论文标题
在Renyi熵功率和Gagliardo-Nirenberg-Sobolev不平等现象上
On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds
论文作者
论文摘要
在本文中，我们证明了与Laplacian和Witten Laplacian相关的非线性扩散方程（NLDE）的Renyi熵能力在紧凑的Riemannian歧管上具有非阴性RICCI曲率或$ CD（0，M）$ - 条件和与时间相关的型和势相关的Meterics和Meterics Meterics和电位。由于Savaré和Toscani \ cite {st}在Euclidean空间上NLDE的凹入性上，我们的结果可以视为结果的自然扩展。此外，我们证明了Renyi熵功率的刚性模型是Einstein或Quasi-Einstein歧管，以及带有Hessian Solitons的特殊$（K，M）$ RICCI流动。受Lu-ni-vazquez-Villani \ cite {lnvv}的启发，我们证明了Aronson-Benilan在紧凑型Riemannian歧管上的nlde估计，带有$ cd（0，m）$ - 条件。我们还证明了NIW公式，该公式表明Renyi Entropy Power $ N_P $，$ p $ -th Fisher Information $ I_P $与与NLDE相关的$ W $ -Entropy的时间衍生物之间的固有关系。最后，我们证明了Renyi熵功率和Gagliardo-Nirenberg-Sobolev的熵等级不平等，这是具有非负RICCI曲率或$ CD（0，M）$ CD（0，M）$的完整Riemannian歧管上的不平等。
In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savaré and Toscani \cite{ST} on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein manifolds and a special $(K,m)$-Ricci flow with Hessian solitons. Inspired by Lu-Ni-Vazquez-Villani \cite{LNVV}, we prove the Aronson-Benilan estimates for NLDE on compact Riemannian manifolds with $CD(0,m)$-condition. We also prove the NIW formula which indicates an intrinsic relationship between the second order derivative of the Renyi entropy power $N_p$, the $p$-th Fisher information $I_p$ and the time derivative of the $W$-entropy associated with NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete Riemannian manifolds with non-negative Ricci curvature or $CD(0, m)$-condition and maximal volume growth condition.
