Magnetic helicity and subsolutions in ideal MHD

2018/01/15 by Faraco, Daniel, Lindberg, Sauli
#76B03 (Secondary) #76W05 (Primary) 35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.04896

Abstract

We show that ideal 2D MHD does not possess weak solutions (or even subsolutions) with compact support in time and non-trivial magnetic field. We also show that the Λ-convex hull of ideal MHD has empty interior in both 2D and 3D; this is seen by finding suitable Λ-convex functions. As a consequence we show that mean-square magnetic potential is conserved in 2D by subsolutions and weak limits of solutions in the physically natural energy space L^∞_t L²_x, and in 3D we show the conservation of magnetic helicity by L³-integrable subsolutions and weak limits of solutions. However, in 3D the Λ-convex hull is shown to be large enough that nontrivial smooth, compactly supported strict subsolutions exist.

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