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

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.


Introduction
Ideal magnetohydrodynamics (MHD for short) couples Maxwell equations with Euler equations to study the macroscopic behaviour of electrically conducting fluids such as plasmas and liquid metals (see [31,50]). The corresponding system of partial differential equations governs the simultaneous evolution of a velocity field u and a magnetic field B which are divergence free. The evolution of u is described by the Cauchy momentum equation with an external force given by the Lorentz force induced by B. The evolution of B is, in turn, described by the induction equation which couples Maxwell-Faraday law with Ohm's law. The ideal MHD equations give a wealth of structure to smooth solutions and several integral quantities are preserved. In 3D, smooth solutions conserve the total energy, but also two other quantities related to the topological invariants of the system are constant functions of time: the cross helicity measures the entanglement of vorticity and magnetic field, and the magnetic helicity measures the linkage and twist of magnetic field lines. Magnetic helicity was first studied by Woltjer [59] and interpreted topologically in the highly influential work of Moffatt [42], see also [4]. In fact, it was recently been proved in [36] that cross helicity and magnetic helicity characterise all regular integral invariants of ideal MHD.
In this paper we are interested in weak solutions of the ideal MHD system, which in some sense describe the infinite Reynolds number limit. As pointed out in [12] such weak solutions should reflect two properties: (i) anomalous dissipation of energy; (ii) conservation of magnetic helicity.
Indeed, just as in the hydrodynamic situation, in MHD turbulence the rate of total energy dissipation in viscous, resistive MHD seems not to tend to zero when the Reynolds number and magnetic Reynolds number tend to infinity. This has been recently verified numerically in 3D; see [20,39,41]. On the other hand simulations, and theoretical results have shown that magnetic helicity is a rather robust conserved quantity even in turbulent regimes, and J.B. Taylor conjectured that magnetic helicity is approximately conserved for small resistivities [57] (unlike subhelicities along Lagrangian subdomains that are magnetically closed at the initial time). Taylor's conjecture is at the core of Woltjer-Taylor relaxation theory which predicts that after an initial turbulent state, various laboratory plasmas relax towards a quiescent state which minimises magnetic energy subject to the constraint of magnetic helicity conservation (see [4,46]).
The conservation of magnetic helicity for weak solutions of ideal MHD was first addressed in [12], and subsequently it was shown in [1,35] that it is conserved if u, B ∈ L 3 x,t , that is, in contrast with energy conservation, no smoothess is required. Moreover, the first and second author recently proved that if a solution in the energy space L ∞ t L 2 x arises as an inviscid limit, then it conserves magnetic helicity (see [29]). In this context, Theorem 2.2 below extends [29,Corollary 1.3], from ideal (that is inviscid, non-resistive) limits of Leray-Hopf solutions to a larger class of possible approximation schemes.
Our main purpose in this paper is to show the existence of nontrivial weak solutions to ideal 3D MHD compatible with both requirements (i)-(ii) above. Theorem 1.1. There exist bounded, compactly supported weak solutions of ideal MHD in R 3 , with both u, B nontrivial, such that neither total energy nor cross helicity is conserved in time, but magnetic helicity vanishes identically.
We note that bounded solutions in particular fall into the subcritical regime of [35] for magnetic helicity, so that for the solutions above magnetic helicity must vanish at all times even though the magnetic field B is not identically zero. Moreover, as a corollary of Theorem 2.2 below, it also holds that for bounded solutions on T 3 , either the initial data has vanishing magnetic helicity or B cannot have compact support in time. Indeed, as noted by Arnold [3], T 3 |B| 2 dx ≥ C T 3 A · B dx at every t ∈ [0, T [, where A is the magnetic potential. It is also worth pointing out that the solutions in Theorem 1.1 have nontrivial cross-helicity.
MHD turbulence in 2D seems to have many similarities with the 3D case (in stark contrast with hydrodynamic turbulence), in particular there is plenty of numerical evidence for anomalous dissipation of energy [6,7]. Nevertheless, we will show in Section 2.2 that in 2D, under very mild conditions, weak solutions with nontrivial magnetic field cannot decay to zero in finite time, in particular solutions as in Theorem 1.1 do not exist in 2D.
Our construction is based on the framework developed in [21] by C. De Lellis and the third author for the construction of weak solutions to the Euler equations. This framework is based on convex integration, which was developed by Gromov [32] following the work of Nash [44], and-in a nutshell-amounts to an iteration procedure whereby one approximates weak solutions via a sequence of subsolutions, in each iteration adding highly oscillatory perturbations designed to cancel the low wavenumber part of the error. In [21] convex integration was used in connection with Tartar's framework to obtain bounded nontrivial weak solutions of the Euler equations which have compact support and violate energy conservation. Such pathological weak solutions were known to exist [49,51] but the method of [21] turned out to be very robust and many equations in hydrodynamics are amenable to it and its ramifications.
Roughly speaking, the development of the theory followed two strands: concerning the Euler equations and in connection with Onsager's conjecture [27,45], an important problem was to push the regularity of such weak solutions beyond mere boundedness to the Onsager-critical regime. This programme, started in [24], finally culminated in Isett's work [33], see also [9]. For a thorough report of these developments and connections to Nash's work on isometric embeddings, we refer to [25]. Another, somewhat independent strand, was to adapt the techniques to other systems of equations, such as compressible Euler system [13], active scalar equations [10,18,52,53] and others [8,14,15,37]. A key point in the technique is a study of the phase-space geometry of the underlying system, to understand the interaction of high-frequency perturbations with the nonlinearity in the equations in the spirit of L. Tartar's compensated compactness. A particularly relevant example to this discussion is the case of 2D active scalar equations, where there seems to be a dichotomy between systems closed under weak convergence such as 2D Euler in vorticity form or SQG, and those with a large weak closure such as IPM [18,52]-see the discussion in [23,Section 8] and [34] of [23,34] in this regard.
Concerning the ideal MHD system, setting the magnetic field b ≡ 0 obviously reduces to the incompressible Euler equations, and thus [21] applies. More generally, in [8] Bronzi, Lopes Filho and Nussenzveig Lopes constructed bounded weak solutions of the symmetry reduced form u( , compactly supported in time and not identically zero. These "2 1 2 -dimensional" solutions were obtained by reducing the symmetry reduced 3D MHD to 2D Euler with a passive tracer, where a modification of the strategy of [21] applied. Nevertheless, such reductions to the Euler system do not seem to be able to capture generic, truly 3-dimensional weak solutions, which-with the simultaneous requirement of properties (i) and (ii) above-seem to lie on the borderline between weakly closed (for example SQG) and non-closed (for example IPM) systems.
The remark above will be explained in more detail in Section 2-for now let us merely point out that whilst the Cauchy momentum equation for the evolution of the velocity u has a large relaxation (the main observation behind all results involving convex integration for the Euler equations), the Maxwell system for the evolution of the magnetic field B is weakly closed (an observation going back to the pioneering work of Tartar [55]). Indeed, our whole philosophy in this paper is to emphasise the role of compensated compactness in connection with conserved quantities-in Section 2 we revisit Taylor's conjecture and conservation of mean-square magnetic potential in this light.
The additional rigidity due to conservation of magnetic helicity is a severe obstruction to applying the available versions of convex integration to MHD: The nonlinear constraint E · B = 0, where E is the electric and B the magnetic field, has to be satisfied not just by the weak solutions in Theorem 1.1 but also along any approximating sequence in the convex integration scheme. In order to ensure this constraint, we use nonlinear potentials and, inspired by Müller and Šverák [43], develop a substantially new version of convex integration directly on differential two-forms (the Maxwell 2-form), consistent with the geometry of full 3D MHD. Indeed, this is the main innovation in this paper, and hence we dedicate Section 4 below to explain the differences to previous schemes in technical terms. We remark in passing that for the special solutions in [8], E · B = 0 is automatic.
We close the introduction by commenting on the recent preprint [5] which we learned about after this paper was completed. In [5], L ∞ t L 2 x weak solutions are constructed which do not preserve magnetic helicity or total energy. The space L ∞ t L 2 x is super-critical with respect to magnetic helicity, c.f. [29,35] and Theorem 2.2 below, and thus the solutions constructed in [5] seem closer in spirit to unbounded, so-called very weak solutions constructed in [2]. The construction in [5] is based on the convex integration scheme developed in [11] for the 3D Navier-Stokes equations, and indeed, we remind the reader that the weak solutions in [11] are super-critical not just in terms of Navier-Stokes regularity, but also in terms of the minimal regularity required for compactness (for example the Leray-Hopf energy space). Thus, the methods of these two papers are completely different, and it would be a very interesting question whether they can be combined-for instance to explore the limiting integrability and smoothness of convex integration solutions to MHD.

