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2022 Volume 44 Issue 3
Article Contents

TAN Qingwei, ZHU Chaosheng. Some Estimates for the 3D Non-autonomous Linearization Kelvin-Voigt-Brinkman-Forchheimer Equations with Singularly Oscillating Forces[J]. Journal of Southwest University Natural Science Edition, 2022, 44(3): 125-129. doi: 10.13718/j.cnki.xdzk.2022.03.015
Citation: TAN Qingwei, ZHU Chaosheng. Some Estimates for the 3D Non-autonomous Linearization Kelvin-Voigt-Brinkman-Forchheimer Equations with Singularly Oscillating Forces[J]. Journal of Southwest University Natural Science Edition, 2022, 44(3): 125-129. doi: 10.13718/j.cnki.xdzk.2022.03.015

Some Estimates for the 3D Non-autonomous Linearization Kelvin-Voigt-Brinkman-Forchheimer Equations with Singularly Oscillating Forces

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  • Corresponding author: ZHU Chaosheng
  • Received Date: 17/01/2021
    Available Online: 20/03/2022
  • MSC: O175.29

  • In this paper, we mainly study the three-dimensional non-autonomous linear Kelvin-Voigt-Brinkman-Forchheimer equation with singular oscillating force. Firstly, the general estimation of the auxiliary linear equation with time-dependent external force was carried out, and then the singular oscillation was derived from the results of these general estimates.
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Some Estimates for the 3D Non-autonomous Linearization Kelvin-Voigt-Brinkman-Forchheimer Equations with Singularly Oscillating Forces

    Corresponding author: ZHU Chaosheng

Abstract: In this paper, we mainly study the three-dimensional non-autonomous linear Kelvin-Voigt-Brinkman-Forchheimer equation with singular oscillating force. Firstly, the general estimation of the auxiliary linear equation with time-dependent external force was carried out, and then the singular oscillation was derived from the results of these general estimates.

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  • Ω$\mathbb{R}^{3}$是一个边界光滑的有界域. 本文主要研究Ω上具有奇异振荡力的三维非自治Kelvin-Voigt-Brinkman-Forchheimer方程[1-4]

    其中:a$\mathbb{R}$b>0,r∈[1,∞),μ>0是流体的运动粘度,α是流体弹性的表征参数,函数u=u(xt)=(u1(xt),u2(xt),u3(xt))表示流体的速度,p=p(xt)表示压力. 当ab=0时,方程(1)为带奇异振荡力的Navier-Stokes-Voigt方程[5-10];当α=0时,方程(1)为带奇异振荡力的Brinkman-Forchheimer方程[11-15];当abα=0时,方程(1)为带奇异振荡力的Navier-Stokes方程[16-17].

    结合方程(1),我们考虑如下平均Kelvin-Voigt-Brinkman-Forchheimer方程:

    记函数

    其中函数f0(xs),f1(xs)∈Lb2($\mathbb{R}$H),Lb2($\mathbb{R}$H)⊆Lloc2(RH)是平移有界函数空间,即有

    其中常数M0M1≥0,定义

    综上有

    引入函数空间

    这里clXS表示S在空间X的闭包,HV是可分的Hilbert空间. 令H′H的对偶空间,V′V的对偶空间,有VH=H′V′,其中嵌入都是连续且稠密的. HV分别具有如下内积和范数:

    用〈·,·〉表示V′V之间的对偶集,用|·|p表示Lp(Ω)空间中的范数,用||·||E表示巴纳赫空间E中的范数. 字母C为常数.

    方程(1)的前两个等式,可以写成如下抽象形式

    A=-PΔ是Stokes算子,P是从L2(Ω)到H的Leray正交投影,有

    BV×VV′是双线性算子,有

    这里

    对于方程(2)的全局解的存在唯一性,可由文献[2]中的标准方法得到如下定理1.

    定理1  假设uτVf(xt)在Lloc2($\mathbb{R}$H)中平移紧,则方程(2)存在唯一解

    我们将考虑具有与时间相关的外力驱动的非自治辅助线性方程,对其进行一系列估计.

    定理2  假设K(t)∈Lloc2($\mathbb{R}$H),则方程(3)存在唯一解

    且满足不等式

      用Galerkin逼近法,可以推出解的存在,将方程(3)与AY(t)作内积,可得

    由不等式(4)可得

    即有

    对不等式(5)在[τt]上积分,得

    易得

    将方程(3)与Y(t)作内积,可得

    对不等式(6)在[tt+1]上积分,再运用Poincaré不等式得

    即有

    定理2证毕.

    定理3  设k(t)∈Lloc2($\mathbb{R}$H),存在常数l满足

    则带奇异振荡力的线性方程

    的解X(t)满足不等式

    其中CK(t)无关.

      首先记

    则由(7)式可推出

    由积分中值定理和定理2可得

    现令

    X(τ)=0,得

    方程(8)在[τt]上积分可得

    综上所述可得

    所以

    由不等式(10)可得不等式(9)成立,定理3证毕.

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