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  北京化工大学学报(自然科学版)  2018, Vol. 45 Issue (1): 119-123  DOI: 10.13543/j.bhxbzr.2018.01.020
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陈肖, 孙颖, 陈亚洲. Bird-Carreau型黏性van der Waals流体周期解的渐近稳定性[J]. 北京化工大学学报(自然科学版), 2018, 45(1): 119-123. DOI: 10.13543/j.bhxbzr.2018.01.020.
CHEN Xiao, SUN Ying, CHEN YaZhou. Asymptotic stability of the periodic solution of bird-carreau type viscous van der waals fluids[J]. Journal of Beijing University of Chemical Technology (Natural Science), 2018, 45(1): 119-123. DOI: 10.13543/j.bhxbzr.2018.01.020.

基金项目

国家自然科学基金(11671027)

第一作者

陈肖, 女, 1992年生, 硕士生

通讯联系人

陈亚洲, E-mail:chenyz@mail.buct.edu.cn

文章历史

收稿日期:2017-06-22
Bird-Carreau型黏性van der Waals流体周期解的渐近稳定性
陈肖 , 孙颖 , 陈亚洲     
北京化工大学 理学院, 北京 100029
摘要:讨论了一维可压缩黏性van der Waals流体系统的渐近稳定性,其中黏性系数为满足Bird-Carreau模型的非线性函数,压力为非凸函数。通过构造能量函数并运用能量估计方法及单调算子理论,证明得出:大黏性条件下初值位于稳定区域时,以及大黏性、小扰动条件下初值位于亚稳定区域时,该类van der Waals流体的解是渐近稳定的。
关键词Bird-Carreau型黏性    van der Waals流体    周期边界    
Asymptotic stability of the periodic solution of Bird-Carreau type viscous van der Waals fluids
CHEN Xiao , SUN Ying , CHEN YaZhou     
Faculty of Science, Beijing University of Chemical Technology, Beijing 100029, China
Abstract: In this paper, the asymptotic stability of a one-dimensional compressible viscous van der Waals fluids system is discussed, where the viscosity coefficient is a nonlinear function that satisfies the Bird-Carreau model, and the pressure is a non-convex function. By constructing the energy function and using the energy estimation method and the monotone operator theory, we prove that:under the condition of large viscosity, the solutions of the non-Newtonian fluid are asymptotically stable when the initial value is either located in the stable region, or located in the metastable region under small disturbance conditions.
Key words: Bird-Carreau type viscosity    van der Waals fluids    periodic boundary    
引言

可压缩黏性流体是偏微分方程和流体动力学研究的重要课题之一,其一维模型解的稳定性和大时间行为对工程实践和偏微分方程理论的完善有着非常重要的意义。对于状态方程为p=Avγ的理想多方气体,Matsumura等[1]最早证明了小扰动情况下行波解的渐近稳定性。此后,Matsumura等[2]、Kawashima等[3]分别证明了一维黏性可压缩流体稀疏波解的全局稳定性以及半空间中黏性非线性波解的渐近稳定性。可压缩黏性流体解的大时间行为的研究中另一重要且困难的问题是,当状态方程满足van der Waals模型,即压力为非凸函数时,不稳定区域的出现导致该问题在物理上不稳定且在数学上不适定。其中,对状态方程满足p=v3v的黏弹性流体,Mei等[4-5]证明了带人工黏性的周期边界问题解的存在性、一致有界性和收敛性。另外,对于一般的van der Waals流体,Hoff等[6]证明了小扰动条件下,时间趋于无穷时一维全空间中可压缩van der Waals流体全局解收敛到稳态解。之后,Hsieh等[7]用数值模拟方法求出了带人工黏性的van der Waals系统的解的性态,并证明当初始密度位于椭圆(不稳定)区域时发生相变。Huang等[8]通过研究可压缩黏性van der Waals系统周期解的渐近稳定性,证明了当初始密度和初始动量足够接近平均密度和平均动量时,方程的解收敛到稳态解。

