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The helicity distribution for the 3D incompressible Euler equations

Marco Inversi, Massimo Sorella

TL;DR

The paper develops a Duchon--Robert-type defect framework for helicity in the 3D incompressible Euler equations, proving that a defect distribution $D[u]$ governs the local helicity balance under mild regularity and providing a precise representation $D[u] = 2 \lim_{\varepsilon \to 0} \nabla \omega_\varepsilon : R_\varepsilon$. It then derives a global helicity balance on bounded domains, showing that the rate of change of total helicity is determined by boundary fluxes involving the normal vorticity, velocity, and pressure, while the total helicity is conserved in the absence of boundary vorticity flux. The results rely on fractional Sobolev and Besov tools, boundary trace theory, and careful mollification arguments to control nonlinear terms and pass to the limit. This work clarifies how local regularity and boundary effects influence helicity conservation or dissipation in weak solutions, connecting local defect measures to global invariants.

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.

The helicity distribution for the 3D incompressible Euler equations

TL;DR

The paper develops a Duchon--Robert-type defect framework for helicity in the 3D incompressible Euler equations, proving that a defect distribution governs the local helicity balance under mild regularity and providing a precise representation . It then derives a global helicity balance on bounded domains, showing that the rate of change of total helicity is determined by boundary fluxes involving the normal vorticity, velocity, and pressure, while the total helicity is conserved in the absence of boundary vorticity flux. The results rely on fractional Sobolev and Besov tools, boundary trace theory, and careful mollification arguments to control nonlinear terms and pass to the limit. This work clarifies how local regularity and boundary effects influence helicity conservation or dissipation in weak solutions, connecting local defect measures to global invariants.

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.
Paper Structure (9 sections, 17 theorems, 90 equations)

This paper contains 9 sections, 17 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Tools
  3. The fractional Sobolev space $H^{1/2}$
  4. Some commutator estimates of Besov vector fields
  5. Boundary trace
  6. Explicit formula for the vorticity
  7. Proofs
  8. Local helicity balance
  9. Total helicity balance

Key Result

Theorem 1.2

Let $\Omega \subset \mathbb{R}^3$ be any open set and let $(u,p)$ be a weak solution to Euler in $\Omega \times (0,T)$ according to d: weak solution with $u\in L^2_{\rm loc} ( (0,T) ; H^{1/2}_{\rm loc} (\Omega)) \cap L^\infty_{\rm loc}( \Omega \times (0,T))$. Then, there exists $D[u] \in \mathcal{D} Furthermore, letting $u_\varepsilon, (u \otimes u)_\varepsilon, \omega_\varepsilon$ be the spatial

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2: Duchon--Robert type helicity distribution
  • Corollary 1.3
  • Definition 1.4: Total helicity
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 25 more