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De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

Francesca Anceschi, Giampiero Palatucci, Mirco Piccinini

TL;DR

The paper advances the nonlocal DGNM program for kinetic equations with fractional diffusion in velocity by introducing a delta-interpolative $L^2$-$L^\infty$ framework that incorporates a local $L^p$ tail in velocity. It establishes a sharp local regularity theory, including a first local $L^2$-$L^\infty$ estimate and a strong Harnack inequality under tail summability, and it provides a geometric Harnack interpretation in the kinetic setting. Central to the approach are hypoelliptic Sobolev gains, a kinetic Caccioppoli inequality with tail, and a Dalalian-style De Giorgi iteration, all adapted to the nonlocal, anisotropic (Galilean) geometry encoded by slanted cylinders $Q_r(z_o)$ and the homogeneous dimension $\mathbf{d}$. The results resolve a longstanding gap in the nonlocal kinetic DGNM theory and align with recent counterexamples while offering a robust tail-based framework that could extend to nonlinear fractional diffusion and mean field type problems.

Abstract

We extend the celebrated De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations naturally arising in kinetic theory, which combine a first-order operator with an elliptic one involving fractional derivatives along only part of the coordinates. Provided that the nonlocal tail in velocity of weak solutions is just $p$-summable along the drift variables, we prove the first local $L^2$-$L^\infty$ estimate for kinetic integral equations. Then, we establish the first strong Harnack inequality under the aforementioned tail summability assumption. The latter is in fact naturally implied in literature, e. g., from the usual mass density boundedness (as for the Boltzmann equation without cut-off), and it reveals to be in clear accordance with the very recent counterexample by Kassmann and Weidner \cite{KW24c}. Armed with the aforementioned results, we are able to provide a geometric characterization of the Harnack inequality in the same spirit of the seminal paper by Aronson and Serrin \cite{AS67} for the (local) parabolic counterpart.

De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

TL;DR

The paper advances the nonlocal DGNM program for kinetic equations with fractional diffusion in velocity by introducing a delta-interpolative - framework that incorporates a local tail in velocity. It establishes a sharp local regularity theory, including a first local - estimate and a strong Harnack inequality under tail summability, and it provides a geometric Harnack interpretation in the kinetic setting. Central to the approach are hypoelliptic Sobolev gains, a kinetic Caccioppoli inequality with tail, and a Dalalian-style De Giorgi iteration, all adapted to the nonlocal, anisotropic (Galilean) geometry encoded by slanted cylinders and the homogeneous dimension . The results resolve a longstanding gap in the nonlocal kinetic DGNM theory and align with recent counterexamples while offering a robust tail-based framework that could extend to nonlinear fractional diffusion and mean field type problems.

Abstract

We extend the celebrated De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations naturally arising in kinetic theory, which combine a first-order operator with an elliptic one involving fractional derivatives along only part of the coordinates. Provided that the nonlocal tail in velocity of weak solutions is just -summable along the drift variables, we prove the first local - estimate for kinetic integral equations. Then, we establish the first strong Harnack inequality under the aforementioned tail summability assumption. The latter is in fact naturally implied in literature, e. g., from the usual mass density boundedness (as for the Boltzmann equation without cut-off), and it reveals to be in clear accordance with the very recent counterexample by Kassmann and Weidner \cite{KW24c}. Armed with the aforementioned results, we are able to provide a geometric characterization of the Harnack inequality in the same spirit of the seminal paper by Aronson and Serrin \cite{AS67} for the (local) parabolic counterpart.
Paper Structure (12 sections, 12 theorems, 165 equations, 2 figures)

This paper contains 12 sections, 12 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The De-Giorgi-Nash-Moser theory: state of the art
  3. Main results
  4. Some further developments
  5. Outline of the paper
  6. Notation and preliminaries
  7. Local gain of integrability and local boundedness estimates
  8. Proof of Theorem \ref{['thm:gain']}
  9. Proof of Theorem \ref{['thm_bdd']}
  10. Strong Harnack inequality and propagation
  11. Proof of Theorem \ref{['thm_strong']}
  12. Geometric Harnack inequality

Key Result

Theorem 1.1

Let $\Omega:= (t_1,t_2)\times \Omega_x \times \Omega_v \subset \mathbb{R}^{1+2n}$ be a domain, $s \in (0,1)$ and let $\bm{d}$ be the homogeneous dimension in def:homo-dim. Assume that $f \in \mathdutchcal{W}$ is a weak subsolution to and let If $\mathop{\mathrm{\rm Tail}}\limits(f_+;B) \in L^{p}_{\rm{loc}}((t_1,t_2)\times\Omega_{x})$ for any $B \Subset \Omega_v$ and $h \in L^{p}_{\rm{loc}}(\Omeg

Figures (2)

  • Figure 1: On the left the cylinder ${Q}_R({0})$ centered at the origin; on the right a slanted cylinder ${Q}_R({\bf z_{\rm o}})\equiv{Q}_R(t_{\rm o},x_{\rm o},v_{\rm o})$ according to the invariant transformation given in \ref{['Q-classico']}.
  • Figure 2: The geometry of the Harnack inequalities for kinetic equations.

Theorems & Definitions (25)

  • Theorem 1.1: The $\delta$-interpolative $L^2$-$L^\infty$ estimate
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Local gain of integrability
  • Theorem 1.5: The Strong Harnack inequality
  • Remark 1.6: Hölder continuity as a corollary
  • Definition 2.1
  • Lemma 2.2: Proposition 1.11 in Fol75
  • Theorem 2.3: Theorem 1.6 in IS20
  • Lemma 2.4
  • ...and 15 more