Proof of Taylor’s Conjecture on Magnetic Helicity Conservation

Original Title: “Proof of Taylor’s Conjecture on Magnetic Helicity Conservation”

Authors: Daniel Faraco (ICMAT-UAM) y Sauli Lindberg (Universidad de Helsinki)

Source: Communications in Mathematical Physics 373 (2019), 707–738

Date of online publication: 2019



The so-called magneto hydrodynamic (MHD) equations govern the behaviour of fluids acted upon by a magnetic field, such as the plasma found in tokamak nuclear reactors, or solar plasma. In order to analyze these equations, physicists and mathematicians use one of their favourite tools: the study of integral quantities that are either conserved or dissipated over time. However, in different problems concerning hydrodynamics with turbulent regimes, these quantities may not be conserved. In 1974, John Bryan Taylor conjectured that the so-called magnetic helicity, which describes the topological behaviour of magnetic field lines, was practically conserved for small resistivity. In the paper under review, Daniel Faraco and Sauli Lindberg prove this conjecture; that is, they rigorously demonstrate that the magnetic helicity is approximately conserved for limits of Leray Hopf solutions. Furthermore, this conjecture is studied in configurations that are topologically complicated, such as in tokamak reactors, which are not simply connected.

This paper forms part of a larger project whose aim is to understand MHD equations in the framework of the convex integration technique and the socalled compensated compactness theorem. Specifically, the analysis of magnetic helicity in these equations is based on the discovery that the time derivative of the helicity is simply the integral of the scalar product of the electric field and the magnetic field.