Talk by Rico Zacher
On March 21st, 2022, Prof. Dr. Rico Zacher (University Ulm) gave a talk about "Li-Yau inequalities for general non-local diffusion equations via reduction to the heat kernel" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
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I will present a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. The approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we obtain a Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-\Delta)^{\beta/2}(\log u)\le C/t$, where $\beta\in (0,2)$. I will also show that this Li-Yau inequality allows to derive a Harnack inequality. The general result is further illustrated with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete graph. This is joint work with Frederic Weber (Münster).
Slides of the talk (PDF 299 KB)