The Ideal MHD System
We recall that the ideal MHD equations in three space dimensions are written as for a velocity field u, magnetic field B and total pressure . In this paper we consider both the full space case R 3 and the periodic setting T 3 . In the latter case the zero-mean condition is added to (2.1)-(2.3), where for notational convenience we write u for the spatial average on T 3 . As usual, weak solutions of (2.1)-(2.3) can be defined in the sense of distributions for u, B ∈ L 2 loc , using the identities An analogous definition is given in the periodic setting on the torus T 3 .

Conserved Quantities
It is well known that there are three classically conserved quantities of ideal 3D MHD on the torus T 3 . For the first two, analogous definitions are available in R 3 . Definition 2.1. Let (u, B) be a smooth solution of (2.1)-(2.3) and let A be a vector potential for B, that is ∇ × A = B. The total energy, cross helicity and magnetic helicity of (u, B) are defined as All three quantities defined above are conserved in time by smooth solutions. The conservation of total energy and cross helicity conservation was studied in [12,35,58,60]. Conservation of the magnetic helicity was shown in [12] for solutions u ∈ C([0, T ]; B α 1 3,∞ ) and B ∈ C([0, T ]; B α 2 3,∞ ) with α 1 + 2α 2 > 0. In [1,35], magnetic helicity conservation is shown under the assumption that u, B ∈ L 3 (T 3 ×]0, T [).
We note in passing that on the whole space R 3 the analogous definitions of total energy and cross helicity lead to conserved quantities for square integrable solutions, but magnetic helicity is not well-defined. This boils down to the scaling properties of the function spaces in question; see Appendix A. However, for square integrable magnetic fields that are compactly supported in space, magnetic helicity is well-defined. Indeed, every B ∈ L 1 (R 3 ) with ∇ · B = 0 satisfies R 3 B(x) dx = B(0) = 0 sinceB(ξ ) · ξ/ |ξ | = 0 for all ξ = 0 andB is continuous.
Following L. Tartar's pioneering work [55] one can understand the system (2.1)-(2.3) as a coupling between linear conservation laws and a set of constitutive laws in form of pointwise constraints. The conservation laws are where in 3D, S is a symmetric 3 × 3 tensor (the Cauchy stress tensor) and E is a vector field (the electric field). Indeed, (2.6) is simply the Maxwell-Faraday law for the electric field. In the periodic case (2.4) is added to (2.5)-(2.7). The constitutive set is then obtained by relating the stress tensor S and the electric field E to velocity, magnetic field and pressure, for example via the ideal Ohm's law: It is easy to verify that the system (2.1)-(2.3) can be equivalently formulated for the state variables (u, S, B, E) as (2.5)-(2.7) together with (u, S, B, E)(x, t) ∈ K a.e. (x, t).
Using this formulation one can easily identify the conservation of magnetic helicity as an instance of compensated compactness following the work of Tartar [55] when applied to the Maxwell system To explain this, we recall the following generalisation of the div-curl lemma from Example 4 in [55]: suppose we have a sequence of magnetic and electric fields (B j , E j ) (B, E) converging weakly in L 2 x,t and such that {∂ t B j + ∇ × E j } and {∇ · B j } are in a compact subset of H −1 . Then B j · E j * B · E in the space of measures. In view of the constitutive law E = B × u we deduce that any reasonable approximation of bounded weak solutions of ideal MHD leads in the limit to a solution (u, S, B, E) of (2.5)-(2.7) with B · E = 0. That is, the state variables are constrained to the relaxed constitutive set (2.10) In turn, perpendicularity of the electric E and magnetic B fields is closely related to conservation of magnetic helicity. Indeed, if A is a magnetic potential (so that ∇ × A = B), adapting the classical computation (for example [6]) shows that More generally, we have the following theorem, establishing the connection between compensated compactness and conservation of magnetic helicity, an issue that has been emphasised by L. Tartar [55,56]: , is a solution of (2.9) and assume B · E = 0 a.e. Then magnetic helicity is conserved.
is a sequence of solutions of (2.9) as in (a), and in addition Then magnetic helicity is conserved.
Part (a) extends in particular the L 3 result of [35]. Indeed, for weak solutions of ideal MHD with u, B ∈ L 3 we have E = B × u ∈ L 3/2 . On the other hand our proof does not rely on a specific regularisation technique as in [17], and merely relies on a weak version of formula (2.11). Part (b) shows that conservation of magnetic helicity holds even beyond the setting of weak continuity of the quantity B · E. As a matter of fact this line of argument furnishes a proof of Taylor conjecture for simply connected domains [29].
We begin the proof of Theorem 2.2 by recalling the following L p Poincaré-type lemma for the Maxwell system (2.9): Indeed, we set A = − −1 (∇ × B) and ϕ = −1 ∇ · (∂ t A + E) and apply standard Calderón-Zygmund estimates for the Laplacian.

