# Magnetic helicity, weak solutions and relaxation of ideal MHD

Daniel Faraco  Departamento de Matemáticas
Universidad Autónoma de Madrid, E-28049 Madrid, Spain; ICMAT CSIC-UAM-
UC3M-UCM, E-28049 Madrid, Spain<EMAIL_ADDRESS>,  Sauli Lindberg
Department of Mathematics and Statistics
University of Helsinki, P.O. Box 68, 00014 Helsingin yliopisto, Finland
<EMAIL_ADDRESS>and  László Székelyhidi, Jr  Institut für
mathematik
Universität Leipzig, Augustusplatz 10, D-04109, Leipzig, Germany
<EMAIL_ADDRESS>
###### Abstract.

We revisit the issue of conservation of magnetic helicity and the Woltjer-
Taylor relaxation theory in magnetohydrodynamics 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 $L^{3}$
integrability condition for magnetic helicity conservation and provide
turbulent bounded solutions for MHD dissipating energy and cross helicity but
with (arbitrary) constant magnetic helicity.

###### Key words and phrases:

Magnetohydrodynamics, convex integration, conservation laws, compensated
compactness

###### 2010 Mathematics Subject Classification:

35Q35, 76W05, 76B03

D.F. acknowledges financial support from the Spanish Ministry of Science and
Innovation through the Severo Ochoa Programme for Centres of Excellence in
R&D(CEX2019-000904-S and by the MTM2017-85934-C3-2-P2. He was also partially
supported by CAM through the Line of excellence for University Teaching Staff
between CM and UAM. D.F. was also partially supported by the ERC Advanced
Grant 834728. S.L. was supported by the ERC Advanced Grant 834728. L. Sz. was
supported by ERC Grant 724298-DIFFINCL.

##  1\. 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

(1) $\begin{split}\partial_{t}u+u\cdot\nabla u-B\cdot\nabla B+\nabla p&=0,\\\
\partial_{t}B+u\cdot\nabla B-B\cdot\nabla u&=0,\\\ \nabla\cdot u=\nabla\cdot
B&=0,\end{split}$

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 $\mathbb{T}^{3}$.
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

(2a) $\displaystyle\partial_{t}u+\textrm{div}(u\otimes u-B\otimes
B+p\,\textrm{Id})$ $\displaystyle=0,$ (2b) $\displaystyle\nabla\cdot u$
$\displaystyle=0$

with the Faraday-Maxwell system

(3a) $\displaystyle\partial_{t}B+\nabla\times E$ $\displaystyle=0,$ (3b)
$\displaystyle\nabla\cdot B$ $\displaystyle=0,$

with constitutive law given by Ohm’s law for perfectly conducting fluids

(4) $E=B\times u.$

In the formulation (2)-(4) it is apparent that solutions in the sense of
distributions can be defined for $u,B\in L^{2}_{loc}$.

###  1.2. Conserved quantities

Formally, the Euler system (2) together with (3b) leads to the balance
equations

(5a)
$\displaystyle\tfrac{1}{2}\partial_{t}\Bigl{(}|u|^{2}+|B|^{2}\Bigr{)}+\textrm{div}\Bigl{(}u\bigl{(}p+\tfrac{1}{2}(|u|^{2}+|B|^{2})\bigr{)}-B(u\cdot
B)\Bigr{)}$ $\displaystyle=0,$ (5b) $\displaystyle\partial_{t}\Bigl{(}u\cdot
B\Bigr{)}+\textrm{div}\Bigl{(}B\bigl{(}p-\tfrac{1}{2}(|u|^{2}+|B|^{2})\bigr{)}+u(u\cdot
B)\Bigr{)}$ $\displaystyle=0.$

In integrated form these imply conservation of total energy and cross-
helicity, where

(6a) $\displaystyle\mathcal{E}(u,B)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{T}^{3}}|u|^{2}+|B|^{2}\,dx,$ (6b)
$\displaystyle\mathcal{W}(u,B)$ $\displaystyle=\int_{\mathbb{T}^{3}}u\cdot
B\,dx.$

Next, define the magnetic vector potential $A=\textrm{curl}^{-1}B$, defined
uniquely on $\mathbb{T}^{3}$ by applying the Biot-Savart law to the system

(7a) $\displaystyle\textrm{curl}A$ $\displaystyle=B,$ (7b)
$\displaystyle\textrm{div}A$ $\displaystyle=0.$

Then formally the Faraday-Maxwell system (3) leads to

(8) $\partial_{t}\Bigl{(}A\cdot B\Bigr{)}+\textrm{div}\Bigl{(}E\times
A-fB\Bigr{)}=-2B\cdot E,$

where $f$ is a scalar function acting as “electric scalar potential” in the
sense that from (3a) we obtain

(9) $\partial_{t}A+E=\nabla f,$

see e.g [FLS21, Lemma 2.3]. Using (4), from (8) we deduce conservation of
magnetic helicity

$\mathcal{H}(B):=\int_{\mathbb{T}^{3}}A\cdot B\,dx.$

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 $u,B\in
C([0,T[;B^{\alpha}_{3,\infty})$ for $\alpha>1/3$, and the end-point result
$u,B\in L^{3}_{t}B^{1/3}_{3,c_{0}}$ is shown in [KL07] by closely following
[CCFS08]. Magnetic helicity, in contrast, is already conserved if $u,B\in
C([0,T];B^{\alpha}_{3,\infty})$ for $\alpha>0$ [CKS97] or $u,B\in
L^{3}(\mathbb{T}^{3}\times[0,T])$ as shown by 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 $L^{\infty}$ 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 $\nabla\times B=\alpha B$ [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 $B\cdot E$ 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]) $\omega\in\Lambda^{2}(\mathbb{R}^{4})$, related to
the magnetic and electric fields $B,E$ via

(10) $\begin{split}\omega=B_{1}&dx_{2}\wedge dx_{3}+B_{2}dx_{3}\wedge
dx_{1}+B_{3}dx_{1}\wedge dx_{2}\\\ &+E_{1}dx_{1}\wedge dt+E_{2}dx_{2}\wedge
dt+E_{3}dx_{3}\wedge dt\end{split}$

as

(11) $d\omega=0.$

The corresponding potential 1-form $\alpha$ can be written as

(12) $\alpha=A_{1}dx_{1}+A_{2}dx_{2}+A_{3}dx_{3}+fdt,$

so that $\omega=d\alpha$ is equivalent to (7a) and (9).

In this formalism $\omega\wedge\omega=0$ is equivalent to orthogonality of the
electric and magnetic fields ([FLS21, Section 5.3] or [Tar05]). Noting that
then $d(\alpha\wedge\omega)=\omega\wedge\omega$, a simple argument using
integration by parts and Sobolev embedding shows that $\omega\wedge\omega$ is
weakly continous in appropriate function spaces. Since in the MHD system we
have $E=B\times u$, the pointwise identity

(13) $B\cdot E=0$

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 $B\cdot E$ below the integrability threshold
where $B\cdot E$ 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 $\mathcal{H}(B)$ is well-defined provided $B\in
L^{3/2}(\mathbb{T}^{3})$. Moreover, recall that whenever $3/2\leq p<\infty$,
solutions $(E,B)$ of the Faraday-Maxwell system (3) with $B\cdot E=0$ satisfy

$(B,E)\in L^{p}\times
L^{p^{\prime}}(\mathbb{T}^{3}\times[0,T])\quad\Longrightarrow\quad\mathcal{H}(B)(t)=\mathcal{H}(B)(0)\textrm{
a.e. }t>0$

[FLS21, Theorem 2.2]. 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:

  1. (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;

  2. (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 $L^{3,\infty}$ 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: $\Omega\subset\mathbb{T}^{3}$ or
$\Omega\subset\mathbb{R}^{3}$ denotes a bounded open spatial subset whose
boundary $\partial\Omega$ has zero Lebesgue measure;

  * •

Space-time domains: $Q\subset\mathbb{T}^{3}\times\mathbb{R}$ or
$Q\subset\mathbb{R}^{3}\times\mathbb{R}$ denotes a bounded open subset of
space-time whose space-time boundary $\partial Q$ has zero 4D Lebesgue
measure;

  * •

Time slices:  Given a space-time domain $Q$, for any fixed time $t$ the set
$Q(t)$ denotes the time-slice of $Q$, i.e. $Q(t)=\\{x:\,(x,t)\in Q\\}$.

  * •

Lebesgue measure: We will frequently use the notation $|A|$ to denote Lebesgue
measure of the set $A$ of appropriate dimension. Thus, $|Q|$ and $|\partial
Q|$ refer to 4D Lebesgue measure, whereas $|\Omega|$, $|Q(t)|$ and
$|\partial\Omega|$ to 3D Lebesgue measure.

  * •

Piecewise constant fields: A vector field $v:Q\to\mathbb{R}^{3}$ is said to be
piecewise constant if there exists a countable family of pairwise disjoint
open subdomains $\\{Q_{i}\\}_{i}$ such that $|Q\setminus\bigcup_{i}Q_{i}|=0$
and $v|_{Q_{i}}$ is constant for each $i$.

  * •

Function spaces: The usual Lebesgue spaces will be denoted by $L^{p}(\Omega)$
or $L^{p}(Q)$, respecting the convention above that $\Omega$ is a spatial
domain and $Q$ a space-time domain. Appropriate Lebesgue spaces of vector
fields will be denoted by $L^{p}(\Omega;\mathbb{R}^{3})$ or
$L^{p}(Q;\mathbb{R}^{3})$. The weak $L^{p}$ spaces of Marcinkiewicz will be
referred to in the Lorentz notation $L^{p,\infty}(\Omega)$.

