varphi ({{{\bf u}}_n} - {{\bf u}}), \nabla v). $

由于在$ {L^2}((0, T);{H^1}(\Omega )) $中$ {{{\bf u}}_n} $弱收敛到$ {{\bf u}} $, 因此

$ \int_0^T {(\varphi ({{{\bf u}}_n} - {{\bf u}}} ), \nabla v){\rm d}t \to 0, $

由空间嵌入定理, 推得

$ {\left| {\int_0^T {(({\varphi _n} - \varphi ){{{\bf u}}_n}, \nabla v){\rm d}t} } \right|} { \le \left\| {\nabla v} \right\|{{\left\| {{{{\bf u}}_n}} \right\|}_{{L^2}(0, T;{H^1})}}{{\left\| {{\varphi _n} - \varphi } \right\|}_{{L^2}(0, T;{L^3})}} \to 0}. $

因此, 非线性项的收敛性得以证明.

下面给出一些估计.

将方程$ (1.3) $乘以$ v = 1 $, 进行分部积分, 可得

$ \int_\Omega {\varphi (t){\rm d}x = \int_\Omega {{\varphi _0}{\rm d}x = {I_0}} } . $

分别将方程$ (1.1) $, $ (1.3) $乘以$ {{\bf u}} $, $ \mu $, 并各自进行积分, 将得到的积分结果相加, 有

$ \frac{\rm d}{{{\rm d}t}} \bigg(\int_\Omega \Big( {\frac{1}{2}{{\left| {\nabla \varphi } \right|}^2} + } F(\varphi )\Big){\rm d}x\bigg) + {\left\| {\nabla \mu } \right\|^2} + {\left\| {\nabla {{\bf u}}} \right\|^2} + {\left\| {{\bf u}} \right\|^2} = 0, $

在区间$ ({\rm{0}}, {\rm{t}}), {\kern 1pt} {\kern 1pt} \forall {\kern 1pt} {\rm{t}} \ge {\rm{0}} $进行积分, 有

$ \int_\Omega \Big( {\frac{1}{2}{{\left| {\nabla \varphi } \right|}^2} + } F(\varphi )\Big){\rm d}x \le \int_\Omega \Big( {\frac{1}{2}{{\left| {\nabla {\varphi _0}} \right|}^2} + F({\varphi _0})\Big){\rm d}x \le c} . $

结合假设条件$ (1.8) $, 有

(2.1) $ \begin{equation} \left\| {\nabla \varphi (t)} \right\| \le \left\| {\nabla {\varphi _0}} \right\| + 2\int_\Omega {F({\varphi _0}){\rm d}x \le c}, \end{equation} $

$ \int_0^t {\left\| {\nabla \mu (s)} \right\|} {\rm d}s \le c, $

$ \int_0^t {\left\| {{\bf u}} \right\|_{{H^1}}^2} {\rm d}t \le c. $

将方程$ (1.4) $乘以$ \nabla \Delta \varphi $, 可得

$ \begin{eqnarray*} {\left\| {\nabla \Delta \varphi } \right\|^2} &\le &c{\left\| {\nabla \mu } \right\|^2} + \frac{1}{4}{\left\| {\nabla \Delta \varphi } \right\|^2} + {\left\| {f(\dot{\varphi} )} \right\|_{{L^3}}}{\left\| {\nabla \varphi } \right\|_{{L^6}}}\left\| {\nabla \Delta \varphi } \right\|\\ &\le &c{\left\| {\nabla \mu } \right\|^2} + \frac{1}{4}{\left\| {\nabla \Delta \varphi } \right\|^2} + C{\left\| {\nabla \varphi } \right\|^{\frac{1}{2}}}{\left\| {\nabla \Delta \varphi } \right\|^{\frac{3}{2}}}, \end{eqnarray*} $

推出

$ \int_0^t {\left\| {\nabla \Delta \varphi } \right\|_{{H^3}}^2} {\rm d}t \le c. $

利用插值不等式, 有

$ \int_0^t {{{\left\| {\Delta \varphi (s)} \right\|}^4}{\rm d}s} \le \int_0^t {{{\left\| {\nabla \varphi (s)} \right\|}^2}{{\left\| {\nabla \Delta \varphi (s)} \right\|}^2}{\rm d}s} \le c\int_0^t {{{\left\| {\nabla \Delta \varphi (s)} \right\|}^2}} {\rm d}s \le C. $

证毕.

下面证明弱解的唯一性.

定理2.2  假设$ {\varphi _0} \in {H^1}(\Omega ) $, 则系统$ \rm (1.1)–(1.7) $存在唯一弱解.

