Li-Yau type differential Harnack inequalities and applications for nonlocal diffusion equations

Li-Yau型微分Harnack不等式及其非局部扩散方程的应用

• 批准号: 355354916
• 负责人: Professor Dr. Rico Zacher
• 金额: --
• 依托单位: Institut für Angewandte Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/355354916?language=en
• 关键词: Li; Yau; type; differential; Harnack

The main objective of the project is to derive Li-Yau type differential Harnack inequalities for a wide class of non-local diffusion equations which includes problems on infinite discrete structures (graphs) where arbitrary long jumps are possible, equations in Euclidean space involving a fractional Laplacian and problems with fractional dynamics. A major difficulty is the failure of the classical chain rule for the involved non-local operators. A crucial part of the project consists in proving new variants of the curvature-dimension (CD) inequality. Our previous work shows that the classical Gamma-calculus is not suitable and suggests to consider CD-inequalities involving a so-called (non-quadratic) CD-function, which depends on the non-local operator resp. the underlying structure. By means of the new Li-Yau type inequalities we want to derive new Harnack inequalities resp. find new and much simpler (purely analytic) proofs for such results (e.g. in the space-fractional case). We also want to study applications of the (new) CD-inequalities in case of a positive lower curvature bound with regard to functional inequalities (Poincare, Sobolev and log-Sobolev inequalites), which in turn will lead to further results on the long-time behaviour of the solutions to the diffusion equations.

该项目的主要目标是为一类广泛的非局部扩散方程导出 Li-Yau 型微分哈纳克不等式，其中包括无限离散结构（图）上的问题，其中任意长跳跃是可能的，欧几里得空间中涉及分数的方程拉普拉斯算子和分数动力学问题。一个主要困难是经典链式法则对于所涉及的非本地运营商的失败。该项目的一个关键部分在于证明曲率维数（CD）不等式的新变体。我们之前的工作表明经典的伽马演算并不合适，并建议考虑涉及所谓（非二次）CD 函数的 CD 不等式，该函数取决于非局部算子。底层结构。通过新的 Li-Yau 型不等式，我们希望推导出新的 Harnack 不等式。为此类结果找到新的且更简单的（纯分析的）证明（例如在空间分数情况下）。我们还想研究（新的）CD 不等式在函数不等式（庞加莱、索博列夫和对数索博列夫不等式）方面的正下曲率界的情况下的应用，这反过来将导致长期的进一步结果扩散方程解的时间行为。