# Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and
Asymptotics

Diogo Arsénio  ,  Zineb Hassainia  and  Haroune Houamed  New York University
Abu Dhabi
Abu Dhabi
United Arab Emirates<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the global analysis of the three-dimensional
axisymmetric Navier–Stokes–Maxwell equations. More precisely, we are able to
prove that, for large values of the speed of light $c\in(c_{0},\infty)$, for
some threshold $c_{0}>0$ depending only on the initial data, the system in
question admits a unique global solution. The ensuing bounds on the solutions
are uniform with respect to the speed of light, which allows us to study the
singular regime $c\rightarrow\infty$ and rigorously derive the limiting
viscous magnetohydrodynamic (MHD) system in the axisymmetric setting.

The strategy of our proofs draws insight from recent results on the two-
dimensional incompressible Euler–Maxwell system to exploit the
dissipative–dispersive structure of Maxwell’s system in the axisymmetric
setting. Furthermore, a detailed analysis of the asymptotic regime
$c\to\infty$ allows us to derive a robust nonlinear energy estimate which
holds uniformly in $c$. As a byproduct of such refined uniform estimates, we
are able to describe the global strong convergence of solutions toward the MHD
system.

This collection of results seemingly establishes the first available global
well-posedness of three-dimensional viscous plasmas, where the electric and
magnetic fields are governed by the complete Maxwell equations, for large
initial data as $c\to\infty$.

###### Key words and phrases:

Incompressible viscous three-dimensional fluids, Maxwell’s system, plasmas,
global well-posedness, axisymmetric structure.

###### Contents

  1. 1 Introduction and main results
  2. 2 The axisymmetric structure, Hardy inequalities and paradifferential calculus
  3. 3 A priori estimates
  4. 4 Asymptotic analysis of electromagnetic fields
  5. 5 Closing the estimates and proof of Theorem 1.1
  6. 6 Convergence and proof of Theorem 1.3

##  1\. Introduction and main results

In this paper, we consider the incompressible Navier–Stokes–Maxwell equations

(1.1) $\begin{cases}\begin{aligned} \text{\tiny(Navier--Stokes's
equation)}&&&\partial_{t}u+u\cdot\nabla u=\nu\Delta u-\nabla p+j\times
B,&\operatorname{div}u=0,&\\\ \text{\tiny(Amp\\`{e}re's
equation)}&&&\frac{1}{c}\partial_{t}E-\nabla\times
B=-j,&\operatorname{div}E=0,&\\\ \text{\tiny(Faraday's
equation)}&&&\frac{1}{c}\partial_{t}B+\nabla\times
E=0,&\operatorname{div}B=0,&\\\ \text{\tiny(Ohm's
law)}&&&j=\sigma\big{(}cE+P(u\times
B)\big{)},&\operatorname{div}j=0,&\end{aligned}\end{cases}$

completed with a divergence-free initial data

$(u,E,B)|_{t=0}=(u_{0},E_{0},B_{0}).$

Here, $u$, $E$, $B$ and $j$ are defined on the whole space
$\mathbb{R}^{+}\times\mathbb{R}^{d}$, with $d\in\\{2,3\\}$, and take their
values in $\mathbb{R}^{3}$. The operator $P\overset{\rm
def}{=}\text{Id}-\nabla\Delta^{-1}\operatorname{div}$ denotes Leray’s
projector on divergence-free vector fields. Note that the divergence-free
condition $\operatorname{div}B=0$ is not a constraint and (1.1) is therefore
not overdetermined. Instead, it is a property propagated by the flow provided
that it holds initially.

The Navier–Stokes–Maxwell equations describe the evolution of a viscous
plasma, a charged gas or an electrically conducting fluid subject to the self-
induced electromagnetic Lorentz force $j\times B$.

As for the physical meaning of the quantities in (1.1), the electric and
magnetic fields are denoted by $E$ and $B$, respectively, while the electric
current is denoted by $j$. Additionally, the positive constants $\nu$,
$\sigma$ and $c$ stand for the fluid viscosity, the electrical conductivity
and the speed of light, respectively. We refer the interested reader to [9,
13] for further details on the underlying physical theories about plasma
modeling and to [6] for a mathematical derivation of (1.1) through the
analysis of the viscous incompressible hydrodynamic regimes of
Vlasov–Maxwell–Boltzmann systems.

The only known a priori global information on solutions of (1.1) is given by
the energy inequality

(1.2)
$\left\|\left(u,E,B\right)(t)\right\|_{L^{2}}^{2}+2\nu\int_{0}^{t}\left\|\nabla
u(\tau)\right\|_{L^{2}}^{2}d\tau+\frac{2}{\sigma}\int_{0}^{t}\left\|j(\tau)\right\|_{L^{2}}^{2}d\tau\leq\mathcal{E}_{0}^{2},$

for all $t\geq 0$, where we denote

$\mathcal{E}_{0}\overset{\rm
def}{=}\left\|(u_{0},E_{0},B_{0})\right\|_{L^{2}}.$

Note that (1.2) is in fact an equality for smooth solutions.

Observe that, at least formally, taking the limit of $c$ to infinity in (1.1)
yields the well-known magneto-hydrodynamic (MHD) system

(MHD) $\begin{cases}\begin{aligned} &\partial_{t}u+u\cdot\nabla u=\nu\Delta
u-\nabla p+B\cdot\nabla B,\\\ &\partial_{t}B+u\cdot\nabla
B=\frac{1}{\sigma}\Delta B+B\cdot\nabla u.\end{aligned}\end{cases}$

Similarly to (1.1), solutions to the system of equations (MHD) enjoy the
energy inequality

(1.3)
$\left\|\left(u,B\right)(t)\right\|_{L^{2}}^{2}+2\nu\int_{0}^{t}\left\|\nabla
u(\tau)\right\|_{L^{2}}^{2}d\tau+\frac{2}{\sigma}\int_{0}^{t}\left\|\nabla
B(\tau)\right\|_{L^{2}}^{2}d\tau\leq\left\|\left(u_{0},B_{0}\right)\right\|_{L^{2}}^{2},$

for any $t\geq 0$.

Although Leray-type weak solutions for (MHD) satisfying (1.3) are known to
exist, this remains unknown for (1.1), even for large values of $c$.

The issue comes from the fact that standard compactness methods, usually used
to prove the existence of weak solutions, raise the following question: Given
two sequences $(j_{n})_{n\in\mathbb{N}}$, $(B_{n})_{n\in\mathbb{N}}$, bounded
in $L^{2}_{t,x}$ and $L^{\infty}_{t}L^{2}_{x}$ and converging weakly to $j$
and $B$, respectively, does it hold that $j_{n}\times B_{n}$ converges to
$j\times B$, at least in the sense of distributions? This remains unsettled as
there is no known phenomenon which prevents the build-up of high frequencies
in $j_{n}\times B_{n}$, as $n$ tends to infinity.

One way to overcome this lack of compactness consists in propagating some
additional regularity on the magnetic field. Based on this idea, the first
attempt to construct a (unique) global solution to the two dimensional
Navier–Stokes–Maxwell system (1.1) is due to Masmoudi [36] who proved that,
given any $s>0$, if $u_{0}\in L^{2}(\mathbb{R}^{2})$ and $(E_{0},B_{0})\in
H^{s}(\mathbb{R}^{2})$, then there is a unique global weak solution to (1.1).
Apart from the energy estimate, the solution constructed therein does not
satisfy any uniform bound with respect to the speed of light $c$.

Later on, the first author, Ibrahim and Masmoudi [5] established a conditional
convergence result which entails the convergence of weak solutions of (1.1) to
(MHD).

Crucial progress was then achieved in [2] by the first author and Gallagher by
showing the persistence of the $H^{s}$-regularity of the electromagnetic
field, uniformly with respect to the speed of light $c$.

As for the inviscid version of (1.1), i.e., the incompressible Euler–Maxwell
system, the first global results on that model were established recently by
the first and the third authors in [3, 4]. In that context, owing to known
results on Euler equations, see for instance [14], working at the level of the
energy (1.2), with $\nu=0,$ is insufficient to ensure the global well-
posedness of the system. One is therefore compelled to study the existence and
uniqueness of weak solutions in a higher-regularity space.

The main contribution from [3] is then the construction of a unique global
solution of the Euler–Maxwell equations (1.1) (with $\nu=0$) in the spirit of
Yudovich’s work [40], where the electromagnetic field has some Sobolev
regularity $H^{s}(\mathbb{R}^{2}),$ $s\in(\frac{7}{4},2)$. Moreover, it is
shown therein that the solution is uniformly bounded with respect to $c$ in
adequate spaces. The other work by the same authors [4] further establishes
the strong convergence of that solution, as $c$ goes to $\infty$. We should
point out that the results in [3, 4], for the Euler–Maxwell system, only hold
under the assumption that $(u,E,B)$ has the two-dimensional normal structure

(1.4) $u(t,x)=\begin{pmatrix}u_{1}(t,x)\\\ u_{2}(t,x)\\\ 0\end{pmatrix},\qquad
E(t,x)=\begin{pmatrix}E_{1}(t,x)\\\ E_{2}(t,x)\\\
0\end{pmatrix}\qquad\text{and}\qquad B(t,x)=\begin{pmatrix}0\\\ 0\\\
b(t,x)\end{pmatrix}.$

Extending the arguments from [3, 4] to general structures is a challenging
open problem. As we shall see later on, a novelty of the present work is to
permit the consideration of a new three-dimensional structure in the context
of plasmas, namely, the axisymmetric structure.

From now on, we are going to focus on the three dimensional case by first
considering the work by Ibrahim and Keraani [27] where it is proved that the
system (1.1) is globally well-posed provided that the initial data
$(u_{0},E_{0},B_{0})$ are small enough in

$\dot{B}_{2,1}^{\frac{1}{2}}\times\dot{H}^{\frac{1}{2}}\times\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3}).$

This smallness condition is slightly weakened in the work of Germain, Ibrahim
and Masmoudi [19], where the initial data lies in the space

$\dot{H}^{\frac{1}{2}}\times\dot{H}^{\frac{1}{2}}\times\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3}).$

Note that the scaling of these spaces is at the critical level of the three-
dimensional Navier–Stokes equations (i.e., in the case $E\equiv B\equiv 0$).
Accordingly, we do not expect to maintain the uniqueness of the solution to
(1.1) if the initial velocity lies in spaces that scale below
$\dot{H}^{\frac{1}{2}}$.

Beside [19, 27], the three-dimensional Navier–Stokes–Maxwell system (1.1) was
studied in several papers, for instance [2, 41, 42, 43]. However, to the best
of our knowledge, none of the existing results provide enough information to
study the regime $c\rightarrow\infty$ in the three-dimensional case.

Thus, an important novelty of our paper is achieved by shedding light on the
regime $c\rightarrow\infty$ and extending the techniques recently developed by
the first and the third authors in [3, 4] to the three-dimensional
axisymmetric setting, which we introduce next.

###  1.1. The axisymmetric structure

Throughout this paper, we assume that the velocity and the electromagnetic
fields are axisymmetric, $u$ and $E$ are without swirl, whereas, $B$ has pure
swirl. That is to say

(1.5) $u(x)=u_{r}(r,z)e_{r}+u_{z}(r,z)e_{z},\quad
E(x)=E_{r}(r,z)e_{r}+E_{z}(r,z)e_{z},\quad B(x)=B_{\theta}(r,z)e_{\theta},$

where $(r,\theta,z)$ denotes the usual cylindrical coordinates and
$(e_{r},e_{\theta},e_{z})$ is the corresponding orthonormal basis (see Section
2 for the explicit definition of these variables). The careful reader may now
notice that this assumption shares some similarity with the two-dimensional
normal structure (1.4). Indeed, in (1.5), the magnetic field $B$ remains
orthogonal to $u$ and $E$.