证  设$ ({\varphi _1}, {{{\bf u}}_1}) $和$ ({\varphi _2}, {{{\bf u}}_2}) $是两个不同的解, 令$ \varphi = {\varphi _1} - {\varphi _2} $, $ {{\bf u}} = {{{\bf u}}_1} - {{{\bf u}}_2} $, $ \mu = {\mu _1} - {\mu _2} $以及$ p = {p_1} - {p_2} $. 将其满足的方程组结合, 得到

$ {\varphi _t} + {{{\bf u}}_1} \cdot \nabla \varphi + {{\bf u}} \cdot \nabla {\varphi _2} = \Delta \mu , $

$ - \Delta {{\bf u}} + {{\bf u}} = - \nabla p - ({\varphi _1}\nabla \mu - \varphi \nabla {\mu _2}), $

$ \mu = - \Delta \varphi + f({\varphi _1}) - f({\varphi _2}). $

将第一个方程乘以$ - \Delta \varphi $, 第二个方程乘以$ {{\bf u}} $, 并对其进行积分, 将其结果结合, 得到

$ \begin{eqnarray*} &&\frac{\rm d}{{{\rm d}t}}\frac{1}{2}{\left\| {\nabla \varphi } \right\|^2} + {\left\| {\nabla \Delta \varphi } \right\|^2} + {\left\| {\nabla {{\bf u}}} \right\|^2} + {\left\| {{\bf u}} \right\|^2}\\ & =& - \int_\Omega {{{{\bf u}}_1}\varphi \nabla \Delta \varphi{\rm d}x - \int_\Omega {{\varphi _2}} \nabla \Delta \varphi {{\bf u}}{\rm d}x - \int_\Omega {(f({\varphi _1}) - f({\varphi _2})){\Delta ^2}\varphi {\rm d}x} } \\ &\le& {\left\| {{{{\bf u}}_1}} \right\|_{{L^3}}}{\left\| \varphi \right\|_{{L^6}}}\left\| {\nabla \Delta \varphi } \right\| + {\left\| {{\varphi _2}} \right\|_{{L^6}}}\left\| {\nabla \Delta \varphi } \right\|{\left\| {{\bf u}} \right\|_{{L^3}}} - \int_\Omega {f(\dot{\varepsilon} )} \varphi {\Delta ^2}\varphi {\rm d}x\\ &\le &c{\left\| {{{{\bf u}}_1}} \right\|_{{H^1}}}\left\| {\nabla \varphi } \right\|\left\| {\nabla \Delta \varphi } \right\| + \left\| {\nabla {\varphi _2}} \right\|\left\| {\nabla \Delta \varphi } \right\|{\left\| {{\bf u}} \right\|_{{H^1}}} + {c_0}\int_\Omega {\varphi {\Delta ^2}\varphi {\rm d}x} \\ & \le &c{\left\| {{{{\bf u}}_1}} \right\|_{{H^1}}}\left\| {\nabla \varphi } \right\|\left\| {\nabla \Delta \varphi } \right\| + \left\| {\nabla {\varphi _2}} \right\|\left\| {\nabla \Delta \varphi } \right\|{\left\| {{\bf u}} \right\|_{{H^1}}} + {c_0}\left\| {\nabla \varphi } \right\|\left\| {\nabla \Delta \varphi } \right\|. \end{eqnarray*} $

将其与上述估计, Young不等式以及Gronwall不等式结合, 得到

$ \left\| {\nabla \varphi } \right\| \le {e^{ct}}\left\| {\nabla {\varphi _0}} \right\|, $

即

$ {\varphi _1} = {\varphi _2}. $

将结果带回以上方程, 可得

$ {\left\| {\nabla {{\bf u}}} \right\|^2} + {\left\| {{\bf u}} \right\|^2} \le 0, $

因此

$ {{{\bf u}}_1} = {{{\bf u}}_2}. $

证毕.

定理3.1  假设$ {\varphi _0} \in {H^1}(\Omega ) $, $ (\varphi , {{\bf u}}) $是系统$ \rm (1.1)–(1.7) $的弱解, 则存在$ T>0 $, 使得当$ t \ge T $时, 满足

$ {\left\| \varphi \right\|_{{H^s}}} \le c, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s = 1, 2, 3, 4. $

证  由方程$ (1.4) $, 得到

$ \int_\Omega {\mu {\rm d}x = \int_\Omega {f(\varphi ){\kern 1pt} {\rm d}x} }. $

通过假设条件$ (1.11) $, 得到

(3.1) $ \begin{equation} \int_\Omega {\mu {\kern 1pt} {\kern 1pt} {\rm d}x} \le c(1 + {\int_\Omega {\left\| {\nabla \varphi } \right\|} ^3}{\rm d}x) \le C. \end{equation} $

将方程$ (1.1) $, $ (1.3) $, $ (1.4) $分别乘以$ {{\bf u}}, \mu , \varphi $, 并各自积分, 将其积分结果相加, 有

(3.2) $ \begin{equation} \frac{\rm d}{{{\rm d}t}} \bigg(\int_\Omega \Big( {\frac{1}{2}{{\left| {\nabla \varphi } \right|}^2} + F(\varphi )\Big){\rm d}x\bigg) + {{\left\| {\nabla \mu } \right\|}^2} + {{\left\| {\nabla {{\bf u}}} \right\|}^2} + {{\left\| {{\bf u}} \right\|}^2} + {{\left\| {\nabla \varphi } \right\|}^2} + \int_\Omega {f(\varphi )\varphi {\rm d}x} } = \int_\Omega {\varphi \mu {\rm d}x}. \end{equation} $