Proof of Theorem 2.2. For part (a) suppose
be a standard spacetime mollification of B. By using Lemma 2.3 and integrating by parts a few times we get is a sequence of solutions of (2.9) with B j · E j = 0 a.e. and assume B j B in L 2 (T 3 ×]0, T [) and sup j∈N E j L 1 < ∞. We intend to use to the Aubin-Lions Lemma (see for example [48,Lemma 7.7]) to get, up to a subsequence, which verifies the assumptions of the Aubin-Lions Lemma.

The 2-Dimensional Case
In comparison to the above analysis, let us briefly look at the 2-dimensional case. Here (2.2) reduces to where we write B ⊥ = (−B 2 , B 1 ) for the vector B = (B 1 , B 2 ), and similarly ∇ ⊥ = (−∂ 2 , ∂ 1 ). The magnetic potential (stream function) of B is a scalar field ψ such that ∇ ⊥ ψ = B. As for conserved quantities, total energy and cross-helicity has analogous expressions, but magnetic helicity is replaced by the mean-square magnetic potential, defined as Mean-square magnetic potential is conserved by smooth solutions, and in [12] the conservation was shown for weak solutions (u, B) with the regularity u ∈ C([0, T ]; B α 1 3,∞ ) and B ∈ C([0, T ]; B α 2 3,∞ ) for α 1 + 2α 2 > 1. Next, observe that (2.3) implies that u· B ⊥ is a div-curl product. Consequently, if we have a sequence of velocity and magnetic fields (u j , B j ) (u, B) converging weakly in L 2 and such that {∇ · u j } and {∇ · B j } are in a compact subset of H −1 , then u j · B ⊥ j * u · B ⊥ in the space of measures. In other words (2.12) is stable under weak convergence in L 2 . Another way of writing (2.12) is by using the stream functions of u and B. Indeed, if we write u = ∇ ⊥ φ and B = ∇ ⊥ ψ, with φ = ψ = 0, then (2.12) becomes where we write, as usual, J (φ, ψ) = ∇φ ·∇ ⊥ ψ for the Jacobian determinant of the mapping (φ, ψ) : T 2 → R 2 . Observe that the same equation appears also for the 2D Euler equations, where we replace ψ by the vorticity ω = ∂ 1 u 2 − ∂ 2 u 1 and φ by the velocity potential v = ∇ ⊥ φ. However, here we do not assume any coupling between φ and ψ, and treat (2.13) as a passive scalar equation. The form (2.13) allows us to prove conservation of the mean-square magnetic potential under very mild conditions as follows: ) is a weak solution of (2.3) and (2.12). Then the mean-square magnetic potential is conserved.
We point out that the analogous result for the 2D Euler equations, namely the conservation of enstrophy 1 2 |ω| 2 dx is well-known [28,40], and the proof is based on the theory of renormalised solutions. Here we give an alternative, short proof, again emphasising that compensated compactness lies at the heart of the matter. For the proof we first recall the H 1 regularity theory of Coifman, Lions, Meyer and Semmes from [16], more precisely the following adaptation of the classical Wente inequality to the torus T 2 (see [30, Theorem A.1]): 14) The left-hand side of (2.14) can be understood in terms of H 1 -BMO duality, but we in fact only require (2.14) where the left-hand side is Lebesgue integrable.
Proof of Theorem 2.4. First let us assume that u and B are smooth. Then, using (2.13), we obtain after integration by parts d dt For the general case note first that under the assumption on (u, B), using the compact ). Then the computation (2.15) can be carried out using standard regularisation of ψ, φ and the uniform bound in (2.14).
Theorem 2.4 implies the following corollary: and (2.12). Then either B ≡ 0 or there exists a constant c > 0 such that

Proof. The proof follows by using the Poincaré inequality at every
Thus, in 2D even if the kinetic and magnetic energies T 2 |u| 2 dx and T 2 |B| 2 dx may fluctuate (and indeed, numerical experiments indicate anomalous dissipation of the total energy even in 2D [7]), by Corollary 2.6 it is impossible for the magnetic energy to dissipate to zero.
Finally, we remark that although it is natural to ask whether an analogue of Theorem 2.4 holds in the whole space R 2 , in fact square integrable divergence-free vector fields do not in general have a square integrable stream function in R 2 . This is shown in Appendix A.