This key observation will allow us to somewhat simplify the equations.
However, this structure will also require some extra attention when performing
estimates in Besov spaces, for the cylindrical basis
$(e_{r},e_{\theta},e_{z})$ depends on the position variables.

Let us now state some crucial properties provided by (1.5). First of all, note
that the vorticity

$\omega\overset{\rm def}{=}\operatorname{curl}u$

is acting only in the direction $e_{\theta}$, that is,

$\omega=\omega_{\theta}\,e_{\theta}\overset{\rm
def}{=}\big{(}\partial_{z}u_{r}-\partial_{r}u_{z}\big{)}e_{\theta}.$

On the other hand, straightforward computations show that

$\nabla\times(j\times B)=B\cdot\nabla j-j\cdot\nabla
B=j_{r}\frac{B}{r}-j\cdot\nabla B.$

Thus, taking the curl of the momentum equation in (1.1) yields

(1.6)
$\partial_{t}\omega_{\theta}+u\cdot\nabla\omega_{\theta}-\nu\big{(}\Delta-\frac{1}{r^{2}}\big{)}\omega_{\theta}=\frac{u_{r}}{r}\omega_{\theta}+j_{r}\frac{B_{\theta}}{r}-j\cdot\nabla
B_{\theta}.$

Then, by introducing

$\Omega\overset{\rm def}{=}\frac{\omega_{\theta}}{r},\qquad\Gamma\overset{\rm
def}{=}\frac{B_{\theta}}{r},$

we further obtain the equation

(1.7)
$\partial_{t}\Omega+u\cdot\nabla\Omega-\nu\big{(}\Delta+\frac{\partial_{r}}{r}\big{)}\Omega=j\cdot\nabla\Gamma.$

Note that the last equation is very similar to the vorticity equation of the
two-dimensional Euler–Maxwell system treated in [3].

An equivalent formulation of (1.6) and (1.7) is obtained by employing Ampère’s
equation to eliminate the current density $j$ and deduce that

(1.8)
$\partial_{t}\omega_{\theta}+u\cdot\nabla\omega_{\theta}-\nu\big{(}\Delta-\frac{1}{r^{2}}\big{)}\omega_{\theta}=\frac{u_{r}}{r}\omega_{\theta}-\partial_{z}\big{(}\Gamma
B_{\theta}\big{)}-\frac{1}{c}\partial_{t}E_{r}\frac{B_{\theta}}{r}+\frac{1}{c}\partial_{t}E\cdot\nabla
B_{\theta},$

and

(1.9)
$\partial_{t}\Omega+u\cdot\nabla\Omega-\nu\big{(}\Delta+\frac{\partial_{r}}{r}\big{)}\Omega=-\partial_{z}\big{(}\Gamma^{2}\big{)}-\frac{1}{c}\partial_{t}E\cdot\nabla\Gamma.$

Furthermore, by virtue of (1.5), combining Ampère and Faraday’s equations
yields

$\frac{1}{c^{2}}\partial_{tt}B+\partial_{t}B+u\cdot\nabla
B-\frac{1}{\sigma}\Delta B=\frac{u_{r}}{r}B,$

and therefore

$\frac{1}{c^{2}}\partial_{tt}\Gamma+\partial_{t}\Gamma+u\cdot\nabla\Gamma-\frac{1}{\sigma}\big{(}\Delta+\frac{\partial_{r}}{r}\big{)}\Gamma=0.$

The key observation behind the global estimates in the axisymmetric setting
when $E\equiv B\equiv 0$ is the fact that the stretching term
$\omega\cdot\nabla u$ is reduced to $\frac{u_{r}}{r}\omega$. As shown in [37],
this term can be controlled by the estimate111$L^{p,q}$ stands for the usual
Lorentz spaces.

(1.10)
$\left\|\frac{u_{r}}{r}\right\|_{L^{\infty}}\lesssim\left\|\Omega\right\|_{L^{3,1}}\lesssim\left\|\Omega\right\|_{L^{2}}^{\frac{1}{2}}\left\|\nabla\Omega\right\|_{L^{2}}^{\frac{1}{2}},$

which then yields a global bound on the velocity in sub-critical spaces.
Indeed, this is due to the fact that $\Omega$ obeys a transport–diffusion
equation (resp. transport equation) in the case of the Navier–Stokes equations
(resp. Euler equations), which can be used to deduce a global bound for
$\Omega$ in $L^{\infty}_{t}L^{2}\cap L^{2}_{t}\dot{H}^{1}$ (resp. in
$L^{\infty}_{t}L^{3,1}$), by means of standard energy estimates. Similar
results can be extended to the MHD equations (see Theorem 1.2, below).

We conclude by pointing out that the axisymmetric structure has been
extensively studied in a variety of fluid models, such as the Navier–Stokes
[11, 12, 33, 34, 35], Euler [15], Boussinesq [1, 16, 17, 22, 23, 25, 26] and
MHD [10, 28, 29, 31, 32, 39] systems with partial or full dissipation.

###  1.2. Aims and main results

We intend to show that the global well-posedness of the three dimensional
system (1.1) holds whenever $c$ is large enough. We shall prove our results
uniformly with respect to $c$ and then derive the MHD equations by studying
the singular limit $c\rightarrow\infty$. We refer to Section 2 for the
definition of all functional spaces.

######  Theorem 1.1 (Global well-posedness).

Let $(u_{0}^{c},E_{0}^{c},B_{0}^{c})_{c>0}$ be a family of divergence-free
axisymmetric initial data such that $u_{0}^{c}$ and $E_{0}^{c}$ are without
swirl, whereas $B_{0}^{c}$ has pure swirl. Assume further that

$(u_{0}^{c},E_{0}^{c},B_{0}^{c})\in\left(H^{1}\times H^{\frac{3}{2}}\times
H^{\frac{3}{2}}\right)(\mathbb{R}^{3}),\qquad
c^{-1}(E_{0}^{c},B_{0}^{c})\in\dot{B}^{\frac{5}{2}}_{2,1}(\mathbb{R}^{3}),$
$\Omega_{0}^{c}\overset{\rm def}{=}\frac{\omega_{0}^{c}}{r}\in
L^{2}(\mathbb{R}^{3}),\qquad j_{0}^{c}\overset{\rm
def}{=}\sigma(cE_{0}^{c}+P(u_{0}^{c}\times B_{0}^{c}))\in
H^{\frac{1}{2}}(\mathbb{R}^{3}),$

uniformly in $c>0$. Then, there is a constant $c_{0}>0$, depending only on the
size of the initial data, such that, for any $c\in(c_{0},\infty)$, there is a
unique global axisymmetric solution $(u^{c},E^{c},B^{c})$ of the three
dimensional Navier–Stokes–Maxwell equations (1.1), with $\nu>0$, such that
$u^{c}$ and $E^{c}$ are without swirl and $B^{c}$ has pure swirl. This
solution enjoys the bounds

$\begin{gathered}u^{c}\in L^{\infty}(\mathbb{R}^{+};H^{1}),\qquad\nabla
u^{c}\in L^{2}(\mathbb{R}^{+};H^{1}),\qquad\Omega^{c}\overset{\rm
def}{=}\frac{\omega_{\theta}^{c}}{r}\in L^{\infty}(\mathbb{R}^{+};L^{2})\cap
L^{2}(\mathbb{R}^{+};\dot{H}^{1}),\\\ (E^{c},B^{c})\in
L^{\infty}(\mathbb{R}^{+};H^{\frac{3}{2}}),\qquad c^{-1}(E^{c},B^{c})\in
L^{\infty}(\mathbb{R}^{+};\dot{B}^{\frac{5}{2}}_{2,1}),\\\ (E^{c},B^{c})\in
L^{2}(\mathbb{R}^{+};\dot{B}^{\frac{5}{2}}_{2,1}),\quad cE^{c}\in
L^{2}(\mathbb{R}^{+};H^{\frac{3}{2}}),\quad B^{c}\in
L^{2}(\mathbb{R}^{+};\dot{H}^{1}),\quad
j^{c}\in\bigcap_{p=2}^{\infty}L^{p}(\mathbb{R}^{+};H^{\frac{1}{2}}),\end{gathered}$

uniformly in $c\in(c_{0},\infty)$. If, moreover,

$(E_{0}^{c},B_{0}^{c})\in\dot{B}^{\frac{5}{2}}_{2,1}(\mathbb{R}^{3}),$

uniformly in $c>0$, then the bound

$(E^{c},B^{c})\in L^{\infty}(\mathbb{R}^{+};\dot{B}^{\frac{5}{2}}_{2,1})$

holds uniformly in $c\in(c_{0},\infty).$

The complete proof of Theorem 1.1 is given in Section 5.

######  Remark.

By setting $E\equiv B\equiv 0$ in the above theorem, observe that we recover a
classical global well-posedness result for the axisymmetric Navier–Stokes
equations (see, for instance, Theorem 10.4 in [30]).

We also wish to understand the singular regime $c\rightarrow\infty$ in (1.1).
The next corollary establishes a well-posedness result for the limiting system
(MHD) by considering the weak limit of solutions constructed in Theorem 1.1.

######  Corollary 1.2 (Weak singular limit).

Let $(u_{0},B_{0})$ be an axisymmetric divergence-free vector field, where
$u_{0}$ is without swirl and $B_{0}$ has pure swirl. Further assume that

$\displaystyle(u_{0},B_{0})\in\left(H^{1}\times
H^{\frac{3}{2}}\right)(\mathbb{R}^{3}),\quad\frac{\omega_{0}}{r}\in
L^{2}(\mathbb{R}^{3}).$

Then, there is a unique axisymmetric solution to (MHD), with $\nu>0$ and
initial data $(u_{0},B_{0})$, enjoying the bounds

$\displaystyle(u,B)\in
L^{\infty}\left(\mathbb{R}^{+};H^{1}(\mathbb{R}^{3})\times
H^{\frac{3}{2}}(\mathbb{R}^{3})\right),\quad(\nabla u,\nabla B)\in
L^{2}\left(\mathbb{R}^{+};H^{1}(\mathbb{R}^{3})\times
B_{2,1}^{\frac{3}{2}}(\mathbb{R}^{3})\right),$

and

$\displaystyle\frac{\omega_{\theta}}{r}\in
L^{\infty}(\mathbb{R}^{+};L^{2}(\mathbb{R}^{3}))\cap
L^{2}(\mathbb{R}^{+};\dot{H}^{1}(\mathbb{R}^{3})).$

###### Proof.

We build global solutions by making use of the bounds from the preceding
theorem and employing standard compactness methods. To that end, let
$(u_{0}^{c_{n}},E_{0}^{c_{n}},B_{0}^{c_{n}})_{n\in\mathbb{N}}$ be any family
of divergence-free vector fields satisfying the assumptions of Theorem 1.1 and
converging to $(u_{0},0,B_{0})$, at least in the sense of distributions. In
particular, by Theorem 1.1, there exists a unique family
$(u^{c_{n}},E^{c_{n}},B^{c_{n}})_{n\in\mathbb{N}}$ of solutions to (1.1),
which obeys the bounds

$\displaystyle(u^{c_{n}},B^{c_{n}})\in
L^{\infty}\left(\mathbb{R}^{+};H^{1}(\mathbb{R}^{3})\times
H^{\frac{3}{2}}(\mathbb{R}^{3})\right),\quad(\nabla u^{c_{n}},\nabla
B^{c_{n}})\in L^{2}\left(\mathbb{R}^{+};H^{1}(\mathbb{R}^{3})\times
B_{2,1}^{\frac{3}{2}}(\mathbb{R}^{3})\right),$

and

$\displaystyle\frac{\omega_{\theta}^{c_{n}}}{r}\in
L^{\infty}(\mathbb{R}^{+};L^{2}(\mathbb{R}^{3}))\cap
L^{2}(\mathbb{R}^{+};\dot{H}^{1}(\mathbb{R}^{3})),\qquad c_{n}E^{c_{n}}\in
L^{2}(\mathbb{R}^{+};H^{\frac{3}{2}}),$

uniformly in $n\in\mathbb{N}$.