注意, 利用Poincar'e不等式, 推出

$ \begin{eqnarray*} \int_\Omega {\varphi \mu {\rm d}x} &= &\int_\Omega {\varphi (\mu - \frac{1}{{\left| \Omega \right|}}} \int_\Omega {\mu {\rm d}x + } \frac{1}{{\left| \Omega \right|}}\int_\Omega {\mu {\rm d}x){\rm d}x} \\ &\le& \left\| {\mu - \frac{1}{{\left| \Omega \right|}}\int_\Omega {\mu {\rm d}x} } \right\|\left\| \varphi \right\| + {I_0}\int_\Omega {\mu {\rm d}x} \\ & \le &{c_1}{\left\| {\nabla \mu } \right\|^2} + \frac{1}{2}{\left\| {\nabla \varphi } \right\|^2} + C{I_0}. \end{eqnarray*} $

在区间$ (0, t) $对方程$ (3.2) $积分, 通过Gronwall不等式, 得到

$ {\int_\Omega {\left| {\nabla \varphi } \right|} ^2} + 2F(\varphi ){\kern 1pt} {\rm d}x \le J({\varphi _0}){e^{ - t}} + c, $

因此, 存在$ {T_1}>0 $, 当$ t \ge {T_1} $, 有

(3.3) $ \begin{equation} \left\| \varphi \right\|_{{H^1}}^2 \le {m_1}, \end{equation} $

(3.4) $ \begin{equation} \int_t^{t + 2} {{{\left\| {\nabla \mu (s)} \right\|}^2} + 2\left\| {{{\bf u}}(s)} \right\|_{{H^1}}^2} {\kern 1pt} {\rm d}s \le {m_2}. \end{equation} $

将方程$ (1.4) $乘以$ - {\Delta ^2}\varphi $, 有

$ (\nabla \mu , \nabla \Delta \varphi ) = - {\left\| {\nabla \Delta \varphi } \right\|^2} + ( f(\dot{\varphi} ) \nabla \varphi , \nabla \Delta \varphi ). $

对方程右边的第二项, 有

$ \begin{eqnarray*} ( {f(\dot{\varphi })}\nabla \varphi , \nabla \Delta \varphi ) &\le &{\left\| {f(\dot{\varphi} )} \right\|_{{L^6}}}{\left\| {\nabla \varphi } \right\|_{{L^3}}}\left\| {\nabla \Delta \varphi } \right\|\\ &\le &c{\left\| {\nabla \varphi } \right\|^{\frac{1}{2}}}{\left\| {\nabla \Delta \varphi } \right\|^{\frac{1}{2}}}\left\| {\nabla \Delta \varphi } \right\| \\ & \le& C + \frac{1}{4}{\left\| {\nabla \Delta \varphi } \right\|^2}, \end{eqnarray*} $

可推出

$ {\left\| {\nabla \Delta \varphi } \right\|^2} \le 2{\left\| {\nabla \mu } \right\|^2} + c. $

将以上方程在区间$ (0, T) $上积分, 结合方程$ (3.2) $, 方程$ (3.3) $以及方程$ (3.4) $, 得到

(3.5) $ \begin{equation} \int_t^{t + 2} {\left\| \varphi \right\|} _{{H^3}}^2{\kern 1pt} {\rm d}x \le c. \end{equation} $

由于插值不等式

$ {\left\| \varphi \right\|_{{H^2}}} \le c\left\| \varphi \right\|_{{H^1}}^{\frac{1}{2}}\left\| \varphi \right\|_{{H^3}}^{\frac{1}{2}}. $

因此

(3.6) $ \begin{equation} \int_0^T {\left\| \varphi \right\|} _{{H^2}}^4 \le c, \end{equation} $

即$ \varphi \in {L^4}(0, T;{H^2}) $.

在方程$ (1.3) $中乘以$ {\Delta ^2}\varphi $, 并进行积分, 有

(3.7) $ \begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\Delta \varphi } \right\|^2} + {\left\| {{\Delta ^2}\varphi } \right\|^2}\\ & = & - \int_\Omega {{\bf u}} \cdot \nabla \varphi {\Delta ^2}\varphi {\rm d}x + \int_\Omega {\Delta f(\varphi ){\Delta ^2}\varphi {\rm d}x} \\ & \le& {\left\| {{\bf u}} \right\|_{{L^6}}}{\left\| {\nabla \varphi } \right\|_{{L^3}}} \left\| {{\Delta ^2}\varphi } \right\| + \left\| {\Delta f(\varphi )} \right\|\left\| {{\Delta ^2}\varphi } \right\| \\ & \le& c{\left\| {{\bf u}} \right\|_{{H^1}}}(\left\| {\nabla \varphi } \right\| + {\left\| {\nabla \varphi } \right\|^{\frac{1}{2}}}{\left\| {\Delta \varphi } \right\|^{\frac{1}{2}}})\left\| {{\Delta ^2}\varphi } \right\|+ \frac{1}{4}{\left\| {{\Delta ^2}\varphi } \right\|^2} + c{\left\| {\Delta f(\varphi )} \right\|^2}. \end{eqnarray} $

注意到

$ \Delta f(\varphi ) = {f(\ddot{\varphi })} {\left| {\nabla \varphi } \right|^2} + {f(\dot{\varphi} )}\Delta \varphi . $