上述可压缩黏性van der Waals流体的研究中均假设黏性系数为常数,本文在前人成果的基础上,研究了具有Bird-Carreau型黏性的压力非凸的一维可压缩流体系统周期解的渐近稳定性。黏性非线性和压力非凸导致渐近状态更加复杂,且更难以得到高阶导数的能量估计,这是本课题的难点。本文通过构造能量函数并结合非线性单调算子方法解决了这一难题。

1 基本模型及其主要定理

考虑拉格朗日坐标系下具有人工黏性的一维可压缩流体的周期边值问题:

$ \left\{ \begin{gathered} {v_t} - {u_x} = \varepsilon {v_{xx}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t > 0,x \in \mathbb{R} \hfill \\ {u_t} + p{\left( v \right)_x} = {\left( {\mu \left( {\frac{{{u_x}}}{v}} \right)\frac{{{u_x}}}{v}} \right)_x}\;\;\;\;\;\;\;\;t > 0,x \in \mathbb{R} \hfill \\ \left( {v,u} \right)\left| {_{t = 0}} \right. = \left( {{v_0},{u_0}} \right)\left( x \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathbb{R} \hfill \\ \left( {v,u} \right)\left( {x,t} \right) = \left( {v,u} \right)\left( {x + 2L,t} \right) \hfill \\ \end{gathered} \right. $ (1)

其中vup分别表示流体的比容、速度和压力,ε为人工黏性,2L为周期常数。流体黏性μ满足Bird-Carreau模型:

$ \mu = {\mu _\infty } + \left( {{\mu _0} - {\mu _\infty }} \right){\left( {1 + {{\left( {{u_x}/v} \right)}^2}} \right)^{\frac{\gamma }{2}}} $ (2)

其中-1≤γ < 0。压力p是关于v的非凸函数,其函数关系由流体状态方程决定(图 1)。

图 1 pv之间的关系 Fig.1 Plot of p against v

设集合K=[v1, v2],其中0 < v1 < α < β < v2。假定存在常数c=c(K),使得对于vK,有

$ \left\{ \begin{array}{l} p'\left( v \right) < 0,\;\;\;\;\;\;v \in \left( { - \infty ,\alpha } \right) \cup \left( {\beta , + \infty } \right)\\ p'\left( v \right) > 0,\;\;\;\;\;\;v \in \left( {\alpha ,\beta } \right)\\ \left| {p\left( v \right)} \right| \le c,\;\;\;\;\;\;\left| {p'\left( v \right)} \right| \le c \end{array} \right. $ (3)

满足式(3)模型的共同点是:都存在v的一个区间,在此区间上pv的增大而增大,即在这个区域中方程组(1)表示的系统是椭圆的,因此这个系统是物理上不稳定和数学上不适定的。

定义初值(v0, u0)在[0, 2L]上的均值为

$ \left\{ \begin{array}{l} \bar v = \frac{1}{{2L}}\int_0^{2L} {{v_0}\left( x \right){\rm{d}}x} \\ \bar u = \frac{1}{{2L}}\int_0^{2L} {{u_0}\left( x \right){\rm{d}}x} \end{array} \right. $ (4)

由此,从方程组(1)中可以得出

$ \left\{ \begin{array}{l} \int_0^{2L} {\left( {v - \bar v} \right){\rm{d}}x} = 0\\ \int_0^{2L} {\left( {u - \bar u} \right){\rm{d}}x} = 0 \end{array} \right. $ (5)

此时构造能量函数

$ \begin{array}{l} \mathit{\Phi }\left( v \right) = \int_{\bar v}^v {\left( {p\left( {\bar v} \right) - p\left( \xi \right)} \right){\rm{d}}\xi } = p\left( {\bar v} \right)\left( {v - \bar v} \right) - \\ \int_{\bar v}^v {p\left( \xi \right){\rm{d}}\xi } \end{array} $ (6)

