Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularity of kinetic Fokker-Planck equations in bounded domains

Yuzhe Zhu

Abstract

We obtain the existence, uniqueness and regularity results for solutions to kinetic Fokker-Planck equations with bounded measurable coefficients in the presence of boundary conditions, including the inflow, diffuse reflection and specular reflection cases.

Regularity of kinetic Fokker-Planck equations in bounded domains

Abstract

We obtain the existence, uniqueness and regularity results for solutions to kinetic Fokker-Planck equations with bounded measurable coefficients in the presence of boundary conditions, including the inflow, diffuse reflection and specular reflection cases.
Paper Structure (33 sections, 21 theorems, 209 equations, 1 figure)

This paper contains 33 sections, 21 theorems, 209 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Phase boundaries
  4. Inflow boundary condition
  5. Nonlocal reflection boundary condition
  6. Specular reflection boundary condition
  7. Statement of the main theorem
  8. Backgrounds and related work
  9. Kinetic boundary value problems
  10. Trace problem in kinetic equations
  11. Hypoellipticity
  12. Characteristic points
  13. De Giorgi's technique
  14. Extension method
  15. Notations
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let the domain $\Omega$ be bounded with $\partial\Omega\in C^{1,1}$, and let $\mathscr{B}\in\{\mathcal{G},{\mathcal{N}\space},\mathcal{R}\}$ be the boundary operator. Assume that H and M hold.

Figures (1)

  • Figure 1: The solid dark region is a section of $\mathcal{W}$. The corresponding section of $\mathcal{W}^\natural$ consists of regions with dark color and checkerboard pattern.

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5: Renormalization formula
  • proof
  • Corollary 2.6: Uniqueness
  • proof
  • ...and 35 more