Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients

TL;DR

This paper proves the well-posedness of Fokker-Planck equations with Sobolev diffusion and BV drift coefficients, addressing an open question in the mathematical analysis of such PDEs.

Contribution

It provides a rigorous proof of existence and uniqueness for these equations under the specified conditions, advancing the theoretical understanding.

Findings

Established well-posedness for Fokker-Planck equations with Sobolev diffusion

Proved existence and uniqueness of solutions with BV drift coefficients

Addressed an open problem from prior research

Abstract

In this paper we give an affirmative answer to an open question mentioned in [Le Bris and Lions, Comm. Partial Differential Equations 33 (2008), 1272--1317], that is, we prove the well-posedness of the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.