Bounded Solutions of Ideal MHD with Compact Support in Space-Time
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Bounded Solutions of Ideal MHD with Compact Support in Space-Time Daniel Faraco, Sauli Lindberg & László Székelyhidi
Jr.
Communicated by V. Šverák
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.
1. Introduction Ideal magnetohydrodynamics (MHD for short) couples Maxwell equations with Euler equations to study the macroscopic behaviour of electrically conducting fluids such as plasmas and liquid metals (see [31,50]). The corresponding system of partial differential equations governs the simultaneous evolution of a velocity field u and a magnetic field B which are divergence free. The evolution of u is described by the Cauchy momentum equation with an external force given by the Lorentz force induced by B. The evolution of B is, in turn, described by the induction equation which couples Maxwell–Faraday law with Ohm’s law. D.F. was partially supported by ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-556, the Grants MTM2014-57769-P-1 and MTM2017-85934-C3-2-P (Spain) and the ERC Grant 307179-GFTIPFD, ERC Grant 834728-QUAMAP. S.L. was supported by the ERC Grant 307179-GFTIPFD and by the AtMath Collaboration at the University of Helsinki. L. Sz. was supported by ERC Grant 724298-DIFFINCL. Part of this work was completed in the Hausdorff Research Institute (HIM) in Bonn during the Trimester Programme Evolution of Interfaces. The authors gratefully acknowledge the warm hospitality of HIM during this time. D.F also thanks the hospitality of the University of Aalto were part of his research took place.
D. Faraco et al.
The ideal MHD equations give a wealth of structure to smooth solutions and several integral quantities are preserved. In 3D, smooth solutions conserve the total energy, but also two other quantities related to the topological invariants of the system are constant functions of time: the cross helicity measures the entanglement of vorticity and magnetic field, and the magnetic helicity measures the linkage and twist of magnetic field lines. Magnetic helicity was first studied by Woltjer [59] and interpreted topologically in the highly influential work of Moffatt [42], see also [4]. In fact, it was recently been proved in [36] that cross helicity and magnetic helicity characterise all regular integral invariants of ideal MHD. In this paper we are interested in weak solutions of the ideal MHD system, which in some sense describe the infinite Reynolds number limit. As pointed out in [12] such weak solutions should reflect two properties: (i) anomalous dissipation of energy; (ii) conservation of magnetic helicity. Indeed, just as in the hydrodynamic situation, in MHD turbulence the rate of total energy dissipation in viscou
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