On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations

Classical solutions of the three-dimensional Euler equations of an ideal incompressible fluid conserve the helicity. We introduce a new weak formulation of the vorticity formulation of the Euler equations in which (by implementing the Bony paradifferential calculus) the advection terms are interpreted as paraproducts for weak solutions with low regularity. Using this approach we establish an equation of local helicity balance, which gives a rigorous foundation to the concept of local helicity density and flux at low regularity. We provide a sufficient criterion for helicity conservation which is weaker than many of the existing sufficient criteria for helicity conservation in the literature. Subsequently, we prove a sufficient condition for the helicity to be conserved in the zero viscosity limit of the Navier-Stokes equations. Moreover, we establish a relation between the defect measure (which is part of the local helicity balance) and a third-order structure function for solutions of the Euler equations. As a byproduct of the approach introduced in this paper, we also obtain a new sufficient condition for the conservation of magnetic helicity in the inviscid MHD equations, as well as for the kinematic dynamo model. Finally, it is known that classical solutions of the ideal (inviscid) MHD equations which have divergence-free initial data will remain divergence-free, but this need not hold for weak solutions. We show that weak solutions of the ideal MHD equations arising as weak-$*$ limits of Leray-Hopf weak solutions of the viscous and resistive MHD equations remain divergence-free in time.

💡 Research Summary

The paper addresses the long‑standing question of under what regularity conditions weak solutions of the three‑dimensional incompressible Euler equations and the inviscid magnetohydrodynamic (MHD) equations conserve helicity and magnetic helicity, respectively. Classical smooth solutions preserve these topological invariants, but for weak solutions the nonlinear terms are not a priori well‑defined, and the conservation may fail.

The authors introduce a novel weak formulation of the vorticity equation based on Bony’s paradifferential calculus. By interpreting the advection terms (u\cdot\nabla\omega) and (\omega\cdot\nabla u) as paraproducts, they are able to give a rigorous meaning to the nonlinearities when the vorticity (\omega) belongs to a negative Sobolev or Besov space (B^{-1/2}_{1,2}) while the velocity (u) lies in (H^{1/2+}). This “functional vorticity solution” is shown to be equivalent to the standard distributional weak solution under the stated regularity, establishing a solid functional‑analytic framework.

Within this framework the authors derive a local helicity balance equation
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