结合假设条件(1.11) 以及Agmon不等式, 得到

$ \left\|f(\ddot{\varphi} ) \left|\nabla \varphi \right|^2 \right\| \le c(1 + {\left\| \varphi \right\|_{{L^\infty }}})\left\| {\nabla \varphi } \right\|_{{L^4}}^2 \le c(1 + {\left\| {\Delta \varphi } \right\|^{\frac{1}{2}}}){\left\| {\Delta \varphi } \right\|^{\frac{3}{2}}}, $

$ \left\| {f(\dot{\varphi} )} \Delta \varphi \right\| \le c(1 + \left\| \varphi \right\|_{{L^\infty }}^2)\left\| {\Delta \varphi } \right\| \le c(1 + \left\| {\Delta \varphi } \right\|)\left\| {\Delta \varphi } \right\|. $

因此, 推得

$ {\left\| {\Delta f(\varphi )} \right\|^2} \le c(1 + {\left\| {\Delta \varphi } \right\|^2}){\left\| {\Delta \varphi } \right\|^3}. $

结合方程$ (3.7) $与Young不等式, 得到

$ \frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\Delta \varphi } \right\|^2} + {\left\| {{\Delta ^2}\varphi } \right\|^2} \le C\left\| {{\bf u}} \right\|_{{H^1}}^2 + \frac{1}{2}{\left\| {{\Delta ^2}\varphi } \right\|^2} + c(1 + {\left\| {\Delta \varphi } \right\|^2}){\left\| {\Delta \varphi } \right\|^3}. $

通过一致Gronwall不等式, 对任意的$ t \ge {T_2}({T_2} = {T_1} + 2) $, 推出

(3.8) $ \begin{equation} {\left\| {\Delta \varphi } \right\|^2} \le {m_3}, \end{equation} $

(3.9) $ \begin{equation} \int_t^{t + 2} {\left\| {{\Delta ^2}\varphi (s)} \right\|} {\kern 1pt} {\rm d}s \le {m_4}. \end{equation} $

由于方程$ (1.4) $, 可得

$ \begin{eqnarray*} \left\| {\nabla \Delta \varphi } \right\| &\le &\left\| {\nabla \mu } \right\| + {\left\| {f(\dot{\varphi} )} \right\|_{{L^\infty }}}\left\| {\nabla \varphi } \right\|\\ &\le &\left\| {\nabla \mu } \right\| + c(1 + \left\| {\Delta \varphi } \right\|)\left\| {\nabla \varphi } \right\| \\ & \le& \left\| {\nabla \mu } \right\| + c{m_1}(1 + {m_3}). \end{eqnarray*} $

将方程$ (1.3) $乘以$ {\mu _t} $, 并在$ \Omega $上进行积分, 有

$ \frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\nabla \mu } \right\|^2} + {\left\| {\nabla {\varphi _t}} \right\|^2} + \int_\Omega {f(\ddot{\varphi})} \varphi _t^2{\rm d}x = - \int_\Omega {{\bf u}} \cdot \nabla \varphi ( - \Delta {\varphi _t} + {f(\dot{\varphi} ){\varphi _t}}) {\rm d}x, $

通过假设条件$ (1.10) $, 得到

(3.10) $ \begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\nabla \mu } \right\|^2} + {\left\| {\nabla {\varphi _t}} \right\|^2} - {c_0}{\left\| {{\varphi _t}} \right\|^2}\\ &\le& \int_\Omega {{\bf u}} \cdot \nabla \varphi \Delta {\varphi _t}{\rm d}x + {c_0}\int_\Omega {{\bf u}} \cdot \nabla \varphi {\varphi _t}{\rm d}x\\ & \le &C{\left\| {\nabla {{\bf u}}} \right\|_{{L^3}}}{\left\| {\nabla \varphi } \right\|_{{L^6}}}\left\| {\nabla {\varphi _t}} \right\| + {c_0}{\left\| {{\bf u}} \right\|_{{L^3}}}{\left\| {\Delta \varphi } \right\|_{{L^6}}}\left\| {\nabla {\varphi _t}} \right\|. \end{eqnarray} $

由于Sobolev嵌入定理

$ {{{\left\| {\nabla {{\bf u}}} \right\|}_{{L^3}}} \le c\left\| {\Delta {{\bf u}}} \right\|}, \; \; \; \; {{{\left\| {{\bf u}} \right\|}_{{L^3}}} \le C\left\| {\nabla {{\bf u}}} \right\|}. $

取方程$ (1.1) $与$ - \Delta {{\bf u}} $的内积, 结合Young不等式以及Agmons不等式, 得到

$ {\left\| {\Delta {{\bf u}}} \right\|^2} + {\left\| {\nabla {{\bf u}}} \right\|^2} \le \left\| {\Delta {{\bf u}}} \right\|{\left\| \varphi \right\|_{{L^\infty }}}\left\| {\nabla \mu } \right\| \le \frac{1}{2}{\left\| {\Delta {{\bf u}}} \right\|^2} + c\left\| \varphi \right\|_{{L^\infty }}^2{\left\| {\nabla \mu } \right\|^2}, $

