不定常度量测度空间上Boltzmann熵和Cheeger能量的梯度流
Gradient flow for the Boltzmann entropy and Cheeger's energy on time-dependent metric measure spaces
摘要：
我们引入了动态梯度流的概念，不定常度量空间以及不定常希尔伯特空间。我们证明了一类不定常能量泛函在这两种情况下解的存在性。特别是我们感兴趣的情况下，当底层空间是度量测度空间和能量泛函是由不定常玻尔兹曼熵或不定常Cheeger的能量。在每个静态空间满足里奇曲率下界的假设下，证明了熵梯度流的存在唯一性，并将其与Kopfer/Sturm在“度量测度空间的里奇流“中引入的前向伴随热流相一致.我们确定的梯度流的不定常Cheeger的能量与Kopfer/Sturm介绍的热流。
英文摘要：
We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time dependent energy functionals in both settings. In particular we are interested in the case when the underlying spaces are metric measure spaces and the energy functional is given by the time-dependent Boltzmann entropy or the time-dependent Cheeger's energy. Under the assumption that each static space satisfies a lower Ricci curvature bound, we prove existence and uniqueness of the entropy gradient flow and we identify it with the forward adjoint heat flow introduced in "Super-Ricci Flows for Metric Measure Spaces" by Kopfer/Sturm. We identify the gradient flow for the time-dependent Cheeger's energy with the heat flow introduced by Kopfer/Sturm.
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