Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
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DOI：
10.1515/anona-2020-0024
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发表时间：
2018-04
影响因子：
4.2
通讯作者：
S. Cooper;A. Savostianov
中科院分区：
文献类型：
--
作者：
S. Cooper;A. Savostianov
对半线性各向异性阻尼波动方程∂t2uε+y∂tuε−divaxε∇uε+f(uε)=g, $\begin{array}{} \displaystyle \partial_t ^2u^\varepsilon + y \partial_t u^\varepsilon-\operatorname{div} \left(a\left( \tfrac{x}{\varepsilon} \right)\nabla u^\varepsilon \right)+f(u^\varepsilon)=g, \end{array}$在有界域Ω≡∈3上的全局𝓐ε和指数𝓜ε吸引子进行了均匀化。建立了轨迹uε(t)与其均匀化轨迹u0(t)之间的有序锐估计。这些估计是根据椭圆算子divaxε∇$\begin{array}{} \displaystyle \operatorname{div}\left(a\left( \tfrac{x}{\varepsilon} \right)\nabla \right) \end{array}$和它的均匀化极限div （ah∇）的解之间的算子范数差给出的。因此，建立了各向异性吸引子与其均质对应物𝓐0和𝓜0之间的豪斯多夫距离的范数分辨估计。这些结果暗示了在X = L2（Ω） × H-1 （Ω）和X = （Cβ（Ω））2的空间中，distX（𝓐ε，𝓐0）≤cε通知和distX(Mε,M0)≤cε通知$\begin{array}{} \displaystyle \operatorname{dist}^s_X(\mathcal M^\varepsilon, \mathcal M^0) \le C \varepsilon^\varkappa \end{array}$的误差估计。在自然能量空间<s:2>:= H01 $\begin{array}{} \displaystyle H^1_0 \end{array}$ （Ω） × L2（Ω）中，建立了误差估计dist <e:1>（𝓐ε, Tε𝓐0）≤cε乳$\begin{array}{} \displaystyle C \sqrt{\varepsilon}^\varkappa \end{array}$和distEs (Mε,Tε m0)≤cε乳$\begin{array}{} \displaystyle \operatorname{dist}^s_\mathcal{E}(\mathcal M^\varepsilon, \text{T}_\varepsilon \mathcal M^0) \le C \sqrt{\varepsilon}^\varkappa \end{array}$，其中Tε是由渐近展开给出的匀化吸引子的一阶修正。我们的结果应用于Dirchlet、Neumann和周期边界条件。
Abstract Homogenisation of global 𝓐ε and exponential 𝓜ε attractors for the damped semi-linear anisotropic wave equation ∂t2uε+y∂tuε−divaxε∇uε+f(uε)=g, $\begin{array}{} \displaystyle \partial_t ^2u^\varepsilon + y \partial_t u^\varepsilon-\operatorname{div} \left(a\left( \tfrac{x}{\varepsilon} \right)\nabla u^\varepsilon \right)+f(u^\varepsilon)=g, \end{array}$ on a bounded domain Ω ⊂ ℝ3, is performed. Order-sharp estimates between trajectories uε(t) and their homogenised trajectories u0(t) are established. These estimates are given in terms of the operator-norm difference between resolvents of the elliptic operator divaxε∇ $\begin{array}{} \displaystyle \operatorname{div}\left(a\left( \tfrac{x}{\varepsilon} \right)\nabla \right) \end{array}$ and its homogenised limit div (ah∇). Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts 𝓐0 and 𝓜0 are established. These results imply error estimates of the form distX(𝓐ε, 𝓐0) ≤ Cεϰ and distXs(Mε,M0)≤Cεϰ $\begin{array}{} \displaystyle \operatorname{dist}^s_X(\mathcal M^\varepsilon, \mathcal M^0) \le C \varepsilon^\varkappa \end{array}$ in the spaces X = L2(Ω) × H–1(Ω) and X = (Cβ(Ω))2. In the natural energy space 𝓔 := H01 $\begin{array}{} \displaystyle H^1_0 \end{array}$(Ω) × L2(Ω), error estimates dist𝓔(𝓐ε, Tε 𝓐0) ≤ Cεϰ $\begin{array}{} \displaystyle C \sqrt{\varepsilon}^\varkappa \end{array}$ and distEs(Mε,TεM0)≤Cεϰ $\begin{array}{} \displaystyle \operatorname{dist}^s_\mathcal{E}(\mathcal M^\varepsilon, \text{T}_\varepsilon \mathcal M^0) \le C \sqrt{\varepsilon}^\varkappa \end{array}$ are established where Tε is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.