即

$ {\left\| {\Delta {{\bf u}}} \right\|^2} + {\left\| {\nabla {{\bf u}}} \right\|^2} \le c\left\| \varphi \right\|_{{L^\infty }}^2{\left\| {\nabla \mu } \right\|^2} \le {c_1}{\left\| \varphi \right\|_{{H^1}}}{\left\| \varphi \right\|_{{H^2}}}{\left\| {\nabla \mu } \right\|^2}. $

将方程$ (1.3) $乘以$ - \Delta \varphi $, 并在$ \Omega $上积分, 结合$ (1.10) $, $ (3.8) $以及Ladyzhenskaya's不等式, 得到

$ \begin{eqnarray*} \frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\nabla \varphi } \right\|^2} + {\left\| {\nabla \Delta \varphi } \right\|^2} &= & - \int_\Omega f(\ddot{\varphi}) \left| {\Delta \varphi } \right|^2{\rm d}x - \int_\Omega f(\ddot{\varepsilon}) \left| {\nabla \varphi } \right|^2\Delta \varphi {\rm d}x + \int_\Omega {\bf u} \cdot \nabla \varphi \Delta \varphi {\rm d}x\\ & \le &{c_0} \left\| {\Delta \varphi } \right\|^2 + \left\| f(\ddot{\varepsilon}) \right\|_{L^\infty } \left\| {\nabla \varphi } \right\|_{{L^4}}^2 \left\| {\Delta \varphi } \right\| + {\left\| {{\bf u}} \right\|_{{L^3}}} {\left\| {\nabla \varphi } \right\|_{{L^6}}}\left\| {\Delta \varphi } \right\|\\ &\le& c\left\| {\nabla {{\bf u}}} \right\| + C. \end{eqnarray*} $

将以上方程在区间$ ({T_2}, t) $上进行积分, 结合方程$ (3.3) $以及方程$ (3.4) $, 得到: 对任意的$ t \ge {T_2} $,

(3.11) $ \begin{equation} {\int_t^{t + 2} {\left\| {\nabla \Delta \varphi (s)} \right\|} ^2}{\rm d}s \le {m_5}, \end{equation} $

通过方程$ (3.10) $, 发现

$ \begin{eqnarray*} \frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\nabla \mu } \right\|^2} + {\left\| {\nabla {\varphi _t}} \right\|^2} &\le& \frac{1}{2}{\left\| {\nabla {\varphi _t}} \right\|^2} + c{\left\| {\Delta {{\bf u}}} \right\|^2} + C{\left\| {\nabla {{\bf u}}} \right\|^2}{\left\| {\nabla \Delta \varphi } \right\|^2}\\ &\le& \frac{1}{2}{\left\| {\nabla {\varphi _t}} \right\|^2} + {C_1}{\left\| {\nabla \mu } \right\|^2}(1 + {\left\| {\nabla \Delta \varphi } \right\|^2}), \end{eqnarray*} $

结合方程$ (3.11) $以及Gronwall不等式, 得到

(3.12) $ \begin{equation} {\left\| {\nabla \mu } \right\|^2} \le {m_6}, \end{equation} $

进一步推出, 对任意的$ t \ge {T_3} = {T_2} + 2 $, 有

(3.13) $ \begin{equation} {\left\| {\nabla {{\bf u}}} \right\|^2} \le {m_7}, \end{equation} $

(3.14) $ \begin{equation} {\left\| {\nabla \Delta \varphi } \right\|^2} \le {m_8}, \end{equation} $

(3.15) $ \begin{equation} {\int_t^{t + 2} {\left\| {\nabla {\varphi _t}} \right\|} ^2} {\rm d}t \le {m_9}, \end{equation} $

令$ {\varphi _t} = \phi $, $ {{{\bf u}}_t} = {{\bf v}} $及$ {p_t} = q $. 将方程$ (1.1) $, $ (1.2) $, $ (1.3) $以及$ (1.4) $关于$ t $求微分, 得到

(3.16) $ \begin{equation} - \Delta {{\bf v}} + {{\bf v}} = - \nabla q - \phi \nabla \mu - \varphi \nabla {\mu _t}, \end{equation} $

(3.17) $ \begin{equation} \nabla \cdot {{\bf v}} = 0, \end{equation} $

(3.18) $ \begin{equation} {\phi _t} + {{\bf v}} \cdot \nabla \varphi + {{\bf u}} \cdot \nabla \phi = \Delta {\mu _t}, \end{equation} $

(3.19) $ \begin{equation} {\mu _t} = - \Delta \phi + {f(\dot{\varphi})} \phi. \end{equation} $

受以下初边值条件约束

$ {{\bf v}}(x, t) = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x, t) \in \partial \Omega \times {R^ + }, $

$ \frac{{\partial \phi }}{{\partial n}} = \frac{{\partial \Delta \phi }}{{\partial n}} = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x, t) \in \partial \Omega \times {R^ + }. $

