Non-conservation of a generalized helicity in the Euler equations

TL;DR

This paper introduces a generalized helicity concept for weak solutions of the 3D Euler equations, demonstrating that helicity conservation can fail at low regularity levels, unlike classical solutions.

Contribution

It defines a generalized helicity applicable to low-regularity solutions and constructs solutions with prescribed generalized helicity and energy.

Findings

Generalized helicity extends classical definitions.

Constructed weak solutions with prescribed helicity.

Helicity conservation does not hold at low regularity.

Abstract

For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to define the helicity, or whether it must be constant in time in the case that there is a clear definition. In this paper, we define a generalized helicity which extends the classical definitions and construct weak solutions of Euler of almost Onsager-critical regularity in $L^3$ with prescribed generalized helicity and kinetic energy.