Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

Abstract

We study the existence of steady solutions of ideal magnetofluid systems (ideal MHD and ideal Euler equations) without continuous Euclidean symmetries. The existence of such solutions can potentially allow for complex shapes in the design of confining magnetic fields. It is shown that all nontrivial magnetofluidostatic solutions are locally symmetric, although the symmetry is not necessarily an Euclidean isometry. Furthermore, magnetofluidostatic equations admit both force-free (Beltrami type) and non-force-free (with finite pressure gradients) solutions that do not exhibit invariance under translations, rotations, or their combination. Examples of smooth solutions without continuous Euclidean symmetries in bounded domains are given. Finally, the existence of square integrable solutions of the tangential boundary value problem without continuous Euclidean symmetries is proved.