不可压微极流方程组的稳定性研究
Research on the Stability of Incompressible Micropolar Equations
DOI: 10.12677/pm.2025.155169, PDF, HTML, XML,   
作者: 侯 洁:成都理工大学数学科学学院,四川 成都
关键词: 微极流方程组稳定性频率分解Micropolar Equations Stability Frequency Decomposition
摘要: 本文研究了不可压微极流方程组的稳定性问题。论文首先介绍了微极流方程组的背景以及它的研究现状,给出了本文的研究内容与结论。其次给出了论文相关的不等式及相关的引理。最后证明了论文的主要定理,通过频率分解方法得到了方程组在平衡态附近扰动后解的稳定性。
Abstract: This paper investigates the stability problem of incompressible micropolar equations. It begins by introducing the background of micropolar equations and their current research status, presenting the research content and conclusions of this study. Subsequently, relevant inequalities and lemmas pertinent to the paper are provided. Finally, the main theorem of the paper is proved, and the stability of solutions to the equations following a perturbation near the equilibrium state is obtained through the method of frequency decomposition.
文章引用:侯洁. 不可压微极流方程组的稳定性研究[J]. 理论数学, 2025, 15(5): 203-211. https://doi.org/10.12677/pm.2025.155169

1. 引言

微极流体是具有微观结构的流体。它们属于一类具有非对称应力张量的流体,我们称之为极性流体,并且作为一种特殊情况,包括经典流体的公认Navier-Stokes模型,我们称其为普通流体。从物理上讲,微极流体可以表示由悬浮在粘性介质中的刚性、随机取向(或球形)颗粒组成的流体,其中流体颗粒的变形被忽略。

微极流模型由Eringen (参见[1])在Electrodynamics of Continua II: Fluids and Complex Media中引入,用于描述物质的微旋转运动和旋转惯性。在物理学中,微极性流体可以表示由悬浮在粘性介质中的刚性、随机取向(或球形粒子)组成的流体,揭示了它们在物理学中的许多潜在应用(参见[2]-[4])。此外,由于粘性微极性流体微观结构的非牛顿性和不对称性,它不能用Navier-Stokes方程来描述(参见[5]-[8])。因此,该模型比Navier-Stokes方程更适合描述由偶极元件组成的各种复杂流体的运动,如悬浮液、动物血液和液晶(参见[9]-[11])。

微极和磁微极方程等密切相关方程的适定性问题最近在数学流体界引起了广泛关注(参见[12]-[17])。

更多的研究集中在具有耗散的微极方程上,该方程自然地将无粘微极方程与完全耗散微极方程联系起来(参见[18]-[22])。

α=β=1 时,董柏青老师和陈志敏老师(参见[23])得到了二维微极方程的整体存在性和唯一性,以及尖锐的代数时间衰减率。基本的数学问题,如解的整体正则性,在高维空间(维度大于或等于3)中得到了广泛的研究,并获得了许多有趣的结果(参见[24]-[28])。三维不可压缩微极流体方程光滑解的整体适应度是一个具有挑战性的问题。叶海龙老师等人(参见[29])研究了仅具有速度耗散的三维微极流体方程解的适应度和大时间行为。通过探索两个相关线性化问题的高阶正则性,研究了整体弱解和正则小解。由于对方程特殊结构的巧妙观察,他们引入了另一种分析论证,并详细描述了弱解和正则小解的大时间衰减率。李洪敏老师(参见[30])讨论了任何三维广义微极流体方程的整体弱解,其中 α1 β>1 对应于任何

L 2 初始数据。事实上,当 α 5 4 β>1 时,如果初始值足够平滑,则它们的弱解是完全经典的解。刘生全老师(参见[31])研究了具有部分粘性和阻尼的三维不可压缩微极性流体方程的柯西问题,并利用一般初值建立了解的整体存在性和唯一性。在他们的研究中,他们旨在证明 α 3 2 的不可压缩微极流方程的整体存在性和唯一性。在他们的研究中,与Navier-Stokes方程相比,他们更关注速度 α 3 2 和微转速 w 之间的耦合。

具有分数耗散和角粘性的n维不可压缩微极方程可以写成

{ u t +uu+( μ+χ ) ( Δ ) α u+P=2χ×w, w t +uw+γ ( Δ ) β w+4χwκw=2χ×u, u=0, u( x,0 )= u 0 ,w( x,0 )= w 0 ,

其中 α β 是非负实参数,未知数 u w 分别表示速度和角速度。 P 代表压力。正常数 μ χ 分别是运动粘度和涡流粘度,而 γ>0 κ>0 是角粘度。

( Δ ) α 根据 n 中的傅里叶变换定义如下:

