WEIGHTED ENERGY-DISSIPATION FUNCTIONALS FOR GRADIENT FLOWS
WEIGHTED ENERGY-DISSIPATION FUNCTIONALS FOR GRADIENT FLOWS
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梯度流的加权能量耗散函数
DOI：
10.1051/cocv/2009043
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
U. Stefanelli
中科院分区：
文献类型：
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作者：
A. Mielke;U. Stefanelli
我们利用Mielke和Ortiz （ESAIM: COCV 14(2008) 494-516）提出的加权能量耗散泛函研究抽象演化的全局时变分方法。我们特别关注希尔伯特空间中的梯度流。主要结果是这些泛函的极小值和近似极小值收敛于梯度流的唯一解。给出了快速收敛速率，并将收敛分析与时间离散化相结合。给出了该理论在各类抛物型偏微分方程问题中的应用。我们特别关注了两个微观结构演变的例子，分别来自（S. Conti）和M. Ortiz， J. Mech。理论物理。固体56(2008)1885-1904。
We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke and Ortiz (ESAIM: COCV 14 (2008) 494-516). In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time- discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from (S. Conti and M. Ortiz, J. Mech. Phys. Solids 56 (2008) 1885-1904.).