Thus, by the Banach–Alaoglu theorem, up to extraction of a subsequence (which
is not distinguished, for simplicity), it holds that

$(u^{c_{n}},E^{c_{n}},B^{c_{n}})\overset{n\rightarrow\infty}{\longrightarrow}(u,0,B),\quad\text{
in }\,\mathcal{D}^{\prime}(\mathbb{R}^{+}\times\mathbb{R}^{3}),$

where $(u,B)$ is in the same functional spaces as $(u^{c_{n}},B^{c_{n}})$.

In fact, one can also show that
$(\partial_{t}u^{c_{n}},\partial_{t}B^{c_{n}})_{n\in\mathbb{N}}$ is uniformly
bounded in $L^{2}_{t,x,\mathrm{loc}}$, which implies, by a classical
compactness result by Aubin and Lions (see [38] for a thorough discussion of
such compactness results and, in particular, Section 9 therein, for convenient
results which are easily applicable to our setting), that
$(u^{c_{n}},B^{c_{n}})_{n\in\mathbb{N}}$ is relatively compact in
$L^{2}_{t,x,\mathrm{loc}}$.

Therefore, by taking the limit in (1.1) in the sense of distributions, and
exploiting the strong compactness of $(u^{c_{n}},B^{c_{n}})_{n\in\mathbb{N}}$
to show the weak stability of nonlinear terms, it is readily seen that $(u,B)$
is a weak solution of (MHD), thereby completing the existence proof.

As for the uniqueness of solutions, we only need to note that it follows
directly from weak–strong uniqueness principles for fluid dynamical models
(for instance, see [6, Section 3.2.3] and [18]). ∎

######  Remark.

The solution constructed in the previous corollary enjoys the refined bound

(1.11)
$B\in\widetilde{L}^{\infty}\big{(}\mathbb{R}^{+};\dot{B}^{\frac{3}{2}}_{2,2}(\mathbb{R}^{3})\big{)}.$

Indeed, by virtue of the bounds from Corollary 1.2, one can show that
$u\cdot\nabla B$ and $B\cdot\nabla u$ belong to
$L^{2}(\mathbb{R}^{+};\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3}))$. Therefore,
standard parabolic regularity estimates applied directly to the heat equation
satisfied by $B$ in (MHD) give the bound above. In particular, this bound
guarantees that

(1.12)
$\lim_{n\rightarrow\infty}\sup_{t\in[0,\infty)}\|\mathds{1}_{\\{|D|\geq\Theta_{n}\\}}B(t)\|_{\dot{H}^{\frac{3}{2}}}=0,$

for any sequence $(\Theta_{n})_{n\geq 1}$ satisfying

$\lim_{n\rightarrow\infty}\Theta_{n}=\infty.$

Further note that the weaker bound

$B\in
L^{\infty}\big{(}\mathbb{R}^{+};\dot{H}^{\frac{3}{2}}(\mathbb{R}^{3})\big{)}$

would not be enough to establish (1.12). This will be important, later on, in
the proofs of our main results.

We state now our second main theorem. It concerns the strong convergence, as
$c\rightarrow\infty$, of the solution given in Theorem 1.1 toward the solution
constructed in Corollary 1.2.

######  Theorem 1.3 (Strong singular limit).

Let $(u_{0}^{c},E_{0}^{c},B_{0}^{c})_{c>0}$ be a family of initial data
satisfying the assumptions in Theorem 1.1 and denote by
$(u^{c},E^{c},B^{c})_{c>c_{0}}$ the corresponding unique global solution to
(1.1) given by the same theorem. Further consider divergence-free vector
fields $(u_{0},B_{0})$ such that

$(u_{0},B_{0})\in\left(H^{1}\times
H^{\frac{3}{2}}\right)(\mathbb{R}^{3}),\quad\frac{\omega_{0}}{r}\in
L^{2}(\mathbb{R}^{3}),$

and assume that

$\lim_{c\rightarrow\infty}\left\|(u_{0}^{c},B_{0}^{c})-(u_{0},B_{0})\right\|_{H^{1}\times
L^{2}}=0.$

Finally, let $(u,B)$ be the unique global solution of (MHD), given in
Corollary 1.2, associated to the data $(u_{0},B_{0})$.

Then $(u^{c},B^{c})$ converges strongly to $(u,B)$, as $c\rightarrow\infty$.
More specifically, it holds that

(1.13)
$\displaystyle\lim_{c\rightarrow\infty}\left(\sup_{t\in[0,\infty)}\|(u^{c}-u)(t)\|_{\dot{H}^{1}}+\int_{0}^{\infty}\|(u^{c}-u)(t)\|_{\dot{H}^{2}}^{2}dt\right)=0,$

and, for all $s\in[0,\frac{3}{2})$,

(1.14)
$\displaystyle\lim_{c\rightarrow\infty}\left(\sup_{t\in[0,\infty)}\|(B^{c}-B)(t)\|_{\dot{H}^{s}}+\int_{0}^{\infty}\|(B^{c}-B)(t)\|_{\dot{H}^{s+1}}^{2}dt\right)=0.$

If, moreover, we assume that

$\lim_{c\rightarrow\infty}\left\|(E_{0}^{c},B_{0}^{c})-(0,B_{0})\right\|_{\dot{H}^{\frac{3}{2}}}=0$

and

(1.15)
$\lim_{c\rightarrow\infty}\left(c^{-1}\|(E^{c}_{0},B^{c}_{0})\|_{\dot{B}^{\frac{5}{2}}_{2,1}}\right)=0,$

then (1.14) holds for $s=\frac{3}{2}$, as well.

The complete proof of Theorem 1.3 is given in Section 6.

######  Remark.

It is possible to quantify (1.13) and (1.14) with a rate of convergence
$O(c^{-\alpha})$, for some $\alpha>0$, provided that the initial data satisfy
a similar algebraic rate of convergence.

######  Remark.

Observe that the initial data in Theorem 1.1 are required to enjoy the
regularity $\dot{B}^{\frac{5}{2}}_{2,1}$. However, this is not needed in the
limiting system obtained in the regime $c\rightarrow\infty$, as reflected in
Corollary 1.2.

Moreover, the growth of the global solution from Theorem 1.1 in that space is
of order $c$, which is consistent with the fact that it is uniformly bounded
in the space $\dot{H}^{\frac{3}{2}}$. Indeed, heuristically, we notice from
our proofs, later on, that each spatial derivative has the same dimension as
the speed of light $c$, i.e.,

$c^{-1}\dot{B}^{\frac{5}{2}}_{2,1}\sim\dot{H}^{\frac{3}{2}},$

which means that the solutions have a comparable size in each space. This is
natural in view of the fact that waves produced by Maxwell’s system have a
characteristic speed $c$.

In conclusion, we supplement our remark with a typical example of data which
summarizes the preceding observations and fulfills the assumptions of Theorem
1.1 and Theorem 1.3. To that end, let $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{3})$ and introduce the standard mollifier

$\varphi_{c}(\cdot)\overset{\rm def}{=}c^{3}\varphi(c\,\cdot).$

Let $(u_{0},B_{0})$ be the initial data from Corollary 1.2 and $E_{0}$ be any
divergence-free profile in $H^{\frac{3}{2}}.$ Then the sequence of data
defined by

(1.16) $(u_{0}^{c},E_{0}^{c},B_{0}^{c})\overset{\rm
def}{=}\varphi_{c}*(u_{0},0,B_{0})$

is suitable and satisfies all the assumptions in Theorem 1.1 and Theorem 1.3,
as long as $c>c_{0}$, where $c_{0}>0$ only depends on the size of $u_{0}$,
$E_{0}$, $B_{0}$ and $\varphi.$

###  1.3. Challenges and ingredients of proof

Here, we provide the reader with a short roadmap that sheds light on our
strategy and main ingredients towards proving our results. We already made it
clear in the introduction that the construction of a (unique global) solution
to (1.1) at the level of the energy (1.2) is an outstanding open problem.
Hence, we seek solutions in higher-regularity spaces.

Notice that one cannot expect to have a better understanding of the Cauchy
problem for (1.1) compared to the Navier–Stokes equations (i.e., the case
$E\equiv B\equiv 0$). In the three-dimensional case, one of the well known
settings where the Navier–Stokes equations are globally well-posed is when the
initial data obey an axisymmetric geometric condition, which, roughly
speaking, reduces the dimension from three to two, in some sense. In this
work, we therefore make the choice of restricting ourselves to the case of
axisymmetric data.

We first clarify the key idea leading to an estimate of the velocity field,
which can be performed on (1.6)–(1.7) or (1.8)–(1.9). We argue now that it is
better to work on (1.8)–(1.9). The reasons behind this choice can be
summarized in two crucial points. The first one is that the terms containing
$\frac{1}{c}\partial_{t}E$ can be seen as an error that should vanish and be
discarded from the system when $c\rightarrow\infty$, at least formally.
Observe that this formal limit yields exactly the equations for the vorticity
in the axisymmetric case, which have been studied in several papers (for
instance, see [24]). Accordingly, we believe that (1.8)–(1.9) are more
suitable to study the equations in the regime $c\rightarrow\infty$. We shall
come back to this limit, later on, to comment on the formal claim

(1.17) $\lim_{c\rightarrow\infty}\frac{1}{c}\partial_{t}E=0.$

The second reason why we choose to consider (1.8)–(1.9) is more technical and
it relies on the fact that the $L^{2}$ energy estimate (1.2) provides us with
weak information in dimension three, unlike the two dimensional case. Indeed,
our alternative choice would be to consider (1.6)–(1.7) and to estimate $j$ in
$L^{2}_{t,x}$, which is the only global bound on $j$ that can be extracted
from the energy estimate. By doing so, one can can only establish an estimate
of $(\omega,\Omega)$ in $L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}\dot{H}^{1}_{x}$, which is linear in terms of the
$L^{2}_{t}B^{\frac{5}{2}}_{2,1}$ norm of $B$. This is comparable to the
techniques from [3] where, in two dimensions, it is shown that such estimates
can be closed. However, in three dimensions, if one goes into the details of
low-frequency estimates, one notices that the interpolation argument used in
[3] to weaken the power of $B$ in some crucial norms will likely not work. It
seems that this method would require a global control of $j$ in
$L^{2}_{t}L^{3}_{x}$, which is not available.

Thus, our main estimate on the velocity field is based on (1.8)–(1.9) and is
given in Proposition 3.1, below, where we show that

$\displaystyle\left\|(\omega,\Omega)\right\|_{L^{\infty}_{t}L^{2}\cap
L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\qquad\lesssim\left(\left\|(\omega_{0},\Omega_{0})\right\|_{L^{2}}+\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}\left\|B\right\|_{L^{\infty}_{t}H^{2}}+\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}\left\|(B,\Gamma)\right\|_{L^{2}_{t}\dot{H}^{1}}\right)\exp\big{(}C\mathcal{E}^{2}_{0}\big{)},$

for some $C>0$. Now, in order to use this bound, we need the following
ingredients:

  1. (1)

An asymptotically vanishing estimate for $\frac{1}{c}\partial_{t}E$ of the
form

$\frac{1}{c}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}\lesssim
c^{-\alpha}F\Big{(}\left\|(u,E,B)\right\|_{X}\Big{)},$

for some $\alpha>\frac{1}{2}$, some (nonlinear) function $F$ and a suitable
functional space $X$.