( Δ ) α f ^ ( ξ )= | ξ | 2α f ^ ( ξ ),

其中 ξ n

傅立叶变换可以通过消去空间导数,将某些类型的偏微分方程转换为常微分方程。这种方法的本质是将方程从物理空间转换到频域,从而简化求解过程。

本文研究的内容是微极流方程组在n维区域上,当初始值足够小时如 ( u 0 , w 0 ) 2 ε 2 ,推出方程的解是否满足 ( u,w ) 2 C ε 2 ,在此基础上研究解的衰减估计。论文结论的证明方法参考了文献[11] [30] [31]。为方便计算,在论文的证明中令方程组系数都为1,即 ( μ+χ )=1 χ= 1 2 γ=κ=1 ,所以本文研究的微极流方程组如下:

{ u t +uu+ ( Δ ) α u+P=×w, w t +uw+ ( Δ ) β w+2ww=×u, u=0, u( x,0 )= u 0 ,w( x,0 )= w 0 .

频率分解方法在微极流方程组的稳定性分析中发挥了重要作用,通过将时域问题转化为频域问题,揭示了系统的内在稳定性机制。该方法在理论研究和工程应用中均具有重要价值,为理解复杂流体的动力学行为提供了有力工具。未来,随着计算技术和数学方法的不断发展,频率分解方法有望在微极流体力学领域发挥更大的作用。

用频率分解的方法我们可以得到本文的结论:

定理1.1在 0<α,β<1 的情况下研究柯西问题即上面的方程组,假设初始值 ( u 0 , w 0 ) L 2 ,并且 u 0 =0 ,那么对于柯西问题,存在一个弱解 ( u,w ) 而且满足

lim t ( u,w )( t ) L 2 =0.

2. 预备知识

2.1. 基本不等式

Cauchy不等式:

ab a 2 2 + b 2 2 , ( a,b ).

Young不等式:令 1<p,q< 1 p + 1 q =1 则有

ab a p p + b q q , ( a,b>0 ).

Hölder不等式:假设 s=θ s 0 +( 1θ ) s 1 s=θ s 0 +( 1θ ) s 1 ,如果 s=θ s 0 +( 1θ ) s 1 wL ,则

| uw |dx u L p w L q .

插值不等式:令 s 0 s s 1 ,则对 s=θ s 0 +( 1θ ) s 1 ,有

Λ s u L 2 Λ s 0 u L 2 θ Λ s 1 u L 2 1θ .

Gronwall不等式:设 η( t ) [ 0,T ] 上的非负绝对连续函数,满足微分不等式

η ( t )ϕ( t )η( t )+ψ( t ),

其中 ϕ( t ) ψ( t ) [ 0,T ] 上的非负可求和函数,那么对于所有 [ 0,T ]

η( t ) e 0 t ϕ( s )ds [ η( 0 )+ 0 t ψ( s )ds ].

Hausdorff-Young不等式:如果 f L p , 1p2 并且 1 p + 1 q =1 ,则 f ^ L q

f ^ L q f L p .

2.2. 基本引理

引理2.1 (交换子估计式)令 1<p<, s>0 并且 1 p = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2 ,则对任意的 f W 1, p 1 W s, q 2 g L p 2 W s1, q 1 ,存在一个正常数 C 使得

Λ s ( fg )fΛg L p C( f L p 1 Λ s1 g L q 1 + g L p 2 Λ s f L q 2 ).

此外,如果 f L p 1 W s, q 2 g L p 2 W s, q 1 ,则存在一个正常数 C 使得

Λ s ( fg ) L p C( f L p 1 Λ s1 g L q 1 + g L p 2 Λ s f L q 2 ).

引理2.2 (Plancherel定理)如果 f L 1 L 2 ,则 f ^ L 2

f ^ L 2 = f L 2 .

3. 稳定性证明

本文研究的方程组为:

{ u t +uu+ ( Δ ) α u+P=×w, w t +uw+ ( Δ ) β w+2ww=×u, u=0, u( x,0 )= u 0 ,w( x,0 )= w 0 . (1)

定理3.1在 0<α,β<1 的情况下研究柯西问题(1),假设初始值 ( u 0 , w 0 ) L 2 ,并且 u 0 =0 ,那么对于柯西问题(1),存在一个弱解 ( u,w ) 而且满足

lim t ( u,w )( t ) L 2 =0. (2)

