AI 中文总结
本文针对全空间带约束的拟线性Fokker-Planck方程，表征了可使Bakry-Émery型熵方法给出指数收敛的可容许熵，将线性情形的研究扩展到非线性情形，得到两类关键结果及相关不等式、收敛估计。
AI 中文摘要
本文研究全空间$\boldsymbol{\text{R}}^d$上带约束的拟线性Fokker-Planck方程的长时间行为，旨在表征所有相对熵泛函，使得Bakry-Émery型熵方法能给出所有解向唯一稳态（与初始条件质量相同）的指数收敛，我们称这类熵为可容许熵。收敛率由约束势的一致凸性参数决定，该研究将线性Fokker-Planck方程的类似研究（Bakry-Émery、Arnold-Markowich-Toscani-Unterreiter）扩展到非线性情形，除Jüngel-Carrillo-Markowich-Toscani-Unterreiter中使用的Ralston-Newman熵外，还为非线性情形推导了额外的泛函。两个关键结果是：表征那些对相应线性Fokker-Planck方程的所有可容许熵泛函均适用的非线性Fokker-Planck方程，反之亦然，即表征给定非线性项的所有可容许熵。对幂律非线性项的研究得到了多孔介质方程的一大类熵，但快扩散方程仅对应Ralston-Newman熵。额外结果包括：为本文的熵泛函推导新的广义Csiszár-Kullback不等式和广义对数Sobolev不等式，以及Fokker-Planck解的矩加权$L^1$收敛估计。
英文摘要
This paper is concerned with the large-time behavior of quasilinear Fokker-Planck equations with confinement on the whole space $\mathbb{R}^d$. It aims at characterizing all relative entropy functionals such that the entropy method à la Bakry-Émery yields exponential convergence of all solutions towards the unique steady state (with the same mass as the initial condition). We call such entropies admissible. The convergence rate is determined by the uniform convexity parameter of the confinement potential. As such, this program extends the analogous study of linear Fokker-Planck equations [Bakry-Émery, Arnold-Markowich-Toscani-Unterreiter] to the nonlinear case, and it derives additional functionals for the nonlinear case --- beyond the Ralston-Newman entropies used in [Jüngel-Carrillo-Markowich-Toscani-Unterreiter]. Two key results are the characterization of those nonlinear Fokker-Planck equations which admit all entropy functionals that are admissible for the corresponding linear Fokker-Planck equation, and vice versa, the characterization of all admissible entropies for a given nonlinearity. The latter quest for power-law nonlinearities yields a large family of entropies for the porous-medium equations, but only the Ralston-Newman entropy for the fast-diffusion equations. Additional results include the derivation of new generalized Csiszár-Kullback and generalized Log-Sobolev inequalities for our entropy functionals as well as moment-weighted $L^1$--convergence estimates for the Fokker-Planck solutions.
