Non-conservation of a generalized helicity in the Euler equations

Abstract

For a $C_{t,x}^1$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))={\int }_{{\mathbb{T}}^3}u\cdot \text{curl}u$ 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.

59 pages