Plane-Wave Analysis
Recall that the ideal MHD system in 3D can be written for a state variable (u, S, B, E) in terms of the conservation laws (2.5)-(2.7) with the constitutive set K , defined in (2.8). The framework introduced by Tartar amounts to an analysis of one-dimensional oscillations compatible with (2.5)-(2.7)-the wave-cone-and then the interaction of the wave-cone with the constitutive set; we carry out this analysis in this section.

The Wave Cone and the Lamination Convex Hull
Plane waves are one-dimensional oscillations of the form ( In what follows, we will write, with a slight abuse of notation, Definition 3.1. The wave cone for ideal MHD is We also denote In the convex integration process we will use instead of 0 , as the requirement ξ x = 0 is crucial to many of the arguments. We next define lamination convex and -convex hulls. Given a set Y ⊂ R 15 we denote Y 0, := Y and define, inductively It is well-known that semiconvex hulls can be expressed by duality in terms of measures, see for example [15,38,47]. The set of laminates of finite order (with respect to ), denoted L(Y ), is the smallest class of atomic probability measures supported on Y with the following properties: Similarly, we write V 1 = λ 1,1 V 1,1 + λ 1,2 V 1,2 . Repeating this process, by induction we arrive at a finite-order laminate with support in Y and barycentre V : In addition to the lamination convex hull, another, potentially larger, hull is used in convex integration theory. In order to define it we recall the notion of -convex functions.
While the lamination convex hull is defined by taking convex combinations, the -convex hull Y of Y ⊂ R 15 is defined as the set of points that cannot be separated from Y by -convex functions.

Normalisations of the Constitutive Set
In order to produce bounded solutions of 3D MHD we consider normalised versions of the constitutive set K . We wish to prescribe both the total energy density (|u| 2 + |B| 2 )/2 and the cross helicity density u · B, but for this aim it is obviously not enough to prescribe |u| and |B|. However, by using the Elsässer variables , and thus it suffices to prescribe z ± . This motivates the normalisation given below; As pointed out in Section 2, the Maxwell system is essentially closed under weak convergence; the scalar product B · E is weakly continuous. As an immediate consequence the -convex hull K r,s has empty interior (in 3D, and also in 2D). Indeed, Tartar's result in Example 4 [55] is based on the fact that the quadratic and consequently, Q is 0 -affine. Then we deduce that where M is the set in (2.10). In 3D, assume now B × u = 0. Then (3.1) implies, up to normalisation, that which is an easier condition to check in the sequel.

Discussion of the Convex Integration Scheme in 3D
The standard way of finding nontrivial compactly supported solutions for equations of fluid dynamics was first presented in [21] and axiomatised in [53]. We describe it briefly in the case of Theorem 1.1.
With a bounded domain ⊂ R 4 fixed, it suffices to find a solution V of the relaxed MHD equations L(V ) = 0 such that V (x, t) ∈ K 2,1 a.e. in (K 2,1 defined in (3.4)) and V (x, t) = 0 a.e. outside . One intends to construct V as a limit of subsolutions, that is, mappings V solving L(V ) = 0 and taking values in K lc, 2,1 . The basic building blocks of the construction are plane waves which oscillate in directions of . In order to prevent harmful interference of the waves and to make the eventual solutions compactly supported, one needs to localise the plane waves. The localisation is customarily carried out by constructing potentials. This causes small error terms, and in order for each V to take values in the lamination convex hull, one hopes to prove that the hull has non-empty interior. The specifics of the convex integration scheme vary (see for example [15,18,21] for three different approaches in fluid dynamics and [38,53] for a more general discussion).
In the case of 3D MHD, the process is more subtle, as K lc, 2,1 has an empty interior, more precisely K lc, 2,1 ⊂ M . Therefore, although we may proceed with the 'symmetric (fluid) part' u and S, the 'anti-symmetric (electromagnetic) part' B and E needs special attention.
As a first step towards overcoming the emptiness of int(K lc, 2,1 ), we construct a pair of non-linear potential operators P B and P E that satisfy (For u and S we simply use the potentials in [21] for the Euler equations.) We add the localised plane waves within P B and P E ; despite their non-linearity, P B and P E have cancellation properties which allow them to map suitable sums of localised plane waves to sums of localised plane waves (up to a small error term).
As a drawback, P B and P E do not allow oscillating plane waves for every -segment-their applicability depends not only on the direction but also on the location of the segment. We consider -segments for which P B and P E give plane waves and call them good -segments or g -segments. This leads us to study K lc, g 2,1 , the restricted lamination convex hull of K 2,1 in terms of g . A priori, g is a rather large subset of -segments. However, even though K lc, 2,1 has non-empty interior relative to M , the electromagnetic part of K lc, g 2,1 is rigid: the constraint E = B × u holds for all (u, S, B, E) ∈ K lc, g 2,1 . Nevertheless, as the in-approximation formulation of convex integration shows, the iterative step happens at relatively open sets and it is a limit procedure which leads to the inclusion in closed sets. Thus, in this case the size of g saves the day; as it turns out, for relatively open subsets U ⊂ M we have U lc, = U lc, g . This eventually allows us to apply the Baire category framework of convex integration in U 2,1 := int M (K lc, 2,1 ). We present useful characterisations of U 2,1 in Theorem 6.7; in particular, int M (K lc, 2,1 ) = ∪ 0≤τ <1 K lc, 2τ,τ . Theorem 6.7 is the most technically difficult part of the paper and the heart of the convex integration scheme. The proof of Theorem 1.1 is then completed in Section 7.
Notice that actually, we do not compute the exact hull K lc, 2,1 . However, the formula int M (K lc, 2,1 ) = ∪ 0≤τ <1 K lc, 2τ,τ turns out to give us enough information about K lc, 2,1 . The formula is used in a similar manner as in [15].

