Weak Solutions of Ideal MHD Which Do Not Conserve Magnetic Helicity

Rajendra Beekie; Tristan Buckmaster; Vlad Vicol

doi:10.1007/s40818-020-0076-1

Weak Solutions of Ideal MHD Which Do Not Conserve Magnetic Helicity

Rajendra Beekie, Tristan Buckmaster, Vlad Vicol

Mathematics

Research output: Contribution to journal › Article › peer-review

37 Scopus citations

Abstract

We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor’s conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.

Original language	English (US)
Article number	1
Journal	Annals of PDE
Volume	6
Issue number	1
DOIs	https://doi.org/10.1007/s40818-020-0076-1
State	Published - Jun 1 2020

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics
Geometry and Topology
Mathematical Physics
General Physics and Astronomy

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abstract = "We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor{\textquoteright}s conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.",

author = "Rajendra Beekie and Tristan Buckmaster and Vlad Vicol",

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AU - Buckmaster, Tristan

AU - Vicol, Vlad

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N2 - We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor’s conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.

AB - We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor’s conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.

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Weak Solutions of Ideal MHD Which Do Not Conserve Magnetic Helicity

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