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.

######  Theorem 1.

Let $(\overline{B},\overline{E})\in L^{\infty}(\mathbb{T}^{3}\times[0,T])$ be
a pair of piecewise constant vector fields solving the Faraday-Maxwell system
(3) in the sense of distributions, and let $p,p^{\prime}$ Hölder-dual
exponents with $3/2<p<\infty$. Then there exist piecewise constant vector
fields $B,E\in L^{1}(\mathbb{T}^{3}\times[0,T])$ solving (3) with

$B\in L^{\infty}(0,T;L^{p,\infty}(\mathbb{T}^{3})),\quad E\in
L^{\infty}(0,T;L^{p^{\prime},\infty}(\mathbb{T}^{3}))$

such that $B\cdot E=0$ for a.e. $(x,t)$ and

$\mathcal{H}(B)(t)=\mathcal{H}(\overline{B})(t)\textrm{ for a.e. }t.$

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 $\overline{B}\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{3})))$ and
$\overline{E}\in L^{\infty}(0,T;L^{1}(\mathbb{T}^{3})))$ be a pair of
piecewise constant vector fields solving the Faraday-Maxwell system (3) in the
sense of distributions and such that $\overline{E}\cdot\overline{B}=0$.

Let $\\{Q_{i}\\}$ be a countable family of pairwise disjoint open sets on each
of which $\overline{B},\overline{E}$ are constant, and let
$\zeta_{+},\zeta_{-}\in L^{\infty}(0,T;L^{1}(\mathbb{T}^{3}))$ such that
$\zeta_{+},\zeta_{-}\in C(\overline{Q}_{i})$ for each $i$ and

$M_{0}(|\overline{B}|^{2}+|\overline{E}|)\leq\min\\{\zeta_{+}^{2},\zeta_{-}^{2}\\}\quad\textrm{
for a.e. }(x,t)\in\mathbb{T}^{3}\times[0,T].$

Then there exists a weak solution $(u,B)\in
L^{\infty}([0,T];L^{2}(\mathbb{T}^{3}))$ of (1) such that

(14a) $|B+u|=\zeta_{+}\textrm{ and }|B-u|=\zeta_{-}\textrm{ a.e. $(x,t)$}$ and
(14b) $\mathcal{H}(B)(t)=\mathcal{H}(\overline{B})(t)\quad\textrm{ a.e. }t.$

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 $h\in\mathbb{R}$ and suppose $e,w\in C([0,T])$ with $e(t)-|w(t)|>M|h|$ for
all $t$. Then there exists a weak solution $(u,B)\in
L^{\infty}(\mathbb{T}^{3}\times[0,T])$ of (1) such that

(15) $\mathcal{E}(u,B)(t)=e(t),\quad\mathcal{W}(u,B)(t)=w(t)\quad\textrm{ for
a.e. t}$

and

(16) $\mathcal{H}(B)(t)=h\quad\textrm{ for a.e. t}.$

Secondly, we show sharpness of the Kang-Lee result.

######  Corollary 2.

There exist weak solutions of ideal MHD with

$u,B\in L^{\infty}(0,T;L^{3,\infty}(\mathbb{T}^{3})),$

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
$L^{\infty}_{t}L^{2}_{x}$ solutions (in fact $L^{\infty}_{t}H^{\beta}_{x}$ for
$0<\beta\ll 1$) of ideal MHD need not preserve magnetic helicity. Corollary 2
solves the flexible part of [BV21, Conjecture 11] (in the $L^{p}$ scale).

##  2\. 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

(17) $\begin{split}\partial_{t}B+\nabla\times E&=0,\\\ \nabla\cdot
B&=0,\end{split}$

and in particular to prove Theorem 1, that is, construct weak solutions
$(B,E)$ with $B\cdot E=0$ 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 $B\cdot
E$ 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
$L^{3,\infty}_{x,t}$ solutions. In order to achieve
$L^{\infty}_{t}L^{3,\infty}_{x}$, 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
$(B,E):Q\to\mathbb{R}^{3}\times\mathbb{R}^{3}$ with $\textrm{div }B=0$ and
$\partial_{t}B+\textrm{curl }E=0$, a typical situation is as follows. Suppose
$Q_{0}\subset Q$ 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 $(B,E):Q_{0}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$ such
that the “glued” vector fields, defined by

$(B^{(1)},E^{(1)})=\begin{cases}(B_{0},E_{0})&\textrm{ outside }Q_{0}\\\
(B,E)&\textrm{ in }Q_{0}\end{cases}$

still satisfy $\textrm{div}B^{(1)}=0$ and
$\partial_{t}B^{(1)}+\textrm{curl}E^{(1)}=0$. It is easy to check that, in
general, a necessary and sufficient condition for this is that

(18) $\textrm{ The extensions
}(\overline{B},\overline{E})=\begin{cases}(B_{0},E_{0})&\textrm{outside
}Q_{0}\\\ (B,E)&\textrm{in }Q_{0}\end{cases}\textrm{ satisfy \eqref{e:Faraday}
in }\mathcal{D}^{\prime}(\mathbb{R}^{4}).$

###  2.1. The basic staircase

We start by constructing a discrete ”staircase” laminate in the plane
$\mathbb{R}^{2}$. Recall that laminates in the plane are defined with respect
to separate convexity, i.e. corresponding to the wave cone
$\\{(x,y)\in\mathbb{R}^{2}:\,x=0\textrm{ or }y=0\\}$.

######  Lemma 1.

Let $1<p<\infty$. For any $n\in\mathbb{N}$ and $\beta>1$ we define

$\mu_{n}=\sum_{k=0}^{n-1}\left[\lambda_{1}^{(k)}\delta_{(0,\beta^{k(p-1)})}+\lambda_{2}^{(k)}\delta_{(\beta^{k+1},0)}\right]+\gamma^{(n)}\delta_{(\beta^{n},\beta^{n(p-1)})},$

where

(19)
$\lambda_{1}^{(k)}=(1-\frac{1}{\beta})\gamma_{k},\quad\lambda_{2}^{(k)}=\frac{1}{\beta}(1-\frac{1}{\beta^{p-1}})\gamma_{k},\quad\gamma^{(k)}=\beta^{-kp}.$

Then $\mu_{n}$ is a laminate on $\mathbb{R}^{2}$ with barycenter $(1,1)$.

###### Proof.

The proof is by induction on $n$. The case $n=0$ is trivial, since
$\mu_{0}=\delta_{(1,1)}$. The inductive step $n\mapsto n+1$ proceeds by the
following splitting procedure:

(20)
$\begin{split}\delta_{(\beta^{n},\beta^{n(p-1)})}\mapsto&(1-\tfrac{1}{\beta})\delta_{(0,\beta^{n(p-1)})}+\tfrac{1}{\beta}\delta_{(\beta^{n+1},\beta^{n(p-1)})}\\\
\mapsto&(1-\tfrac{1}{\beta})\delta_{(0,\beta^{n(p-1)})}+\tfrac{1}{\beta}(1-\tfrac{1}{\beta^{p-1}})\delta_{(\beta^{n+1},0)}+\tfrac{1}{\beta^{p}}\delta_{(\beta^{n+1},\beta^{(n+1)(p-1)})}.\end{split}$

Thus, we obtain $\gamma^{(n+1)}=\tfrac{1}{\beta^{p}}\gamma^{(n)}$. Combined
with $\gamma_{1}=1$ we obtain $\gamma^{(n)}=\beta^{-np}$. The expressions for
$\lambda^{(k)}_{1}$ and $\lambda_{2}^{(k)}$ are analogous. ∎

Now given vectors $B_{0},E_{0}$ we can embed the two dimensional laminate in
the plane spanned by them.

######  Corollary 3.

For any $1<p<\infty$, any $\beta>1$, any
$(B_{0},E_{0})\in\mathbb{R}^{3}\times\mathbb{R}^{3}$ and any $n\in\mathbb{N}$
the probability measure

$\sum_{k=0}^{n-1}\left[\lambda_{1}^{(k)}\delta_{(0,\beta^{k(p-1)}E_{0})}+\lambda_{2}^{(k)}\delta_{(\beta^{k+1}B_{0},0)}\right]+\gamma^{(n)}\delta_{(\beta^{n}B_{0},\beta^{n(p-1)}E_{0})},$

with $\lambda_{1}^{(k)}$, $\lambda_{2}^{(k)}$, $\gamma^{(k)}$ defined as in
(19), is a laminate on $\mathbb{R}^{3}\times\mathbb{R}^{3}$ with respect to
the wave cone $\Lambda=\\{(B,E):B\cdot E=0\\}$, with barycenter
$(B_{0},E_{0})$.

###  2.2. 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
$\textrm{Lip}_{0}(Q)$ the set of Lipschitz functions on $\overline{Q}$ such
that $f=0$ on $\partial Q$.

######  Lemma 2.

Let $B_{1},E_{1},B_{2},E_{2}\in\mathbb{R}^{3}$ with
$(B_{2}-B_{1})\cdot(E_{2}-E_{1})=0$ and $\lambda_{1},\lambda_{2}\in(0,1)$ with
$\lambda_{1}+\lambda_{2}=1$. For any open bounded domain
$Q\subset\mathbb{R}^{4}$ with $|\partial Q|=0$ and any $r,\varepsilon>0$ there
exist piecewise constant vector fields $B,E\in L^{\infty}(Q;\mathbb{R}^{3})$
satisfying (17) and the boundary conditions given by

$(B_{0},E_{0})=\lambda_{1}(B_{1},E_{1})+\lambda_{2}(B_{2},E_{2})$

in the sense of (18), with the following properties:

  * •

$Q$ admits a pairwise disjoint decomposition

(21a) $Q=Q^{(1)}\cup Q^{(2)}\cup Q^{(error)}\cup\mathcal{N}$

where $\mathcal{N}$ a nullset, $Q^{(1)},Q^{(2)}$ and $Q^{(error)}$ are open
sets where $(B,E)$ is locally constant, and such that $(B,E)=(B_{i},E_{i})$ in
$Q_{i}$, $i=1,2$ and $|B-B_{0}|+|E-E_{0}|<r$ in $Q^{(error)}$.