显然,p′(v)>0所对应的区域与Φ″(v) < 0所对应的区域一致。麦克斯韦结构由v1 < v < v2所表示的整个区域构成(图 1中虚线)。其中v1v2满足

$ \left\{ \begin{array}{l} p\left( {{v_1}} \right)\left( {{v_2} - {v_1}} \right) = \int_{{v_1}}^{{v_2}} {p\left( \xi \right){\rm{d}}\xi } \\ p\left( {{v_1}} \right) = p\left( {{v_2}} \right) \end{array} \right. $ (7)

图 1所示,初始状态分离成AB1B2C3个区域,分别对应不稳定区域、亚稳定区域和稳定区域。从Φ的定义可知:

$ \mathit{\Phi }\left( v \right)\left\{ \begin{array}{l} \ge 0\;\;\;\bar v \in 区域\;C\\ < 0\;\;\;\bar v \in 区域\;A \end{array} \right. $ (8)
$ \mathit{\Phi }\left( v \right)\left\{ \begin{array}{l} \ge 0\;\;\;v \le {v_ * },\bar v \in 区域\;{B_1}\\ < 0\;\;\;v > {v_ * },\bar v \in 区域\;{B_1} \end{array} \right. $ (9)
$ \mathit{\Phi }\left( v \right)\left\{ \begin{array}{l} \ge 0\;\;\;v \ge {v^* },\bar v \in 区域\;{B_2}\\ < 0\;\;\;v < {v^* },\bar v \in 区域\;{B_2} \end{array} \right. $ (10)

其中v*和v*分别满足$\int_{\bar v}^{{v_*}} {\left( {p\left( {\bar v} \right)-p\left( \xi \right)} \right){\rm{d}}\xi {\rm{ = 0}}} $$\int_{{v^*}}^{\bar v} {\left( {p\left( {\bar v} \right)-p\left( \xi \right)} \right){\rm{d}}\xi {\rm{ = 0}}} $

记函数空间如式(11)~(13):

$ L_{{\rm{per}}}^2 = \left\{ {f\left( x \right)\left| {f\left( x \right)} \right. = f\left( {x + 2L} \right);f\left( x \right) \in {L^2}\left( {0,2L} \right)} \right\} $ (11)

其范数表示为$\left\| f \right\| = {\left( {\int_0^{2L} {{f^2}\left( x \right){\rm{d}}x} } \right)^{\frac{1}{2}}} $

$ H_{{\rm{per}}}^2 = \left\{ {f\left( x \right)\left| {f\left( x \right)} \right. \in L_{{\rm{per}}}^2,\partial _x^if \in L_{{\rm{per}}}^2\left( \mathbb{R} \right);i = 1, \cdots ,k} \right\} $ (12)

其范数表示为${\left\| f \right\|_k} = {\left( {\sum\limits_{i = 0}^k {\int_0^{2L} {{{\left| {\partial _x^if\left( x \right)} \right|}^2}{\rm{d}}x} } } \right)^{\frac{1}{2}}} $

$ H_{{\rm{per,0}}}^k = \left\{ {f\left( x \right)\left| {f\left( x \right) \in H_{{\rm{per}}}^k,\int_0^{2L} {f\left( x \right){\rm{d}}x} = 0} \right.} \right\} $ (13)

其中k=1, 2, …。

定义‖(f, g)‖2=‖f2+‖g2和‖(f, g)‖k2=‖fk2+‖gk2

定理1  假设初值(v0, u0)∈Hper1,且存在正数mM,使得$\mathop {\inf }\limits_{x \in \mathbb{R}} {v_0}\left( x \right) \ge m, {\left\| {\left( {{v_0}-\bar v, {u_0}-\bar u} \right)} \right\|_1} \le M $成立,且满足

