AI 中文总结
研究具有迁移率的非线性福克 - 普朗克方程的瓦瑟斯坦梯度流，通过建立对数索伯列夫不等式、HWI不等式等，利用加权齐次索伯列夫范数与$W_h$度量关系及吉拉诺夫定理，得出自由能泛函收敛速率等结果。
AI 中文摘要
对于具有迁移率的非线性福克 - 普朗克方程，瓦瑟斯坦梯度流结构由广义相对熵作为能量泛函以及修正的瓦瑟斯坦度量$W_h$作为相关度量结构来描述。本文研究了由迁移率引起的非线性效应并建立了相应不等式。对于非线性扩散，建立了对数索伯列夫不等式，得出自由能泛函的收敛速率和塔拉格兰德不等式。通过进一步利用加权齐次索伯列夫范数与$W_h$度量之间的关系，推导了一个将相对熵、$W_h$度量和费希尔信息联系起来的HWI不等式。在迁移率依赖漂移和线性扩散的情况下，应用吉拉诺夫定理也得到了$W_h$度量下的收敛速率。
英文摘要
For nonlinear Fokker-Planck equations with mobility, the Wasserstein gradient flow structure is described by the generalized relative entropy as the energy functional and the modified Wasserstein metric $W_h$ as the associated metric structure. This work investigates the nonlinear effects induced by mobility and establishes the corresponding inequalities. For nonlinear diffusion, we establish a logarithmic Sobolev inequality, which yields the convergence rate of the free energy functional and the Talagrand inequality. By further exploiting the relationship between the weighted homogeneous Sobolev norm and the $W_h$ metric, we derive an HWI inequality relating the relative entropy, the $W_h$ metric, and the Fisher information. In the case of mobility dependent drift and linear diffusion, the convergence rate in the $W_h$ metric is also obtained by applying the Girsanov theorem.
