Magnetic helicity, weak solutions and relaxation of ideal MHD

We revisit the issue of conservation of magnetic helicity and the Woltjer‐Taylor relaxation theory in magnetohydrodynamics (MHD) in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by‐products we answer two open questions in the field: We show the sharpness of the L3 integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for ideal MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.


Introduction
In this paper we consider the system of ideal magnetohydrodynamics (MHD in short), which couples the incompressible Euler equations with the Faraday-Maxwell system via Ohm's law.The MHD system, with nonzero viscosity and magnetic resistivity, is used in modelling electrically conducting fluids such as plasmas and liquid metals (see [GLBL06] and [ST83]).The ideal MHD system, where kinematic viscosity and magnetic diffusivity are set to zero, contains a wealth of mathematical structure [AK98] and can be written as Moreover, in analogy with the role of the incompressible Euler equations for hydrodynamical turbulence [On49,CET94], the ideal system is relevant in the inviscid, irresitive "turbulent" limit in the context of weak solutions [CKS97,E15].
A key question concerning weak solutions is to understand the correct space in which to formulate the problem.This question is closely related to the issue of anomalous dissipation and conservation of energy.Let us recall the conserved quantities.It is well known that energy and cross helicity are conserved by smooth solutions whereas magnetic helicity is preserved by turbulent solutions (see section 1.2 for the precise function spaces).To avoid technical issues concerning the topology of the domain and boundary conditions, we will work in the 3D periodic setting T 3 .Identical arguments are valid for simply connected magnetically closed domains.For a definition of magnetic helicity in domains with non-trivial topology see [MV19].
1.1.Weak solutions.The ideal MHD system is obtained by combining the Euler system for ideal incompressible fluids driven by a magnetic field with the Faraday-Maxwell system with constitutive law given by Ohm's law for perfectly conducting fluids (4) E = B × u.
In the formulation (2)-(4) it is apparent that solutions in the sense of distributions can be defined for u, B ∈ L 2 loc .
1.2.Conserved quantities.Formally, the Euler system (2) together with (3b) leads to the balance equations In integrated form these imply conservation of total energy and cross-helicity, where Next, define the magnetic vector potential A = curl −1 B, defined uniquely on T 3 by applying the Biot-Savart law to the system curlA = B, (7a) divA = 0. (7b) Then formally the Faraday-Maxwell system (3) leads to (8) where f is a scalar function acting as "electric scalar potential" in the sense that from (3a) we obtain (9) ∂ t A + E = ∇f, see e.g [FLS21, Lemma 2.3].Using (4), from (8) we deduce conservation of magnetic helicity Indeed, in manifolds with no boundary helicity is a gauge invariant property of the n − 1 form induced by B-see the book [AK98] and [A86] for the precise topological meaning of helicity in relation with the asymptotic Hopf invariant.
Concerning weak solutions, an analogous development to the pure hydrodynamic case (B = 0) has led to the following Onsager-type criteria for conservation, formulated in terms of spatial Besov spaces: in [CKS97] is is shown that energy and cross helicity are conserved as long as Kang and Lee in [KL07].There is also a wealth of function spaces in which the regularity is distributed differently in u, B for which cross helicity or energy is preserved.Such conservation results lead to the natural flexible side of Onsager-type conjectures for Hölder continuous solutions, see for example [BV21, Conjecture 2.10].In this regard our Corollary 4 below shows the flexibility in the realm of L ∞ solutions (for previous works see [BLL15] and [FLS21] for null magnetic helicity).The remarkable robustness of magnetic helicity as a conserved quantity is reflected in simulations and experiments, as we next briefly discuss.1.3.Magnetic helicity, Woltjer-Taylor relaxation and magnetic reconnection.Woltjer proposed magnetic helicity conservation as an explanation of the observation that various astrophysical plasmas tend to evolve toward a force-free state ∇ × B = αB [W58].A similar relaxation process also occurs in many laboratory settings, even if the inital state of the system is turbulent [OS93].Woltjer suggested the variational problem of minimizing total energy under the constraint that magnetic helicity is fixed, and he computed formally that the minimizers are, indeed, force-free (see also [LA91]).Moffatt [M69] interpreted magnetic helicity topologically and noted that, furthermore, subhelicities over magnetically closed Lagrangian subvolumes are conserved (by smooth solutions).This abundance of conserved quantities is, however, seemingly at odds with the observed relaxation of turbulent plasmas [OS93].