  2. (2)

A bound of the type

$\left\|B\right\|_{L^{\infty}_{t}H^{2}}\lesssim
c^{\frac{1}{2}}F\Big{(}\left\|(u,E,B)\right\|_{X}\Big{)}.$

  3. (3)

An asymptotically global estimate for $B$ in $L^{2}_{t}\dot{H}^{1}$ and
$\Gamma$ in $L^{\infty}_{t}L^{3}\cap L^{2}_{t}\dot{H}^{1}$ of the form

$\left\|B\right\|_{L^{2}_{t}\dot{H}^{1}}+\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}\cap
L^{2}_{t}\dot{H}^{1}}\leq
C_{0}+c^{-\beta}F\Big{(}\left\|(u,E,B)\right\|_{X}\Big{)},$

for some $\beta>0$ and $C_{0}>0$ depending only on the initial data.

The first bound above is inspired by the results from [4, Section 3], whereas
the third one is obtained in the spirit of [3, Section 3.6].

The precise proof of these claims is the subject of Section 4 where we will
build on the arguments from [3, 4]. For simplicity, the reader can think of
the space $X$ in the foregoing estimates as a combination of the spaces
appearing in the statement of Theorem 1.1. The precise construction of $X$ is
also detailed in Section 4.

The next step in our strategy is to study the Maxwell system

(1.18) $\begin{cases}\begin{aligned} \frac{1}{c}\partial_{t}E-\nabla\times
B+\sigma cE&=-\sigma P(u\times B),\\\ \frac{1}{c}\partial_{t}B+\nabla\times
E&=0,\\\
\operatorname{div}u=\operatorname{div}B=\operatorname{div}E&=0.\end{aligned}\end{cases}$

The relevant estimates for $(E,B)$ with a general forcing term have recently
been established in [3]. They are reproduced in Lemmas 3.2 and 3.3.
Nevertheless, some refinements in the analysis of (1.18) are required in order
to obtain adequate bounds for the electromagnetic field which are compatible
with the asymptotic behavior of $\frac{1}{c}\partial_{t}E$, $B$ and $\Gamma$.

As a starting point, we will be facing the problem of estimating products of
the form

$\left\|P(u\times B)\right\|_{B^{s}_{p,q}},$

with $s>\frac{3}{p}$, where the regularity of $u$ is restricted to
$L^{\infty}_{t}\dot{H}^{1}\cap L^{2}_{t}\dot{H}^{2}$. Note that a similar
issue was overcome in [3] by exploiting the two dimensional normal structure
(1.4), which is not valid in our context. Here, we show that a similar
analysis can be performed in the axisymmetric case, by using the fact that $B$
remains orthogonal to $E$ and $u$. Accordingly, in Lemma 2.3, we provide a
general result refining the classical paraproduct estimates in a framework
which covers the axisymmetric setting.

The ability to prove Theorem 1.1 further requires a precise understanding of
the damping phenomenon in (1.18), which is obtained by studying the
parabolic–hyperbolic properties of fields, their interaction and their
behavior relatively to $c$. This analysis is laid out in detail in Section
3.2.

With the above ingredients, we are then in a position to conclude Theorem 1.1.
Thus, in Section 5.3, we gather all the estimates to establish a nonlinear
bound of the form

$\left\|(u,E,B)\right\|_{X}\leq
C_{0}+\mathcal{P}\left(\left\|(u,E,B)\right\|_{X}\right),$

where $\mathcal{P}$ is a polynomial whose coefficients vanish as
$c\rightarrow\infty$. The conclusion of the global estimates is then a
straightforward application of the abstract Lemma 5.2 which ensures that

$\left\|(u,E,B)\right\|_{X}\leq 2C_{0},$

as long as $c$ is larger than some power of $C_{0}$.

Note that the proof of Theorem 1.1 is divided into two parts. The first one
deals with the case that we call rough profiles, in the spirit of the example
of initial data given in (1.16). In that case, the initial electromagnetic
field $(E_{0}^{c},B_{0}^{c})$ can have a $\dot{B}^{\frac{5}{2}}_{2,1}$ norm
growing at most like $c$. In the second part of the proof, in the case of
regular profiles, we show that if, moreover, the corresponding
$\dot{B}^{\frac{5}{2}}_{2,1}$ norm of $(E_{0}^{c},B_{0}^{c})$ does not blow up
as $c$ goes to infinity, then, the $\dot{B}^{\frac{5}{2}}_{2,1}$ norm of
$(E^{c}(t),B^{c}(t))$ remains bounded, for any $t\in[0,\infty)$. The second
case will be proved with a kind of bootstrap argument and it can be seen as
the result of persistence of initial regularity.

Our second main result (Theorem 1.3) establishes the convergence toward (MHD)
of the solution constructed in Theorem 1.1. It is to be emphasized that a
fundamental ingredient in the proof of Theorem 1.3 hinges upon the
understanding of the limit (1.17). This is crucial to obtain a convergence
result in the whole domain $[0,\infty)\times\mathbb{R}^{3}$.

The proof of Theorem 1.3 will be done in several steps. Firstly, we prove the
convergence in the $L^{2}$ energy space by performing adequate stability
estimates. Subsequently, by interpolation, it follows that the convergence of
velocities and magnetic fields holds in $L^{\infty}_{t}\dot{H}^{s}\cap
L^{2}_{t}\dot{H}^{s+1}$ and $L^{\infty}_{t}\dot{H}^{\frac{3s}{2}}\cap
L^{2}_{t}\dot{H}^{\frac{3s}{2}+1}$, respectively, for any $s\in[0,1)$.

Therefore, the convergence of velocity fields in the endpoint space
$L^{\infty}_{t}\dot{H}^{1}\cap L^{2}_{t}\dot{H}^{2}$ will be achieved by an
energy estimate in $\dot{H}^{1}$ and by making use of the stability results
from the previous step. Here, it is important to mention that the validity of
(1.17) in $L^{2}_{t}\dot{H}^{\frac{1}{2}}$ (see Proposition 4.1) is crucial.

On the other hand, the convergence of the magnetic fields in the endpoint
space $L^{\infty}_{t}\dot{H}^{\frac{3}{2}}\cap
L^{2}_{t}\dot{H}^{\frac{3}{2}+1}$ will be established by a different approach,
for a standard $\dot{H}^{\frac{3}{2}}$ energy estimate would require the
validity of (1.17) in $L^{2}_{t}\dot{H}^{\frac{3}{2}}$ which is not available
from Proposition 4.1. More precisely, the convergence of magnetic fields in
that space follows from extrapolation compactness techniques, introduced and
utilized by the first and third authors in [4].

In summary, the proof of convergence of magnetic fields in the energy space of
$\dot{H}^{\frac{3}{2}}$ is split into two main steps. In the first step, we
treat the convergence of frequencies that are localized in a ball whose radius
grows as the speed of light increases. By suitably choosing that radius, the
convergence of low frequencies then follows as a direct consequence of the
convergence in the $L^{2}$ energy space. In the second step, by exploiting the
assumption (1.15), we take care of the remaining high frequencies by building
on the refined analysis of Maxwell equations laid out in Section 3.2. The
combination of these ideas eventually leads to the completion of all proofs.

###  1.4. Notation

All definitions and basic properties of functional spaces utilized throughout
the paper are introduced in Section 2.

Furthermore, the letter $C$ will often denote a universal (possibly large)
constant that is independent of the variables of the problem, and which is
also allowed to change from one line to the next.

For simplicity, we will also be using $A\lesssim B$ instead of $A\leq CB.$
Moreover, when needed, in order to specify the dependence of some estimates on
some parameters, we will occasionally utilize $A\lesssim_{s}B$ to insist on
the fact that the generic constant $C$ might depend on a parameter $s$.

##  2\. The axisymmetric structure, Hardy inequalities and paradifferential
calculus

In this section, we establish several lemmas which shed light on crucial
features of the axisymmetric structure, which are similar to the properties of
the two-dimensional normal structure which is exploited in [3, 4]. In
particular, this structure will be employed to obtain useful improvements on
the classical paradifferential product laws and will serve in our main a
priori estimates, later on.

First of all, we recall that a vector field
$F:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is axisymmetric if it has the form

$F(x_{1},x_{2},x_{3})=F_{r}(r,z)e_{r}+F_{\theta}(r,z)e_{\theta}+F_{z}(r,z)e_{z},$

where the triple $(r,\theta,z)$ denotes the usual cylindrical coordinates
defined by the relations

$x_{1}=r\cos\theta,\quad x_{2}=r\sin\theta,\quad x_{3}=z,$

and $(e_{r},e_{\theta},e_{z})$ is the corresponding cylindrical orthonormal
basis

$e_{r}=\Big{(}\frac{x_{h}}{r},0\Big{)},\quad
e_{\theta}=\Big{(}\frac{{x_{h}^{\perp}}}{r},0\Big{)},\quad
e_{\theta}=(0,0,1).$

Here, the index $h$ is used to refer to the horizontal components

$x_{h}\overset{\rm def}{=}(x_{1},x_{2}),\quad x_{h}^{\perp}\overset{\rm
def}{=}(-x_{2},x_{1}).$

Thus, axisymmetry is characterized by the property that the components
$(F_{r},F_{\theta},F_{z})$ are independent of $\theta$. In other words, the
field $F$ is axisymmetric if and only if it satisfies $R\circ F=F\circ R$, for
all rotations $R$ around the $z$-axis.

Furthermore, we say that the vector field $F:\mathbb{R}^{3}\to\mathbb{R}^{3}$
is axisymmetric without swirl if it has the form

$F(x_{1},x_{2},x_{3})=F_{r}(r,z)e_{r}+F_{z}(r,z)e_{z},$

and that it is axisymmetric with pure swirl if it can be represented as

$F(x_{1},x_{2},x_{3})=F_{\theta}(r,z)e_{\theta}.$

Observe that an axisymmetric field with pure swirl is always divergence free,
which follows from a straightforward computation.

We take some time now to carefully introduce the functional spaces which we
use in this article and some related notation. To that end, we first consider
the Littlewood–Paley decomposition

$\sum_{k\in\mathbb{Z}}\Delta_{k}f=f$

of a tempered distribution modulo polynomials
$f\in\mathcal{S}^{\prime}/\mathcal{P}(\mathbb{R}^{d})$, in any dimension
$d\geq 1$, where the operator $\Delta_{k}$ is the classical frequency
truncation which restricts the support of the Fourier transform

$\mathcal{F}f\left(\xi\right)=\hat{f}(\xi)\overset{\rm
def}{=}\int_{\mathbb{R}^{d}}e^{-i\xi\cdot x}f(x)dx$

to the set $\\{2^{k-1}\leq|\xi|\leq 2^{k+1}\\}$, for each $k\in\mathbb{Z}$.
Recall that the space $\mathcal{S}^{\prime}/\mathcal{P}(\mathbb{R}^{d})$ is
isomorphic to the space $\mathcal{S}_{0}^{\prime}(\mathbb{R}^{d})$ of tempered
distribution restricted to the subspace $\mathcal{S}_{0}(\mathbb{R}^{d})$,
which is made up of all Schwartz functions
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$ such that

$\int_{\mathbb{R}^{d}}x^{\alpha}\varphi(x)dx=0,$

for every mutli-index $\alpha\in\mathbb{N}^{d}$. It is always possible to
construct $\Delta_{k}$ so that it acts on tempered distributions through a
convolution with a radial smooth function in
$\mathcal{S}_{0}(\mathbb{R}^{d})$, which is appropriately dilated by a factor
$2^{k}$.