  * •

For $i=1,2$ and any $t\in\mathbb{R}$

(21b)
$|Q^{(error)}(t)|+\frac{1}{\lambda_{i}}|Q^{(i)}(t)|\leq(1+\varepsilon)|Q(t)|$

and moreover

(21c) $|Q^{(error)}|\leq\varepsilon|Q|.$

  * •

There exists a vector potential $\tilde{A}\in Lip_{0}(Q)$ with

(21d) $B_{0}+\textrm{curl}\tilde{A}=B\textrm{ and
}|\tilde{A}|\leq\varepsilon.$

###### Proof.

Let $\bar{B}=B_{2}-B_{1}$ and $\bar{E}=E_{2}-E_{1}$. Since
$\bar{B}\cdot\bar{E}=0$, there exist vectors $\xi,\eta\in\mathbb{R}^{3}$ with
$|\xi|=|\eta|=1$ such that

$\bar{B}=|\bar{B}|\xi\times\eta,\quad\bar{E}=|\bar{E}|\xi.$

Let us now consider first a polyhedral spatial domain $\Omega$ and space-time
domain of the form

(22) $Q=\Omega\times(t_{0},t_{1}).$

Define

$\displaystyle E(x,t)$ $\displaystyle=E_{0}+\tilde{E}=E_{0}+|\bar{E}|\nabla
f_{N}(x,t)-|\bar{B}|\partial_{t}[\eta f(x,t)],$ $\displaystyle B(x,t)$
$\displaystyle=B_{0}+\tilde{B}=B_{0}+|\bar{B}|\textrm{curl }(\eta
f_{N}(x,t))=B_{0}+|\bar{B}|\nabla f(x,t)\times\eta,$

where

$\displaystyle f_{N}(x,t)$
$\displaystyle=\min\left\\{r\textrm{dist}(x,\partial\Omega),f_{N}^{p}(x,t)\right\\}$
$\displaystyle f_{N}^{p}(x,t)$
$\displaystyle=\min\left\\{r(t-t_{0})_{+},r(t_{1}-t)_{+},\frac{1}{N}h(Nx\cdot\xi)\right\\},$

and $h:\mathbb{R}\to\mathbb{R}$ is a 1-periodic non-negative Lipschitz
function with $h^{\prime}(s)\in\\{-\lambda_{2},\lambda_{1}\\}$ for a.e.
$s\in\mathbb{R}$. Observe that for every $t$ the function $x\mapsto
f^{p}_{N}(x,t)$ is a periodic piecewise affine Lipschitz function such that

(23) $\nabla
f_{N}^{p}(x,t)\in\left\\{0,-\lambda_{2}\xi,\lambda_{1}\xi\right\\}\quad\textrm{
a.e. }x$

with respective volume fractions
$\mu(t),\lambda_{1}(1-\mu(t)),\lambda_{2}(1-\mu(t))$, relative to one period.

Since $\Omega$ is a polygonal domain, $f_{N}$ is piecewise affine. Thus, by
definition, there exists an open subset $\mathring{Q}\subset Q$ such that
$|Q\setminus\mathring{Q}|=0$ and $f_{N}$ is locally affine (as well as $\nabla
f_{N}$ is locally constant) in $\mathring{Q}$. Let us define
$\mathcal{N}=Q\setminus\mathring{Q}$,

$\displaystyle Q^{(1)}$ $\displaystyle=\left\\{(x,t)\in\mathring{Q}:\,\nabla
f_{N}(x,t)=-\lambda_{2}\xi\right\\},$ $\displaystyle Q^{(2)}$
$\displaystyle=\left\\{(x,t)\in\mathring{Q}:\,\nabla
f_{N}(x,t)=\lambda_{1}\xi\right\\},$ $\displaystyle Q^{(error)}$
$\displaystyle=\left\\{(x,t)\in\mathring{Q}:\,\nabla
f_{N}(x,t)\notin\\{-\lambda_{2}\xi,\lambda_{1}\xi\\}\right\\}.$

It then follows that for all $t$

$\displaystyle\left|Q^{(1)}(t)\right|$
$\displaystyle\leq\lambda_{1}(1-\mu(t))|Q(t)|,$
$\displaystyle\left|Q^{(2)}(t)\right|$
$\displaystyle\leq\lambda_{2}(1-\mu(t))|Q(t)|,$
$\displaystyle\left|Q^{(error)}(t)\right|$
$\displaystyle\leq(\mu(t)+O(\tfrac{1}{N}))|Q(t)|,$

the latter following from the pointwise bound $|f_{N}^{p}|\leq\frac{1}{N}$.
Then, after eliminating $\mu(t)$ and from the above relationships and choosing
$N$ sufficiently large in terms of $\varepsilon$, we obtain (21b). To deduce
(21c) we may again use the pointwise bound on $f_{N}^{p}$ to see that in fact

$Q^{(error)}\subset\left\\{(x,t)\in Q:\,r\textrm{dist}\left((x,t),\partial
Q\right)\leq\frac{1}{N}\right\\},$

so that (21c) follows by choosing $N$ sufficiently large. Finally, observe
that $\tilde{A}=|\bar{B}|\eta f_{N}$, so that (21d) also follows from choosing
$N$ sufficiently large. This concludes the proof for the case of a cylindrical
polyhedral set $Q=\Omega\times(t_{0},t_{1})$.

For a general open space-time domain $Q$ we find a pairwise disjoint countable
family of cylindrical polyhedral sets $Q_{k}\subset Q$ such that
$|Q\setminus\bigcup_{k}Q_{k}|=0$, apply the above to obtain
$(\tilde{B}_{k},\tilde{E}_{k})$ in $Q_{k}$ with estimates (21b)-(21d).
Defining

$Q^{(1)}=\bigcup_{k}Q^{(1)}_{k},\,Q^{(2)}=\bigcup_{k}Q^{(2)}_{k},\,Q^{(error)}=\bigcup_{k}Q^{(error)}_{k}$

as well as
$\mathcal{N}=(Q\setminus\bigcup_{k}Q_{k})\cup\bigcup_{k}\mathcal{N}_{k}$ leads
to the required properties. ∎

######  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 $\tilde{A}=|\bar{B}|f\eta$. We denote
$Q(t)\mathrel{:\mkern-0.25mu=}\\{x\in\mathbb{R}^{3}:(x,t)\in Q\\}$ for
$t\in\mathbb{R}$. By integrating by parts and using the facts that
$\tilde{A}|_{\partial Q}=0$ and $\tilde{A}\cdot\tilde{B}=0$ we get

$\displaystyle\int_{\mathbb{R}^{3}}[(A_{0}+\tilde{A})\cdot B-A_{0}\cdot
B_{0}]$ $\displaystyle=\int_{Q(t)}\tilde{A}\cdot
B_{0}+\int_{Q(t)}A_{0}\cdot\tilde{B}=2\int_{Q(t)}\tilde{A}\cdot B_{0}$
$\displaystyle=2|\bar{B}|\eta\cdot B_{0}\int_{Q(t)}f.$

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
$B_{0},E_{0}$, $\beta>1$ and set

$B_{n}=\beta^{n}B_{0},\quad E_{n}=\beta^{(p-1)n}E_{0}.$

######  Lemma 3 (Approximation of Steps).

For any $n\in\mathbb{N}$, any open bounded domain $Q\subset\mathbb{R}^{4}$
with $|\partial Q|=0$ and any $r,\varepsilon>0$ there exist piecewise constant
vector fields $B,E\in L^{\infty}(Q;\mathbb{R}^{3})$ 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

(24a) $Q=Q^{(good)}\cup Q^{(inductive)}\cup Q^{(error)}\cup\mathcal{N}$

where $\mathcal{N}$ a nullset, $Q^{(good)},Q^{(inductive)}$ and $Q^{(error)}$
are open sets where $(B,E)$ is locally constant with

(24b) $\begin{split}|B||E|&=0\textrm{ in }Q^{(good)},\\\
(B,E)&=(B_{n+1},E_{n+1})\textrm{ in }Q^{(inductive)},\\\
\textrm{dist}(B,\\{B_{n},B_{n+1}\\})&+|E-E_{n}|<r\textrm{ in
}Q^{(error)}.\end{split}$

  * •

For all $t$ we have

(24c)
$|Q^{(error)}(t)|+\beta^{p}|Q^{(inductive)}(t)|\leq(1+\varepsilon)|Q(t)|$

and

(24d) $|Q^{(error)}|<\varepsilon.$

  * •

There exists a vector potential $\tilde{A}\in Lip_{0}(Q)$ such that, for any
vector potential $A_{0}$ of $B_{0}$,

(24e) $B_{0}+\textrm{curl}\tilde{A}=B,\;|\tilde{A}|\leq\varepsilon\textrm{ and
}\int_{\mathbb{R}^{3}}[(A_{0}+\tilde{A})\cdot B-A_{0}\cdot
B_{0}]\,dx=0\textrm{ a.e. }t\in\mathbb{R}.$

###### Proof.

We may assume without loss of generality that $\varepsilon<1$. In the first
step we apply Lemma 2 with the elementary splitting

