Bounded solutions of ideal MHD with compact support in space-time
Daniel Faraco, Sauli Lindberg, L\'aszl\'o Sz\'ekelyhidi Jr
TL;DR
This paper demonstrates the existence of infinitely many bounded, compactly supported solutions in 3D ideal MHD that violate some conservation laws but preserve magnetic helicity, contrasting with the 2D case.
Contribution
It constructs explicit examples of compactly supported solutions in 3D ideal MHD and proves non-existence in 2D energy space.
Findings
Existence of infinitely many bounded, compactly supported solutions in 3D MHD.
Solutions violate energy and cross helicity conservation.
No nontrivial compactly supported solutions exist in 2D energy space.
Abstract
We show that in 3-dimensional ideal magnetohydrodynamics there exist infinitely many bounded solutions that are compactly supported in space-time and have non-trivial velocity and magnetic fields. The solutions violate conservation of total energy and cross helicity, but preserve magnetic helicity. For the 2-dimensional case we show that, in contrast, no nontrivial compactly supported solutions exist in the energy space.
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