Magnetic helicity, weak solutions and relaxation of ideal MHD

2021/09/19 by Faraco, Daniel, Lindberg, Sauli, Székelyhidi, László · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Plasma Physics (physics.plasm-ph)

paper · doi:10.48550/arxiv.2109.09106

Abstract

We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the L³ integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.

Cited by

• Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth
• Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion

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