$ \left\{ \begin{array}{l} \bar v \in 区域\;C\\ 或\;\bar v \in 区域\;{B_1}\;且\;M \le \frac{{{v_ * } - \bar v}}{{2\sqrt {2L} }}\\ 或\;\bar v \in 区域\;{B_2}\;且\;M \le \frac{{\bar v - {v^ * }}}{{2\sqrt {2L} }} \end{array} \right. $ (14)

若人工黏性ε满足

$ \varepsilon > \frac{{{2^{\frac{5}{2}}}{L^{\frac{1}{2}}}\left( {{L^2} + {{\rm{ \mathsf{ π} }}^2}} \right)M}}{{{\mu _\infty }{{\rm{ \mathsf{ π} }}^2}}}\mathop {\sup }\limits_{v \in 区域A} \left| {p'\left( v \right)} \right| $ (15)

则存在δ>0,使得‖v0v, u0u2δ时,方程组(1)存在唯一的全局解

$ \left( {v,u} \right)\left( {x,t} \right) \in C\left( {0, + \infty ;H_{{\rm{per}}}^1} \right) $

并满足能量不等式(16)(C为常数)

$ \begin{array}{l} \left\| {\left( {v - \bar v,u - \bar u} \right)\left( t \right)} \right\|_1^2 + \int_0^{ + \infty } {\left\| {\left( {v - \bar v,u - } \right.} \right.} \\ \left. {\left. {\bar u} \right)} \right\|_2^2{\rm{d}}t \le C\left\| {\left( {{v_0} - \bar v,{u_0} - \bar u} \right)} \right\|_1^2 \end{array} $ (16)

进一步有

$ \mathop {\lim }\limits_{t \to + \infty } \left\| {v - \bar v,u - \bar u} \right\| = 0 $ (17)
2 问题重构

令(φ, ψ)=(v, u)-(v, u),方程组(1)可化为方程组(18):

$ \left\{ \begin{array}{l} {\phi _t} - {\psi _x} = \varepsilon {\phi _{xx}}\\ {\psi _t} = {\left( {p\left( {\phi + \bar v} \right) - p\left( {\bar v} \right)} \right)_x} = {\left( {\mu \frac{{{\psi _x}}}{{\phi + \bar v}}} \right)_x}\\ \left( {\phi ,\psi } \right)\left( {x,t} \right) = \left( {\phi ,\psi } \right)\left( {x + 2L,t} \right)\\ \left( {\phi ,\psi } \right)\left| {_{t = 0}} \right. = \left( {{v_0} - \bar v,{u_0} - \bar u} \right)\left( x \right)\\ \int_0^{2L} {\left( {\phi ,\psi } \right)\left( {x,t} \right){\rm{d}}x} = \left( {0,0} \right) \end{array} \right. $ (18)

其中$\mu = \mu \left( {\frac{{{\psi _x}}}{{\phi + \bar v}}} \right) $

对于时间区间I=[τ, τ+t0]∈[0, +∞),定义其解空间Xm, M(I)为

$ \begin{gathered} {X_{m,M}}\left( I \right) = \left\{ {\left( {\phi ,\psi } \right)\left| {\left( {\phi ,\psi } \right) \in C\left( {I;H_{{\rm{per,0}}}^1} \right),\left( {\phi ,} \right.} \right.} \right. \hfill \\ \left. \psi \right) \in {L^2}\left( {I;H_{{\rm{per,0}}}^2} \right),\mathop {\sup }\limits_{t \in I} {\left\| {\left( {\phi ,\psi } \right)\left( t \right)} \right\|_2} \leqslant M,\mathop {\inf }\limits_{t \in I,x \in \mathbb{R}} \left( {\bar v + } \right. \hfill \\ \left. {\left. \phi \right)\left( {x,t} \right) \geqslant m} \right\} \hfill \\ \end{gathered} $ (19)

将方程组(1)解的渐近稳定性问题转化为考虑方程组(18)的稳态问题,其稳态解满足方程组(20)

