度量空间中梯度流的Kurdyka-Lojasiewicz-Simon不等式
Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces
摘要：
本文致力于在熵和度量意义下研究度量空间中梯度流的平衡趋势，建立梯度流的衰减率和有限灭绝时间，以及刻画Lyapunov稳定平衡点，提供新的工具和方法.此外，我们概述了著名的熵熵生产不等式中使用的动力学理论是小于一个整体Kurdyka-Lojasiewicz-Simon不等式。这将两个不同的领域，即代数几何与动力学理论。作为本文中开发的工具的应用，我们得到了以下结果：-新的上界的消亡时的梯度流与总变流。- 如果度量空间是p-Wasserstein空间，则新的HWI-，Talagrand-，和对数Sobolev不等式得到与非线性扩散问题建模漂移，潜在的和相互作用现象的泛函。- 结果表明，这些不等式等价于Kurdyka-Lojasiewicz-Simon不等式，因此，它们意味着具有衰减率或在有限时间内到达的梯度流趋于平衡.
英文摘要：
This paper is dedicated to providing new tools and methods for studying the trend to equilibrium of gradient flows in metric spaces in the entropy and metric sense, to establish decay rates, finite time of extinction, and to characterize Lyapunov stable equilibrium points. In addition we outline that the celebrated Entropy-Entropy production inequality used in kinetic theory is nothing less than a global Kurdyka-Lojasiewicz-Simon inequality. This links two different areas, namely, algebraic geometry with kinetic theory. As an application of the tools developed in this paper, we obtain the following results: - New upper bounds on the extinction time of gradient flows associated with the total variational flow. - If the metric space is the p-Wasserstein space, then new HWI-, Talagrand-, and logarithmic Sobolev inequalities are obtained for functionals associated with nonlinear diffusion problems modeling drift, potential and interaction phenomena. - It is shown that these inequalities are equivalent to the Kurdyka-Lojasiewicz-Simon inequality and hence, they imply trend to equilibrium of the gradient flows with decay rates or arrival in finite time.
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