(25)
$\delta_{(\beta^{n}B_{0},\beta^{n(p-1)}E_{0})}\mapsto(1-\tfrac{1}{\beta})\delta_{(0,\beta^{n(p-1)}E_{0})}+\tfrac{1}{\beta}\delta_{(\beta^{n+1}B_{0},\beta^{n(p-1)}E_{0})}.$

We obtain $(B^{(1)},E^{(1)})$ and the decomposition

$Q=Q^{(1)}\cup Q^{(2)}\cup Q^{(error,1)}\cup\mathcal{N}^{(1)},$

where $\mathcal{N}^{(1)}$ is a null-set,

$(B^{(1)},E^{(1)})=\begin{cases}(0,\beta^{n(p-1)}E_{0})&\textrm{ in
}Q^{(1)},\\\ (\beta^{n+1}B_{0},\beta^{n(p-1)}E_{0})&\textrm{ in }Q^{(2)}\\\
\end{cases}$

and for all $t$

$\displaystyle\left|Q^{(error,1)}(t)\right|+\beta\left|Q^{(2)}(t)\right|$
$\displaystyle\leq(1+\varepsilon/4)|Q(t)|,$
$\displaystyle\left|Q^{(error,1)}\right|$ $\displaystyle\leq\varepsilon/2|Q|.$

Furthermore, $B^{(1)}=B_{0}+\textrm{curl}A^{(1)}$, with
$|A^{(1)}|\leq\varepsilon/2$. We then use Remark 1 to compute the change of
magnetic helicity. In the elementary splitting (25),
$(\beta^{n+1}B_{0},0)=(\bar{B},\bar{E})=|\bar{B}|\xi\times\eta,0)$ for some
$\xi,\eta\in\mathbb{R}^{3}$ with $|\xi|=|\eta|=1$, and so Remark 1 gives

(26) $\int_{\mathbb{R}^{3}}[(A_{0}+A^{(1)})\cdot B^{(1)}-A_{0}\cdot
B_{0}]\,dx=0.$

Then we apply Lemma 2 in $Q^{(2)}$ with the second elementary splitting

(27)
$\delta_{(\beta^{n+1}B_{0},\beta^{n(p-1)}E_{0})}\mapsto(1-\tfrac{1}{\beta^{p-1}})\delta_{(\beta^{n+1}B_{0},0)}+\tfrac{1}{\beta^{p-1}}\delta_{(\beta^{n+1}B_{0},\beta^{(n+1)(p-1)}E_{0})}.$

We obtain $(B,E)$ and the decomposition

$Q^{(2)}=Q^{(3)}\cup Q^{(4)}\cup Q^{(error,2)}\cup\mathcal{N}^{(2)},$

where $\mathcal{N}^{(2)}$ is a null-set,

$(B,E)=\begin{cases}(\beta^{n+1}B_{0},0)&\textrm{ in }Q^{(3)},\\\
(\beta^{n+1}B_{0},\beta^{(n+1)(p-1)}E_{0})&\textrm{ in }Q^{(4)}\\\
\end{cases}$

and for all $t$

$\displaystyle\left|Q^{(error,2)}(t)\right|+\beta^{p-1}\left|Q^{(4)}(t)\right|$
$\displaystyle\leq(1+\varepsilon/4)|Q^{(2)}(t)|,$
$\displaystyle\left|Q^{(error,2)}\right|$
$\displaystyle\leq\varepsilon/2|Q^{(2)}(t)|.$

Furthermore, $B=B^{(1)}+\textrm{curl}A^{(2)}$, with
$|A^{(2)}|\leq\varepsilon/2$. Set

(28) $Q^{(inductive)}:=Q^{(4)},\quad Q^{(error)}=Q^{(error,1)}\cup
Q^{(error,2)}.$

Then for every $t$

$\displaystyle\left|Q^{(error)}(t)\right|+\beta^{p}\left|Q^{(inductive)}(t)\right|$
$\displaystyle\leq\left|Q^{(error,1)}(t)\right|+\beta\left|Q^{(error,2)}(t)\right|+\beta^{p}\left|Q^{(4)}(t)\right|$
$\displaystyle\leq\left|Q^{(error,1)}(t)\right|+\beta(1+\varepsilon/4)\left|Q^{(2)}(t)\right|$
$\displaystyle\leq(1+\varepsilon/4)^{2}\left|Q(t)\right|$
$\displaystyle\leq(1+\varepsilon)\left|Q(t)\right|$

and

(29)
$\left|Q^{(error)}\right|\leq\left|Q^{(error,1)}\right|+\left|Q^{(error,2)}\right|\leq\varepsilon|Q|$

as required. Moreover, $B=B_{0}+\textrm{curl}(A^{(1)}+A^{(2)})$, with
$|A^{(1)}+A^{(2)}|\leq\varepsilon$. In the elementary splitting (27) we can
write
$(0,\beta^{(n+1)(p-1)}E_{0})=(\bar{B},\bar{E})=(|\bar{B}|\xi\times\eta,|\bar{E}|\xi)$
for $\xi=\bar{E}/|\bar{E}|$ and any $\eta\in\mathbb{R}^{3}$ with $|\eta|=1$.
By letting $\eta\cdot B_{0}=0$, Remark 1 and (26) give

$\int_{\mathbb{R}^{3}}[(A_{0}+A^{(1)}+A^{2})\cdot B-A_{0}\cdot
B_{0}]\,dx=0\quad\textrm{a.e. }t\in\mathbb{R}.$

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).

######  Proposition 1.

Let $B_{0},E_{0}\in\mathbb{R}^{3}$, $1<p<\infty$ and $Q\subset\mathbb{R}^{4}$
an open bounded domain with $|\partial Q|=0$. For any $\beta>1$ and
$\varepsilon>0$ there exist piecewise constant vector fields $B,E\in
L^{\infty}(Q;\mathbb{R}^{3})$ 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

(30a) $Q=Q^{(good)}\cup Q^{(error)}\cup\mathcal{N}$

where $\mathcal{N}$ 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

(30b)
$\mathcal{I}(s,t)\leq\beta^{2(p+1)}|Q(t)|\min(|B_{0}|^{p}+|E_{0}|^{p^{\prime}},s),$

where

$\mathcal{I}(s,t):=\int_{Q^{(error)}(t)}\min\\{|B|^{p}+|E|^{p^{\prime}},s\\}\,dx+s\left|\\{x\in
Q^{(good)}(t):|B|^{p}+|E|^{p^{\prime}}>s\\}\right|$

and $p^{\prime}$ is the Hölder dual of $p$.

  * •

Furthermore,

(30c) $\int\int_{Q^{(error)}}|B|^{p}+|E|^{p^{\prime}}\,dxdt\leq\varepsilon.$

  * •

There exists a vector potential $\tilde{A}\in Lip_{0}(Q)$ with

(30d) $B_{0}+\textrm{curl}\tilde{A}=B,\;|\tilde{A}|\leq\varepsilon\textrm{ and
}\int_{\mathbb{R}^{3}}[(A_{0}+\tilde{A})\cdot B-A_{0}\cdot
B_{0}]\,dx=0\textrm{ a.e. }t\in\mathbb{R}.$

###### Proof.

Based on Lemma 3 we define inductively a sequence $B^{(n)},E^{(n)}\in
L^{\infty}(Q;\mathbb{R}^{3})$ satisfying (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

(31) $Q=\bigcup_{k=0}^{n-1}(Q_{k+1}^{(error)}\cup Q_{k+1}^{(good)})\cup
Q_{n}^{(inductive)}\cup\mathcal{N}_{n},$

where $\mathcal{N}_{n}$ is a null-set, $Q_{k}^{(good)}$, $Q_{n}^{(inductive)}$
and $Q_{k}^{(error)}$ are open sets where $(B_{n},E_{n})$ is locally constant,
such that

$\displaystyle|B^{(n)}||E^{(n)}|$ $\displaystyle=0\textrm{ in }Q_{n}^{(good)}$
$\displaystyle(B^{(n)},E^{(n)})$
$\displaystyle=(\beta^{n}B_{0},\beta^{n(p-1)}E_{0})\textrm{ in
}Q_{n}^{(inductive)}.$

Secondly, $B^{(n+1)}=B^{(n)}+\textrm{curl}A^{(n)}$ with
$|A^{(n)}|\leq\varepsilon 2^{-n-1}$.