Potentials in 3D
We wish to find potentials corresponding to -segments. For the fluid variables (u, S), we simply use the potentials of [21,22] for the Euler equations. In the case of the electromagnetic variables (B, E), the question about existence of potentials is more subtle because of the non-linear constraint B · E = 0 that the potentials need to obey. This issue is studied in Sections 5.3-5.8.
sym,0 ) with the following properties: This lemma essentially follows from the proof of [21,Proposition 3.2]. For the convenience of the reader we sketch a simplified proof, following the exposition in [54]: Proof. As noted in [22,54], a matrix-valued quadratic homogeneous polynomial P : R 4 → R 4×4 gives rise to a differential operator P(∂) as required in the lemma, if P = P(η) satisfies Elementary examples satisfying the first 3 conditions above are given by P(η) = 1 2 (Rη⊗Qη+Qη⊗ Rη) for antisymmetric 4×4 matrices R, Q such that Re 4 = 0. In particular, for any a, b ⊥ ξ with a ⊥ e 4 , set R = a⊗ξ −ξ ⊗a and Q = b⊗ξ −ξ ⊗b, to obtain P a,b (η). One quickly verifies that P a,b (ξ ) sym,0 with U ξ = 0 can be written as a linear combination for vectors a i , b i ∈ R 4 with a i · ξ = b i · ξ = 0 and a i · e 4 = 0, we obtain P U as required in the lemma as

Wave Cone Conditions on u, B and E
It will turn out that when we choose which -directions to use, we have much more freedom in the choice of S than the three other variables u, B and E. Recall that in 3D, the wave cone conditions are We can typically first find u, B, E, ξ satisfying (5.2) and (5.4) and afterwards choose S satisfying (5.3). This motivates the following observation: The following conditions are equivalent.

Maxwell Two-Forms
Our aim in the rest of this chapter is to find potentials for the variables B and E. We carry out this task using the formalisms of two-forms and bivectors in R 4 . In electromagnetics, it is customary to express (B, E) ∈ R 3 × R 3 as a unique bivector ω ∈ 2 (R 4 ) via the identification (see [26]). We write ω ∼ = (B, E). Then, Gauss' law and Maxwell-Faraday law are written concisely via differential forms: that is, ω is an exact two-form called Maxwell two-form or electromagnetic twoform.
Recall that in addition to (5.6), we also need E and B to satisfy B · E = 0. We express the latter condition in the language of bivectors: (where the last equality showing that ω is simple will be proved in the forthcoming If such a ξ is found, in view of Lemma 5.2 it can be modified to verify ξ x · u = 0 as well. Thus it only remains to verify the condition involving S, that is, (5.3). It turns out that the interaction of (5.7) and (5.8) is very neat with the forms formalism. This is the content of the next section.

-Segments in Terms of Simple Bivectors
The following well-known proposition collects characterisations equivalent to the condition B · E = 0 (The Plücker identity for the bivector ω): The following conditions are equivalent: Proof. The equivalence of (i) and (iii) was already noted, and (ii) clearly implies (i). Suppose then (iii) holds; our aim is to prove (ii).
Using Proposition 5.3, we formulate some useful further characterisations of (5.8).
The following conditions are equivalent: Recall that every -segment is contained in M . We give equivalent characterisations for this condition.
is also clear, and we complete the proof by showing that (ii) ⇒ (iii). The case ω 0 = 0 being clear, we assume that ω 0 = 0.

Clebsch Variables
Now (5.6) means that ω is closed and thus, by Poincaré lemma, exact: ω = dα. Here the so-called electromagnetic four-potential α is of course not unique. We specify a choice of α below. Recall from (5.7) that our potential α is required to satisfy dα ∧ dα = 0. This fact, among other things, motivates us to set α = ϕ dψ which leads to ω = dα = dϕ ∧ dψ; here φ, ψ ∈ C ∞ (R 4 ) are traditionally called Clebsch variables or Euler potentials.
With the Clebsch variables at our disposal we make a natural Ansatz on the electromagnetic side of the localised plane waves.
Use the simplicity of ω 0 to write ω 0 = v 0 ∧ w 0 , and recall the operator P U given by Lemma 5.1, with U given in (5.1).
Fix a cube Q ⊂ R 4 and a cutoff function χ ∈ C ∞ c (Q). Given h ∈ C ∞ (R) and ∈ N, our aim is to find φ , ϕ and ψ such that ). The choice of φ is specified in Lemma 5.1. For the electromagnetic part we define Clebsch variables of the form In (5.11)-(5.12), we use h instead of h in order to be consistent with the scaling on the fluid part. The Ansatz (5.11)-(5.12) yields (5.14) This raises the question whether (5.14) can be solved for c 1 , c 2 ∈ R. Notice that if ω 0 = 0, the answer is independent of the factors v 0 , w 0 of ω 0 = v 0 ∧ w 0 . It turns out that given general ω 0 , ω with ω 0 ∧ ω = 0, such c 1 , c 2 do not always exist. (The canonical bad case is ω 0 = v ∧ ξ, ω = w ∧ ξ , as then (c 1 v + c 2 ξ) ∧ ξ = c 1 v ∧ ξ = w ∧ ξ if and only if v is parallel to w). Essentially, when (5.14) holds, the segment defined by V 0 and V is good (the case ω 0 = 0 yielding some additional cases). Remark 5.7. In (5.13), we use crucially the cancellation properties of the wedge product dϕ ∧ dψ to overcome the nonlinearity of P B and P E . In fact, dϕ ∧ dψ arises, up to a term O(1/ ), as pullbacks of the bivector v 0 ∧ w 0 . In other words, Note that the class of simple two-forms is closed under taking pull-backs with ∈ C ∞ (R 4 ; R 4 ), as a consequence of the formula * (v ∧ w) = * v ∧ * w.

