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The nonlinear fast diffusion equation on smooth metric measure spaces: Hamilton-Souplet-Zhang estimates and a Ricci-Perelman super flow

Ali Taheri, Vahideh Vahidifar

TL;DR

This work analyzes the nonlinear fast diffusion equation $\partial_t u - \Delta_f u^p = \mathscr N(t,x,u)$ on smooth metric measure spaces, allowing time-dependent metrics and potentials and coupling to Bakry–Émery curvature ${\mathscr Ric}^m_f(g)$. It develops two families of Hamilton–Souplet–Zhang gradient estimates for positive solutions in the fast-diffusion range $0<p<1$, each tied to a different pressure-transform $v$ and evolution operator $\mathscr L$, and both yielding local and global forms. A key novelty is integrating a Ricci–Perelman type super-flow for the evolving geometry, yielding additional gradient bounds and parabolic Liouville-type results, along with characterizations of ancient solutions. The results demonstrate that the maximal exponent range for the gradient estimates is robust to the geometric evolution and curvature bounds, linking nonlinear diffusion, geometric analysis, and Liouville-type rigidity in a unified framework.

Abstract

This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the $f$-Laplacian. The gradient estimates of interest are mainly of Hamilton-Souplet-Zhang or elliptic type and are proved using different set of methods and techniques. Various implications notably to parabolic Liouville type results and characterisation of ancient solutions are given. The problem is considered in the general setting where the metric and potential evolve under a super flow involving the Bakry-Émery $m$-Ricci curvature tensor. The remarkable interplay between geometry, nonlinearity, and evolution -- and their intricate roles in the estimates and the maximum exponent range of fast diffusion -- is at the core of the investigation.

The nonlinear fast diffusion equation on smooth metric measure spaces: Hamilton-Souplet-Zhang estimates and a Ricci-Perelman super flow

TL;DR

This work analyzes the nonlinear fast diffusion equation on smooth metric measure spaces, allowing time-dependent metrics and potentials and coupling to Bakry–Émery curvature . It develops two families of Hamilton–Souplet–Zhang gradient estimates for positive solutions in the fast-diffusion range , each tied to a different pressure-transform and evolution operator , and both yielding local and global forms. A key novelty is integrating a Ricci–Perelman type super-flow for the evolving geometry, yielding additional gradient bounds and parabolic Liouville-type results, along with characterizations of ancient solutions. The results demonstrate that the maximal exponent range for the gradient estimates is robust to the geometric evolution and curvature bounds, linking nonlinear diffusion, geometric analysis, and Liouville-type rigidity in a unified framework.

Abstract

This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the -Laplacian. The gradient estimates of interest are mainly of Hamilton-Souplet-Zhang or elliptic type and are proved using different set of methods and techniques. Various implications notably to parabolic Liouville type results and characterisation of ancient solutions are given. The problem is considered in the general setting where the metric and potential evolve under a super flow involving the Bakry-Émery -Ricci curvature tensor. The remarkable interplay between geometry, nonlinearity, and evolution -- and their intricate roles in the estimates and the maximum exponent range of fast diffusion -- is at the core of the investigation.

Paper Structure

This paper contains 11 sections, 23 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. A Hamilton-Souplet-Zhang estimate in the range $p_c(m)< p < 1$
  3. Evolution inequalities ${\bf I}$: $\mathscr L = \partial_t -(1-p) v \Delta_f$ acting on $w = |\nabla v|^2/v^{\beta}$
  4. Localisation in space-time cylinders $Q_{R,T}$ and proof of Theorem \ref{['thm2-FDE']}
  5. Ancient solutions to $\partial_t u - \Delta_f u^p = {\mathscr N}(u)$ ${\bf I}$: Proof of Theorem \ref{['ancient-1']}
  6. The Ricci-Perelman super flow ${\bf I}$: $\partial_t g + 2(1-p) v {\mathscr Ric}^m_f(g) \ge -2\mathsf{k}g$
  7. A Hamilton-Souplet-Zhang estimate in the range $p_0(m)< p < 1$
  8. Evolution inequalities ${\bf II}$: ${\mathscr L} = \partial_t- pv^{2(p-1)/(2p-1)} \Delta_f$ acting on $w = |\nabla v|^2$
  9. Localisation in space-time cylinders $Q_{R,T}$ and proof of Theorem \ref{['thm-6.1-EQ-FDE']}
  10. Ancient solutions to $\partial_t u - \Delta_f u^p = {\mathscr N}(u)$ ${\bf II}$: Proof of Theorem \ref{['ancient-2']}
  11. The Ricci-Perelman super flow ${\bf II}$: $\partial_t g + 2p v^{2(p-1)/(2p-1)} {\mathscr Ric}^m_f(g) \ge - 2\mathsf{k}g$

Key Result

Theorem 2.1

Let $(M, g, d\mu)$ be a complete smooth metric measure space with $d\mu=e^{-f} dv_g$. Assume the metric and potential are time dependent, of class $\mathscr{C}^2$ and that for suitable constants $k, h \ge 0$ and $m \ge n$ satisfy ${\mathscr Ric}_f^m (g) \ge -(m-1)k g$, $\partial_t g \ge -2hg$ in the

Theorems & Definitions (43)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 33 more