A Li-Yau inequality for the fractional heat equation
A talk in the Nonlocal Equations: Analysis and Numerics series by
Frederic Weber from Ulm
| Abstract: | $$\text{(joint work with Adrian Spener and Rico Zacher)}$$
The celebrated Li-Yau inequality states that $- \Delta (\log u) \leq \frac{d}{2t}$ holds for any positive solution $u$ to the diffusion equation $\partial_t u = \Delta u$ on $(0,\infty)\times \mathbb{R}^d$. This talk is motivated by the following question: Does a Li-Yau inequality also holds for the fractional heat equation on $\mathbb{R}^d$? In the classical situation there exist two approaches to deduce the Li-Yau inequality. One uses the Bakry-Émery theory and the fact that the Laplacian satisfies the curvature-dimension condition $CD(0,d)$. The other makes use of integral representation of positive solutions to the heat equation and reduces the problem to the heat kernel. In this talk we discuss both approaches from the non-local perspective. We observe that the Bakry-Émery curvature-dimension condition with finite dimension term fails to be true for the fractional Laplacian. On the other hand, we establish a reduction principle that consists in reducing the answer to our motivating question to the fractional heat kernels. As a result, we obtain a Li-Yau inequality for the fractional heat equation with the same time-behavior as in the classical case. Within the CRC this talk is associated to the project(s): A7 |