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Optimal regularity for kinetic Fokker-Planck equations in domains

Xavier Ros-Oton, Marvin Weidner

TL;DR

This work resolves a long-standing question on the boundary regularity of linear kinetic Fokker-Planck equations with specular reflection by identifying optimal smoothness up to the boundary: solutions are $C^ abla$-smooth away from the grazing set and belong to $C^{4,1}_{\ell}$ up to the grazing set, with counterexamples proving that $C^{5}_{\ell}$ is not generically attainable. The authors develop a robust framework based on kinetic cylinders, blow-up/compactness, and Liouville theorems in full and half-spaces, and then extend the analysis to general domains via a specular-preserving boundary flattening. They also construct a precise Tricomi-type correction at grazing boundary points, which is essential for the optimality of regularity. The results yield classical regularity up to the boundary (e.g., $C^1_t C^1_x$ in time and space and $C^2_v$ in velocity) and provide a foundation for further sharp regularity results for kinetic equations with various boundary conditions.

Abstract

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $Ω\subset \mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\partial_t f +v\cdot\nabla_x f-Δ_v f=h$. Our main results establish the following: - Solutions are always $C^\infty$ in $t,v,x$ away from the grazing set $\{x\in\partialΩ,\ v\cdot n_x=0\}$. - They are $C^{4,1}_{\text{kin}}$ up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not $C^5_{\text{kin}}$. These results show for the first time that solutions are classical up to boundary, i.e. $C^1_{t,x}$ and $C^2_v$.

Optimal regularity for kinetic Fokker-Planck equations in domains

TL;DR

This work resolves a long-standing question on the boundary regularity of linear kinetic Fokker-Planck equations with specular reflection by identifying optimal smoothness up to the boundary: solutions are -smooth away from the grazing set and belong to up to the grazing set, with counterexamples proving that is not generically attainable. The authors develop a robust framework based on kinetic cylinders, blow-up/compactness, and Liouville theorems in full and half-spaces, and then extend the analysis to general domains via a specular-preserving boundary flattening. They also construct a precise Tricomi-type correction at grazing boundary points, which is essential for the optimality of regularity. The results yield classical regularity up to the boundary (e.g., in time and space and in velocity) and provide a foundation for further sharp regularity results for kinetic equations with various boundary conditions.

Abstract

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection condition, including Kolmogorov's equation . Our main results establish the following: - Solutions are always in away from the grazing set . - They are up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not . These results show for the first time that solutions are classical up to boundary, i.e. and .
Paper Structure (32 sections, 48 theorems, 532 equations)

This paper contains 32 sections, 48 theorems, 532 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Regularity estimates for general kinetic Fokker-Planck equations
  4. Strategy of the proof
  5. Flat domains
  6. General domains
  7. Organization of the paper
  8. Acknowledgments
  9. Preliminaries
  10. Scaling properties of kinetic equations
  11. Kinetic distance and Hölder spaces
  12. Generalized kinetic polynomials
  13. Flattening the boundary
  14. Reflecting kinetic equations
  15. Definition of weak solutions
  16. ...and 17 more sections

Key Result

theorem 1

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain, and let $h \in C^\infty((0,T) \times \Omega \times \mathbb{R}^n)$. Let $f$ be any solution of the Kolmogorov equation eq:Kolmogorov0 with specular reflection specular-ref and assume that both, $f$ and $h$ have fast decay as $|v|\to\infty$ Moreover, this regularity is optimal, i.e. there exist bounded smooth domains $\Omega$ and function

Theorems & Definitions (100)

  • theorem 1
  • theorem 2
  • theorem 3
  • remark 4
  • lemma 5
  • proof
  • lemma 6
  • proof
  • lemma 7
  • proof
  • ...and 90 more