We start by defining

$(B^{(1)},E^{(1)}):\equiv(B_{0},E_{0}),\quad Q^{(inductive)}_{0}:=Q,\quad
Q^{(good)}_{0}=Q^{(error)}_{0}=\mathcal{N}_{0}=\emptyset.$

To obtain $(B^{(n+1)},E^{(n+1)})$ we apply Lemma 3 to $(B^{(n)},E^{(n)})$ in
$Q^{(inductive)}_{n}$ with small parameters $r_{n},\varepsilon_{n}>0$ chosen
below, with $\varepsilon_{n}<\varepsilon 2^{-n-1}$. Then we obtain

$Q^{(inductive)}_{n}=Q^{(good)}_{n+1}\cup Q^{(error)}_{n+1}\cup
Q^{(inductive)}_{n+1}\cup\mathcal{N}^{\prime}_{n+1}$

with

$(B^{(n+1)},E^{(n+1)})\approx_{r_{k}}(\beta^{k}B_{0},\beta^{k(p-1)}E_{0})\textrm{
or }(\beta^{k+1}B_{0},\beta^{k(p-1)}E_{0})\textrm{ in }Q_{k+1}^{(error)}$

for all $k<n+1$, where in the last line $\approx_{r_{k}}$ means that the norm
of the difference is bounded by $r_{k}$. In particular we may ensure by the
choice of $r_{k}$ that

$(|B_{0}|^{p}+|E_{0}|^{p^{\prime}})\beta^{kp-1}\leq|B|^{p}+|E|^{p^{\prime}}\leq(|B_{0}|^{p}\beta^{p}+|E_{0}|^{p^{\prime}}\beta)\beta^{kp}\textrm{
in }Q^{(error)}_{k+1}.$

Then we have for all $t$

(32)
$\left|Q_{n+1}^{(error)}(t)\right|+\beta^{p}\left|Q_{n+1}^{(inductive)}(t)\right|\leq\left|Q_{n}^{(inductive)}(t)\right|+\varepsilon_{n}|Q(t)|,$

and furthermore

$\left|Q_{n+1}^{(error)}\right|\leq\varepsilon_{n}.$

The parameters $\varepsilon_{n}>0$, $n=0,1,2,\dots$ will be chosen below, for
the moment let us merely specify that they satisfy

(33) $\sum_{n=0}^{\infty}\beta^{np}\varepsilon_{n}<\beta-1.$

Such condition and (32) immediately imply that, for all $t$,

(34) $|Q_{n}^{(inductive)}(t)|\leq\beta^{-np}|Q(t)|.$

Also, by construction, for any $n\geq k$

(35) $(B^{(n)},E^{(n)})=(B^{(k)},E^{(k)})\,\textrm{ outside
}Q^{(inductive)}_{k}.$

Since the measure of $Q^{(inductive)}_{k}$ 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

$Q^{(good)}=\bigcup_{k=0}^{\infty}Q^{(good)}_{k},\quad
Q^{(error)}=\bigcup_{k=0}^{\infty}Q^{(error)}_{k}$

the decomposition (30a) holds.

We turn to estimate (30b). From (32) we obtain, using (33),

(36)
$\begin{split}\beta^{np}\left|Q_{n}^{(inductive)}(t)\right|+\sum_{k=0}^{n-1}\beta^{kp}\left|Q_{k+1}^{(error)}(t)\right|&\leq\left|Q(t)\right|+\sum_{k=0}^{n}\beta^{kp}\varepsilon_{k}|Q(t)|\\\
&\leq\beta|Q(t)|.\end{split}$

Now let $n\in\mathbb{N}$ and let

(37) $s_{n}=(|B_{0}|^{p}\beta^{p}+|E_{0}|^{p^{\prime}}\beta)\beta^{np}.$

Then

$\displaystyle\left\\{(x,t)\in
Q^{(error)}:\,|B|^{p}+|E|^{p^{\prime}}<s_{n}\right\\}$
$\displaystyle=\bigcup_{k=0}^{n-1}Q_{k+1}^{(error)}\,,$
$\displaystyle\left\\{(x,t)\in Q:\,|B|^{p}+|E|^{p^{\prime}}>s_{n}\right\\}$
$\displaystyle=Q_{n}^{(inductive)}\,.$

Therefore, for every $t$

$\displaystyle\mathcal{I}(s_{n},t)$ $\displaystyle=\int_{\\{x\in
Q^{(error)}(t):|B|^{p}+|E|^{p^{\prime}}\leq
s_{n}\\}}|B|^{p}+|E|^{p^{\prime}}\,dx+s_{n}\left|\\{x\in
Q(t):|B|^{p}+|E|^{p^{\prime}}>s_{n}\\}\right|$
$\displaystyle\leq\sum_{k=0}^{n-1}\int_{Q_{k+1}^{(error)}(t)}|B|^{p}+|E|^{p^{\prime}}\,dx+s_{n}\left|Q^{(inductive)(t)}_{n}\right|$
$\displaystyle\leq(|B_{0}|^{p}\beta^{p}+|E_{0}|^{p^{\prime}}\beta)\left(\sum_{k=0}^{n-1}\beta^{kp}|Q_{k+1}^{(error)}|+\beta^{np}|Q_{n}^{(inductive)}(t)|\right)$
$\displaystyle\leq(|B_{0}|^{p}\beta^{p}+|E_{0}|^{p^{\prime}}\beta)\beta|Q(t)|$
$\displaystyle\leq\beta^{2+p}(|B_{0}|^{p}+|E_{0}|^{p^{\prime}})|Q(t)|.$

where we have used the definition of $s_{n}$, (36) and that on
${Q_{k+1}^{(error)}(t)}$ $|B|^{p}+|E|^{p^{\prime}}$ is bounded by $s_{k}$.

Next, let $s>s^{\prime}$. From the elementary inequality
$\min(|B|^{p}+|E|^{p^{\prime}},s)\leq\tfrac{s}{s^{\prime}}\min(|B|^{p}+|E|^{p^{\prime}},s^{\prime})$
we easily deduce that $I(s,t)\leq\tfrac{s}{s^{\prime}}I(s^{\prime},t)$ for
every $t$. Now, for any $s\geq s_{0}$ (the latter defined in (37) with $n=0$)
there exists $n\in\mathbb{N}$ such that $s_{n}\leq s\leq s_{n}\beta^{p}$.
Consequently

$I(s,t)\leq\frac{s}{s_{n}}I(s_{n},t)\leq\beta^{2(p+1)}(|B_{0}|^{p}+|E_{0}|^{p^{\prime}})|Q(t)|,$

as claimed in (30b). On the other hand for $1\leq s\leq s_{0}$ we may use the
trivial estimate $I(s,t)\leq s|Q(t)|$, from which (30b) also follows.

The estimate (30d) follows from

$\displaystyle\int\int_{Q^{(error)}}|B|^{p}+|E|^{p^{\prime}}\,dxdt$
$\displaystyle\leq(|B_{0}|^{p}\beta^{p}+|E_{0}|^{p^{\prime}}\beta)\sum_{k=0}^{\infty}|Q_{k+1}^{(error)}|\beta^{kp}$
$\displaystyle\leq(|B_{0}|^{p}\beta^{p}+|E_{0}|^{p^{\prime}}\beta)\sum_{k=0}^{\infty}\varepsilon_{k}\beta^{kp},$

and an appropriate choice of $\varepsilon_{n}>0$, $n=0,1,2,\dots$.

Finally, observe that (30b) implies uniform-in-time weak
$L^{p}-L^{p^{\prime}}$ bounds for $(B,E)$, which also clearly hold for the
sequence $(B^{(n)},E^{(n)})$ and $p,p^{\prime}>1$. We deduce that in fact
$(B^{(n)},E^{(n)})\to(B,E)$ strongly in $L^{1}(Q)$. This in turn implies that
$\textrm{div}B=0$, $\partial_{t}B+\textrm{curl}E=0$ 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 $\Omega^{error}$.

######  Theorem 3.

Let $B_{0},E_{0}\in\mathbb{R}^{3}$, $1<p<\infty$ and $Q\subset\mathbb{R}^{4}$
an open bounded domain with $|\partial Q|=0$. For any $\varepsilon>0$ there
exist piecewise constant vector fields $B,E\in L^{\infty}(Q;\mathbb{R}^{3})$
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. $(x,t)\in Q$;

  * •

For all $t$ and any $s>1$

(38) $\left|\left\\{x\in
Q(t):\,|B|^{p}+|E|^{p^{\prime}}>s\right\\}\right|\leq\frac{2}{s}|Q(t)|\min(|B_{0}|^{p}+|E|_{0}|^{p^{\prime}},s),$

so that, in particular, $B\in L^{\infty}_{t}L^{p,\infty}_{x}$ and $E\in
L^{\infty}_{t}L^{p^{\prime},\infty}_{x}$.

  * •

There exists a vector potential $\tilde{A}\in Lip_{0}(Q)$ with

(39) $B_{0}+\textrm{curl}\tilde{A}=B,\;|\tilde{A}|\leq\varepsilon\textrm{ and
}\int_{\mathbb{R}^{3}}[(A_{0}+\tilde{A})\cdot B-A_{0}\cdot
B_{0}]\,dx=0\textrm{ a.e. }t\in\mathbb{R}.$

###### 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

$Q=Q^{(good)}_{q}\cup Q^{(error)}_{q}\cup\mathcal{N}_{q}$

where $\mathcal{N}_{q}$ is a nullset, $Q_{q}^{(good)}$ and $Q_{q}^{(error)}$
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

(40)
$\displaystyle\int\int_{Q^{error}_{q+1}}1+|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}}\,dxdt$
$\displaystyle\leq\frac{1}{2}\int\int_{Q^{error}_{q}}1+|B_{q}|^{p}+|E_{q}|^{p^{\prime}}\,dxdt\,,$
(41) $\displaystyle I_{q+1}(s,t)$
$\displaystyle\leq\beta_{q+1}^{2(p+1)}I_{q}(s,t)\textrm{ for all $t$ and
$s\geq 1$},$

where

$\mathcal{I}_{q}(s,t):=\int_{Q_{q}^{(error)}(t)}\min\\{|B_{q}|^{p}+|E_{q}|^{p^{\prime}},s\\}\,dx+s\left|\\{x\in
Q_{q}^{(good)}(t):|B_{q}|^{p}+|E_{q}|^{p^{\prime}}>s\\}\right|\,.$

Furthermore, $B_{q+1}=B_{q}+\textrm{curl}A_{q}$ with $|A_{q}|\leq\varepsilon
2^{-q-1}$. We start with the constant maps $(B_{0},E_{0})$ and set
$Q_{0}^{error}=Q$.