Nevertheless, an important aspect of ideal MHD, or MHD with very low resistivity, is that the regime is consistent with turbulence and in particular magnetic reconnection can cause the non-conservation of subhelicities.Taylor conjectured in [T74] that in the presence of slight resistivity, subhelicity conservation would break down but magnetic helicity in the whole domain would nevertheless be approximately preserved (see [B84,FL20,FLMV21] for mathematical confirmation of the latter hypothesis).
Taylor's ensuing relaxation theory (an archetypical example of self-organization [Has85]) revisits Woltjer's variational problem for magnetic energy emphasizing that magnetic reconnection would be responsible for energy dissipation but magnetic helicity should be kept fixed.It is a matter of discussion in the physics literature ([E15] for example) what patterns of the MHD equation should prevail in the macroscopic variables compatible with magnetic reconnection and indeed under which circumstances Taylor relaxation theory is valid.We look at this issue from the perspective of mathematical relaxation, which we next describe.1.4.Relaxation.In the context of nonlinear PDE arising in continuum physics, mathematical relaxation has become an indispensable tool in large part due to the pioneering work of L. Tartar in the 1970-80s (e.g.[Tar77,Tar05]).A key point in Tartar's program is to study the behaviour of (in general nonlinear) constitutive relations under weak convergence in combination with differential constraints arising from conservation laws.Weak limits can be interpreted as a deterministic analogue of averaging or coarse-graining, and thus, in many cases of interest, one is able to obtain 'averaged' constitutive relations.An indispensable and powerful tool in this program is compensated compactness.
A standard example, treated for instance in [Tar05], is the Maxwell system of electromagnetism.In particular for the Faraday-Maxwell system (Maxwell equations in vacuum) it is shown that B • E is a weakly continuous quantity where E, B have the natural integrability conditions.A particularly elegant way of seeing this is by using space-time differential forms -Tartar attributes this observation to J. Robbin: The Faraday-Maxwell system (3) can be equivalently formulated for the Faraday 2-form (mistakenly called the Maxwell 2-form in [FLS21]) ω ∈ Λ 2 (R 4 ), related to the magnetic and electric fields B, E via The corresponding potential 1-form α can be written as so that ω = dα is equivalent to (7a) and (9).In this formalism ω ∧ ω = 0 is equivalent to orthogonality of the electric and magnetic fields ([FLS21, Section 5.3] or [Tar05]).Noting that then d(α∧ω) = ω ∧ω, a simple argument using integration by parts and Sobolev embedding shows that ω ∧ ω is weakly continous in appropriate function spaces.Since in the MHD system we have E = B × u, the pointwise identity (13) B • E = 0 must be satisfied in the relaxation of MHD (the set of weak limits of solutions to MHD).
A central aspect of this paper is to understand what happens with the compensated compactness quantity B • E below the integrability threshold where B • E is weakly continuous.The recent constructions of irregular Jacobians (determinants of gradient maps and a prototype of differential forms) are particularly relevant for us, see e.g.[AA96,H11,LM16,FMO18].
Note that (13) is a consequence of Ohm's law (4) but not vice versa -indeed, a key observation of our analysis is that (13) can be thought of as a suitable relaxation of (4), sufficiently strong to retain conservation of magnetic helicity but consistent with scenarios of magnetic reconnection and Woltjer-Taylor relaxation (compare again with [E15]).
In the context of weak solutions, it is easy to see using Sobolev embedding that magnetic helicity H(B) is well-defined provided B ∈ L 3/2 (T 3 ).Moreover, recall that whenever 3/2 ≤ p < ∞, solutions (E, B) of the Faraday-Maxwell system (3) with Our first result, Theorem 1 below, in this paper shows the sharpness of this statement.Combining this result with the techniques introduced in [FLS21], we are able to 'lift' solutions of the Faraday-Maxwell system to the full MHD system in two ways: (i) First, in the context of bounded weak solutions we show the existence of weak solutions with arbitrary (constant in time) magnetic helicity and at the same time arbitrary time development of energy and cross-helicity -see Corollary 1; (ii) Secondly, we show the sharpness of the criteria for magnetic helicity conservation by Kang-Lee [KL07]; namely, the existence of weak solutions uniformly-in-time in the spatial Lorentz space L 3,∞ which do not conserve magnetic helicity -see Corollary 2.