Then, for any $s\in\mathbb{R}$ and $1\leq p,q,r\leq\infty$, we define the
homogeneous Besov space $\dot{B}^{s}_{p,q}\left(\mathbb{R}^{d}\right)$ and the
homogeneous Chemin–Lerner space
$\widetilde{L}^{r}\left([0,T);B^{s}_{p,q}\left(\mathbb{R}^{d}\right)\right)$,
with $T\in(0,\infty]$, as the subspaces of tempered distributions modulo
polynomials $\mathcal{S}^{\prime}/\mathcal{P}$ endowed with the respective
norms

$\displaystyle\left\|f\right\|_{\dot{B}^{s}_{p,q}\left(\mathbb{R}^{d}\right)}$
$\displaystyle\overset{\rm
def}{=}\left(\sum_{k\in\mathbb{Z}}2^{ksq}\left\|\Delta_{k}f\right\|_{L^{p}\left(\mathbb{R}^{d}\right)}^{q}\right)^{\frac{1}{q}},$
$\displaystyle\left\|f\right\|_{\widetilde{L}^{r}\left([0,T);B^{s}_{p,q}\left(\mathbb{R}^{d}\right)\right)}$
$\displaystyle\overset{\rm
def}{=}\left(\sum_{k\in\mathbb{Z}}2^{ksq}\left\|\Delta_{k}f\right\|_{L^{r}\left([0,T);L^{p}\left(\mathbb{R}^{d}\right)\right)}^{q}\right)^{\frac{1}{q}},$

if $q<\infty$, and with the usual modifications if $q=\infty$.

We will also employ the homogeneous Sobolev spaces
$\dot{W}^{s,p}\left(\mathbb{R}^{d}\right)\subset\mathcal{S}^{\prime}/\mathcal{P}$
which are defined by the semi-norms

$\left\|f\right\|_{\dot{W}^{s,p}\left(\mathbb{R}^{d}\right)}\overset{\rm
def}{=}\big{\|}|D|^{s}f\big{\|}_{L^{p}\left(\mathbb{R}^{d}\right)},$

where $|D|^{s}$ is the Fourier multiplier corresponding to the symbol
$|\xi|^{s}$, with $s\in\mathbb{R}$ and $1<p<\infty$. When $p=q=2$, note that
the Besov space $\dot{B}^{s}_{2,2}$ is equivalent to the homogeneous Sobolev
space classically denoted by

$\dot{H}^{s}\overset{\rm def}{=}\dot{W}^{s,2}.$

Nonhomogeneous versions of these spaces are also defined in a similar way.

Finally, employing truncated Littlewood–Paley decompositions, one can show
that the subspace $\mathcal{S}_{0}$ is dense in $\dot{B}^{s}_{p,q}$ and
$\dot{W}^{s,p}$, for any $s\in\mathbb{R}$ and $1\leq p,q<\infty$, and that a
similar statement holds for Chemin–Lerner spaces with suitable modifications.

We refer to [3, Appendix A] for some more details and properties of Besov and
Chemin–Lerner spaces in the same notation as in this article, and to [7, 21]
for a comprehensive introduction to the subject of Littlewood–Paley
decompositions and functional spaces.

We move on now to the main results of this section. Prior to presenting the
paradifferential features of the axisymmetric structure, we establish the
following version of Hardy’s inequality which will be useful in our analysis,
below.

######  Lemma 2.1.

Fix the dimension $d\geq 2$. Let $f$ be a smooth function satisfying, for all
$x^{\prime}\in\mathbb{R}^{d-1}$, that

$f(0,x^{\prime})=0.$

Then, it holds that

$\left\|\frac{f}{x_{1}}\right\|_{\dot{W}^{s,p}(\mathbb{R}^{d})}\lesssim_{s,p}\left\|f\right\|_{\dot{W}^{s+1,p}(\mathbb{R}^{d})},$

for any $p\in(1,\infty)$ and all $s>\frac{1}{p}-1$.

###### Proof.

By virtue of the vanishing assumption on $f$, we can write, for any
$(x_{1},x^{\prime})\in\mathbb{R}^{*}\times\mathbb{R}^{d-1}$, that

$\frac{f(x_{1},x^{\prime})}{x_{1}}=\int_{0}^{1}(\partial_{1}f)(\lambda
x_{1},x^{\prime})d\lambda.$

It then follows that

$\displaystyle\left\|\frac{f}{x_{1}}\right\|_{\dot{W}^{s,p}}$
$\displaystyle\leq\int_{0}^{1}\big{\|}|\partial_{1}f(\lambda\cdot,\cdot)\big{\|}_{\dot{W}^{s,p}}d\lambda$
$\displaystyle=\int_{0}^{1}\big{\|}|D|^{s}(\partial_{1}f)(\lambda\cdot,\cdot)\big{\|}_{L^{p}}d\lambda$
$\displaystyle=\int_{0}^{1}\lambda^{-\frac{1}{p}}\big{\|}|D|^{s}m_{\lambda,s}(D)(\partial_{1}f)\big{\|}_{L^{p}}d\lambda$
$\displaystyle=\int_{0}^{1}\lambda^{-\frac{1}{p}}\big{\|}m_{\lambda,s}(D)\partial_{1}f\big{\|}_{\dot{W}^{s,p}}d\lambda,$

where $m_{\lambda,s}(D)$ is the Fourier multiplier operator given, for any
$\lambda\in(0,1)$ and $s\in\mathbb{R}$, by

$m_{\lambda,s}(\xi)=\left(\frac{\sqrt{|\lambda\xi_{1}|^{2}+|\xi^{\prime}|^{2}}}{|\xi|}\right)^{s}.$

Now, we claim that the multiplier norm of $m_{\lambda,s}(D)$ over $L^{p}$ is
bounded by a constant multiple of $\max\\{1,\lambda^{s}\\}$, which leads to

$\displaystyle\left\|\frac{f}{x_{1}}\right\|_{\dot{W}^{s,p}}$
$\displaystyle\lesssim\int_{0}^{1}\lambda^{-\frac{1}{p}}\max\\{1,\lambda^{s}\\}d\lambda\left\|\partial_{1}f\right\|_{\dot{W}^{s,p}}$
$\displaystyle\lesssim\max\left\\{\frac{p}{p-1},\frac{1}{s+1-\frac{1}{p}}\right\\}\left\|f\right\|_{\dot{W}^{s+1,p}},$

provided that $s>\frac{1}{p}-1$, thereby establishing the main estimate of the
lemma.

In order to justify the boundedness of $m_{\lambda,s}(D)$ over $L^{p}$, it is
sufficient, by the Marcinkiewicz–Mikhlin multiplier theorem (see [20, Theorem
6.2.4]), to show that

(2.1)
$\sup_{\xi\in\mathbb{R}^{d}}\left|\xi^{\alpha}\partial_{\xi}^{\alpha}m_{\lambda,s}(\xi)\right|\lesssim\max\\{1,\lambda^{s}\\},$

for all multi-indices $\alpha\in\\{0,1\\}^{d}$. To that end, we introduce the
notation

$\ell_{i,\lambda}(\xi)\overset{\rm
def}{=}\frac{\xi_{i}^{2}}{\lambda^{2}|\xi_{1}|^{2}+|\xi^{\prime}|^{2}}$

and compute that

$\displaystyle\xi_{1}\partial_{\xi_{1}}m_{\lambda,s}(\xi)$
$\displaystyle=s\ell_{1,1}(\xi)\Big{(}\lambda^{2}m_{\lambda,s-2}(\xi)-m_{\lambda,s}(\xi)\Big{)},$
$\displaystyle\xi_{i}\partial_{\xi_{i}}m_{\lambda,s}(\xi)$
$\displaystyle=s\left(\ell_{i,\lambda}(\xi)-\ell_{i,1}(\xi)\right)m_{\lambda,s}(\xi),$
$\displaystyle\xi_{1}\partial_{\xi_{1}}\ell_{j,\lambda}(\xi)$
$\displaystyle=2\left(\delta_{1j}-\lambda^{2}\ell_{1,\lambda}(\xi)\right)\ell_{j,\lambda}(\xi),$
$\displaystyle\xi_{i}\partial_{\xi_{i}}\ell_{j,\lambda}(\xi)$
$\displaystyle=2\left(\delta_{ij}-\ell_{i,\lambda}(\xi)\right)\ell_{j,\lambda}(\xi),$

for any integers $i\in[2,d]$ and $j\in[1,d]$, where $\delta_{ij}$ denotes the
usual Kronecker delta. Therefore, iterating the preceding calculations and
observing that

$\left|m_{\lambda,s}(\xi)\right|+\left|\lambda^{2}m_{\lambda,s-2}(\xi)\right|\lesssim\max\\{1,\lambda^{s}\\},\qquad\left|\lambda^{2}\ell_{1,\lambda}\right|+\left|\ell_{i,\lambda}(\xi)\right|\lesssim
1,$

for all integers $i\in[2,d]$, it is readily seen that (2.1) holds true, which
establishes that the operator norm of $m_{\lambda,s}(D)$ is controlled by
$\max\\{1,\lambda^{s}\\}$. This completes the proof of the lemma. ∎

######  Remark.

Note that the assumption on the smoothness of the function $f$ in Lemma 2.1
can in fact be relaxed. Indeed, as described, for instance, in Theorem 6.6.1
from [8], the trace operator

$\mathrm{Tr}:\mathcal{S}_{0}(\mathbb{R}^{d})\rightarrow\mathcal{S}_{0}(\mathbb{R}^{d-1})$

defined by

$(\operatorname{Tr}f)(x_{1},x^{\prime})\overset{\rm
def}{=}f(0,x^{\prime}),\quad\text{where
}(x_{1},x^{\prime})\in\mathbb{R}\times\mathbb{R}^{d-1},$

has a well defined extension into a bounded operator from
$W^{s+1,p}(\mathbb{R}^{d})$ into
$B^{s+1-\frac{1}{p}}_{p,p}(\mathbb{R}^{d-1})$, for any $p\in(1,\infty)$, as
soon as the condition $s>\frac{1}{p}-1$ is satisfied. Hence, for $f\in
W^{s+1,p}(\mathbb{R}^{d})$, the condition $f(0,x^{\prime})\equiv 0$ in Lemma
2.1 can be replaced by $\operatorname{Tr}f=0.$

The next lemma is a variant of Lemma 2.1 for vector fields with an
axisymmetric structure. It will be employed to control specific quantities
involving electromagnetic fields, later on. The statement of the lemma below
is written in the notation introduced at the beginning of this section.

######  Lemma 2.2.

Let $E$ be an axisymmetric divergence-free vector field without swirl and $B$
be an axisymmetric vector field with pure swirl.

Then, it holds that

$\left\|\frac{\nabla\times
E}{r}\right\|_{L^{p}}\lesssim\left\|E\right\|_{\dot{W}^{2,p}}$

and

$\left\|\frac{B_{\theta}}{r}\right\|_{\dot{W}^{s,p}}\lesssim_{s}\left\|B\right\|_{\dot{W}^{s+1,p}},$

for any $p\in(1,\infty)$ and all $s>\frac{1}{p}-1$.

######  Remark.

For any given axisymmetric vector field $E=E_{r}e_{r}+E_{z}e_{z}$ with no
swirl, a direct computation gives that the curl $\nabla\times
E=(\partial_{z}E_{r}-\partial_{r}E_{z})e_{\theta}$ is axisymmetric with pure
swirl. Similarly, given an axisymmetric vector field $B=B_{\theta}e_{\theta}$
with pure swirl, another straightforward computation gives that the curl
$\nabla\times
B=-\partial_{z}B_{\theta}e_{r}+(\partial_{r}B_{\theta}+\frac{1}{r}B_{\theta})e_{z}$
is axisymmetric with no swirl.

###### Proof.

The bound on $B_{\theta}$ is a consequence of Lemma 2.1. To see this, let us
first assume that $B$ is smooth and has pure swirl. Then, we can write that

$(B_{1},B_{2},0)=B_{\theta}e_{\theta}=B_{\theta}\left(\frac{-x_{2}}{r},\frac{x_{1}}{r},0\right)$

to deduce that

$B_{1}|_{x_{2}=0}\equiv 0,\qquad B_{2}|_{x_{1}=0}\equiv 0,$

and

$\frac{B_{\theta}}{r}=-\frac{B_{1}}{x_{2}}=\frac{B_{2}}{x_{1}},$

as soon as $r\neq 0.$ Therefore, it is enough to estimate
$\frac{B_{1}}{x_{2}}$ or $\frac{B_{2}}{x_{1}}$.