To obtain $(B_{q+1},E_{q+1})$ from $(B_{q},E_{q})$ we consider the
decomposition

$Q_{q}^{(error)}=\bigcup_{i}Q_{q,i}$

with constant values $(B_{q},E_{q})\equiv(B_{q}^{i},E_{q}^{i})$ on $Q_{q,i}$.
In each $Q_{q,i}$ we replace $(B_{q}^{i},E_{q}^{i})$ by the construction from
Proposition 1 with $(B_{0},E_{0})$ given by $(B_{q}^{i},E_{q}^{i})$ and small
parameters $\beta_{q+1}>1,\varepsilon_{q+1}>0$ still to be fixed. We obtain
for each $i$ a new pairwise disjoint decomposition

$Q_{q,i}=Q_{q,i}^{(good)}\cup Q_{q,i}^{(error)}\cup\mathcal{N}_{q,i}$

with associated estimates, corresponding to (30b)-(30c):

$\displaystyle\int_{Q_{q,i}^{(error)}(t)}\min\\{|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}},s\\}\,dx+$
$\displaystyle\qquad+s\left|\\{x\in
Q^{(good)}_{q,i}(t):|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}}>s\\}\right|$
$\displaystyle\leq\beta_{q+1}^{2(p+1)}|Q_{q,i}|\min(|B_{q,i}|^{p}+|E_{q,i}|^{p^{\prime}},s)\,,$
$\displaystyle\int\int_{Q^{(error)}_{q,i}}1+|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}}\,dxdt\leq\frac{1}{2}(|B_{q,i}|^{p}+|E_{q,i}|^{p^{\prime}})|Q_{q,i}|,$

where the latter is obtained by an appropriate choice of $\varepsilon_{q+1}$
in (30c) . We set

$Q_{q+1}^{(error)}:=\bigcup_{i}Q_{q+1,i}^{(error)},\quad
Q_{q+1}^{(good)}=Q_{q}^{(good)}\cup\bigcup_{i}Q_{q+1,i}^{(good)},\quad\mathcal{N}_{q+1}=\mathcal{N}_{q}\cup\bigcup_{i}\mathcal{N}_{q+1,i}.$

Then we obtain

$\displaystyle\int\int_{Q^{(error)}_{q+1}}|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}}\,dxdt$
$\displaystyle\leq\sum_{i}\int\int_{Q^{(error)}_{q+1,i}}|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}}\,dxdt$
$\displaystyle\leq\frac{1}{2}\sum_{i}|Q_{q,i}|(|B_{q,i}|^{p}+|E_{q,i}|^{p^{\prime}})$
$\displaystyle=\frac{1}{2}\int\int_{Q_{q}^{(error)}}|B_{q}|^{p}+|E_{q}|^{p^{\prime}}\,dxdt,$

so that (40) is satisfied. To obtain (41) we calculate

$\displaystyle\mathcal{I}_{q+1}(s,t)$
$\displaystyle=\sum_{i}\int_{Q_{q,i}^{(error)}(t)}\min\\{|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}},s\\}\,dx+$
$\displaystyle\qquad+\sum_{i}s\left|\\{x\in
Q^{(good)}_{q,i}(t):|B_{q+1}|^{p}+|E_{q+1}|^{p^{\prime}}>s\\}\right|$
$\displaystyle\qquad+s\left|\\{x\in
Q^{(good)}_{q}(t):|B_{q}|^{p}+|E_{q}|^{p^{\prime}}>s\\}\right|$
$\displaystyle\leq\beta_{q+1}^{2(p+1)}\sum_{i}|Q_{q,i}|\min(|B_{q,i}|^{p}++|E_{q,i}|^{p^{\prime}},s)$
$\displaystyle\qquad+s\left|\\{x\in
Q^{(good)}_{q}(t):|B_{q}|^{p}+|E_{q}|^{p^{\prime}}>s\\}\right|$
$\displaystyle=\beta_{q+1}^{2(p+1)}\int_{Q_{q}^{(error)}(t)}\min\\{|B_{q}|^{p}+|E_{q}|^{p^{\prime}},s\\}\,dx+$
$\displaystyle\qquad+s\left|\\{x\in
Q^{(good)}_{q}(t):|B_{q}|^{p}+|E_{q}|^{p^{\prime}}>s\\}\right|$
$\displaystyle\leq\beta_{q+1}^{2(p+1)}\mathcal{I}_{q}(s,t).$

This completes the inductive step with estimates (40)-(41).

Now observe that, because of (40), in particular $|Q_{q}^{(error)}|\to 0$ as
$q\to\infty$. Since $(B_{q+1},E_{q+1})=(B_{q},E_{q})$ outside
$Q_{q}^{(error)}$, we deduce that the sequence $(B_{q},E_{q})$ converges
almost everywhere to _piecewise constant_ vector fields $(B,E)$. Furthermore,
for any $q\in\mathbb{N}$, $s>1$ and $t$

$\mathcal{I}_{q}(s,t)\leq\left(\prod_{k=1}^{q}\beta_{k}\right)^{2(p+1)}|Q(t)|\min(|B_{0}|^{p}+|E|_{0}|^{p^{\prime}},s).$

Thus, choosing $\beta_{q}>1$ in such a way that
$\prod_{k=1}^{q}\beta_{k}^{2(p+1)}\leq 2$, we obtain the uniform bounds (38).
We note furthermore that, since $p,p^{\prime}>1$, 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.

∎

##  3\. 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 $\bar{V}\cong(0,0,\bar{B},\bar{E})$ 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 $\bar{V}$ takes values in the hull

Our first goal is to ensure that pointwise a.e.,
$\bar{V}=(0,0,\bar{B},\bar{E})$ belongs to the relative interior of the hull
determined by $\zeta_{+}$ and $\zeta_{-}$. The precise statement is as
follows:

######  Proposition 2.

We have $(0,0,\bar{B},\bar{E})\in\mathcal{U}_{z_{+}(x,t),z_{-}(x,t)}$ for
every $(x,t)\in\bar{Q}_{i}$ and every $i\in\mathbb{N}$.

In fact, we prove the following more general statement where one can set
$\delta_{0}=1/M_{0}$, $r=\zeta_{+}$ and $s=\zeta_{-}$ to get Proposition 2:

######  Proposition 3.

There exists $\delta_{0}>0$ with the following property: whenever $r,s>0$, we
have

$\left\lvert u\right\rvert^{2}+\left\lvert B\right\rvert^{2}+\left\lvert
S\right\rvert+\left\lvert
E\right\rvert\leq\delta_{0}\min\\{r^{2},s^{2}\\},\;B\cdot
E=0\quad\Longrightarrow\quad(u,B,S,E)\in\mathcal{U}_{r,s}.$

We recall from [FLS21] some notions that are relevant to the computations
below. The relaxed Elsässer variables
$(z^{+},z^{-},M)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3\times
3}$ are defined via $(u,S,B,E)$ as follows:

$z^{\pm}\mathrel{:\mkern-0.25mu=}u\pm b,\qquad M\xi\equiv S\xi+\xi\times E.$

The wave cone conditions are written in terms of $(u,B,S,E)$ and
$(z^{+},z^{-},M)$ as

$\displaystyle
u\cdot\xi_{x}=B\cdot\xi_{x}=0,\quad\xi_{t}u+S\xi_{x}=0,\quad\xi_{t}B+\xi_{x}\times
E=0,$ $\displaystyle
z^{\pm}\cdot\xi=0,\quad\xi_{t}z^{+}+M\xi_{x}=0,\quad\xi_{t}z^{-}+M^{T}\xi_{x}=0.$

When $B\times u\neq 0$, it suffices to check the conditions

(42) $S(B\times u)+(E\cdot u)u=0,\qquad B\cdot E=0.$

Recall that

$\displaystyle\mathscr{M}\mathrel{:\mkern-0.25mu=}\\{(u,B,S,E)\colon B\cdot
E=0\\},$ $\displaystyle K\mathrel{:\mkern-0.25mu=}\\{(u,B,S,E)\colon E=B\times
u,\,S=S_{u,B}+\Pi I,\,\Pi\in\mathbb{R}\\},$ $\displaystyle
K_{r,s}\mathrel{:\mkern-0.25mu=}\\{(u,B,S,E)\in
K\colon|u+B|=r,\,|u-B|=s,\,\left\lvert\Pi\right\rvert\leq rs\\},$
$\displaystyle\mathcal{U}_{r,s}\mathrel{:\mkern-0.25mu=}\textup{int}_{\mathscr{M}}(K_{r,s}^{lc,\Lambda}),$

where $S_{u,B}\mathrel{:\mkern-0.25mu=}u\otimes u-B\otimes B$.

We divide the proof of Proposition 3 into several steps. We refer to [FLS21]
when possible. The first lemma restates [FLS21, Lemma 6.10].

######  Lemma 4.

If $\left\lvert z^{+}\right\rvert\leq r$, $\left\lvert z^{-}\right\rvert\leq
s$ and $\left\lvert\Pi\right\rvert\leq rs$, then

$(u,B,S_{u,B}+\Pi I,B\times u)\in K_{r,s}^{lc,\Lambda}.$

Below, all the parameters $\delta_{i}>0$ are independent of $r$ and $s$. The
next lemma adds a primitive metric $e\otimes e$ to the $S$-component. It is
proved by following the proof of [FLS21, Lemma 6.11] verbatim.

######  Lemma 5.

There exists $\delta_{1}>0$ such that if
$|u|^{2}+|B|^{2}+|e|^{2}+|\Pi|\leq\delta_{1}\min\\{r^{2},s^{2}\\}$, then

$(u,S_{u,B}+e\otimes e+\Pi I,B,B\times u)\in K_{r,s}^{lc,\Lambda}.$

The next lemma replaces the primitive metric $e\otimes e$ by a general small
symmetric matrix $S$. The lemma is proved by making obvious, slight
modifications to the proof of [FLS21, Lemma 6.12].

