Laplacian with singular drift in a critical borderline case

TL;DR

This paper proves well-posedness and regularity for a parabolic diffusion equation with a critically singular drift, using Orlicz spaces to handle the borderline singularities.

Contribution

It introduces a novel approach to analyze equations with critical singularities by employing Orlicz spaces, extending previous results beyond traditional $L^p$ frameworks.

Findings

Established well-posedness in critical singularity cases

Demonstrated regularity results for the solutions

Utilized Orlicz space techniques for borderline cases

Abstract

We establish well-posedness and regularity for parabolic diffusion equation in the case when the singularities of a general drift reach the critical magnitude. The latter dictates the need to work in an Orlicz space situated between all $L^p$ and $L^\infty$.