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Quantitative De Giorgi methods in kinetic theory for non-local operators

Amélie Loher

Abstract

We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies Hölder continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate. This permits to quantitatively prove the intermediate value lemma for the full range of non-locality parameter $s \in (0, 1)$. Our results recover the results from Imbert and Silvestre [22] for the inhomogeneous Boltzmann equation in the non-cutoff case. The paper is self-contained.

Quantitative De Giorgi methods in kinetic theory for non-local operators

Abstract

We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies Hölder continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate. This permits to quantitatively prove the intermediate value lemma for the full range of non-locality parameter . Our results recover the results from Imbert and Silvestre [22] for the inhomogeneous Boltzmann equation in the non-cutoff case. The paper is self-contained.
Paper Structure (23 sections, 14 theorems, 262 equations, 2 figures)

This paper contains 23 sections, 14 theorems, 262 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Contribution
  4. Ellipticity Class of $K$
  5. Non-cutoff Boltzmann equation
  6. Invariant transformations
  7. Further remarks
  8. Structure of the article
  9. Weak Formulation
  10. Notation
  11. Bilinear Form
  12. Integral Estimates
  13. Kolmogorov's fundamental solutions
  14. Energy estimates
  15. First Lemma of De Giorgi
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $f$ be a non-negative super-solution to eq:1.1-eq:1.2 in $[-3, 0] \times B_1 \times B_1$ with a non-negative kernel $K$ satisfying eq:coercivity-eq:cancellation2 for $\bar{R} = 2$. Then there is $C$ and $\zeta > 0$ depending on $s, d, \lambda,\Lambda$ such that for $r_0 < \frac{1}{3}$ the Weak H where $\tilde{Q}_{\frac{r_0}{2}}^-:= Q_{\frac{r_0}{2}}((-\frac{5}{2}r_0^{2s} + \frac{1}{2} (\frac{r

Figures (2)

  • Figure 1: The barriers $\varphi_0, \varphi_1, \varphi_2$ in the Intermediate Value Lemma, Theorem \ref{['thm:IVL']}, for $r_0 < \frac{1}{3}$, $\mu < 1$ and fixed $x \in B_{(3r_0)^{1+2s}}$.
  • Figure 2: For a solution in $Q_1$, we prove Hölder continuity \ref{['thm:holder']} in $Q_{\frac{1}{2}}$. The Intermediate Value Lemma, Theorem \ref{['thm:IVL']}, relates $Q_{r_0}$ to $Q^-_{r_0}$ in the past. The Weak Harnack inequality, Theorem \ref{['thm:weakH']}, relates $Q_{\frac{r_0}{2}}$ with $\tilde{Q}^-_{\frac{r_0}{2}}$ and the Not-so-Strong Harnack inequality, Theorem \ref{['thm:not-so-strong-H']}, relates $Q_{\frac{r_0}{4}}$ with $\tilde{Q}^-_{\frac{r_0}{4}}$ in the past. Note that all cylinders depend on $s \in (0, 1)$.

Theorems & Definitions (28)

  • Theorem 1.1: Weak Harnack inequality
  • Theorem 1.2: Hölder continuity
  • Theorem 1.3: Not-so-Strong Harnack inequality
  • Theorem 1.4: Intermediate Value Lemma
  • Theorem 1.5: Imbert-Silvestre
  • Remark 1.6
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 18 more