将方程$ (3.16) $乘以$ {{\bf v}} $并进行积分, 有

(3.20) $ \begin{eqnarray} {\left\| {\nabla {{\bf v}}} \right\|^2} + {\left\| {{\bf v}} \right\|^2} &= &- \int_\Omega {{\bf v}} \cdot \phi \nabla \mu {\rm d}x - \int_\Omega {{{\bf v}} \cdot \nabla \varphi {\mu _t}} {\rm d}x\\ & \le &{\left\| {{\bf v}} \right\|_{{L^3}}}{\left\| \phi \right\|_{{L^6}}}\left\| {\nabla \mu } \right\| + {\left\| {{\bf v}} \right\|_{{L^6}}}{\left\| {\nabla \varphi } \right\|_{{L^3}}}\left\| {{\mu _t}} \right\|\\ &\le &\frac{1}{2}{\left\| {\nabla {{\bf v}}} \right\|^2} + c({\left\| {\Delta \phi } \right\|^2} + {\left\| {\nabla \phi } \right\|^2} + {\left\| \phi \right\|^2}) + C. \end{eqnarray} $

取方程$ (3.18) $与$ \phi $的内积, 结合Young不等式以及Ladyzhenskaya's不等式, 得到

$ \begin{eqnarray*} &&\frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| \phi \right\|^2} + {\left\| {\Delta \phi } \right\|^2}\\ & = &- \int_\Omega {{\bf v}} \cdot \nabla \varphi \phi {\rm d}x - \int_\Omega {{{\bf u}} \cdot \nabla \phi \phi {\rm d}x - \int_\Omega f(\ddot{\varepsilon}) \phi {{\left| {\nabla \phi } \right|}^2}{\rm d}x} - \int_\Omega f(\ddot{\varphi}) \Delta \phi \nabla \phi {\rm d}x\\ & \le& {\left\| {{\bf v}} \right\|_{{L^3}}}{\left\| {\nabla \varphi } \right\|_{{L^6}}}\left\| \phi \right\| + {\left\| {{\bf u}} \right\|_{{L^3}}}{\left\| {\nabla \phi } \right\|_{{L^6}}}\left\| \phi \right\| + \left\| f(\ddot{\varepsilon}) \right\|\left\| \phi \right\|\left\| {\nabla \phi } \right\|_{{L^4}}^2 + {c_0}\left\| {\Delta \phi } \right\|\left\| {\nabla \phi } \right\|\\ & \le& \frac{1}{2}{\left\| {\Delta \phi } \right\|^2} + c{\left\| \phi \right\|^2} + {c_1}{\left\| {\nabla \phi } \right\|^2}. \end{eqnarray*} $

利用方程$ (3.15) $以及一致Gronwall不等式, 推出: 对任意的$ t \ge {T_4} = {T_3} + 2 $, 有

(3.21) $ \begin{equation} {\left\| {{\varphi _t}} \right\|^2} \le {m_{10}}. \end{equation} $

将方程$ (3.18) $乘以$ - \Delta \phi $并积分, 有

$ \begin{eqnarray*} &&\frac{1}{2}\frac{\rm d}{{{\rm d}t}}{\left\| {\nabla \phi } \right\|^2} + {\left\| {\nabla \Delta \phi } \right\|^2}\\ & =& \int_\Omega {{{\bf v}} \cdot \nabla \varphi \Delta \phi {\rm d}x + \int_\Omega {{\bf u}} } \cdot \nabla \phi \Delta \phi {\rm d}x + \int_\Omega f(\ddot{\varepsilon}) \nabla \phi \nabla \Delta \phi {\rm d}x\\ & \le& \left\| {{\bf v}} \right\|_{L^3} \left\| {\nabla \varphi } \right\|{\left\| {\nabla \Delta \phi } \right\|_{{L^6}}} + {\left\| {{\bf u}} \right\|_{{L^3}}}{\left\| \phi \right\|_{{L^6}}}{\left\| {\Delta \phi } \right\|_{{L^2}}} + \left\| f(\ddot{\varepsilon}) \right\|_{L^\infty } \left\| {\nabla \phi } \right\|\left\| {\nabla \Delta \phi } \right\|. \end{eqnarray*} $

结合方程$ (3.3) $, $ (3.13) $, $ (3.20) $, $ (3.21) $以及Gronwall不等式, 有

(3.22) $ \begin{equation} {\left\| {\nabla {\varphi _t}} \right\|^2} \le {m_{11}}. \end{equation} $

将方程$ (1.3) $乘以$ \Delta \mu $并进行积分, 可得

$ \begin{eqnarray*} {\left\| {\Delta \mu } \right\|^2} &\le& \left\| {{\varphi _t}} \right\|\left\| {\Delta \mu } \right\| + {\left\| {{\bf u}} \right\|_{{L^3}}}{\left\| {\nabla \varphi } \right\|_{{L^6}}}\left\| {\Delta \mu } \right\|\\ & \le &\frac{1}{2}{\left\| {\Delta \mu } \right\|^2} + c({\left\| {{\varphi _t}} \right\|^2} + {\left\| {\nabla {{\bf u}}} \right\|^2}{\left\| {\Delta \varphi } \right\|^2}). \end{eqnarray*} $

结合方程$ (3.8) $, $ (3.13) $以及$ (3.21) $, 推出

(3.23) $ \begin{equation} {\left\| {\Delta \mu } \right\|^2} \le {m_{12}}. \end{equation} $