$ \left\{ \begin{array}{l} - {U_x} = \varepsilon {V_{xx}}\\ {\left( {p\left( {V + \bar v} \right) - p\left( {\bar v} \right)} \right)_x} = {\left( {\mu \left( {\frac{{{U_x}}}{{V + \bar v}}} \right)\frac{{{U_x}}}{{V + \bar v}}} \right)_x}\\ \left( {V,U} \right)\left( x \right) = \left( {V,U} \right)\left( {x + 2L} \right)\\ \int_0^{2L} {\left( {V,U} \right)\left( x \right){\rm{d}}x} = \left( {0,0} \right) \end{array} \right. $ (20)

定义解空间Ym, M如式(21)

$ \begin{gathered} {Y_{m,M}} = \left\{ {\left( {V,U} \right)\left| {\left( {V,U} \right) \in {H_{{\rm{per,0}}}},{{\left\| {\left( {V,U} \right)} \right\|}_1}} \right.} \right. \hfill \\ \left. { \leqslant M,\mathop {\inf }\limits_{x \in \mathbb{R}} \left( {V + \bar v} \right)\left( x \right) \geqslant m > 0} \right\} \hfill \\ \end{gathered} $ (21)

引理1  令f(x)∈Hper, 01,则有

$ \left\{ \begin{array}{l} \left\| f \right\| \le \frac{L}{{\rm{ \mathsf{ π} }}}\left\| {{f_x}} \right\|\\ {\sup _{\left[ {0,2L} \right]}}\left| f \right| \le 2\;\;\;\sqrt {2L} \left\| {{f_x}} \right\| \end{array} \right. $ (22)

此外,若fLperfxLper,则有

$ {\sup _{\left[ {0,2L} \right]}}\left| f \right| \le 2\int_0^{2L} {\left| {f'} \right|{\rm{d}}x} + \frac{1}{{2L}}\int_0^{2L} {\left| f \right|{\rm{d}}x} $ (23)

证明  式(22)第1个不等式文献[9]已证明。

式(22)第2个不等式的证明:因为$f\left( x \right) = f\left( 0 \right) + \int_0^x {f'\left( s \right){\rm{d}}s} $,故将其在[0, 2L]上积分得

$ f\left( 0 \right) = \frac{1}{{2L}}\int_0^{2L} {\int_0^x {f'\left( s \right){\rm{d}}s{\rm{d}}x} } $

进而有

$ \begin{array}{l} \sup \left| {f\left( x \right)} \right| \le \left| {\int_0^x {f'\left( s \right){\rm{d}}s} } \right| + \left| {\frac{1}{{2L}}\int_0^{2L} {\int_0^x {f'\left( s \right)} } } \right.\\ \left. {{\rm{d}}s{\rm{d}}x} \right| \le 2\sqrt {2L} \left\| {{f_x}} \right\| \end{array} $

由此式(22)第2个不等式得证。同理可得不等式(23)。证毕。

引理2  若人工黏性$ \varepsilon > \frac{{{L^2}\sup \left| {p'} \right|M}}{{{\mu _\infty }{\pi ^2}}}$,则方程组(20)在解空间Ym, M中有唯一平凡解(V, U)(x)=(0, 0)。

证明  显然,方程组(20)至少有一个解(0, 0),且一般情况下解的存在性都可以由不动点定理得到,因而不再详述。此处仅证明(V, U)的唯一性。令方程组(22)的第1个等式乘以V、第2个等式乘以U,并分别在[0, 2L]上积分,得

$ \left\{ \begin{array}{l} \varepsilon \int_0^{2L} {V_x^2{\rm{d}}x} = \int_0^{2L} {V{U_x}{\rm{d}}x} \\ \int_0^{2L} {\mu \frac{{U_x^2}}{{V + \bar v}}{\rm{d}}x} = \int_0^{2L} {\left( {p\left( {V + \bar v} \right) - p\left( {\bar v} \right)} \right){U_x}{\rm{d}}x} \end{array} \right. $ (24)