1.5.Main results.In order to be able to precisely state our main results, we fix some basic terminology and notation that will be used throughout the paper.
• Spatial domains: • Function spaces: The usual Lebesgue spaces will be denoted by L p (Ω) or L p (Q), respecting the convention above that Ω is a spatial domain and Q a space-time domain.Appropriate Lebesgue spaces of vector fields will be denoted by L p (Ω; R 3 ) or L p (Q; R 3 ).The weak L p spaces of Marcinkiewicz will be referred to in the Lorentz notation L p,∞ (Ω).
Our first main result shows that below the critical integrability, we can restore condition (13) for arbitrary piecewise constant solutions to the Faraday-Maxwell system without being forced to have constant magnetic helicity.
) be a pair of piecewise constant vector fields solving the Faraday-Maxwell system (3) in the sense of distributions, and let p, p ′ Hölder-dual exponents with 3/2 < p < ∞.Then there exist piecewise constant such that B • E = 0 for a.e.(x, t) and The proof, presented in Section 2, relies on an anisotropic version of convex integration in L p through staircase laminates from [F03,AFS08] which might be of independent interest.
Our second main theorem states that orthogonal solutions of the Faraday-Maxwell system can be "lifted" to weak solutions of the full ideal MHD system (1) whilst preserving integrability.
Theorem 2. There exists a geometric constant M 0 > 0 with the following property. Let ) be a pair of piecewise constant vector fields solving the Faraday-Maxwell system (3) in the sense of distributions and such that E • B = 0.
Let {Q i } be a countable family of pairwise disjoint open sets on each of which B, E are constant, and let and The proof, presented in Section 3, is an extension of our previous paper [FLS21], where we adapted the convex integration scheme from [DLS09] to be compatible with the non-linear constraint (13).
Theorem 2 suggests that, at least at the level of merely bounded weak solutions, the dynamics of ideal MHD is determined entirely by the behaviour of the Faraday-Maxwell system together with the condition (13) replacing (4) -thus providing a satisfactory mathematical relaxation of the full ideal MHD system.
There are two particular consequences of this result: First, we have a natural extension of the main result from [FLS21] to initial data with arbitrary (a fortiori constant) magnetic helicity: Corollary 1.There exists a geometric constant M > 0 with the following property.
Let h ∈ R and suppose e, w ∈ C([0, T ]) with e(t) − |w(t)| > M |h| for all t.Then there exists a weak solution such that neither magnetic helicity, nor energy or cross-helicity are conserved in time.
The proofs of these corollaries are quite simple, and will be presented in Section 4. Corollary 2 should be compared with [BBV20,FL20].In [FL20] it was shown that magnetic helicity is conserved by those solutions of ideal MHD which arise as inviscid, non-resistive weak limits of Leray-Hopf solutions.Opposite to this result, in [BBV20] it was shown that general x for 0 < β ≪ 1) of ideal MHD need not preserve magnetic helicity.Corollary 2 solves the flexible part of [BV21, Conjecture 11] (in the L p scale).

Weak solutions of the Faraday-Maxwell system
The purpose of this section is to develop a version of convex integration for the Faraday-Maxwell system and in particular to prove Theorem 1, that is, construct weak solutions (B, E) with B • E = 0 a.e., which do not conserve magnetic helicity.Convex integration in L p in relation with integrability issues was introduced in [AFS08], based on the staircase laminates from [F03].Such constructions have turned out be useful in a number of problems [CFM05,F04] particularly to obtain lower bounds for singular integrals [BSV13].As B • E is a compensated compactness quantity, our result is inspired by construction of gradients of homeomorphisms with vanishing Jacobian determinant.In fact, those were inspired by the construction of very weak solutions to elliptic equations [F04,AFS08].Notice that such constructions can only exist in function spaces where the corresponding compensated compactness quantity is no longer weakly continuous.Thus, the dichotomy between weak compactness and rigidity versus lack of compactness and flexible convex integration solution arises once more.Let us further emphasize that the construction here is anisotropic, and thus is based in a curvy staircase laminate which is a new feature in the literature.The known convex integration constructions applied to such curvy laminates yield L 3,∞ x,t solutions.In order to achieve x , it is needed to control what happens at almost every time slice.The innovations introduced in the paper to deal with this issue are also of potential use elsewhere.
