Finite-energy weak solutions to ideal MHD exist with time-varying magnetic helicity, proving they are not attainable in the infinite-conductivity zero-viscosity limit.
Nonuniqueness of weak solutions to the SQG equation
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abstract
We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Sz\'ekelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.
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Weak solutions of ideal MHD which do not conserve magnetic helicity
Finite-energy weak solutions to ideal MHD exist with time-varying magnetic helicity, proving they are not attainable in the infinite-conductivity zero-viscosity limit.