运用引理1的结论可得

$ \int_0^{2L} {\left( {\frac{{\varepsilon {{\rm{ \mathsf{ π} }}^2}}}{{{L^2}}}{V^2} - 2V{U_x} + \frac{{{\mu _\infty }}}{{M\sup \left| {p'} \right|}}U_x^2} \right){\rm{d}}x} \le 0 $ (25)

由此可推知引理2。证毕。

3 主要定理的证明

通过第2章的问题重构,方程组(1)解的渐近稳定性的证明就转化成了方程组(18)解的全局存在性和渐近行为的证明。由经典能量方法可知,只需得到解的局部存在性和一致估计,即可完成证明。

命题1(局部存在性)  假设(ϕ0, ψ0)∈Hper, 01,对任意的m>0和M>0,如果‖(ϕ0, ψ0)‖1M,且$\mathop {\inf }\limits_{\mathbb{R} \times \left[{0, {t_0}} \right]} \left| {{\phi _0} + \bar v} \right| \ge m $,则存在一个正常数t0,使得方程(18)存在唯一解$\left( {\phi, \psi } \right) \in {X_{\frac{1}{2}m, 2M}}\left( {\tau, \tau + {t_0}} \right) $

证明  不失一般性,假设τ=0。设$A\left( {{\psi ^{\left( n \right)}}} \right): = \mu \left( {\frac{{\psi _x^{\left( n \right)}}}{{{\phi ^{\left( {n-1} \right)}} + \bar v}}} \right)\frac{{\psi _x^{\left( n \right)}}}{{{\phi ^{\left( {n-1} \right)}} + \bar v}} $,用单调算子法由A的单调性可证明对所有的${\left( {\phi, \psi } \right)^{\left( {n- 1} \right)}} \in {X_{\frac{1}{2}m, 2M}}\left( {\left[{0, {t_0}} \right]} \right) $,方程组(26)存在唯一解(ϕ, ψ)(n)

$ \left\{ \begin{array}{l} \phi _t^{\left( n \right)} - \varepsilon \phi _{xx}^{\left( n \right)} = \psi _x^{\left( n \right)}\\ \psi _t^{\left( n \right)} - {\left( {\mu \left( {\frac{{\psi _x^{\left( n \right)}}}{{{\phi ^{\left( {n - 1} \right)}} + \bar v}}} \right)\frac{{\psi _x^{\left( n \right)}}}{{{\phi ^{\left( {n - 1} \right)}} + \bar v}}} \right)_x} = - p_x^{\left( n \right)}\\ \left( {{\phi ^{\left( n \right)}},{\psi ^{\left( n \right)}}} \right)\left| {_{t = 0}} \right. = \left( {{v_0} - \bar v,{u_0} - \bar u} \right)\left( x \right)\\ \left( {{\phi ^{\left( n \right)}},{\psi ^{\left( n \right)}}} \right)\left( {x,t} \right) = \left( {{\phi ^{\left( n \right)}},{\psi ^{\left( n \right)}}} \right)\left( {x + 2L,t} \right) \end{array} \right. $ (26)

其中px(n)=px(ϕ(n-1)+v)。由不动点定理可知:存在t0>0且$\left( {\phi, \psi } \right) \in {X_{\frac{1}{2}m, 2M}}\left( {\left[{0, {t_0}} \right]} \right) $,当τ=0时方程组(18)成立。证毕。

命题2(先验估计)  对于(ϕ, ψ)∈Xm, M(I),当v满足式(14)且ε满足式(15)时,对t∈[0, +∞),有式(27)成立

$ \begin{array}{l} \left\| {\left( {\phi ,\psi } \right)\left( t \right)} \right\|_1^2 + \int_0^t {\left\| {\left( {\phi ,\psi } \right)\left( \tau \right)} \right\|_2^2{\rm{d}}\tau } \le C\\ \left( {\left\| {{\phi _0}} \right\|_1^2 + \left\| {{\psi _0}} \right\|_1^2} \right) \end{array} $ (27)