证明:首先,对方程组(1)应用Fourier变换,可以得到

u ^ t ( ξ,t )+ | ξ | 2α u ^ ( ξ,t )= ×w ^ ( ξ,t ) uu ^ ( ξ,t ) P ^ ( ξ,t ) G 1 ( ξ,t ) (3)

w ^ t ( ξ,t )+( | ξ | 2β +2+ξξ ) w ^ ( ξ,t )= ×u ^ ( ξ,t ) uw ^ ( ξ,t ) G 2 ( ξ,t ). (4)

因为 sup ξ n | f ^ ( ξ ) | f L 1 ,因此可以估计 G 1 ( ξ,t ) G 2 ( ξ,t ) 的有界情况:

| ×w ^ ( ξ,t ) |C| ξ || w ^ |, (5)

| uu ^ ( ξ,t ) |=| ( uu ) ^ ( ξ,t ) |C| ξ | uu L 1 C| ξ | u L 2 2 . (6)

接下来估计 P ^ ( ξ,t ) ,对方程组(1)的第一个方程应用散度算子,可以得到

P= ( Δ ) 1 ( ×wuu ) = ( Δ ) 1 ( ×w ) ( Δ ) 1 ( uu ) = ( Δ ) 1 ( ×w ) ( Δ ) 1 [ ( uu ) ], (7)

因此得到

| P ^ ( ξ,t ) |C( | ξ || w ^ |+| ξ | uu L 1 )C| ξ |( | w ^ |+ u L 2 2 ). (8)

用类似的方法, G 1 ( ξ,t ) G 2 ( ξ,t ) 剩余的项的有界情况为:

| ×u ^ ( ξ,t ) |C| ξ || u ^ |, (9)

| uw ^ ( ξ,t ) |C| ξ |( u L 2 2 + w L 2 2 ). (10)

然后对方程(3)和(4)在 [ 0,t ] 上积分,可以得到

u ^ ( ξ,t )= u ^ 0 ( ξ ) e | ξ | 2α t + 0 t e | ξ | 2α ( tτ ) G 1 ( ξ,τ )dτ (11)

w ^ ( ξ,t )= w ^ 0 ( ξ ) e ( |ξ | 2β +2|ξ | 2 )t + 0 t e ( |ξ | 2β +2|ξ | 2 )( tτ ) G 2 ( ξ,τ )dτ . (12)

方程组(1)的基本能量估计式为

d dt ( u,w )( t ) L 2 2 + ( Λ α u, Λ β w ) L 2 2 + w L 2 2 + w L 2 2 0, (13)

积分可以得到

( u,w )( t ) L 2 2 ( u 0 , w 0 ) L 2 2 . (14)

把上面 G 1 ( ξ,t ) 的估计式代入(11),可以得到

| u ^ ( ξ,t ) || u ^ 0 ( ξ ) | e | ξ | 2α t + 0 t e | ξ | 2α ( tτ ) | G 1 ( ξ,τ ) |dτ | u ^ 0 ( ξ ) |+ 0 t e | ξ | 2α ( tτ ) C| ξ |( | w ^ |+ u L 2 2 + w L 2 2 )dτ | u ^ 0 ( ξ ) |+C| ξ |( u 0 L 2 2 + w 0 L 2 2 ) 0 t e | ξ | 2α ( tτ ) dτ +C| ξ | 0 t e | ξ | 2α ( tτ ) | w ^ |dτ | u ^ 0 ( ξ ) |+Ct| ξ |+C| ξ | 0 t | w ^ |dτ . (15)

类似的,把 G 2 ( ξ,t ) 的估计式代入(12),可以得到

| w ^ ( ξ,t ) || w ^ 0 ( ξ ) |+Ct| ξ |+C| ξ | 0 t | u ^ |dτ . (16)

另外,利用Plancherel定理,可以得到

Λ α u L 2 2 = n | ξ | 2α | u ^ ( ξ,t ) | 2 dξ |ξ|ρ | ξ | 2α | u ^ ( ξ,t ) | 2 dξ ρ 2α |ξ|ρ | u ^ ( ξ,t ) | 2 dξ = ρ 2α ( u L 2 2 |ξ|ρ | u ^ ( ξ,t ) | 2 dξ ) (17)

Λ β w L 2 2 θ 2β ( w L 2 2 |ξ|θ | w ^ ( ξ,t ) | 2 dξ ). (18)