Then, an application of Lemma 2.1 yields that

$\left\|\frac{B_{\theta}}{r}\right\|_{\dot{W}^{s,p}}=\left\|\frac{B_{2}}{x_{1}}\right\|_{\dot{W}^{s,p}}\lesssim_{s}\left\|B_{2}\right\|_{\dot{W}^{s+1,p}},$

thereby establishing the desired bound on $B_{\theta}$ in the case of a smooth
vector field. The general nonsmooth case is then obtained by a standard
approximation argument.

As for the bound on $E$, it will follow from a remarkable identity valid for
axisymmetric divergence-free vector fields with no swirl. Indeed, the fact
that $E$ is axisymmetric without swirl allows us to write that

$\displaystyle\nabla\times E$
$\displaystyle=\left(\partial_{z}E_{r}-\partial_{r}E_{z}\right)e_{\theta},$
$\displaystyle e_{r}\cdot\nabla\left(\nabla\times E\right)$
$\displaystyle=\left(\partial_{r}\partial_{z}E_{r}-\partial_{r}^{2}E_{z}\right)e_{\theta},$

which, when combined with the divergence-free condition for axisymmetric
fields

$\partial_{r}E_{r}+\frac{1}{r}E_{r}+\partial_{z}E_{z}=0$

leads to

$\displaystyle\frac{\nabla\times E}{r}+e_{r}\cdot\nabla\left(\nabla\times
E\right)$
$\displaystyle=\left(\frac{1}{r}\partial_{z}E_{r}-\frac{1}{r}\partial_{r}E_{z}+\partial_{r}\partial_{z}E_{r}-\partial_{r}^{2}E_{z}\right)e_{\theta}$
$\displaystyle=-\left(\partial_{r}^{2}E_{z}+\frac{1}{r}\partial_{r}E_{z}+\partial_{z}^{2}E_{z}\right)e_{\theta}.$

Then, identifying the action of the Laplacian on axisymmetric functions to
deduce that

$\Delta
E_{z}=\partial_{r}^{2}E_{z}+\frac{1}{r}\partial_{r}E_{z}+\partial_{z}^{2}E_{z},$

we arrive at the expression

$\frac{\nabla\times E}{r}=-e_{r}\cdot\nabla\left(\nabla\times E\right)-(\Delta
E_{z})e_{\theta}.$

The bound on $E$ therefore follows from a direct estimate in $L^{p}$ on the
preceding identity, which completes the proof of the lemma. ∎

We conclude this section with a lemma that extends the range of parameters in
the classical paraproduct laws by exploiting a geometric condition which is
satisfied by the axisymmetric structure. These extended paradifferential
estimates will be employed to obtain important a priori estimates for the
Navier–Stokes–Maxwell system (1.1), later on.

######  Lemma 2.3.

Let $F,G:[0,T)\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ be such that
$\operatorname{div}F=0$ and

(2.2) $\int_{\mathbb{R}^{3}}\varphi(x-y)\nabla\times
F(t,y)dy\quad\text{and}\quad\int_{\mathbb{R}^{3}}\psi(x-y)G(t,y)dy\quad\text{are
colinear,}$

for all $t\in[0,T)$, $x\in\mathbb{R}^{3}$, and any radially symmetric
$\varphi,\psi\in\mathcal{S}_{0}(\mathbb{R}^{3})$ (i.e., such that $\varphi(x)$
and $\psi(x)$ only depend on $|x|$). Further consider parameters in
$[1,\infty]$ such that

$\frac{1}{a}=\frac{1}{a_{1}}+\frac{1}{a_{2}}\qquad\text{and}\qquad\frac{1}{c}=\frac{1}{c_{1}}+\frac{1}{c_{2}}.$

Then, recalling that $P=(-\Delta)^{-1}\operatorname{curl}\mathrm{curl}$
denotes Leray’s projector onto solenoidal vector fields, one has the product
estimate

(2.3) $\left\|P(F\times
G)\right\|_{\widetilde{L}^{a}([0,T);\dot{B}^{s+\eta-\frac{3}{2}}_{2,c})}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}([0,T);\dot{B}^{s}_{2,c_{1}})}\left\|G\right\|_{\widetilde{L}^{a_{2}}([0,T);\dot{B}^{\eta}_{2,c_{2}})},$

for any $s\in(-\infty,\frac{3}{2})$ and $\eta\in(-\infty,\frac{5}{2})$ with
$s+\eta>0$.

Furthermore, in the endpoint case $s=\frac{3}{2}$, one has that

(2.4) $\left\|P(F\times
G)\right\|_{\widetilde{L}^{a}([0,T);\dot{B}^{\eta}_{2,c})}\lesssim\left\|F\right\|_{L^{a_{1}}([0,T);L^{\infty})\cap\widetilde{L}^{a_{1}}([0,T);\dot{B}^{\frac{3}{2}}_{2,\infty})}\left\|G\right\|_{\widetilde{L}^{a_{2}}([0,T)\dot{B}^{\eta}_{2,c})},$

for any $\eta\in(-\frac{3}{2},\frac{5}{2})$.

Finally, in the case $\eta=\frac{5}{2}$, it holds that

(2.5) $\left\|P(F\times
G)\right\|_{\widetilde{L}^{a}([0,T);\dot{B}^{s+1}_{2,c})}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}([0,T);\dot{B}^{s}_{2,c})}\left\|G\right\|_{\widetilde{L}^{a_{2}}([0,T);\dot{B}^{\frac{5}{2}}_{2,1})},$

as soon as $s\in(-\frac{5}{2},\frac{3}{2})$, and the case $s=\frac{3}{2}$ is
allowed provided that $c=1$.

######  Remark.

The preceding result also holds for vector fields which are independent of
time in classical Besov spaces (without any norm in time). Accordingly, the
lemma also holds in the case of Besov-space-valued Lebesgue spaces. This means
that removing the tildes in (2.3), (2.4) and (2.5) produces valid estimates.

######  Remark.

Consider an axisymmetric field $H:\mathbb{R}^{3}\to\mathbb{R}^{3}$. For any
radially symmetric $\varphi\in\mathcal{S}_{0}$ and any rotation $R$ around the
$z$-axis, we find that

$\varphi*H(Rx)=\int_{\mathbb{R}^{3}}\varphi(R(x-y))H(Ry)dy=\int_{\mathbb{R}^{3}}\varphi(x-y)RH(y)dy=R\varphi*H(x),$

thereby showing that $\varphi*H$ is axisymmetric, too. If, furthermore, the
field $H$ has pure swirl, then, employing that $S_{x}(y)\overset{\rm
def}{=}y-2(y\cdot e_{\theta}(x))e_{\theta}(x)$, for any $x\neq 0$, is an
isometry and satisfies that

$\frac{1}{2}\big{(}e_{\theta}(y)+e_{\theta}(S_{x}y)\big{)}=e_{\theta}(y)-\big{(}e_{\theta}(y)\cdot
e_{r}(x)\big{)}e_{r}(x)=\big{(}e_{\theta}(y)\cdot
e_{\theta}(x)\big{)}e_{\theta}(x),$

we compute that

$\displaystyle\int_{\mathbb{R}^{3}}\varphi(x-y)H_{\theta}(y)e_{\theta}(y)dy$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}\big{(}\varphi(x-y)H_{\theta}(y)e_{\theta}(y)+\varphi(x-S_{x}y)H_{\theta}(S_{x}y)e_{\theta}(S_{x}y)\big{)}dy$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}\varphi(x-y)H_{\theta}(y)\big{(}e_{\theta}(y)+e_{\theta}(S_{x}y)\big{)}dy$
$\displaystyle=\left(\int_{\mathbb{R}^{3}}\varphi(x-y)H_{\theta}(y)\big{(}e_{\theta}(y)\cdot
e_{\theta}(x)\big{)}dy\right)e_{\theta}(x),$

which establishes that $\varphi*H$ has pure swirl, as well.

######  Remark.

The hypothesis (2.2) is satisfied by axisymmetric divergence-free vector
fields such that $F$ has no swirl and $G$ has pure swirl. Indeed, as
previously emphasized, the curl of an axisymmetric vector field with no swirl
is axisymmetric with pure swirl. Therefore, in this situation, it holds that
$\nabla\times F$ and $G$ both have pure swirls, which implies, according to
the preceding remark, that their convolutions with radial functions remain
axisymmetric with pure swirl and are thus colinear.

######  Remark.

Note that Lemma 2.3 above is an extension of [3, Lemma 3.4] to the three-
dimensional setting. In particular, its significance lies in the fact that it
allows us to cover the range of parameters $\eta\in[\frac{3}{2},\frac{5}{2})$.
Indeed, without the geometric assumption (2.2) on $F$ and $G$, the
paradifferential estimates remain valid but may need to be restricted to
parameters satisfying $\eta<\frac{3}{2}$.

###### Proof.

We follow the method of proof of Lemma 3.4 from [3] and write Bony’s
decomposition

$F\times G=T_{F}G-T_{G}F+R(F,G),$

where the paraproducts are defined by

$T_{F}G=\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j-k<-2\end{subarray}}\Delta_{j}F\times\Delta_{k}G,\qquad
T_{G}F=\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j-k<-2\end{subarray}}\Delta_{j}G\times\Delta_{k}F=-\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j-k>2\end{subarray}}\Delta_{j}F\times\Delta_{k}G,$

and the remainder is given by

$R(F,G)=\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\ |j-k|\leq
2\end{subarray}}\Delta_{j}F\times\Delta_{k}G,$

to deduce that a direct application of classical paraproduct estimates on
homogeneous Besov spaces (see [3, Appendix A] or [7], for instance), combined
with the fact that $P$ is bounded over Besov spaces, yields the validity of
(2.3) for parameters $s\in(-\infty,\frac{3}{2})$ and
$\eta\in(-\infty,\frac{3}{2})$ with $s+\eta>0$, and the validity of (2.4) for
parameters $\eta\in(-\frac{3}{2},\frac{3}{2})$.

It is important to emphasize here that the restriction $\eta<\frac{3}{2}$
comes solely from the estimate of $T_{G}F$. Thus, in order to establish the
validity of (2.3) and (2.4) for the full range of parameters, we only need to
show now that

(2.6)
$\left\|P(T_{G}F)\right\|_{\widetilde{L}^{a}([0,T);\dot{B}^{s+\eta-\frac{3}{2}}_{2,c})}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}([0,T);\dot{B}^{s}_{2,c_{1}})}\left\|G\right\|_{\widetilde{L}^{a_{2}}([0,T);\dot{B}^{\eta}_{2,c_{2}})},$

for any $s\in\mathbb{R}$ and $\eta\in(-\infty,\frac{5}{2})$, as a consequence
of the divergence-free structure of $F$ and the geometric assumption (2.2).