证明  方程组(18)第1个等式乘以ϕ再在[0, 2L]上积分得

$ \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {{\phi ^2}{\rm{d}}x} + \varepsilon \int_0^{2L} {\phi _x^2{\rm{d}}x} = \int_0^{2L} {\phi {\psi _x}{\rm{d}}x} $ (28)

方程组(18)第2个等式乘以ψ再在[0, 2L]上积分得

$ \begin{array}{l} \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {{\psi ^2}{\rm{d}}x} + \int_0^{2L} {\left( {\left( {p\left( {\bar v} \right) - p\left( {\phi + \bar v} \right)} \right){\psi _x} + } \right.} \\ \left. {\frac{{\mu \psi _x^2}}{{\phi + \bar v}}} \right){\rm{d}}x = 0 \end{array} $ (29)

将方程组(18)第1个等式带入式(29),由式(6)中的能量函数Φ的定义可得

$ \begin{array}{l} \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right){\rm{d}}x} + \int_0^{2L} {\left( {\frac{\mu }{\phi } + } \right.} \\ \left. {\frac{{\psi _x^2}}{{\bar v}}} \right){\rm{d}}x - \varepsilon \int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x = 0} \end{array} $ (30)

由引理1知$\mathop {\sup }\limits_{x \in \left[{0, 2L} \right]} \left| {\phi + \bar v} \right| \le 2\sqrt {2L} \left\| {{{\left( {\phi + \bar v} \right)}_x}} \right\| \le 2\sqrt {2L} M $,又注意到μμμ0,故式(30)可转化为式(31)

$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right){\rm{d}}x} + \frac{{{\mu _\infty }}}{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}\int_0^{2L} {\psi _x^2{\rm{d}}x} - \\ \varepsilon \int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \le 0 \end{array} $ (31)

式(31)乘以$\frac{{{2^{\frac{3}{2}}}{L^{\frac{5}{2}}}M}}{{{\mu _\infty }{\pi ^2}\varepsilon }} $,再加上式(28)得

$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left[ {\frac{{{2^{\frac{3}{2}}}{L^{\frac{5}{2}}}M}}{{{\mu _\infty }{\mathit{\pi }^2}\varepsilon }}\left( {\frac{{{\psi ^2}}}{2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right) + \frac{1}{2}{\phi ^2}} \right]{\rm{d}}x} + \\ \int_0^{2L} {\left( {\frac{{{L^2}}}{{2{{\rm{ \mathsf{ π} }}^2}\varepsilon }}\psi _x^2 + \frac{\varepsilon }{2}\phi _x^2} \right){\rm{d}}x} + \int_0^{2L} {\left( {\frac{{{L^2}}}{{2{{\rm{ \mathsf{ π} }}^2}\varepsilon }}\psi _x^2 - {\psi _x}\phi + } \right.} \\ \left. {\frac{{\varepsilon {{\rm{ \mathsf{ π} }}^2}}}{{2{L^2}}}{\phi ^2}} \right){\rm{d}}x \le \frac{{{2^{\frac{3}{2}}}{L^{\frac{5}{2}}}M}}{{{\mu _\infty }{\mathit{\pi }^2}}}\int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \end{array} $ (32)

方程组(18)第1个等式乘以ϕxx再在[0, 2L]上积分得

$ \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\phi _x^2{\rm{d}}x} + \varepsilon \int_0^{2L} {\phi _{xx}^2{\rm{d}}x} = - \int_0^{2L} {{\phi _{xx}}{\psi _x}{\rm{d}}x} $ (33)

式(31)乘以$\frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}{{{\mu _\infty }\varepsilon }} $再加上式(33)得