States in Clebsch Variables
As a matter of fact, when we iterate the construction and apply convex integration we will be modifying dϕ and dψ instead of dϕ ∧ dψ. We will therefore use a separate notation in which we keep track of the factors forming a bivector: 15) The case ω 0 = 0 is special as we will be able to construct potentials only when we interpret 0 = 0 ∧ 0.
We define a lamination convex hull in terms of g -segments.
Definition 5.11. Let Y ⊂ M . We define the sets Y k, g , k ∈ N 0 , as follows: is a good -segment whose endpoints belong to Y k−1, g .
We also give a related notion for finite-order laminates; recall Remark 3.4.

Localised Plane Waves Along g Segments
To every g -segment there corresponds a potential, and thus we can localise the plane waves.

Furthermore, W is locally constant in
For the proof we first specify the oscillating functions that we intend to use. Their first derivatives can be chosen to be mollifications of 1-periodic sawtooth functions.
Lemma 5.14. Suppose 0 < λ < 1 and ε > 0. Then there exists h ∈ C ∞ (R) with the following properties: In (iii) let ε > 0, fix a cubeQ ⊂ Q with |Q| > (1 − ε/3) |Q| and choose χ such that χ = 1 inQ. CoverQ, up to a set of measure ε |Q| /3, by cubes Q 1 , . . . , Q N with one of the sides parallel to ξ . We wish to show that {y ∈ Q k : h ( y · ξ) = λ} ≥ |Q k | (1 − ε/3)(1 − λ) for every large enough ∈ N; in (iii) we may then choose A 1 = ∪ N k=1 {y ∈ Q k : h ( y · ξ) = λ}. Similarly, Choose an orthonormal basis { f 1 , f 2 , f 3 , f 4 } of R 4 such that f 1 = ξ/ |ξ | and for some ζ ∈ R 4 . In order to switch to coordinates where Q k has sides parallel to coordinate axes, define L := 4 j=1 e j ⊗ f j ∈ R 4×4 , so that L f j = e j for j = 1, . . . , 4 and therefore L ∈ O(4). Then, denoting z = Ly, To finish the proof of (iii), write Without loss of generality, assumev ·w = 0. Let ε > 0 and chooseQ ⊂ Q and χ as above. Then have the sought properties for all large enough ∈ N. (5.20) CoverQ up to a small set by cubes Q 1 , . . . , Q N with two sides parallel tov x and w x ; recall thatv x ·w x = 0. 2 as in the previous proof, and a similar inequality holds for the other three cases of (5.20). This completes the proof of the case λ = 1/2. We then cover the case λ = 1/2. Let 0 < δ < min{λ, 1 − λ}. Using the case above, we choose dφ , dψ satisfying claims (i)-(iv) for [V 0 − δV , V 0 + δV ] and ε/2. Note that dφ ∧ dψ = δω = 0 in A 1 and dφ ∧ dψ = −δω = 0 in A 2 . We then cover the sets A 1 and A 2 by cubes up to a small set and apply the case (5.18) in the cubes. (The last claim of (iii) is clear.) Remark 5.15. We have looked for solutions of (5.10) of the form (5.11)-(5.12), and in some special cases, a solution does not exist. It is conceivable that another Ansatz would satisfy (5.10) in some of the cases excluded by (5.11)-(5.12). This would essentially require a degenerate Darboux Theorem with a Dirichlet boundary condition-more concretely, However, such theorems are remarkably difficult to prove and to the authors' knowledge, a suitable existence result is not available at this point; we refer to [19,Section 14] and the references contained therein.

Characterisations of the Relative Interior of the Lamination Convex Hull
Our next task is to find a suitable (relatively open) set U r,s ⊂ int M (K lc, r,s ) in state space, which will serve the purpose of defining subsolutions-see Section 7.1 below. Since we have only constructed potentials for g -segments, we would like to produce U r,s by using g -segments only. The choice of U r,s is, however, nontrivial, as discussed in Section 6.1. Nevertheless, eventually the following simple definition turns out to suffice. Definition 6.1. We denote U r,s := int M (K lc, r,s ). In the main result of this chapter, Theorem 6.7, we give several characterisations of U r,s and show, in particular, that 0 ∈ U r,s .

A Rigidity Result on the Good -Hull
Initially, it appears natural to choose some set U r,s ⊂ K lc, g r,s for strict subsolutions. However, K lc, g r,s turns out to be rather small; in fact, M is a g -segment, and assume that V 1 := V 0 + λV and V 2 : The proof consists of two parts. First, the g -conditions and the assumption E j = B j × u j lead to the conclusion (B 1 − B 2 ) × (u 1 − u 2 ) = 0. Then a bit of algebra gives At first sight, Proposition 6.2 seems to prevent convex integration unless potentials are found for bad -segments. However, this rigidity disappears once one considers g -convex hulls of relatively open sets. Indeed, whenever U is bounded and relatively open in M , we have U lc, g = U lc, (see Proposition 6.6). The basic reason behind this phenomenon is the fact that, loosely speaking, bad -segments become good when translated to almost any direction.

