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Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients

Yuzhe Zhu

Abstract

This article addresses the local boundedness and Hölder continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part $\partial_t+b(v)\cdot\nabla_x$ is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot and Vasseur in the case $b(v)= v$, that combines the elliptic De Giorgi-Nash-Moser theory with velocity averaging lemmas.

Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients

Abstract

This article addresses the local boundedness and Hölder continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot and Vasseur in the case , that combines the elliptic De Giorgi-Nash-Moser theory with velocity averaging lemmas.

Paper Structure

This paper contains 22 sections, 13 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work and contributions
  4. Motivation and background
  5. Physical interpretation
  6. Nondegenerate condition and velocity averaging
  7. Hypoelliptic structure
  8. Notation
  9. Organization of the paper
  10. Velocity averaging
  11. Classical averaging theory
  12. Hypoelliptic transfer of regularity
  13. Local boundedness
  14. Moser's approach
  15. De Giorgi's approach
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $f$ be a subsolution to FP satisfying H and nond in $Q_1:=(-1,0]\times B_1\times B_1$, and in addition, for some $q>\frac{(4+\alpha)(4+d_2)(1+{d_1}+{d_2})}{2\alpha}$. Then, the positive part $f^+$ of the subsolution is bounded in $Q_{\rm int}:=(-\frac{1}{2},0]\times B_\frac{1}{2}\times B_\frac{1}{2}$. More precisely, there exists some positive constant $C$ only depending on $\lambda,\Lambda,d

Figures (1)

  • Figure :

Theorems & Definitions (28)

  • Theorem 1.1: local boundedness
  • Theorem 1.2: Hölder regularity
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • proof
  • ...and 18 more