$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left[ {\frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}{{{\mu _\infty }\varepsilon }}\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right) + \frac{1}{2}\phi _x^2} \right]{\rm{d}}x} + \\ \frac{1}{{2\varepsilon }}\int_0^{2L} {\psi _x^2{\rm{d}}x} + \frac{\varepsilon }{2}\int_0^{2L} {\phi _{xx}^2{\rm{d}}x} + \int_0^{2L} {\left( {\frac{1}{{2\varepsilon }}\psi _x^2 - {\psi _x}{\phi _{xx}} + } \right.} \\ \left. {\frac{\varepsilon }{2}\phi _{xx}^2} \right){\rm{d}}x \le \frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}{{{\mu _\infty }}}\int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \end{array} $ (34)

将式(32)、(34)相加得

$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M\left( {{L^2} + {\pi ^2}} \right)}}{{{\mu _\infty }{\mathit{\pi }^2}\varepsilon }}\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right)} + \\ \left. {\frac{1}{2}\left( {\phi _x^2 + {\phi ^2}} \right)} \right]{\rm{d}}x + \frac{{{L^2} + {\pi ^2}}}{{2{\mathit{\pi }^2}\varepsilon }}\int_0^{2L} {\psi _x^2{\rm{d}}x} + \frac{\varepsilon }{2}\int_0^{2L} {\left( {\phi _x^2 + } \right.} \\ \left. {\phi _{xx}^2} \right){\rm{d}}x \le \frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M\left( {{L^2} + {\pi ^2}} \right)}}{{{\mu _\infty }{\mathit{\pi }^2}}}\int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \end{array} $ (35)

由式(8)~(10)可知v∈区域C、或v∈区域B1vv*、或v∈区域B2vv*时,有Φ(ϕ+v)≥0;因vv*,故ϕv*-v。又由引理1可知$\mathop {\sup }\limits_{\left[{0, 2L} \right]} \left| \phi \right| \le 2\sqrt {2L} \left\| {{\phi _x}} \right\| \le 2\sqrt {2L} M $,因此v∈区域B1$M \le \frac{{{v_*}-\bar v}}{{2\sqrt {2L} }} $时,有Φ(ϕ+v)≥0;同理,v∈区域B2$ M \le \frac{{\bar v-{v^*}}}{{2\sqrt {2L} }}$时,有Φ(ϕ+v)≥0。因此当vε分别满足式(14)、(15)时,有

$ \begin{array}{l} {\left\| {\psi \left( t \right)} \right\|^2} + \left\| {\phi \left( t \right)} \right\|_1^2 + \int_0^{ + \infty } {\left( {\left\| {\psi} \right\|_1^2 + } \right.} \\ \left. {\left\| {\phi} \right\|_2^2} \right){\rm{d}}x \le C\left( {\left\| {{\phi _0}} \right\|_1^2 + {{\left\| {{\psi _0}} \right\|}^2}} \right) \end{array} $ (36)

类似地,方程组(18)第2个等式乘以ψxx再在[0, 2L]上积分得

$ \begin{array}{l} {\left\| {{\psi _x}\left( t \right)} \right\|^2} + \int_0^{ + \infty } {\left( {{{\left\| {{\psi _{xx}}} \right\|}^2}} \right){\rm{d}}x} \le C\left( {\left\| {\left( {{\phi _0}} \right)} \right\|_1^2 + } \right.\\ \left. {\left\| {\left( {{\psi _0}} \right)} \right\|_1^2} \right) \end{array} $ (37)

联立式(36)、(37),得到先验估计式(27)。

由局部存在定理和先验估计可得到方程(18)的全局解的存在性。类似于文献[8]中的证明过程,最终得到解的渐近行为式(17)。综上所述,定理1得证。

4 结束语

本文运用经典能量方法,通过构造能量函数并结合非线性单调算子方法,解决了方程本身的压力非凸性和黏性非线性的问题,得到了解的全局估计和渐近行为。

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