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Non-conservation of a generalized helicity in the Euler equations

Vikram Giri, Hyunju Kwon, Matthew Novack

TL;DR

The paper constructs non-smooth weak solutions to the 3D Euler equations with prescribed kinetic energy and a generalized helicity, demonstrating that generalized helicity need not be conserved for low-regularity flows. Using a refined convex integration framework with intermittent helical building blocks, Mikado flows, and a shortened transport time-scale, the authors inject controlled helicity while keeping the solutions almost Onsager-critical in $L^3$. The key innovations include a robust definition of generalized helicity that remains meaningful under mollification and shell decompositions, and the introduction of helical pipe bundles to precisely adjust helicity without spoiling energy or regularity. The results imply non-uniqueness and show that enforcing generalized helicity conservation does not constrain the energy dynamics, highlighting the nuanced interplay between topology and analysis in turbulent-like flows.

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.

Non-conservation of a generalized helicity in the Euler equations

TL;DR

The paper constructs non-smooth weak solutions to the 3D Euler equations with prescribed kinetic energy and a generalized helicity, demonstrating that generalized helicity need not be conserved for low-regularity flows. Using a refined convex integration framework with intermittent helical building blocks, Mikado flows, and a shortened transport time-scale, the authors inject controlled helicity while keeping the solutions almost Onsager-critical in . The key innovations include a robust definition of generalized helicity that remains meaningful under mollification and shell decompositions, and the introduction of helical pipe bundles to precisely adjust helicity without spoiling energy or regularity. The results imply non-uniqueness and show that enforcing generalized helicity conservation does not constrain the energy dynamics, highlighting the nuanced interplay between topology and analysis in turbulent-like flows.

Abstract

For a solution to the incompressible 3D Euler equations, the helicity 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 with prescribed generalized helicity and kinetic energy.
Paper Structure (55 sections, 42 theorems, 242 equations)

This paper contains 55 sections, 42 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Generalized helicity and main result
  4. Ideas of the proof
  5. Mikado flows have zero helicity
  6. Shortening of the time-scale
  7. Using regularity to deal with lower order terms
  8. New 'helical' building blocks in order to inject a desired helicity
  9. Inductive assumptions
  10. General notations and parameters
  11. Relaxed equations
  12. Inductive assumptions for velocity cutoff functions
  13. Inductive bounds on the Reynolds stress
  14. Dodging inductive hypotheses
  15. Inductive velocity bounds
  16. ...and 40 more sections

Key Result

Theorem 1.4

Let $0<\beta<1/2$ and $T>0$. Suppose that smooth functions $e:[0,T]\rightarrow(0,\infty)$ and $h:[0,T]\rightarrow(-\infty,\infty)$ are given. Then there exist weak solutions $u:[0,T]\times\mathbb{T}^3\rightarrow\mathbb{R}^3$ to the 3D Euler equations, belonging to $C([0,T]; (H^\beta\cap L^{\frac{1}{

Theorems & Definitions (90)

  • Definition 1.1: Generalized helicity
  • Remark 1.2: Computing helicity shell-by-shell
  • Remark 1.3: Comparison with the classical definitions of helicity
  • Theorem 1.4
  • Remark 2.1: Space-time norms
  • Remark 2.2: Space-time balls
  • Remark 2.5: Upgrading material derivatives for velocity and velocity cutoffs
  • Proposition 2.10: Iterative proposition
  • proof : Proof of Proposition \ref{['prop:main']}
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 80 more