Bounded Solutions of Ideal MHD with Compact Support in Space-Time

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.

Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system

“…These solutions have finite energy (they are in fact H β , for some β > 0) but do not preserve magnetic helicity [4]. We also refer to [15] for the existence of bounded solutions that violate conservation of energy and cross helicity. Generally speaking, this means that "reasonable" solutions must be of some regularity.…”

Weak stability and closure in turbulence

“…Applying the Aubin-Lions lemma, one concludes strong convergence A k → Ā in L 2 , and consequently that H(B k ) → H( B) (see [50,51], where the physically more relevant problem of the inviscid limit is treated). In particular, magnetic helicity remains stable under the weak closure from S to S. More generally and analogously to the passage from (3.1) to (3.3), we might ask for the characterization of the Reynolds-type term R as well as the limiting electric field Ē in the system…”

Lamination Convex Hull of Stationary Incompressible Porous Media Equations

“…As a direct reflection of this, bounded solutions conserve the meansquare magnetic potential in 2D and the magnetic helicity in 3D. This rules out solutions with a nontrivial, compactly supported magnetic field in 2D but, perhaps surprisingly, not in 3D [11]. By Tartar's Theorem (see [24,Theorem 11]), quadratic \Lambda -affine functions are weakly continuous, and as such, they also aid the understanding of various asymptotic regimes such as weak limits of (sub)solutions or the inviscid limit; see also [4, p. 58].…”

“…Indeed, one would need to find potentials satisfying the extremely rigid constraint m \equiv [\rho -(1 -\rho 2 )v 2 / | v| 2 ]v in the rigid region where (\rho , v) \in pr(X 3 ) \subset \BbbR \times \BbbR 2 or avoid the (rather large) rigid region altogether. In 3D MHD, by contrast, the lamination convex hull is, loosely, speaking, 1-codimensional, which left enough room for running convex integration in [11]…”