To that end, assuming first that $F$ and $G$ are smooth, we compute that

$\nabla\times(F\times G)=\nabla(F\cdot G)-G\times(\nabla\times
F)-F\times(\nabla\times G)-2F\cdot\nabla
G+F\operatorname{div}G-G\operatorname{div}F.$

Thus, if $\nabla\times F$ and $G$ are colinear and $F$ is divergence free, we
arrive at

(2.7) $P(F\times G)=(-\Delta)^{-1}\nabla\times\big{(}-F\times(\nabla\times
G)-2F\cdot\nabla G+F\operatorname{div}G\big{)}.$

Now, applying the same reasoning to $\Delta_{k}F$ and $\Delta_{j}G$, instead
of $F$ and $G$, and utilizing (2.2) to deduce that $\nabla\times\Delta_{k}F$
and $\Delta_{j}G$ are colinear, we obtain that

(2.8)
$P(T_{G}F)=(-\Delta)^{-1}\nabla\times\Bigg{(}\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j-k<-2\end{subarray}}\Big{(}\Delta_{k}F\times(\nabla\times\Delta_{j}G)+2\Delta_{k}F\cdot\nabla\Delta_{j}G-\Delta_{k}F\operatorname{div}\Delta_{j}G\Big{)}\Bigg{)}.$

Therefore, applying classical paraproduct estimates (see [3, Appendix A] or
[7]), we conclude, for any $s\in\mathbb{R}$ and $\eta<\frac{5}{2}$, that

$\displaystyle\left\|P(T_{G}F)\right\|_{\widetilde{L}^{a}\dot{B}^{s+\eta-\frac{3}{2}}_{2,c}}$
$\displaystyle\qquad\lesssim\Bigg{\|}\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j-k<-2\end{subarray}}\Big{(}\Delta_{k}F\times(\nabla\times\Delta_{j}G)+2\Delta_{k}F\cdot\nabla\Delta_{j}G-\Delta_{k}F\operatorname{div}\Delta_{j}G\Big{)}\Bigg{\|}_{\widetilde{L}^{a}\dot{B}^{s+\eta-\frac{5}{2}}_{2,c}}$
$\displaystyle\qquad\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c_{1}}}\left\|\nabla
G\right\|_{\widetilde{L}^{a_{2}}\dot{B}^{\eta-\frac{5}{2}}_{\infty,c_{2}}}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c_{1}}}\left\|\nabla
G\right\|_{\widetilde{L}^{a_{2}}\dot{B}^{\eta-1}_{2,c_{2}}}$
$\displaystyle\qquad\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c_{1}}}\left\|G\right\|_{\widetilde{L}^{a_{2}}\dot{B}^{\eta}_{2,c_{2}}},$

which establishes (2.6), thereby completing the proof of (2.3) and (2.4).

The justification of (2.5) is similar. Indeed, the classical paradifferential
estimates apply directly to the paraproduct $T_{F}G$ and the remainder
$R(F,G)$ in the range of parameters desribed in (2.5). Thus, we see that (2.5)
will follow from the justification of the paraproduct estimate

(2.9)
$\left\|P(T_{G}F)\right\|_{\widetilde{L}^{a}([0,T);\dot{B}^{s+1}_{2,c})}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}([0,T);\dot{B}^{s}_{2,c})}\left\|G\right\|_{\widetilde{L}^{a_{2}}([0,T);\dot{B}^{\frac{5}{2}}_{2,1})},$

for any $s\in\mathbb{R}$. As before, in order to prove (2.9), we apply
classical paraproduct estimates to (2.8). This leads to

$\displaystyle\left\|P(T_{G}F)\right\|_{\widetilde{L}^{a}\dot{B}^{s+1}_{2,c}}$
$\displaystyle\lesssim\Bigg{\|}\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j-k<-2\end{subarray}}\Big{(}\Delta_{k}F\times(\nabla\times\Delta_{j}G)+2\Delta_{k}F\cdot\nabla\Delta_{j}G-\Delta_{k}F\operatorname{div}\Delta_{j}G\Big{)}\Bigg{\|}_{\widetilde{L}^{a}\dot{B}^{s}_{2,c}}$
$\displaystyle\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c}}\left\|\nabla
G\right\|_{\widetilde{L}^{a_{2}}L^{\infty}}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c}}\left\|\nabla
G\right\|_{\widetilde{L}^{a_{2}}\dot{B}^{0}_{\infty,1}}$
$\displaystyle\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c}}\left\|G\right\|_{\widetilde{L}^{a_{2}}\dot{B}^{1}_{\infty,1}}\lesssim\left\|F\right\|_{\widetilde{L}^{a_{1}}\dot{B}^{s}_{2,c}}\left\|G\right\|_{\widetilde{L}^{a_{2}}\dot{B}^{\frac{5}{2}}_{2,1}},$

for all $s\in\mathbb{R}$, which establishes the validity of (2.5) and
concludes of the proof of the lemma. ∎

######  Remark.

If, instead of (2.2), one merely assumes that $\nabla\times F$ and $G$ are
colinear, in the sense that $G\times(\nabla\times F)=0$, then the
paradifferential estimate (2.3) remains valid for any parameters
$s\in(-\infty,\frac{3}{2})$ and $\eta\in(-\infty,\frac{5}{2})$ in the range
$s+\eta>1$. This follows from applying Bony’s decomposition to each term of
(2.7) and then estimating the resulting paraproducts and remainders as in the
proof above. The restriction $s+\eta>1$ is then a consequence of the estimates
of the remainders. Similarly, under the assumption that $\nabla\times F$ and
$G$ are colinear, one can show that (2.4) and (2.5) remain valid in the
respective ranges $\eta\in(-\frac{1}{2},\frac{5}{2})$ and
$s\in(-\frac{3}{2},\frac{3}{2})$.

##  3\. A priori estimates

Here, we establish a priori estimates on smooth solutions of the
Navier–Stokes–Maxwell system (1.1). The ensuing bounds will be employed to
prove existence of global solutions to that system. For simplicity, from now
on, we take $\nu=1$ in (1.1). However, we emphasize that all estimates below
hold for any $\nu>0$.

We recall first that the only available global bound for smooth solutions of
(1.1) corresponds to the $L^{2}$-energy estimate

(3.1) $\left\|u\right\|_{L^{\infty}_{t}L^{2}\cap
L^{2}_{t}\dot{H}^{1}}+\left\|(E,B)\right\|_{L^{\infty}_{t}L^{2}}+\left\|j\right\|_{L^{2}_{t}L^{2}}\lesssim\mathcal{E}_{0}\overset{\rm
def}{=}\left\|(u_{0},E_{0},B_{0})\right\|_{L^{2}}.$

As explained in the introduction, this bound does not seem to be enough to
construct any kind of solutions. Therefore, we shall aim to propagate some
higher regularity for $u$, $E$ and $B$.

In what follows, we recall that are using the notation

$\Omega\overset{\rm
def}{=}\frac{\omega_{\theta}}{r}\qquad\text{and}\qquad\Gamma\overset{\rm
def}{=}\frac{B_{\theta}}{r},$

where

$\omega_{\theta}=\omega\cdot e_{\theta}=\left(\nabla\times u\right)\cdot
e_{\theta}\qquad\text{and}\qquad B_{\theta}=B\cdot e_{\theta}.$

###  3.1. Controlling the velocity field

In order to control higher regularities for the velocity field, we shall
exploit the axisymmetric structure and perform an energy estimate on the
equations describing the evolution of $\omega_{\theta}$ and $\Omega$, which we
derived in Section 1.1. For convenience, we recall here that the equation for
$\omega_{\theta}$ can be written as

(3.2)
$\partial_{t}\omega_{\theta}+u\cdot\nabla\omega_{\theta}-\big{(}\Delta-\frac{1}{r^{2}}\big{)}\omega_{\theta}=\frac{u_{r}}{r}\omega_{\theta}-\partial_{z}\big{(}\Gamma
B_{\theta}\big{)}-\frac{1}{c}\partial_{t}E_{r}\frac{B_{\theta}}{r}+\frac{1}{c}\partial_{t}E\cdot\nabla
B_{\theta},$

while the equation for $\Omega$ reads as

(3.3)
$\partial_{t}\Omega+u\cdot\nabla\Omega-\big{(}\Delta+\frac{\partial_{r}}{r}\big{)}\Omega=-\partial_{z}\big{(}\Gamma^{2}\big{)}-\frac{1}{c}\partial_{t}E\cdot\nabla\Gamma.$

The following proposition provides a control on the velocity field in terms of
some suitable norms of electromagnetic fields. This will be combined with the
estimates from Section 3.2 to obtain uniform global bounds in Section 5, later
on.

######  Proposition 3.1.

Let $T\in(0,\infty]$ and $(u,E,B)$ be a smooth axisymmetric solution of (1.1)
on $[0,T)$, where $u$ and $E$ have no swirl and $B$ has pure swirl. Then,
there is a universal constant $C>0$ such that

$\displaystyle\left\|(\omega,\Omega)\right\|_{L^{\infty}_{t}L^{2}\cap
L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\qquad\lesssim\left(\left\|(\omega_{0},\Omega_{0})\right\|_{L^{2}}+\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}\left\|B\right\|_{L^{\infty}_{t}H^{2}}+\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}\left\|(B,\Gamma)\right\|_{L^{2}_{t}\dot{H}^{1}}\right)\exp\big{(}C\mathcal{E}^{2}_{0}\big{)},$

where all the norms are taken over the time interval $[0,T)$.

######  Remark.

Note that the right-hand side in the bound above does not exhibit any time
growth, provided that the norms of the electromagnetic fields remain bounded
on any time interval $[0,T)$. This is crucial and necessary to prove the
global existence of solutions in Theorem 1.1.

###### Proof.

Firstly, we observe that a straightforward computation relying on the fact
$\omega$ is axisymmetric without swirl yields that

$e_{r}\cdot\nabla\omega=(\partial_{r}\omega_{\theta})e_{\theta},\qquad
e_{\theta}\cdot\nabla\omega=-\frac{\omega_{\theta}}{r}e_{r}=-\Omega
e_{r},\qquad e_{z}\cdot\nabla\omega=(\partial_{z}\omega_{\theta})e_{\theta}.$

In particular, this implies that

(3.4)
$|\omega|=|\omega_{\theta}|\qquad\text{and}\qquad|\nabla\omega|\sim\left|(\partial_{r}\omega_{\theta},\partial_{z}\omega_{\theta})\right|+\left|\Omega\right|,$

which will allow us to estimate $(\omega_{\theta},\Omega)$ instead of
$(\omega,\Omega)$.

Now, multiplying (3.2) by $\omega_{\theta}$, integrating in time and space,
and then using the divergence-free condition on $E$ and $u$ yields, for all
$t\in[0,T)$, that

(3.5) $\displaystyle\frac{1}{2}\left\|\omega_{\theta}(t)\right\|_{L^{2}}^{2}$
$\displaystyle+\left\|\nabla\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}+\left\|\frac{\omega_{\theta}}{r}\right\|_{L^{2}_{t}L^{2}}^{2}$
$\displaystyle\leq\frac{1}{2}\left\|\omega_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\left\|\frac{u_{r}(\tau)}{r}\right\|_{L^{\infty}}\left\|\omega_{\theta}(\tau)\right\|_{L^{2}}^{2}d\tau+\left|\int_{0}^{t}\int_{\mathbb{R}^{3}}\Gamma(\tau)B(\tau)\partial_{z}\omega_{\theta}(\tau)dxd\tau\right|$
$\displaystyle\quad+\int_{0}^{t}\frac{1}{c}\left\|(\partial_{t}E)B(\tau)\right\|_{L^{2}}\Big{(}\left\|\nabla\omega_{\theta}(\tau)\right\|_{L^{2}}+\left\|\frac{\omega_{\theta}}{r}(\tau)\right\|_{L^{2}}\Big{)}d\tau,$

where the norms in time are taken over the interval $[0,t)$.

In order to estimate the second term in the right-hand side above, we make use
of (1.10) to obtain, for any $\varepsilon>0$, that

(3.6)
$\displaystyle\int_{0}^{t}\left\|\frac{u_{r}(\tau)}{r}\right\|_{L^{\infty}}$
$\displaystyle\left\|\omega_{\theta}(\tau)\right\|_{L^{2}}^{2}d\tau$
$\displaystyle\lesssim\int_{0}^{t}\left\|\Omega(\tau)\right\|_{L^{2}}^{\frac{1}{2}}\left\|\nabla\Omega(\tau)\right\|_{L^{2}}^{\frac{1}{2}}\left\|\omega_{\theta}(\tau)\right\|_{L^{2}}^{2}d\tau$
$\displaystyle\leq\varepsilon\int_{0}^{t}\left\|\Omega(\tau)\right\|_{L^{2}}\left\|\nabla\Omega(\tau)\right\|_{L^{2}}d\tau+C_{\varepsilon}\int_{0}^{t}\left\|\omega_{\theta}(\tau)\right\|_{L^{2}}^{4}d\tau$
$\displaystyle\leq\varepsilon\Big{(}\left\|\frac{\omega_{\theta}}{r}\right\|_{L^{2}_{t}L^{2}}^{2}+\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}\Big{)}+C_{\varepsilon}\int_{0}^{t}\left\|u(\tau)\right\|_{\dot{H}^{1}}^{2}\left\|\omega_{\theta}(\tau)\right\|_{L^{2}}^{2}d\tau,$

where we utilized the celebrated Biot–Savart estimate in the last line.

