Some remarks on the antidynamo theorem

V. I. Arnol'd

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (1982), pp. 50-57

Résumé

We study the evolution of differential $k$-forms on a compact Riemannian $m$-manifold without boundary (due to the transport by the flow of a given vector field and due to the diffusion). Every forjn evolves into a stationary one (which is unique in its cohomology class) if either the diffusion is fast enough or $k=0,m$. We prove that the number of independent stationary forms is at least the $k$-th Betti number (which does not depend on the rate of the diffusion). The $2$-dimensional magnetic fields $(k=1,m=2)$ are proved to evolve into cohomologous stationary fields. Examples show the non-uniqueness of stationary magnetic fields in a given cohomology class on $3$-manifolds $(k=2, m=3)$ and (the existence of fields growing exponentially with time and, in particular, of periodic fields in the usual $3$-space.