Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl

TL;DR

This paper establishes the optimal Hardy-Leray inequality for three-dimensional divergence-free vector fields with axisymmetric swirl, advancing understanding of inequalities in fluid dynamics and vector calculus.

Contribution

It derives the sharp Hardy-Leray inequality for solenoidal fields with axisymmetric swirl, improving previous results by partially relaxing symmetry conditions.

Findings

Proves the Hardy-Leray inequality with the best constant for specific vector fields.

Extends previous work by considering only the swirl component for axisymmetry.

Provides a mathematical foundation for analyzing divergence-free fields in fluid mechanics.

Abstract

In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.