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Yu-Zhao Wang

Publications and source records attributed to Yu-Zhao Wang.

7 recordsLinked to original sources
arXiv2026

Differential Harnack Estimates for the Filtration Equations on Riemannian Manifolds via Nash--Moser Iteration

We prove local differential Harnack estimates for smooth positive solutions of the Filtration Equations $u_t=\Delta F(u)$ on complete Riemannian manifolds in uniformly parabolic ranges. A Bochner--discriminant argument gives a coercive positive-part inequality, which is closed by Nash--Moser iteration. We derive Harnack and Liouville consequences, recover the Aronson-B\'enilan estimates for porous medium equation and fast diffusion equation, and describe a class of genuinely non-power filtration laws.

math.DG↗
arXiv2026

Gradient estimates and Liouville theorems for the \(\Phi\)-Laplacian equations on Riemannian manifolds

This paper establishes gradient estimates and Liouville-type theorems for the \(\Phi\)-Laplacian equation \(\Delta_{\Phi}(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = \Lambda_{\Phi}(u)\) on compact Riemannian manifolds. Using a nonlinear \(\Phi\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds under the lower bound assumption of Ricci curvature and suitable conditions on \(\Phi\) and \(G\), which leads to Liouville theorems for global solutions. For the parabolic case, we employ the maximum principle to derive gradient estimates on compact Riemannian manifolds, and subsequently obtain Liouville-type results. Our work provides a unified framework that generalizes prior results for \(p\)-harmonic functions and other quasilinear equations.

math.DG↗
arXiv2025

$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds

We first prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space, where ${1\over p}+{1\over q}=1$, $1< p, q<\infty$. The local existence, uniqueness and regularity of the Langevin deformation on the $L^q$-Wasserstein space over the Euclidean space and a compact Riemannian manifold are proved for $q\in [2, \infty)$. We further prove the $W$-entropy-information formula and the rigidity theorem for the Langevin deformation on the $L^q$-Wasserstein space over an $n$-dimensional complete Riemannian manifold with non-negative Ricci curvature, where $q\in (1,\infty)$.

math.PR↗
arXiv2023

Rigidity results for the $p$-Laplace type equations on compact Riemannian manifolds

In this paper, we obtain two rigidity results for $p$-Laplace type equation and $n$-Laplace equation with exponential nonlinearity on $n$-dimensional compact Riemannian manifolds by using of nonlinear flow and the carré du champ methods, respectively, where rigidity means that the PDE has only constant solution when a parameter is in a certain range. Moreover, an interpolation inequality is derived as an application.

math.AP↗
arXiv2019

The concavity of $p$-Rényi entropy power for doubly nonlinear diffusion equations and $L^p$-Gagliardo-Nirenberg-Sobolev inequalities

We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on $\mathbb{R}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on $\mathbb{R}^n$. Moreover, two improvement of $L^p$-Gagliardo-Nirenberg inequalities are derived.

math.AP↗
arXiv2019

The concavity of $p$-entropy power and applications in functional inequalities

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improvement of $L^p$-Logarithmic Sobolev inequality is derived.

math.AP↗