Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kinetic Sobolev Spaces

Pascal Auscher, Lukas Niebel

Abstract

We define and study homogeneous kinetic Sobolev spaces adapted to the Kolmogorov equation. We consider both local and non-local diffusion. The spaces are built from the Lebesgue spaces L p for all integrability exponents p $\in$ (1, $\infty$) with regularity assumptions in the transport and diffusive directions according to the scaling of the Kolmogorov equation. The regularity scale accommodates weak and strong solutions. We prove that the proposed spaces satisfy sharp embeddings quantifying the transfer-ofregularity {à} la Bouchut-H{ö}rmander, continuity-in-time in the spirit of Lions and the gainof-integrability of Sobolev and Hardy-Littlewood-Sobolev type. A core tool are mapping properties of the Kolmogorov operator, given by the fundamental solution, established between anisotropic homogeneous Sobolev spaces. To achieve this, we prove L^p boundedness of related singular integral operators, for which we deduce novel kernel estimates by a Littlewood-Paley decomposition and geometric considerations. Moreover, we provide a new uniqueness criterion which allows us to show well-posedness of the Cauchy problem.

Kinetic Sobolev Spaces

Abstract

We define and study homogeneous kinetic Sobolev spaces adapted to the Kolmogorov equation. We consider both local and non-local diffusion. The spaces are built from the Lebesgue spaces L p for all integrability exponents p (1, ) with regularity assumptions in the transport and diffusive directions according to the scaling of the Kolmogorov equation. The regularity scale accommodates weak and strong solutions. We prove that the proposed spaces satisfy sharp embeddings quantifying the transfer-ofregularity {à} la Bouchut-H{ö}rmander, continuity-in-time in the spirit of Lions and the gainof-integrability of Sobolev and Hardy-Littlewood-Sobolev type. A core tool are mapping properties of the Kolmogorov operator, given by the fundamental solution, established between anisotropic homogeneous Sobolev spaces. To achieve this, we prove L^p boundedness of related singular integral operators, for which we deduce novel kernel estimates by a Littlewood-Paley decomposition and geometric considerations. Moreover, we provide a new uniqueness criterion which allows us to show well-posedness of the Cauchy problem.
Paper Structure (43 sections, 63 theorems, 445 equations)

This paper contains 43 sections, 63 theorems, 445 equations.

Table of Contents

  1. Introduction
  2. Kinetic spaces and Kolmogorov operators
  3. Main functional spaces
  4. Homogeneous Sobolev spaces
  5. Homogeneous anisotropic Sobolev spaces
  6. Homogeneous anisotropic Besov spaces
  7. Duality pairings
  8. Kinetic spaces
  9. Forward and backward Kolmogorov operators of order $\beta$
  10. Towards the $\operatorname{L}^p$ boundedness for Kolmogorov operators in the kinetic scale
  11. Integrated kernel estimates
  12. Rescaled variables
  13. Decomposition of the kernel into a good and a bad part
  14. Estimates for the bad part
  15. A kinetic Littlewood--Paley lemma
  16. ...and 28 more sections

Key Result

Lemma 2.1

Let $\gamma\in \mathbb{R}$, $p \in (1,\infty)$. Let $g_{j}\in \mathcal{S}'(\mathbb{R}^d) \cap \operatorname{C}^\infty(\mathbb{R}^d)$, $j\in \mathbb{Z}$, with Fourier transform $\hat{g}_{j}$ supported in $C_{j}=\{\xi\in \mathbb{R}^d; 2^{j- 1} < |\xi|< 2^{j+1}\}$. If $\|(\sum_{j=-\infty}^\infty |2^{j

Theorems & Definitions (143)

  • Lemma 2.1
  • Definition 2.2: Homogeneous Sobolev space
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Definition 2.5: Homogeneous anisotropic Sobolev space
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 133 more