######  Lemma 6.

There exists $\delta_{2}>0$ such that whenever $r,s>0$ and $\left\lvert
u\right\rvert^{2}+\left\lvert B\right\rvert^{2}+\left\lvert
S\right\rvert+\left\lvert\Pi\right\rvert\leq\delta_{2}\min\\{r^{2},s^{2}\\}$,
we have

$(u,S_{u,B}+S+\Pi I,B,B\times u)\in K_{r,s}^{lc,\Lambda}.$

We next get rid of the terms $S_{u,B}$ and $\Pi$.

######  Corollary 4.

There exists $\delta_{3}>0$ such that whenever $\left\lvert
u\right\rvert^{2}+\left\lvert B\right\rvert^{2}+\left\lvert
S\right\rvert\leq\delta_{3}\min\\{r^{2},s^{2}\\}$, we have

$(u,S,B,B\times u)\in K_{r,s}^{lc,\Lambda}.$

###### Proof.

If $\left\lvert u\right\rvert^{2}+\left\lvert B\right\rvert^{2}+\left\lvert
S\right\rvert\leq\delta_{2}\min\\{r^{2}/2,s^{2}/2\\}$, we denote
$\tilde{S}\mathrel{:\mkern-0.25mu=}S-S_{u,b}$. Now $(u,S,B,B\times
u)=(u,S_{u,B}+\tilde{S},B,B\times u)$ satisfies the assumptions of Lemma 6. ∎

In order to finish the proof of Proposition 3 we relax the condition
$E=B\times u$ to $B\cdot E=0$. This corresponds to [FLS21, Lemma 6.13], but
some of the details are different, and we therefore present a proof for
completeness.

######  Lemma 7.

There exists $\delta_{4}>0$ such that whenever $\left\lvert
u\right\rvert^{2}+\left\lvert B\right\rvert^{2}+\left\lvert
S\right\rvert+\left\lvert E\right\rvert\leq\delta_{4}\min\\{r^{2},s^{2}\\}$
and $B\cdot E=0$, we have

$(u,S,B,E)\in K_{r,s}^{lc,\Lambda}.$

###### Proof.

We first assume that $B\neq 0$ and that $\left\lvert
u\right\rvert^{2}+\left\lvert B\right\rvert^{2}+\left\lvert
S\right\rvert\leq\delta_{3}\min\\{r^{2}/2,s^{2}/2\\}$. We start by considering
$E=B\times u+B\times v$, where $0<\left\lvert B\times
v\right\rvert\leq\delta_{3}\min\\{r^{2}/2,s^{2}/2\\}$; the proof of the case
$B\neq 0$ is then finished by arguing as in Corollary 4. Without loss of
generality, we assume that $B\cdot v=0$. Then $\left\lvert B\times
v\right\rvert=\left\lvert B\right\rvert\left\lvert v\right\rvert$.

We denote $c\mathrel{:\mkern-0.25mu=}(\left\lvert B\right\rvert/\left\lvert
v\right\rvert)^{1/2}$ so that

(43) $\left\lvert cv\right\rvert=|c^{-1}B|=(\left\lvert
B\right\rvert\left\lvert v\right\rvert)^{1/2}=\left\lvert B\times
v\right\rvert^{1/2}\leq\delta_{3}\min\\{r^{2}/2,s^{2}/2\\}.$

We then write $(u,S,B,B\times u+B\times v)$ as the middle point of a suitable
$\Lambda$-segment:

$\displaystyle(u,S,B,B\times(u+v))$ $\displaystyle=$
$\displaystyle\frac{1}{2}(u+cv,S+\bar{S},B+c^{-1}B,(1+c^{-1})B\times(u+cv))$
$\displaystyle+$
$\displaystyle\frac{1}{2}(u-cv,S-\bar{S},B-c^{-1}B,(1-c^{-1})B\times(u-cv)),$

where

$\bar{S}=\frac{cu\cdot B\times v}{\left\lvert B\times
v\right\rvert^{2}}(B\times v\otimes v+v\otimes B\times v).$

Notice that thanks to (43) we can apply Corollary 4 to deduce that the
endpoints lie in $K_{r,s}^{lc,\Lambda}$. (In particular, $|\bar{S}|\leq
2c\left\lvert u\right\rvert\left\lvert
v\right\rvert\leq\delta_{3}\min\\{r^{2}/2,s^{2}/2\\}$.) The direction of the
$\Lambda$-segment is

$\bar{V}=\left(2cv,2\bar{S},2c^{-1}B,2\left(B\times cv+c^{-1}B\times
u\right)\right),$

which belongs to $\Lambda$ since (42) is satisfied. The case $B\neq 0$,
$E=B\times u+B\times v$ is now proved. The general case $B\neq 0$, $E=B\times
v$ is obtained as in the proof of Corollary 4.

Suppose then $B=0$. Choose any tiny $\tilde{B}\neq 0$ with $\tilde{B}\cdot
E=0$. We may then write $E=\tilde{B}\times v$ as above. Now

$(u,S,0,E)=\frac{1}{2}(u,S,\tilde{B},\tilde{B}\times
v)+\frac{1}{2}(u,S,-\tilde{B},-\tilde{B}\times(-v)\mathrel{=\mkern-0.25mu:}\frac{1}{2}V_{1}+\frac{1}{2}V_{2},$

where $V_{1},V_{2}\in K_{r,s}^{lc,\Lambda}$ by the case $B\neq 0$ and
$V_{1}-V_{2}=(0,0,2\tilde{B},0)\in\Lambda$. ∎

###  3.2. Modifying $\bar{V}$ in a single set $Q_{i}$

As the main building block of the proof of Theorem 2, we perturb
$\bar{V}=(0,0,\bar{B},\bar{E})$ in a single set $Q_{i}$, where
$\\{Q_{i}\\}_{i\in\mathbb{N}}$ is the family of disjoint open sets
corresponding to $\bar{B},\bar{E}$ in the definition of piecewise constant
fields.

######  Proposition 4.

Under the assumptions of Theorem 2, fix $i\in\mathbb{N}$. There exists
$V=(u,S,B,E)$ with $u,B\in L^{\infty}([0,T];L^{2}(\mathbb{T}^{3}))$ and
$S,E\in L^{\infty}(0,T;L^{1}(\mathbb{T}^{3}))$ such that $V$ solves relaxed
MHD equations in $\mathbb{T}^{3}$, $|B\pm u|=\zeta_{\pm}$ a.e. $(x,t)\in
Q_{i}$, $V$ solves (1) in $Q_{i}$ and $V=\bar{V}$ a.e. outside $Q_{i}$.

Proposition 4 follows from slight modifications of the results of [FLS21, §7],
but below, we indicate the main ideas. In order to keep the notation
consistent with [FLS21] we denote $\Omega=Q_{i}$.

By assumption, $\bar{E}$ and $\bar{B}$ are locally constant in $\bar{\Omega}$
with $\bar{B}\cdot\bar{E}=0$. Furthermore, $\zeta_{+},\zeta_{-}\in
C(\bar{\Omega})$ satisfy

$M_{0}(|\bar{B}|^{2}+|\bar{E}|)\leq\min\\{\zeta_{+}^{2},\zeta_{-}^{2}\\}\quad\textup{for
all }(x,t)\in\bar{\Omega}.$

Following [FLS21] we write
$(\bar{B},\bar{E})\cong\bar{\omega}=d\bar{\varphi}\wedge d\bar{\psi}$, where
$\bar{\varphi}$ and $\bar{\psi}$ are linear. Our aim is to replace
$\bar{V}=(0,0,\bar{\omega})$ in $\bar{\Omega}$ by $V=(u,S,\omega)$ which
satisfies the ideal MHD equations and $\left\lvert B\pm
u\right\rvert=\zeta_{\pm}$ a.e. in $\Omega$.

In analogy to [FLS21, Definition 7.1] we define a class of subsolutions

$\displaystyle X_{0}$
$\displaystyle\mathrel{:\mkern-0.25mu=}\\{(V=(u,S,\omega)\in
C^{\infty}(\bar{\Omega},\mathbb{R}^{15})\colon\textup{there exist
}\varphi,\psi\in C^{\infty}(\bar{\Omega},\mathbb{R}^{15})\textup{ such that }$
$\displaystyle\omega=d\varphi\wedge
d\psi,\,\mathcal{L}(V)=0,\;\textup{supp}(u,S,\varphi-\bar{\varphi},\psi-\bar{\psi})\subset\Omega\textup{
and }$ $\displaystyle
V(x,t)\in\mathcal{U}_{\zeta_{+}(x,t),\zeta_{-}(x,t)}\,\forall(x,t)\in\bar{\Omega}\\}.$

With
$C^{\pm}\mathrel{:\mkern-0.25mu=}\max_{(x,t)\in\bar{Q}_{i}}\zeta_{\pm}(x,t)$
we denote the weak sequential closure of $X_{0}$ in
$L^{2}(\mathbb{T}^{3}\times[0,T];\overline{\textup{co}}(K_{C^{+},C^{-}})$ by
$X$. Now $X\ni\\{\bar{V}\\}$ is a compact metrisable space, and we denote a
metric by $d_{X}$.

We state the main step of the proof of Proposition 4.