In modifying piecewise constant vector fields (B, E) : Q → R 3 ×R 3 with div B = 0 and ∂ t B + curl E = 0, a typical situation is as follows.Suppose Q 0 ⊂ Q is a subdomain where (B (0) , E (0) ) = (B 0 , E 0 ) are constant.We wish to "replace" the constant value (B 0 , E 0 ) by another pair of vector fields (B, E) : Q 0 → R 3 × R 3 such that the "glued" vector fields, defined by still satisfy divB (1) = 0 and ∂ t B (1) + curlE (1) = 0.It is easy to check that, in general, a necessary and sufficient condition for this is that 2.1.The basic staircase.We start by constructing a discrete "staircase" laminate in the plane R 2 .Recall that laminates in the plane are defined with respect to separate convexity, i.e. corresponding to the wave cone {(x, y) ∈ R 2 : x = 0 or y = 0}.

Basic construction.
Here we recall and appropriately adapt the basic so called "roof-construction" for localized plane-waves, see e.g [K03]. in the following we denote by Lip 0 (Q) the set of Lipschitz functions on Q such that f = 0 on ∂Q.
For any open bounded domain Q ⊂ R 4 with |∂Q| = 0 and any r, ε > 0 there exist piecewise constant vector fields B, E ∈ L ∞ (Q; R 3 ) satisfying (17) and the boundary conditions given by in the sense of (18), with the following properties: where N a nullset, Q (1) , Q (2) and Q (error) are open sets where (B, E) is locally constant, and such that (B, and moreover • There exists a vector potential Ã ∈ Lip 0 (Q) with Let us now consider first a polyhedral spatial domain Ω and space-time domain of the form where and h : R → R is a 1-periodic non-negative Lipschitz function with h ′ (s) ∈ {−λ 2 , λ 1 } for a.e.s ∈ R. Observe that for every t the function x → f p N (x, t) is a periodic piecewise affine Lipschitz function such that with respective volume fractions µ(t), λ 1 (1 − µ(t)), λ 2 (1 − µ(t)), relative to one period.
Since Ω is a polygonal domain, f N is piecewise affine.Thus, by definition, there exists an open subset Q ⊂ Q such that |Q \ Q| = 0 and f N is locally affine (as well It then follows that for all t the latter following from the pointwise bound |f p N | ≤ 1 N .Then, after eliminating µ(t) and from the above relationships and choosing N sufficiently large in terms of ε, we obtain (21b).To deduce (21c) we may again use the pointwise bound on f p N to see that in fact so that (21c) follows by choosing N sufficiently large.Finally, observe that Ã = | B|ηf N , so that (21d) also follows from choosing N sufficiently large.This concludes the proof for the case of a cylindrical polyhedral set Q = Ω × (t 0 , t 1 ).
For a general open space-time domain Q we find a pairwise disjoint countable family of cylindrical polyhedral sets Remark 1.We compute explicitly the change of magnetic helicity in Lemma 2. In the proof above, let us denote by f the piecewise affine Lipschitz function which vanishes outside the polyhedral sets Q k and is of the form f = f N in each Q k , so that Ã = | B|f η.We denote Q(t) := {x ∈ R 3 : (x, t) ∈ Q} for t ∈ R. By integrating by parts and using the facts that Ã| ∂Q = 0 and Ã • B = 0 we get Next, we intend to implement the basic splitting (20) in the construction of the staircase laminate in Corollary 3. To this end we fix vectors B 0 , E 0 , β > 1 and set Lemma 3 (Approximation of Steps).For any n ∈ N, any open bounded domain Q ⊂ R 4 with |∂Q| = 0 and any r, ε > 0 there exist piecewise constant vector fields B, E ∈ L ∞ (Q; R 3 ) satisfying (17) and the boundary conditions given by (B n , E n ) in the sense of (18), with the following properties: • Q admits a pairwise disjoint decomposition where N a nullset, Q (good) , Q (inductive) and Q (error) are open sets where (B, E) is locally constant with • For all t we have • There exists a vector potential Ã ∈ Lip 0 (Q) such that, for any vector potential A 0 of B 0 , Proof.We may assume without loss of generality that ε < 1.In the first step we apply Lemma 2 with the elementary splitting . We obtain (B (1) , E (1) ) and the decomposition and for all t Furthermore, B (1) = B 0 + curlA (1) , with |A (1) | ≤ ε/2.We then use Remark 1 to compute the change of magnetic helicity.In the elementary splitting (25), ( Then we apply Lemma 2 in Q (2) with the second elementary splitting . We obtain (B, E) and the decomposition and for all t Then for every t This concludes the proof.