Laminates of Relatively Open Sets in M
We start the proof of Proposition 6.6 by showing that the class of relatively open sets in M is closed with respect to taking laminates:

Proposition 6.3. Suppose U is relatively open in M . Then U lc, g is relatively open in M .
Before beginning the proof of Proposition 6.3 we describe the main difficulty. The proof proceeds by induction.
It is, however, not guaranteed that the endpointsṼ 0 + λV,Ṽ 0 − (1 − λ)V lie on the nonlinear manifold M ! Therefore, we need to perturbṼ 0 +λV andṼ 0 −(1−λ)V in order to place an entire g -segment on M . This is in stark contrast to equations of fluid dynamics where the lamination convex hull has non-empty interior. Again, the two-form formalism comes to the rescue.
We overcome the difficulties via the following lemma which allows us to choose the factors v, w ∈ R 4 of a simple two-form v ∧ w in a continuous way. Henceforth, we denote ω := max | f |=|g|=1 ω( f, g) for every ω ∈ 2 (R 4 ).
We also need a lemma which gives a solution of a matrix equation with a natural norm estimate. Lemma 6.5. If x ∈ R 3 \{0} and y ∈ R 3 , then sym satisfies Sx = y and |S| ≤ 3 |y| / |x|.
Proof of Proposition 6.3. We need to show for every k ∈ N 0 that if V 0 ∈ U k, g , then there exists δ > 0 such that B M (V 0 , δ) ⊂ U lc, g . The claim holds for k = 0 by assumption, so assume, by induction, it holds for k.

Equivalence of Hulls of Relatively Open Sets
We may assume that w x = 0 (by possibly adding a constant multiple of ξ x = 0 to w x ). This time select a basis {ξ, w, f, g} of R 4 with f x = 0 and w x · f x = 0. Select W = (0, 0, ε w ∧ f ) with ε = 0 small. Arguing as in the previous paragraph, Finally assume ω 0 = 0 and ω = kω 0 for k ∈ {−1/λ, 1/(1 − λ)}. We may thus

Formulation of the Characterisations
Proposition 6.6 allows us to use the whole wave cone in computations on hulls of relatively open sets. In order to exploit this, in Theorem 6.7 we characterise U r,s := int M (K lc, r,s ) via different relatively open sets. Our main aim is twofold: first, U r,s = ∪ τ ∈[0,1) (B M (K τr,τ s , ε τ )) lc, g whenever the constants ε τ > 0 are small enough, and secondly, 0 ∈ U r,s . We prove the first one via the (a priori) easier equality U r,s = ∪ τ ∈[0,1) (B M (K τr,τ s , ε τ )) lc, and Proposition 6.6.
In order to prove both of our two aims in a unified manner, we introduce some further terminology. For every u, B ∈ R 3 we denote sym , and for every c > 0 we define relatively open sets Note that given c > 0 we have 0 ∈ V r,s,c and B M (K r,s ,c) ⊂ V r,s,c for every small enoughc > 0.

Elsässer Variables in Relaxed MHD
In some of the computations on relaxed MHD it will be convenient to replace the variables (u, S, B, E) by Elsässer variables and a matrix component, (z + , z − , M), which satisfy The main advantage is that the constraint set obtains the particularly simple form The wave cone conditions (3.1)-(3.3) are written in Elsässer formalism as 6.6. The Proof of Proposition 6.8 Proposition 6.8 gives our first estimation on the hull K lc, r,s . Below, we further divide the proof of Proposition 6.8 into five steps.
Proof. First we cover the case where S = −e ⊗ e. Choose an orthogonal basis {e, f, g} of R 3 , where |e| = | f | = |g|. Write I = |e| −2 (e ⊗ e + f ⊗ f + g ⊗ g) which, in combination with Lemma 6.11, yields the -direction isV = (0, 2 f ⊗ f − 2g ⊗ g, 0, 0) and we may choose (ξ x , ξ t ) = (e, 0). By noting that (0, e ⊗ e ± f ⊗ f, 0, 0) ∈ for every e, f ∈ R 3 and iterating, we obtain the case for any unit vectors f i ∈ S 2 and c i ∈ R with N i=1 |c i | < ε τ . The proof is finished by noting that every S ∈ R 3×3 sym with |S| < ε τ can be written in the form (6.2). Indeed, whenever f and g are unit vectors, we may write We have now covered the case where V differs from an element of K by the perturbation S of the symmetric matrix part. Our next aim, in the following two lemmas, is to allow E = B × u in V = (u, S, B, E). Recall that K lc, is a necessary condition. We will see next that it is also sufficient with the appropriate size normalisations. We will make use of the formula Finally, recall that in view of Proposition 6.2, we are forced to use bad -segments.
(6.4) we can apply Lemma 6.12 to deduce that the endpoints lie in K lc, r,s . The direction of the segment is which belongs to since (3.5) is satisfied.

The Proof of Proposition 6.9
Recall that Proposition 6.9 states the inclusion U r,s ⊂ ∪ τ 0 <τ <1 K lc, τr,τ s and completes the proof of Theorem 6.7.

The Proof of Theorem 1.1
This chapter is dedicated to proving Theorem 1.1. In Section 7.1 we define the set of subsolutions that we use in the proof, and the main steps of the proof are listed in Section 7.2. The proof itself is carried out in the rest of the chapter.

Restricted Subsolutions
We intend to prove Theorem 1.1 by using subsolutions that take values in U r,s and whose B and E components arise via P B and P E . For this, recall the notations Fix a non-empty bounded domain ⊂ R 3 × R, and let r, s > 0, r = s.
Definition 7.1. The set of restricted subsolutions is defined as We define X to be the weak sequential closure of X 0 in L 2 (R 4 ; co(K r,s )), where co(K r,s ) denotes the closed convex hull of the constitutive set K r,s defined in Definition 3.7. Observe that K r,s ⊂ R 15 is compact so that the weak topology of L 2 on X is metrizable. Now X {0} is a compact metrisable space, and we denote a metric by d X .