######  Proposition 5.

There exists $C>0$ with the following property. If $V=(u,S,d\varphi\wedge
d\psi)\in X_{0}$, then there exist
$V_{\ell}=(u_{\ell},S_{\ell},d\varphi_{\ell}\wedge d\psi_{\ell})\in X_{0}$
such that $d_{X}(V_{\ell},V_{0})\to 0$ and

$\displaystyle\int_{\Omega}(\left\lvert
u_{\ell}(x,t)\right\rvert^{2}+\left\lvert
B_{\ell}(x,t)\right\rvert^{2}-\left\lvert u(x,t)\right\rvert^{2}-\left\lvert
B(x,t)\right\rvert^{2})\,dx\,dt$ $\displaystyle\geq\;$ $\displaystyle
C\int_{\Omega}\left(\frac{\zeta_{+}(x,t)^{2}+\zeta_{-}(x,t)^{2}}{2}-\left\lvert
u(x,t)\right\rvert^{2}-\left\lvert B(x,t)\right\rvert^{2}\right)\,dx\,dt.$

In fact, [FLS21, Proposition 7.2] is the special case of Proposition 5 where
$\zeta_{+}$ and $\zeta_{-}$ 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,
pp. 38–39].

###  3.3. Adding the perturbations

We finish the proof of Theorem 2 by iterating Proposition 4 and ensuring that
the magnetic helicity does not change along the iteration.

######  Proposition 6.

There exist $u,B\in L^{\infty}([0,T];L^{2}(\mathbb{T}^{3}))$ that satisfy the
assertions of Theorem 2.

###### Proof.

Given $N\in\mathbb{N}$, we use Proposition 4 to get a solution
$V_{N}=(u_{N},S_{N},B_{N},E_{N})$ of the relaxed MHD equations that solves (1)
in $\cup_{i=1}^{N}Q_{i}$ and satisfies $V_{N}=\bar{V}$ a.e. outside
$\cup_{i=1}^{N}Q_{i}$. The a.e. limit $V(x,t)=\lim_{N\to\infty}V_{N}(x,t)$
satisfies
$\lim_{N\to\infty}\|u_{N}-u\|_{L^{2}}=\lim_{N\to\infty}\|B_{N}-B\|_{L^{2}}=0$
and is a solution of (1) with $|B\pm u|=\zeta_{\pm}$ a.e.
$(x,t)\in\mathbb{T}^{3}\times[0,T]$. Thus $B,u\in
L^{\infty}([0,T];L^{2}(\mathbb{T}^{3}))$.

We then show that $\mathcal{H}(B)(t)=\mathcal{H}(\bar{B})(t)$ a.e.
$t\in(0,T)$. Since $B_{N}\to B$ in $L^{2}(\mathbb{T}^{3}\times[0,T])$, by a
standard argument it suffices to show that
$\mathcal{H}(B_{N})(t)=\mathcal{H}(\bar{B})(t)\text{ a.e. }t\in(0,T)\text{ for
every }N\in\mathbb{N}$. In fact, by setting $\Omega=\cup_{i=1}^{N}Q_{i}$ and
defining $X_{0}$ and $X$ as in §3.2, we approximate $B_{N}$ weakly in
$L^{2}(\mathbb{T}^{3}\times[0,T])$ by a sequence of subsolutions $B_{N}^{k}\in
X_{0}$. Once we show that

(44) $\mathcal{H}(B_{N}^{k})(t)=\mathcal{H}(\bar{B})(t)\text{ a.e.
}t\in(0,T)\text{ for every }N,k\in\mathbb{N},$

we get $\mathcal{H}(B_{N}^{k})(t)=\mathcal{H}(B_{N})(t)$ a.e. $t\in(0,T)$ (see
[FLS21, Theorem 2.2] and its proof).

Fix, therefore, $N,k\in\mathbb{N}$ and recall that for each
$i\in\\{1,\ldots,N\\}$ we can write

$B_{N}^{k}=\nabla\varphi_{i}\times\nabla\psi_{i}\text{ and
}\bar{B}=\nabla\bar{\varphi}_{i}\times\nabla\bar{\psi}_{i}\quad\text{in
}Q_{i},$

where $\varphi_{i}$ and $\psi_{i}$ are smooth, $\bar{\varphi}_{i}$ and
$\bar{\psi}_{i}$ are linear and
$\textup{supp}\,(\varphi_{i}-\bar{\varphi}_{i},\psi_{i}-\bar{\psi}_{i})\subset
Q_{i}$. Given a vector potential $\bar{A}$ of $\bar{B}$, one vector potential
of $B_{N}^{k}$ is therefore given by

$A_{N}^{k}\mathrel{:\mkern-0.25mu=}\bar{A}+\sum_{i=1}^{N}(\varphi_{i}\nabla\psi_{i}-\bar{\varphi}_{i}\nabla\bar{\psi}_{i}).$

For a.e. $t\in(0,T)$ and each $i\in\\{1,\ldots,N\\}$ we denote
$Q_{i}(t)\mathrel{:\mkern-0.25mu=}\\{x\in\mathbb{T}^{3}\colon(x,t)\in
Q_{i}\\}$. By integrating by parts first on $\mathbb{T}^{3}$ and then on each
$Q_{i}(t)$ we get

$\displaystyle\int_{\mathbb{T}^{3}}A_{N}^{k}\cdot B_{N}^{k}$
$\displaystyle=\int_{\mathbb{T}^{3}}\bar{A}\cdot\bar{B}+\int_{\mathbb{T}^{3}}(A_{N}^{k}-\bar{A})\cdot(B_{N}^{k}+\bar{B})$
$\displaystyle=\int_{\mathbb{T}^{3}}\bar{A}\cdot\bar{B}+\sum_{i=1}^{N}\int_{Q_{i}(t)}(\varphi_{i}\nabla\psi_{i}-\bar{\varphi}_{i}\nabla\bar{\psi}_{i})\cdot(\nabla\varphi_{i}\times\nabla\psi_{i}+\nabla\bar{\varphi}_{i}\times\nabla\bar{\psi}_{i})$
$\displaystyle=\int_{\mathbb{T}^{3}}\bar{A}\cdot\bar{B}+\sum_{i=1}^{N}\int_{Q_{i}(t)}(\varphi_{i}\nabla\psi_{i}\cdot\nabla\times(\bar{\varphi}_{i}\nabla\bar{\psi}_{i})-\bar{\varphi}_{i}\nabla\bar{\psi}_{i}\cdot\nabla\times(\varphi_{i}\nabla\psi_{i}))$
$\displaystyle=\int_{\mathbb{T}^{3}}\bar{A}\cdot\bar{B},$

which proves (44) and completes the proof of Theorem 2. ∎

##  4\. Proof of Corollaries 1 and 2

###### Proof of Corollary 1.

We need different initial vector fields for the case $h>0$ and $h<0$. We
select them uniformly for the sake of having unified constants.

We first select piecewise constant vector fields $\bar{B}_{1},\bar{B}_{2}\in
L^{\infty}(\mathbb{T}^{3})$ such that $\textrm{div}\bar{B}_{i}=0$ and
$\mathcal{H}(\bar{B}_{1})>0>\mathcal{H}(\bar{B}_{2})$. This can be done e.g.
by considering piecewise affine vector potentials
$\bar{A}=(\bar{A}_{1},\bar{A}_{2},0)\in
L^{\infty}(\mathbb{T}^{3};\mathbb{R}^{3})$ such that
$\partial_{3}\bar{A}_{1}=1$ in $\textrm{supp}(\bar{A}_{2})$ with
$\int_{\mathbb{T}^{3}}\bar{A}_{2}\,dx>0$ for $\bar{B}_{1}$ and
$\int_{\mathbb{T}^{3}}\bar{A}_{2}\,dx<0$ for $\bar{B}_{2}$. We then fix $M>0$
such that $2M|\mathcal{H}(\bar{B}_{i})|\geq
M_{0}\int_{\mathbb{T}^{3}}|\bar{B}_{i}(y)|^{2}\,dy$ for $i=1,2$, where
$M_{0}>0$ is the geometric constant of Theorem 2.

Let $h>0$ (The case $h=0$ can be proven by minor modifications to [FLS21].)

After scaling, we may assume that $\mathcal{H}(\bar{B}_{1})=h$. In Theorem 2,
set $\bar{B}=\bar{B}_{1}$, $\bar{E}=0$ and
$|\zeta_{\pm}(x,t)|^{2}=M_{0}|\bar{B}(x)|^{2}+2[e(t)\pm
w(t)]-M_{0}\int_{\mathbb{T}^{3}}|\bar{B}(y)|^{2}\,dy$. Our assumptions give
$2[e(t)\pm w(t)]>2M|h|\geq M_{0}\int_{\mathbb{T}^{3}}|\bar{B}(y)|^{2}\,dy$ for
all $t\in[0,T]$, and therefore
$M_{0}(|\bar{B}|^{2}+|\bar{E}|)\leq\min\\{\zeta_{+}^{2},\zeta_{-}^{2}\\}$ for
a.e. $(x,t)\in\mathbb{T}^{3}\times[0,T]$. Theorem 2 now yields a weak solution
$(u,B)\in L^{\infty}(\mathbb{T}^{3}\times[0,T])$ of (1) such that
$\mathcal{E}(u,B)(t)=4^{-1}\int_{\mathbb{T}^{3}}(|\zeta_{+}|^{2}+|\zeta_{-}|^{2})\,dx=e(t)$,
$\mathcal{W}(u,B)(t)=4^{-1}\int_{\mathbb{T}^{3}}(|\zeta_{+}|^{2}-|\zeta_{-}|^{2})\,dx=w(t)$
and $\mathcal{H}(B)(t)=h$ for a.e. $t$. ∎

###### Proof of Corollary 2.

The proof follows immediately by combining Theorem 1 with $p=3$ and Theorem 2.
∎

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