2.3.The staircase construction.The basic construction above has the following structure: up-to an "error set" Q (error) the distribution of values of the pair of vector fields (B, E) agrees with the probability measure (laminate) arising in (20).
There are two types of control on the size of the error set: small space-time measure (24d) on the one hand, and on the other hand control uniformly in time by the proportion of mass moved to the inductive set (24c).
In the following we will iterate the basic construction, to inductively "push" the mass in Q (inductive) to infinity.The balance between the error created at each step and the amount mass pushed inductively further will be quantified by estimate (30b).
an open bounded domain with |∂Q| = 0.For any β > 1 and ε > 0 there exist piecewise constant vector fields B, E ∈ L ∞ (Q; R 3 ) satisfying (17) and the boundary conditions given by (B 0 , E 0 ) in the sense of (18), with the following properties: • Q admits a decomposition where N is a nullset, (B, E) is locally constant in the open sets Q (good) and Q (error) and such that |B||E| = 0 in Q (good) .
• For all t and all s > 1 we have and p ′ is the Hölder dual of p. • Furthermore, (30c) • There exists a vector potential Ã ∈ Lip 0 (Q) with Proof.Based on Lemma 3 we define inductively a sequence 17) and the boundary conditions given by (B 0 , E 0 ) in the sense of (18), with the following properties: Firstly, we have the pairwise disjoint decomposition where are open sets where (B n , E n ) is locally constant, such that We start by defining To obtain (B (n+1) , E (n+1) ) we apply Lemma 3 to ( with small parameters r n , ε n > 0 chosen below, with ε n < ε2 −n−1 .Then we obtain for all k < n + 1, where in the last line ≈ r k means that the norm of the difference is bounded by r k .In particular we may ensure by the choice of r k that Then we have for all t (32) and furthermore The parameters ε n > 0, n = 0, 1, 2, . . .will be chosen below, for the moment let us merely specify that they satisfy (33) Such condition and (32) immediately imply that, for all t, (34) Also, by construction, for any n ≥ k Since the measure of tends to 0, it follows that the sequence (B (n) , E (n) ) converges to a limit (B, E) for almost every (x, t).In the following we derive properties of this limit.To start with we observe that, declaring We turn to estimate (30b).From (32) we obtain, using (33), Now let n ∈ N and let (37) Therefore, for every t where we have used the definition of s n , (36) and that on Q Next, let s > s ′ .From the elementary inequality min(|B| p +|E| p ′ , s) ≤ s s ′ min(|B| p + |E| p ′ , s ′ ) we easily deduce that I(s, t) ≤ s s ′ I(s ′ , t) for every t.Now, for any s ≥ s 0 (the latter defined in (37) with n = 0) there exists n ∈ N such that s n ≤ s ≤ s n β p .Consequently as claimed in (30b).On the other hand for 1 ≤ s ≤ s 0 we may use the trivial estimate I(s, t) ≤ s|Q(t)|, from which (30b) also follows.
The estimate (30d) follows from and an appropriate choice of ε n > 0, n = 0, 1, 2, . . . .Finally, observe that (30b) implies uniform-in-time weak L p − L p ′ bounds for (B, E), which also clearly hold for the sequence (B (n) , E (n) ) and p, p ′ > 1.We deduce that in fact (B (n) , E (n) ) → (B, E) strongly in L 1 (Q).This in turn implies that divB = 0, ∂ t B + curlE = 0 in Q, and the required boundary conditions hold in the sense of (18).This concludes the proof.
