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Harnack estimates for nonlocal drift-diffusion equations

Naian Liao

TL;DR

This work establishes pointwise Harnack-type estimates for nonlocal drift-diffusion equations and, simultaneously, refines Hölder regularity under a De Giorgi-type scheme. By using energy estimates once and a suite of new measure-theoretic lemmas, the authors derive nonlocal parabolic weak Harnack inequalities that account for both local positivity and long-range tail effects, even in the presence of a divergence-free drift. They introduce a robust, tail-aware De Giorgi framework with measure propagation and measure shrinking, enabling local estimates that do not require global solvability. The elliptic counterpart is treated via the same approach, yielding novel weak Harnack estimates with an improved integral exponent and avoiding classical covering arguments. Overall, the results advance the understanding of nonlocal drift-diffusion regularity and provide tools to analyze initial traces, global behavior, and fundamental solution behavior in a unified, robust framework.

Abstract

A set of pointwise estimates are established for local solutions to nonlocal diffusion equations with a drift term. In particular, our Harnack estimates are the first ones for such equations, and our Hölder regularity refines certain known result in several aspects. The approach is measure theoretical in the spirit of DeGiorgi classes. It yields novel nonlocal weak Harnack estimates in the elliptic case as well.

Harnack estimates for nonlocal drift-diffusion equations

TL;DR

This work establishes pointwise Harnack-type estimates for nonlocal drift-diffusion equations and, simultaneously, refines Hölder regularity under a De Giorgi-type scheme. By using energy estimates once and a suite of new measure-theoretic lemmas, the authors derive nonlocal parabolic weak Harnack inequalities that account for both local positivity and long-range tail effects, even in the presence of a divergence-free drift. They introduce a robust, tail-aware De Giorgi framework with measure propagation and measure shrinking, enabling local estimates that do not require global solvability. The elliptic counterpart is treated via the same approach, yielding novel weak Harnack estimates with an improved integral exponent and avoiding classical covering arguments. Overall, the results advance the understanding of nonlocal drift-diffusion regularity and provide tools to analyze initial traces, global behavior, and fundamental solution behavior in a unified, robust framework.

Abstract

A set of pointwise estimates are established for local solutions to nonlocal diffusion equations with a drift term. In particular, our Harnack estimates are the first ones for such equations, and our Hölder regularity refines certain known result in several aspects. The approach is measure theoretical in the spirit of DeGiorgi classes. It yields novel nonlocal weak Harnack estimates in the elliptic case as well.
Paper Structure (28 sections, 26 theorems, 184 equations)

This paper contains 28 sections, 26 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Weak Harnack estimates
  3. Local boundedness
  4. Harnack's estimates
  5. Hölder regularity
  6. Technical innovation
  7. Definitions and notation
  8. Notion of solution
  9. Notation
  10. Organization
  11. Energy estimates
  12. Preparatory materials
  13. DeGiorgi-type lemmata
  14. A measure propagation lemma
  15. Measure shrinking lemmata
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $u$ be a local, weak super-solution to Eq:1:1 in $E_T$, such that $u\ge0$. Assume that $b$ satisfies Eq:b:1. There exist constants $\varepsilon,\,\eta\in(0,1)$ depending on the data $\{s, N, C_o, C_1\}$ and also on $C_2$, such that for almost all provided that

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.1
  • ...and 31 more