Flexibility and rigidity of free boundary MHD equilibria

Abstract

We study stationary free boundary configurations of an ideal incompressible magnetohydrodynamic fluid possessing nested flux surfaces. In 2D simply connected domains, we prove that if the magnetic field and velocity field are never commensurate, the only possible domain for any such equilibria is a disk, and the velocity and magnetic field are circular. We give examples of non-symmetric equilibria occupying a domain of any shape when either the velocity and magnetic field equal only on the boundary or by imposing an external magnetic field generated by a singular current sheet charge distribution (external coils). Some results carry over to 3D axisymmetric solutions. These results highlight the importance of flow velocity to support unforced asymmetric equilibria, thereby shedding some light on Grad's conjecture as it applies to free boundary solutions.

Singularity formation in the incompressible Euler equation in finite and infinite time

“…In a similar spirit to Theorem 3.3, there is the recent work [101] which establishes radial symmetry for all compactly supported single signed vorticity distributions on R 2 , as well as for more singular vortex sheet configurations [102]. Free boundary fluid bodies also exhibit strong forms of rigidity [105,47].…”

“…We remark that for higher eigenfunctions ψ " sinpx 1 q sinpmx 2 q in the highly oscillatory regime with m " 1, unstable eigenfunctions can be constructed using averaging methods together with the Meshalkin-Sinai continued fraction technique [90]. A compelling piece of evidence for nonlinear instability of (47) is that, from the work of Shvydkoy and Latushkin [193], we know that the spectrum of the linearized Euler in H 1 and H ´1 is the full band |Rez| ď 1 and so it is very unstable linearly. It is unknown whether or not there is an embedded eigenvalue; if so, the instability would follow from the work of Lin [149].…”

“…See e.g. Lemma 5 of [47]. Clearly ψ cannot change sign in M (it otherwise would contradict ψ having a unique critical point).…”

“…They ensure that the steady state is either a minimum or a maximum of the energy (the action) on the orbit of a vorticity distribution in the group of volume preserving diffeomorphims [8]. The degenerate case of F pψq " pconst.q 6 We remark in passing that it is an intriguing open issue to establish nonlinear instability of the steady state (47) in L 2 of vorticity (or, in fact, velocity). This would represent a large-scale instability, unlike the small-scale instability/singularity formation which is the subject of §3.3 and in contrast to the stability Theorem 3.5 below.…”

Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2D Euler equation with finite energy

“…In the case of 2D Euler, Hamel-Nadirashvili in [54,52] proved that any stationary solution without a stagnation point must be a shear flow whenever the domain is a strip and also in [53] proved the corresponding rigidity (radial symmetry) result whenever the domain is a two-dimensional bounded annulus, an exterior circular domain, a punctured disk or a punctured plane. Constantin-Drivas-Ginsberg [29,28] obtained rigidity and flexibility results for Euler and other equations (such as MHD) in both 2D and 3D. Coti-Zelati-Elgindi-Widmayer [32] constructed stationary solutions close to the Kolmogorov and Poiseuille flows in T 2 .…”