The helicity distribution for the 3D incompressible Euler equations

TL;DR

This paper investigates the helicity in solutions of the 3D incompressible Euler equations, establishing conditions for defining local and global helicity balances and analyzing boundary contributions.

Contribution

It introduces a defect distribution for local helicity balance and derives global helicity conservation laws under regularity assumptions.

Findings

Defined a defect distribution for local helicity balance

Established global helicity balance with boundary contributions

Provided conditions on velocity field regularity

Abstract

This paper is concerned with the helicity associated to solutions of the 3D incompressible Euler equations. We show that under mild conditions on the regularity of the velocity field of an incompressible ideal fluid it is possible to define a defect distribution describing the local helicity balance. Under suitable regularity assumptions, we also provide the global helicity balance on bounded domains in terms of the boundary contributions of the vorticity, velocity and pressure.