结合(15)和(16),可以估计出(17)和(18)的右边最后一项:

|ξ|ρ | u ^ ( ξ,t ) | 2 dξ |ξ|ρ ( | u ^ 0 ( ξ ) |+Ct| ξ |+C| ξ | 0 t | w ^ |dτ ) 2 dξ |ξ|ρ | u ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 ρ n+2 +C ρ n+2 |ξ|ρ ( 0 t | w ^ |1dτ ) 2 dξ |ξ|ρ | u ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 ρ n+2 +Ct ρ n+2 0 t |ξ|ρ | w ^ | 2 dξ dτ |ξ|ρ | u ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 ρ n+2 +Ct ρ n+2 0 t w L 2 2 dτ |ξ|ρ | u ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 ρ n+2 +C ( 1+t ) 2 ρ n+2 |ξ|ρ | u ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 ρ n+2 (19)

|ξ|θ | w ^ ( ξ,t ) | 2 dξ |ξ|θ | w ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 θ n+2 . (20)

最后,把上述估计式代入不等式(13)中可以得出

d dt ( u,w )( t ) L 2 2 + ρ 2α u( t ) L 2 2 + θ 2β w( t ) L 2 2     ρ 2α |ξ|ρ | u ^ ( ξ,t ) | 2 dξ + θ 2β |ξ|θ | w ^ ( ξ,t ) | 2 dξ     ρ 2α |ξ|ρ | u ^ 0 ( ξ ) | 2 dξ + θ 2β |ξ|θ | w ^ 0 ( ξ ) | 2 dξ +C ( 1+t ) 2 ( ρ n+2+2α + θ n+2+2β ). (21)

ρ= ( k ( 1+t )ln( 1+t ) ) 1 2α (22)

θ= ( k ( 1+t )ln( 1+t ) ) 1 2β , (23)

其中 k>max{ 1+ n+2 2α ,1+ n+2 2β }.

将(22)和(23)代入(21),可以得到

d dt ( ln k ( 1+t ) ( u,w )( t ) L 2 2 )     k ln k1 ( 1+t ) ( 1+t ) |ξ|max{ ρ,θ } ( | u ^ 0 ( ξ ) | 2 + | w ^ 0 ( ξ ) | 2 )dξ         +C ln k ( 1+t ) ( 1+t ) 2 [ ( k ( 1+t )ln( 1+t ) ) 1+ n+2 2α + ( k ( 1+t )ln( 1+t ) ) 1+ n+2 2β ]     H 1 ( t )+ H 2 ( t )+ H 3 ( t ). (24)

将上述有界估计式与 ln k ( 1+t ) 做积,可以得到

对(24)在 [ 0,t ] 上积分,可以得到

( u,w ) L 2 2 ln k ( 1+t ) ( u 0 , w 0 ) L 2 2 + ln k ( 1+t ) 0 t H 1 ( τ )dτ   + ln k ( 1+t ) 0 t H 2 ( τ )dτ + ln k ( 1+t ) 0 t H 3 ( τ )dτ . (25)

显然,

lim t ln k ( 1+t ) ( u 0 , w 0 ) L 2 2 =0. (26)

对于(25)右边的第二项,可以估计得出

0 t H 1 ( τ )dτ =( ln k ( 1+t )1 ) |ξ|max{ ρ,θ } ( | u ^ 0 ( ξ ) | 2 + | w ^ 0 ( ξ ) | 2 )dξ . (27)

如果 t t θ 会趋于0。所以在定理3.1的条件下,可以得到

lim t |ξ|max{ ρ,θ } ( | u ^ 0 ( ξ ) | 2 + | w ^ 0 ( ξ ) | 2 )dξ =0, (28)

这表明

lim t ln k ( 1+t ) 0 t H 1 ( τ )dτ =0. (29)

对于(25)右边的第三项,可以估计得出

0 t H 2 ( τ )dτ C 0 t ln k ( 1+τ ) ( 1+τ ) 2 ( k ( 1+t )ln( 1+t ) ) 1+ n+2 2α dτ C 0 t ln k1 n+2 2α ( 1+τ ) ( 1+τ ) n+2 2α 1 dτ C 0 t ln k1 n+2 2α ( 1+τ ) 1+τ dτ C( ln k n+2 2α ( 1+t )1 ), (30)

所以

lim t ln k ( 1+t ) 0 t H 2 ( τ )dτ =0. (31)

同样的,可以估计出最后一项为

lim t ln k ( 1+t ) 0 t H 3 ( τ )dτ =0. (32)

最后,结合上述的估计式,可以从(25)得到(2),即

lim t ( u,w )( t ) L 2 =0.

到此,完成了定理3.1的证明。

致 谢

由衷感谢我的导师马老师。从论文选题到框架构建,从方法推导到结果验证,马老师始终以深厚的学术造诣和严谨的治学态度为我指明方向。在科研陷入瓶颈时,他总能以独到的视角启发我突破思维定式;在写作遇到困惑时,他耐心细致的批注与建议让我受益匪浅。师恩如海,铭记于心。

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