Two-scale homogenization of systems of nonlinear parabolic equations
Two-scale homogenization of systems of nonlinear parabolic equations
复制标题
非线性抛物型方程组的二尺度齐次化
DOI：
10.18452/17385
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Sina Reichelt
中科院分区：
文献类型：
--
作者：
Sina Reichelt
本文的目的是推导两种不同类型的非线性抛物方程系统的均匀化结果，即不同扩散长度尺度的反应-扩散系统和cahn - hilliard型方程。所考虑的抛物方程的系数函数以周期ε周期性振荡，其中参数ε表示特征微观和宏观长度尺度之间的比值。此外，系数可能以不连续的方式依赖于宏观尺度，因此允许真正的非均质性。考虑到更大的结构洞察力和更少的计算量，我们的目标是在ε趋于零时严格推导有效方程，从而使原始模型的解收敛于有效模型的解。为了考虑周期性微观结构以及不同的扩散长度尺度，我们采用了通过周期性展开的双尺度收敛方法。在论文的第一部分中，我们考虑了反应-扩散系统，其中某些物质的扩散长度尺度为O(1)阶，而另一些物质的扩散长度尺度为O（ε）阶。反应项是全局利普希茨连续的，但它们一般不是给定势的梯度。不同的扩散率伴随着紧性的损失，使得我们不能直接通过非线性项的极限，因为ε趋于零。基于强双尺度收敛的概念，我们证明了有效模型是一个依赖于宏观和微观尺度的双尺度反应扩散系统。在证明的第一步中，我们推导出带有误差项的gronwall型估计，在第二步中，我们控制这些误差，使ε趋于零。我们的方法提供了原始模型解收敛到有效模型解的显式速率。在第二部分中，我们考虑了具有位置相关移动和一般势的cahn - hilliard型方程。众所周知，经典的Cahn-Hilliard方程承认一个由λ-凸能量泛函和二次耗散势组成的梯度结构。利用这种梯度结构，我们利用变分不等式或能量耗散原理重新表述抛物方程。基于能量和耗散势的Γ-convergence，我们证明了相关梯度系统的演化Γconvergence，短e收敛性，从而在ε趋于零的极限下得到了有效（均质）系数的Cahn-Hilliard方程。此外，我们提供了一个示例势，使得相关的能量泛函不是λ-凸的，但我们通过能量耗散原理证明了e收敛性。
The aim of this thesis is to derive homogenization results for two different types of systems of nonlinear parabolic equations, namely reaction-diffusion systems involving different diffusion length scales and Cahn–Hilliard-type equations. The coefficient functions of the considered parabolic equations are periodically oscillating with period ε, where the parameter ε denotes the ratio between the charactersitic microscopic and macroscopic length scales. In addition, the coefficients depend in a possibly discontinuous manner on the macroscopic scale so that real heterogeneities are allowed. In view of greater structural insight and less computational effort, it is our aim to rigorously derive effective equations as ε tends to zero such that solutions of the original model converge to solutions of the effective model. To account for the periodic microstructure as well as for the different diffusion length scales, we employ the method of two-scale convergence via periodic unfolding. In the first part of the thesis, we consider reaction-diffusion systems, where for some species the diffusion length scale is of order O(1) and for other species it is of order O(ε). The reaction terms are globally Lipschitz continuous, however, they are in general not the gradient of a given potential. The different diffusivities accompany a loss of compactness such that we cannot pass directly to the limit as ε tends to zero with the nonlinear terms. Based on the notion of strong two-scale convergence, we prove that the effective model is a two-scale reaction-diffusion system depending on the macroscopic and the microscopic scale. In the first step of the proof, we derive Gronwall-type estimates with error terms, and in the second step, we control these errors as ε tends to zero. Our approach supplies explicit rates for the convergence of the solution of the original model to the solution of the effective model. In the second part, we consider Cahn–Hilliard-type equations with position-dependent mobilities and general potentials. It is well-known that the classical Cahn–Hilliard equation admits a gradient structure which consits of a λ-convex energy functional and a quadratic dissipation potential. Using this gradient structure, we reformulate the parabolic equation via variational inequalities or via the energy-dissipation principle. Based on the Γ-convergence of the energies and the dissipation potentials, we prove evolutionary Γconvergence, short E-convergence, for the associated gradient systems such that we obtain in the limit as ε tends to zero a Cahn–Hilliard equation with effective (homogenized) coefficients. Moreover, we provide one exemplary potential such that the associated energy functional is not λ-convex and yet we prove E-convergence via the energy-dissipation principle.
慢扩散非线性反应扩散系统的二尺度均质化
DOI：
10.3934/nhm.2014.9.353
发表时间：
2014
期刊：
Networks Heterog. Media
影响因子：
--
作者：
Mielke;Alexander;Reichelt;Thomas;Marita
通讯作者：
Marita
影响因子：
2.9
作者：
Kyrychko, Y. N.;Blyuss, K. B.;Schoell, E.
通讯作者：
Schoell, E.