对方程$ (1.4) $关于空间变量求二阶导数, 有

$ \begin{eqnarray*} \left\| {{\Delta ^2}\varphi } \right\| &\le &\left\| {\Delta \mu } \right\| + \left\| f(\ddot{\varphi}) \right\|_{L^\infty } \left\| {\Delta \varphi } \right\| + \left\| f(\ddot{\varepsilon}) \right\|_{L^\infty }\left\| {\nabla \varphi } \right\|_{{L^4}}^2\\ &\le& \left\| {\Delta \mu } \right\| + c\left\| {\Delta \varphi } \right\| + C\left\| {\nabla \varphi } \right\|\left\| {\Delta \varphi } \right\|. \end{eqnarray*} $

利用方程$ (3.3) $, $ (3.8) $以及$ (3.23) $, 得到: 对任意的$ t \ge {T_5} = {T_4} + 2 $, 有

(3.24) $ \begin{equation} \left\| {{\Delta ^2}\varphi } \right\| \le {m_{13}}. \end{equation} $

证毕.

接下来, 证明在$ {H^1}(\Omega ), {H^2}(\Omega ), {H^3}(\Omega ) $和$ {H^4}(\Omega ) $中全局吸引子的存在性. 对于定值$ {I_0} \in R $, 在$ X = \left\{ {\varphi \in {H^i}(\Omega ):\int_\Omega {\varphi {\rm d}x = {I_0}} } \right\} $上定义一个连续半群$ {\left\{ {S(t)} \right\}_{t \ge 0}}, S(t):{\varphi _0} \to \varphi $, 其中$ i = 1, 2, 3, 4 $. 首先给出在$ {H^1}(\Omega ), {H^2}(\Omega ), {H^3}(\Omega ) $上的一些证明.

引理3.1^[20]   若$ S(t) $具有有界吸收集$ {\beta _1} $, 并且$ {\beta _1} $在$ X $中是相对紧性的, 则$ S(t) $存在全局吸引子$ {\rm A} $.

定理3.2 半群$ {\left\{ {S(t)} \right\}_{t \ge 0}} $存在全局吸引子$ {{\rm A}_m} $$ (m = 1 $, $ 2 $, $ 3) $.

证  定义集合

$ {\beta _1} = \left\{ {\varphi :{{\left\| \varphi \right\|}_{{H^1}}} \le c} \right\}, \; \; \; \; {\beta _2} = \left\{ {\varphi :{{\left\| \varphi \right\|}_{{H^2}}} \le c} \right\}, \; \; \; \; {\beta _3} = \left\{ {\varphi :{{\left\| \varphi \right\|}_{{H^3}}} \le c} \right\}, $

即$ {\beta _1} $, $ {\beta _2} $, $ {\beta _3} $是有界集.

由于方程$ (3.3) $, $ (3.8) $, $ (3.14) $以及$ (3.24) $, 得到$ {\beta _1} $, $ {\beta _2} $, $ {\beta _3} $是$ S(t) $的有界吸收集. 利用Sobolev嵌入定理: $ {H^2}\hookrightarrow {H^1} $, $ {H^3}\hookrightarrow{H^2} $, $ {H^4}\hookrightarrow{H^3} $, 推出$ {\beta _1} $在$ {H^2} $中是相对紧性的, 结合引理3.1, 得到半群$ S(t) $在$ {H^1} $中存在全局吸引子$ {{\rm A}_1} $.

重复上述步骤, 可得到$ S(t) $在$ {H^2} $与$ {H^3} $中的全局吸引子的存在性. 证毕.

引理3.2^[20]  若$ \left\{ {{\varphi _{0, n}}} \right\}_{n = 1}^\infty $在$ {H^1}(\Omega ) $中有界, $ \left\{ {S(t){\varphi _{0, n}}} \right\}_{n = 1}^\infty $在$ {H^4}(\Omega ) $中有收敛的子列, 则$ \big\{ {S{{(t)}_{t \ge 0}}} \big\} $在$ {H^4}(\Omega ) $中是渐近紧性的.

引理3.3^[20]  若$ {\left\{ {S(t)} \right\}_{t \ge 0}} $满足条件

$ (1) $$ {\left\{ {S(t)} \right\}_{t \ge 0}} $在$ {H^4}(\Omega ) $中具有有界吸收集$ {\rm A} $,

$ (2) $$ {\left\{ {S(t)} \right\}_{t \ge 0}} $在$ {H^4}(\Omega ) $中是渐近紧性的,

则$ {\left\{ {S(t)} \right\}_{t \ge 0}} $在$ {H^4}(\Omega ) $中存在全局吸引子$ {\rm A} $.

最后, 证明在$ {H^4}(\Omega ) $中全局吸引子的存在性.

定理3.3  假设$ {\varphi _0} \in {H^1}(\Omega ) $, 若非线性项满足额外假设条件

$ \left| {f({\ddot{s}_1})} - {f({\ddot{s}_2})} \right| \le c\left| {{s_1} - {s_2}} \right|(1 + {\left| {{s_1}} \right|^2} + {\left| {{s_2}} \right|^2}), $

则$ \big\{ {S{{(t)}_{t \ge 0}}} \big\} $在$ {H^4}(\Omega ) $中存在全局吸引子.

