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Nonlinear Diffusion, and Geometric and Functional Inequalities on Smooth Metric Measure spaces

Ali Taheri

TL;DR

This work surveys nonlinear diffusion on smooth metric measure spaces (SMMS) and develops a unified geometric-analytic framework that links isoperimetric and functional inequalities to diffusion processes. By employing the phi-Laplacian $\Delta_\varphi$ and the Bakry–Émery Ricci tensors under curvature-dimension conditions $CD(k,m)$, the author connects classical comparisons (e.g., Gaussian and Levy–Gromov) to nonlinear diffusion phenomena, deriving gradient estimates and Liouville-type results for porous medium and fast diffusion equations. The contributions include a coherent set of a priori estimates for $p>1$ and $0<p<1$ on SMMS, together with Liouville-type nonexistence results for ancient solutions under growth restrictions, highlighting a deep interplay between curvature-dimension bounds and nonlinear diffusion. This framework advances the analytic machinery for inverse problems and PDEs on weighted manifolds by clarifying how geometric constraints influence diffusion, entropy, and functional inequalities.

Abstract

This extended abstract is based on a talk given at the workshop and summer school ``Direct and Inverse Problems with Applications" in Ghent Analysis and PDE Centre in August 2024. It focuses on nonlinear diffusion equations of slow and fast types and their links with some geometric and functional inequalities in the framework of smooth metric measure spaces. The article presents some introduction in a summer school style as well as several new results.

Nonlinear Diffusion, and Geometric and Functional Inequalities on Smooth Metric Measure spaces

TL;DR

This work surveys nonlinear diffusion on smooth metric measure spaces (SMMS) and develops a unified geometric-analytic framework that links isoperimetric and functional inequalities to diffusion processes. By employing the phi-Laplacian and the Bakry–Émery Ricci tensors under curvature-dimension conditions , the author connects classical comparisons (e.g., Gaussian and Levy–Gromov) to nonlinear diffusion phenomena, deriving gradient estimates and Liouville-type results for porous medium and fast diffusion equations. The contributions include a coherent set of a priori estimates for and on SMMS, together with Liouville-type nonexistence results for ancient solutions under growth restrictions, highlighting a deep interplay between curvature-dimension bounds and nonlinear diffusion. This framework advances the analytic machinery for inverse problems and PDEs on weighted manifolds by clarifying how geometric constraints influence diffusion, entropy, and functional inequalities.

Abstract

This extended abstract is based on a talk given at the workshop and summer school ``Direct and Inverse Problems with Applications" in Ghent Analysis and PDE Centre in August 2024. It focuses on nonlinear diffusion equations of slow and fast types and their links with some geometric and functional inequalities in the framework of smooth metric measure spaces. The article presents some introduction in a summer school style as well as several new results.
Paper Structure (5 sections, 12 theorems, 42 equations)

This paper contains 5 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Isoperimetric inequality and the Isoperimetric profile
  3. The Gauss Space $(\mathbb R^n, g_{{\mathbb R}^n}, d\gamma_n)$
  4. Smooth metric measure spaces $({\mathscr M}, g, d\mu)$
  5. Nonlinear diffusion equations on smooth metric measure spaces

Key Result

Theorem 4.1

The following conditions for $k>0$ are equivalent:

Theorems & Definitions (16)

  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Remark 4.5
  • Theorem 4.6
  • Theorem 5.1
  • Theorem 5.2
  • ...and 6 more