2.4.Iterating the staircase construction.In this section we iterate the staircase construction in order to successively remove the error in Ω error .Theorem 3. Let B 0 , E 0 ∈ R 3 , 1 < p < ∞ and Q ⊂ R 4 an open bounded domain with |∂Q| = 0.For any ε > 0 there exist piecewise constant vector fields B, E ∈ L ∞ (Q; R 3 ) satisfying (17) and the boundary conditions given by (B 0 , E 0 ) in the sense of (18) with the following conditions: • |B||E| = 0 for a.e.(x, t) ∈ Q; • For all t and any s > 1 • There exists a vector potential Ã ∈ Lip 0 (Q) with Proof.We construct inductively a sequence of piecewise constant vector fields B q , E q satisfying (17) and the boundary conditions given by (B 0 , E 0 ) in the sense of (18), and with following properties: there exists a pairwise disjoint decomposition where N q is a nullset, Q (good) q and Q (error) q are open sets where B q , E q are locally constant, and |B q ||E q | = 0 in Q (good) .This will be complemented with the inductive estimates where Furthermore, B q+1 = B q + curlA q with |A q | ≤ ε2 −q−1 .We start with the constant maps (B 0 , E 0 ) and set Q error 0 = Q.
To obtain (B q+1 , E q+1 ) from (B q , E q ) we consider the decomposition with constant values (B q , E q ) ≡ (B i q , E i q ) on Q q,i .In each Q q,i we replace (B i q , E i q ) by the construction from Proposition 1 with (B 0 , E 0 ) given by (B i q , E i q ) and small parameters β q+1 > 1, ε q+1 > 0 still to be fixed.We obtain for each i a new pairwise disjoint decomposition with associated estimates, corresponding to (30b)-(30c): where the latter is obtained by an appropriate choice of ε q+1 in (30c) .We set Then we obtain so that (40) is satisfied.To obtain (41) we calculate This completes the inductive step with estimates (40)-( 41).Now observe that, because of (40), in particular , we deduce that the sequence (B q , E q ) converges almost everywhere to piecewise constant vector fields (B, E).Furthermore, for any q ∈ N, s > 1 and t Thus, choosing β q > 1 in such a way that q k=1 β 2(p+1) k ≤ 2, we obtain the uniform bounds (38).We note furthermore that, since p, p ′ > 1, these estimates imply uniform L q bounds for some q > 1 on both sequences {B q } and {E q }, which, together with pointwise convergence implies strong L 1 convergence.Therefore (17) and the boundary conditions given by (B 0 , E 0 ) in the sense of (18) remain valid in the limit.This completes the proof.

Proof of Theorem 2
Our next task is to prove Theorem 2. We divide the proof into several propositions and refer to [FLS21] for many of the proofs.We will modify the piecewise constant map V ∼ = (0, 0, B, Ē) in each space-time domain Q i separately and then superimpose the perturbations to get the solution (u, B) whose existence is claimed in Theorem 2.
3.1.Ensuring that V takes values in the hull.Our first goal is to ensure that pointwise a.e., V = (0, 0, B, Ē) belongs to the relative interior of the hull determined by ζ + and ζ − .The precise statement is as follows: Proposition 2. We have (0, 0, B, Ē) ∈ U z+(x,t),z−(x,t) for every (x, t) ∈ Qi and every i ∈ N.
3.2.Modifying V in a single set Q i .As the main building block of the proof of Theorem 2, we perturb V = (0, 0, B, Ē) in a single set Q i , where {Q i } i∈N is the family of disjoint open sets corresponding to B, Ē in the definition of piecewise constant fields.With C ± := max (x,t)∈ Qi ζ ± (x, t) we denote the weak sequential closure of X 0 in L 2 (T 3 × [0, T ]; co(K C + ,C − ) by X.Now X ∋ { V } is a compact metrisable space, and we denote a metric by d X .
We state the main step of the proof of Proposition 4.
given vectors B 0 , E 0 we can embed the two dimensional laminate in the plane spanned by them.
[FLS21,, [FLS21, Proposition 7.2] is the special case of Proposition 5 where ζ + and ζ − are constant in (x, t), and the proof of [FLS21, Proposition 7.2] applies with relatively minor changes.With Proposition 5 in hand, the proof of Proposition 4 is completed by standard methods, see[FLS21,.