证  假设$ \left\{ {{\varphi _{0, n}}} \right\}_{n = 1}^\infty $在$ {H^1}(\Omega ) $中有界, 接下来证明$ \left\{ {S({t_n}){\varphi _{0, n}}} \right\}_{n = 1}^\infty $在$ {H^4}(\Omega ) $中有收敛子列. 令

$ {\varphi _n}(t) = S(t){\varphi _{0, n}}. $

通过方程$ (3.8) $, $ (3.13) $, $ (3.14) $, $ (3.22) $, $ {H^4}\hookrightarrow{H^3} $, $ {H^3}\hookrightarrow{H^2} $以及$ {H^1}\hookrightarrow{L^2} $, 推出

$ \begin{eqnarray*} && {在 {H^3}(\Omega ) 中, {\rm{ }}\nabla \Delta {\varphi _n} 强收敛到 \nabla \Delta \varphi .} \\ && {在 {L^2}(\Omega ) 中, {({\varphi _n})_t} 强收敛到 {\varphi _t} . } \\ && {在 {L^2}(\Omega ) 中, {{{\bf u}}_n} 强收敛到 {{\bf u}} . } \\ && {在 {H^1}(\Omega ) 中, \nabla {\varphi _n} 强收敛到 \nabla \varphi . } \\ && {在 {L^2}(\Omega ) 中, {\varphi _n} 强收敛到 \varphi .} \end{eqnarray*} $

利用方程$ (1.3) $, $ {\varphi _n} $, $ {{{\bf u}}_n} $满足

$ \frac{\rm d}{{{\rm d}t}}{\varphi _n} + {{{\bf u}}_n} \cdot \nabla {\varphi _n} + {\Delta ^2}{\varphi _n} - \Delta f({\varphi _n}) = 0. $

对于$ \left\{ {\nabla {\varphi _n}} \right\} $和$ \left\{ {{{{\bf u}}_n}} \right\} $, 得到

$ \begin{eqnarray*} \left\| {{{{\bf u}}_n} \cdot \nabla {\varphi _n} - {{\bf u}} \cdot \nabla \varphi } \right\| &= &\left\| {{{{\bf u}}_n} \cdot \nabla {\varphi _n} - {{{\bf u}}_n} \cdot \nabla \varphi + {{{\bf u}}_n} \cdot \nabla \varphi - {{\bf u}} \cdot \nabla \varphi } \right\|\\ & \le& \left\| {{{{\bf u}}_n} \cdot (\nabla {\varphi _n} - \nabla \varphi )} \right\| + \left\| {\nabla \varphi \cdot ({{{\bf u}}_n} - {{\bf u}})} \right\| \to 0. \end{eqnarray*} $

对于$ \left\{ {{{({\varphi _n})}_t}} \right\} $, 推出

$ \left\| {{{({\varphi _n})}_t} - {\varphi _n}} \right\| \to 0. $

对于$ \Delta f(\varphi ) $, 结合非线性项假设条件, 得到

$ \begin{eqnarray*} &&\left\| {\Delta f({\varphi _n}) - \Delta f(\varphi )} \right\|\\ & =& \left\| f({\ddot{\varphi} _n}) (\nabla {\varphi _n})^2 - f(\ddot{\varphi} ) (\nabla \varphi )^2 + f(\dot{\varphi} _n) \Delta {\varphi _n} - f(\dot{\varphi}) \Delta \varphi \right\|\\ &=& \left| \left| f({\ddot{\varphi} _n}) {{(\nabla {\varphi _n})}^2} - {f({\ddot{\varphi} _n})} {{(\nabla \varphi )}^2} + {f({\ddot{\varphi }_n})} (\nabla \varphi )^2 - f(\ddot{\varepsilon}) (\nabla \varphi )^2 \right. \right.\\ &&\left. \left. + f(\dot{\varphi} _n) \Delta {\varphi _n} - f(\dot{\varepsilon}_n) \Delta \varphi + f(\dot{\varepsilon}_n) \Delta \varphi - f(\dot{\varphi}) \Delta \varphi \right| \right|\\ & \le& \left\| f(\ddot{\varphi} _n) (\nabla {\varphi _n} - \nabla \varphi )(\nabla {\varphi _n} + \nabla \varphi ) \right\| + c\left\| ({\varphi _n} - \varphi )(1 + \left| \varphi _n \right|^2 + \left| \varphi \right|^2) (\nabla \varphi )^2 \right\|\\ && + \left\| f(\dot{\varphi} _n) (\Delta {\varphi _n} - \Delta \varphi ) \right\| + \left\| f(\ddot{\varepsilon}) ({\varphi _n} - \varphi )\Delta \varphi \right\| \to 0. \end{eqnarray*} $

因此, 推出

$ {\Delta ^2}{\varphi _n} = - {({\varphi _n})_t} - {{{\bf u}}_n} \cdot \nabla {\varphi _n} + \Delta f({\varphi _n}) \to - \varphi - {{\bf u}} \cdot \nabla \varphi + \Delta f(\varphi ), $

即$ {\left\{ {S(t)} \right\}_{t \ge 0}} $在$ {H^4}(\Omega ) $中是渐近紧性的.

因此, 半群$ \big\{ {S{{(t)}_{t \ge 0}}} \big\} $在$ {H^4}(\Omega ) $中存在全局吸引子. 证毕.