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Sharp regularity near the grazing set for kinetic Fokker-Planck equations

Kyeongbae Kim, Marvin Weidner

Abstract

We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp $C^{1/2}$ regularity for either diffuse reflection or prescribed in-flow boundary conditions. Previously, in this setting, it was only known that solutions are $C^α$ for some small $α> 0$. Second, we provide a complete characterization of the solution behavior near the grazing set by proving higher order expansions beyond the critical regularity threshold of $\frac{1}{2}$. These results demonstrate for the first time that solutions maintain higher smoothness up to the grazing set near the incoming boundary.

Sharp regularity near the grazing set for kinetic Fokker-Planck equations

Abstract

We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp regularity for either diffuse reflection or prescribed in-flow boundary conditions. Previously, in this setting, it was only known that solutions are for some small . Second, we provide a complete characterization of the solution behavior near the grazing set by proving higher order expansions beyond the critical regularity threshold of . These results demonstrate for the first time that solutions maintain higher smoothness up to the grazing set near the incoming boundary.
Paper Structure (43 sections, 46 theorems, 617 equations)

This paper contains 43 sections, 46 theorems, 617 equations.

Table of Contents

  1. Introduction
  2. Optimal regularity estimates
  3. Higher order estimates near the grazing set
  4. Strategy of the proof
  5. Higher order expansions at $\gamma_0$
  6. Liouville theorem
  7. Equations with coefficients
  8. Diffuse reflection
  9. Outline
  10. Acknowledgments
  11. Preliminaries
  12. Weak solution concept
  13. Interior regularity
  14. A Liouville theorem for stationary solutions in 1D
  15. Harnack's type inequality for the stationary case
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and let $A,B,F \in C^{\infty}((0,T) \times \overline{\Omega} \times \mathbb{R}^n)$ and $A$ be uniformly elliptic. Let $\mathcal{M} \in C^{\infty}(\gamma_-)$. Let $f$ be a weak solution to eq:FPK with diffuse reflection condition eq:diffuse

Theorems & Definitions (111)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Remark 3
  • Theorem 1.3
  • Theorem 1.4
  • Remark 4
  • Remark 5
  • Theorem 1.5
  • ...and 101 more