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A regularity result for the Fokker-Planck equation with non-smooth drift and diffusion

Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa

TL;DR

The paper addresses the Fokker-Planck equation with non-smooth drift and diffusion by developing a unified, commutator-based framework that links distributional and parabolic solutions. It introduces and leverages new commutator estimates, including controls in negative Sobolev spaces, to prove local and global regularity results and to establish uniqueness in the parabolic solution class under sharp integrability and growth assumptions. By comparing and extending existing results (notably F08 and Le Bris–Lions), the work provides a cohesive view of well-posedness in irregular coefficient regimes and demonstrates how spatial Sobolev regularity of the diffusion, combined with integrable drift, yields $H^1$-type regularity for distributional solutions. These results deepen understanding of FP dynamics in fluid and stochastic models where coefficients are rough, offering precise conditions under which uniqueness and regularity hold and informing stability analyses in applied contexts.

Abstract

The goal of this paper is to study weak solutions of the Fokker-Planck equation. We first discuss existence and uniqueness of weak solutions in an irregular context, providing a unified treatment of the available literature along with some extensions. Then, we prove a regularity result for distributional solutions under suitable integrability assumptions, relying on a new, simple commutator estimate in the spirit of DiPerna-Lions' theory of renormalized solutions for the transport equation. Our result is somehow transverse to Theorem 4.3 of [15]: on the diffusion matrix we relax the assumption of Lipschitz regularity in time at the price of assuming Sobolev regularity in space, and we prove the regularity (and hence the uniqueness) of distributional solutions to the Fokker-Planck equation.

A regularity result for the Fokker-Planck equation with non-smooth drift and diffusion

TL;DR

The paper addresses the Fokker-Planck equation with non-smooth drift and diffusion by developing a unified, commutator-based framework that links distributional and parabolic solutions. It introduces and leverages new commutator estimates, including controls in negative Sobolev spaces, to prove local and global regularity results and to establish uniqueness in the parabolic solution class under sharp integrability and growth assumptions. By comparing and extending existing results (notably F08 and Le Bris–Lions), the work provides a cohesive view of well-posedness in irregular coefficient regimes and demonstrates how spatial Sobolev regularity of the diffusion, combined with integrable drift, yields -type regularity for distributional solutions. These results deepen understanding of FP dynamics in fluid and stochastic models where coefficients are rough, offering precise conditions under which uniqueness and regularity hold and informing stability analyses in applied contexts.

Abstract

The goal of this paper is to study weak solutions of the Fokker-Planck equation. We first discuss existence and uniqueness of weak solutions in an irregular context, providing a unified treatment of the available literature along with some extensions. Then, we prove a regularity result for distributional solutions under suitable integrability assumptions, relying on a new, simple commutator estimate in the spirit of DiPerna-Lions' theory of renormalized solutions for the transport equation. Our result is somehow transverse to Theorem 4.3 of [15]: on the diffusion matrix we relax the assumption of Lipschitz regularity in time at the price of assuming Sobolev regularity in space, and we prove the regularity (and hence the uniqueness) of distributional solutions to the Fokker-Planck equation.
Paper Structure (15 sections, 23 theorems, 130 equations)

This paper contains 15 sections, 23 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Distributional and parabolic solutions
  3. Regularity and uniqueness of distributional solutions
  4. Structure of the paper
  5. Commutator estimates
  6. Commutator estimates in $L^1$
  7. Commutator estimates on negative Sobolev norms
  8. Well-posedness of the Cauchy problem
  9. The equation in divergence form
  10. Comparison with LBL_CPDE
  11. The Fokker-Planck equation
  12. Further comments and remarks
  13. The Regularity Theorems
  14. Local regularity Theorem and uniqueness
  15. Extensions with suitable growth conditions

Key Result

Theorem 1.1

Let $u_0\in L^2\cap L^q(\mathbb{R}^d)$ be a given initial datum and assume that Then, there exists a unique parabolic solution $u$ to eq:fp.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 38 more