The Main Steps of the Proof
Following [21], our main aim is to prove Proposition 7.2 below. Once Proposition 7.2 is proved, Theorem 1.1 follows rather easily in Section 7.5. Proposition 7.2. There exists C = C r,s > 0 with the following property.
For the proof of Proposition 7.2 we need a so-called perturbation property, formulated in our setting in Proposition 7.4. To motivate the formulation of Proposition 7.4, we note that Theorem 6.7 implies the following proposition where we choose any ε τ = ε τ,r,s such that recall from Theorem 6.7 that for any τ 0 ∈ (0, 1) we have Proposition 7.3. Let V 0 ∈ U r,s . Then for every large enough τ ∈ (0, 1) there exists Therefore, (7.1) follows by choosing τ ∈ (0, 1) large enough.
Whereas Proposition 5.13 says, roughly speaking, that every good -segment can be approximated by oscillating mappings with certain properties, Proposition 7.4 makes an analogous claim about good laminates.

Proposition 7.4. Let Q ⊂ R 4 be a cube, and let
For every ε > 0 there exist with the following properties: and dφ and dψ are locally constant in A j .
Condition (iii) says, in particular, that at every ( We will also need the estimate on W (x, t). Proposition 7.4 is proved by a standard induction via Proposition 5.13; we sketch the main ideas.
Proof. The proof follows by iteratively modifying the sequence at the sets where it is locally affine (via Proposition 5.13) and using a diagonal argument. Namely, if N = 1, the result follows from Proposition 5.13. Suppose now that 1 ≤ k ≤ N − 1 and that we have costructed a sequence of mappings W k = (u k , S k , dϕ k , dψ k ) which satisfies (i), (iii) and (iv) and furthermore (ii) holds with the condition j ∈ {1, 2} N replaced by j ∈ {1, 2} k . Fix j ∈ {1, 2} k . We cover A j by disjoint cubes up to a set of small measure and modify W k in each cube via Proposition 5.13. This gives rise to a new sequence which we again modify at the sets A j , j ∈ {1, 2} k+1 , where it is locally affine. Note that we can use Proposition 5.13 iteratively because in each A j , claim (iii) of Proposition 5.13 implies that either dϕ k ∧ dψ k = 0 or dϕ k = dψ k = 0. Finally a standard diagonal argument provides the norm bounds.
Since W is absolutely continuous, we cover by all the cubes Q i ⊂ with centers (x i , t i ) and the following properties: Such cubes exist for every (x i , t i ) ∈ , and therefore they form a Vitali cover of . By the Vitali Covering Theorem, we may choose a finite, pairwise disjoint subcollection so that g(x, t) = (x i , t i ) is constant in R and g = id near ∂ Q i . Setφ := ϕ • g and ψ := ψ • g so that x,t g∇ x,t ϕ • g, ∇ x,tψ = D T x,t g∇ x,t ψ • g. (7.2) Thus |{(x, t) ∈ Q i : dφ(x, t) = dψ(x, t) = 0}| ≥ (1 − ε/2) |Q i |. The claim will be proved once we show that dφ ∧ dψ < ε/2 in Q i ; then V −Ṽ L ∞ < ε. To this end, we fix (x, t) ∈ Q i and estimate and |dϕ(g(x, t)) ∧ dψ(g(x, t))| ≤ |dϕ(g(x, t)) ∧ (dψ(g(x, t)) − dψ(g(x i , t i ))| By Lemma 7.5 and a standard diagonal argument, it suffices to prove Proposition 7.2 for every ε > 0 and every mapping V ∈ X 0 such that

Proof of Proposition 7.2
Assuming that (7.3)-(7.4) hold, we wish to construct the mappings V = p(W ) of Proposition 7.2 by suitably modifying a discretisation argument from [21]. Given V = p(W ) ∈ X 0 we cover˜ , up to a small set, by cubes Q i ⊂˜ with center (x i , t i ) such that W varies very little in Q i . We then approximate W by W (x i , t i ) in each Q i . Now V (x i , t i ) =ν for some ν = j∈{1,2} N μ j δ V j ∈ L g (O τ ) with τ close to 1, and we set W : where (ū ,S , dφ , dψ ) is given by Proposition 7.4.
On one hand, the discretisation needs to be fine enough that V = p(W ) does not take values outside U r,s , and on the other hand, the cubes need to cover a substantial proportion of˜ . Both properties are ensured by the following application of the Vitali Covering Theorem. Lemma 7.6. Suppose ε > 0 and V = p(W ) ∈ X 0 satisfies (7.3)-(7.4). Let γ > 0. Then there exist pairwise disjoint cubes Q i ⊂˜ with centers (x i , t i ) and parameters δ i > 0 with the following properties: (i) For every i ∈ N, there exists τ i ∈ (0, 1) and ν = j∈{1, Proof. Let (x, t) ∈˜ . Since V (x, t) ∈ U r,s , by Theorem 6.7 there exist τ ∈ (0, 1) and ν = j∈{1,2} N μ j δ j ∈ L g (O τ ) with barycentreν = V (x, t). Furthermore, there exists δ > 0 such that B M ([V j ,1 , V j ,2 ]), δ) ⊂ U r,s whenever j ∈ {1, 2} k , 1 ≤ k ≤ N − 1. Since W is continuous, there exists a cube Q ⊂˜ with center (x, t) such that sup (x ,t )∈Q W (x , t ) − W (x, t) < γ δ i .
The collection of cubes chosen above forms a Vitali cover of˜ , and therefore, by the Vitali Covering Theorem, there exists a countable, pairwise disjoint subcollection {Q i } i∈N with |˜ \ ∪ ∞ i=1 Q i | = 0.