On the other hand, the estimate of the third term in the right-hand side of
(3.5) is obtained by employing Hölder’s inequality, again, followed by the
Sobolev embedding $\dot{H}^{1}\hookrightarrow L^{6}(\mathbb{R}^{3})$ to write,
for any $\varepsilon>0$, that

(3.7)
$\displaystyle\left|\int_{0}^{t}\int_{\mathbb{R}^{3}}\Gamma(\tau)B(\tau)\partial_{z}\omega_{\theta}(\tau)dxd\tau\right|$
$\displaystyle\leq C_{\varepsilon}\int_{0}^{t}\left\|\Gamma
B(\tau)\right\|_{L^{2}}^{2}d\tau+\varepsilon\left\|\partial_{z}\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}$
$\displaystyle\leq
C_{\varepsilon}\int_{0}^{t}\left\|\Gamma(\tau)\right\|_{L^{3}}^{2}\left\|B(\tau)\right\|_{L^{6}}^{2}d\tau+\varepsilon\left\|\nabla\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}$
$\displaystyle\leq
C_{\varepsilon}\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}^{2}\left\|B\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}+\varepsilon\left\|\nabla\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}.$

Finally, the last integral in (3.5) can be easily controlled by similar
arguments to obtain that

(3.8)
$\displaystyle\int_{0}^{t}\frac{1}{c}\left\|(\partial_{t}E)B(\tau)\right\|_{L^{2}}$
$\displaystyle\Big{(}\left\|\nabla\omega_{\theta}(\tau)\right\|_{L^{2}}+\left\|\frac{\omega_{\theta}}{r}(\tau)\right\|_{L^{2}}\Big{)}d\tau$
$\displaystyle\leq\frac{C_{\varepsilon}}{c^{2}}\left\|\partial_{t}E\right\|_{L^{2}_{t}L^{3}}^{2}\left\|B\right\|_{L^{\infty}_{t}L^{6}}^{2}+\varepsilon\Big{(}\left\|\nabla\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}+\left\|\frac{\omega_{\theta}}{r}\right\|_{L^{2}_{t}L^{2}}^{2}\Big{)}$
$\displaystyle\leq\frac{C_{\varepsilon}}{c^{2}}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}^{2}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{1}}^{2}+\varepsilon\Big{(}\left\|\nabla\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}+\left\|\frac{\omega_{\theta}}{r}\right\|_{L^{2}_{t}L^{2}}^{2}\Big{)}.$

All in all, incorporating (3.6), (3.7) and (3.8) into (3.5), and choosing
$\varepsilon$ small enough yields that

(3.9)
$\displaystyle\left\|\omega(t)\right\|_{L^{2}}^{2}+\left\|\nabla\omega_{\theta}\right\|_{L^{2}_{t}L^{2}}^{2}$
$\displaystyle+\left\|\frac{\omega_{\theta}}{r}\right\|_{L^{2}_{t}L^{2}}^{2}$
$\displaystyle\leq\left\|\omega_{0}\right\|_{L^{2}}^{2}+\frac{1}{4}\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}+C\int_{0}^{t}\left\|u(\tau)\right\|_{\dot{H}^{1}}^{2}\left\|\omega(\tau)\right\|_{L^{2}}^{2}d\tau$
$\displaystyle\quad+C\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}^{2}\left\|B\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}+\frac{C}{c^{2}}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}^{2}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{1}}^{2},$

where we have utilized (3.4).

Now, we show how to estimate $\Omega$ in the right-hand side above. To that
end, we first perform an $L^{2}$-energy estimate on (3.3) followed by a
standard application of paraproduct laws to find, for any $t\in[0,T)$, that

$\displaystyle\begin{aligned}
\frac{1}{2}\left\|\Omega(t)\right\|_{L^{2}}^{2}+\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}&\leq\frac{1}{2}\left\|\Omega_{0}\right\|_{L^{2}}^{2}+\left\|\Gamma^{2}\right\|_{L^{2}_{t}L^{2}}\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}+\frac{1}{c}\left\|\partial_{t}E\cdot\nabla\Gamma\right\|_{L^{2}_{t}\dot{H}^{-1}}\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}\\\
&\leq\frac{1}{2}\left\|\Omega_{0}\right\|_{L^{2}}^{2}+C\left\|\Gamma\right\|_{L^{4}_{t}L^{4}}^{4}+\frac{C}{c^{2}}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}^{2}\left\|\Gamma\right\|_{L^{\infty}_{t}\dot{H}^{1}}^{2}+\frac{1}{2}\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2},\end{aligned}$

for some universal constant $C>0$. Therefore, by further employing Hölder’s
and embedding inequalities, we obtain that

$\displaystyle\begin{aligned}
\left\|\Omega(t)\right\|_{L^{2}}^{2}+\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}&\leq\left\|\Omega_{0}\right\|_{L^{2}}^{2}+C\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}^{2}\left\|\Gamma\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}+\frac{C}{c^{2}}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}^{2}\left\|\Gamma\right\|_{L^{\infty}_{t}\dot{H}^{1}}^{2}.\end{aligned}$

Hence, recalling that

$\Gamma=\frac{B_{\theta}}{r},$

and employing Lemma 2.2, we arrive at the bound

(3.10)
$\displaystyle\left\|\Omega(t)\right\|_{L^{2}}^{2}+\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}\leq\left\|\Omega_{0}\right\|_{L^{2}}^{2}+C\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}^{2}\left\|\Gamma\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}+\frac{C}{c^{2}}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}^{2}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{2}}^{2}.$

At last, summing (3.9) and (3.10), and utilizing (3.4), we arrive at the
estimate

$\displaystyle\left\|\omega(t)\right\|_{L^{2}}^{2}+\left\|\omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}$
$\displaystyle+\left\|\Omega(t)\right\|_{L^{2}}^{2}+\left\|\Omega\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}$
$\displaystyle\lesssim\left\|\omega_{0}\right\|_{L^{2}}^{2}+\left\|\Omega_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\left\|u(\tau)\right\|_{\dot{H}^{1}}^{2}\left\|\omega(\tau)\right\|_{L^{2}}^{2}d\tau$
$\displaystyle\quad+\left(\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}\left\|B\right\|_{L^{\infty}_{t}H^{2}}+\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}\left\|(B,\Gamma)\right\|_{L^{2}_{t}\dot{H}^{1}}\right)^{2}.$

Finally, an application of the classical Grönwall inequality yields that

$\displaystyle\left\|(\omega,\Omega)\right\|_{L^{\infty}_{t}L^{2}\cap
L^{2}_{t}\dot{H}^{1}}$
$\displaystyle\lesssim\left(\left\|(\omega_{0},\Omega_{0})\right\|_{L^{2}}+\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}\left\|B\right\|_{L^{\infty}_{t}H^{2}}+\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}\left\|(B,\Gamma)\right\|_{L^{2}_{t}\dot{H}^{1}}\right)$
$\displaystyle\quad\times\exp\left(C\int_{0}^{t}\left\|u(\tau)\right\|_{\dot{H}^{1}}^{2}d\tau\right),$

which, in view of the energy inequality (3.1), concludes the proof of the
proposition. ∎

In view of the estimates on the velocity field given in the preceding
proposition, we now need to establish the following bounds on the
electromagnetic field:

  * •

An asymptotically vanishing bound for $\frac{1}{c}\partial_{t}E$ of the form

(3.11)
$\frac{1}{c}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{\frac{1}{2}}}\lesssim
c^{-\alpha}F\Big{(}\left\|(u,E,B)\right\|_{X}\Big{)},$

for some $\alpha>0$, a (nonlinear) function $F$ and a suitable functional
space $X$.

  * •

An asymptotically global bound for $B$ in $L^{2}_{t}\dot{H}^{1}$ and $\Gamma$
in $L^{\infty}_{t}L^{3}\cap L^{2}_{t}\dot{H}^{1}$ of the form

(3.12)
$\left\|B\right\|_{L^{2}_{t}\dot{H}^{1}}+\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}\cap
L^{2}_{t}\dot{H}^{1}}\leq
C_{0}+c^{-\beta}F\Big{(}\left\|(u,E,B)\right\|_{X}\Big{)},$

for some $\beta>0$ and $C_{0}>0$ depending only on the initial data.

The complete justification of these bounds will be the subject of Section 4,
later on.

###  3.2. Controlling the electromagnetic field

Here, we establish several estimates combining the refined study of the
dispersive properties of Maxwell’s equations from [3] with the axisymmetric
structure. The principal results of this part of the article are obtained in
Sections 3.2.1 and 3.2.2, below.

The ensuing bounds will be further combined with the results from the previous
section to arrive at a global control of solutions to (1.1) in Section 5,
later on.

For the sake of clarity, we recall first the essential results from [3] on
Strichartz estimates and maximal parabolic regularity for the three-
dimensional damped Maxwell system which are useful in the present work.

######  Lemma 3.2.

[3, Corollary 2.12] Consider a solution
$(E,B):[0,T)\times\mathbb{R}^{3}\to\mathbb{R}^{6}$ of the damped Maxwell
system

(3.13) $\begin{cases}\begin{aligned} \frac{1}{c}\partial_{t}E-\nabla\times
B+\sigma cE&=G,\\\ \frac{1}{c}\partial_{t}B+\nabla\times E&=0,\\\
\operatorname{div}B&=0,\end{aligned}\end{cases}$

for some initial data $(E,B)(0,x)=(E_{0},B_{0})(x)$, where $\sigma>0$ and
$c>0$.

For any exponent pairs
$(q,r),(\tilde{q},\tilde{r})\in[1,\infty]\times[2,\infty)$ which are
admissible in the sense that

$\frac{1}{q}+\frac{1}{r}\geq\frac{1}{2}\qquad\text{and}\qquad\frac{1}{\tilde{q}}+\frac{1}{\tilde{r}}\geq\frac{1}{2},$

and such that

$\frac{1}{q}+\frac{1}{\tilde{q}}\leq 1,$

one has the high-frequency estimate

$\displaystyle 2^{-j\left(1-\frac{2}{r}\right)}$
$\displaystyle\left\|\Delta_{j}(PE,B)\right\|_{L^{q}([0,T);L^{r})}$
$\displaystyle\lesssim
c^{\frac{1}{2}-\frac{1}{r}-\frac{2}{q}}\left\|\Delta_{j}(PE_{0},B_{0})\right\|_{L^{2}}+c^{2-\frac{1}{r}-\frac{1}{\tilde{r}}-\frac{2}{q}-\frac{2}{\tilde{q}}}2^{j\left(1-\frac{2}{\tilde{r}}\right)}\left\|\Delta_{j}PG\right\|_{L^{\tilde{q}^{\prime}}([0,T);L^{\tilde{r}^{\prime}})},$

for all $j\in\mathbb{Z}$ with $2^{j}\geq\sigma c$, and the low-frequency
estimates

$\displaystyle 2^{-j\left(\frac{3}{2}-\frac{3}{r}\right)}$
$\displaystyle\left\|\Delta_{j}PE\right\|_{L^{q}([0,T);L^{r})}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|\Delta_{j}PE_{0}\right\|_{L^{2}}+c^{-1}2^{j\left(1-\frac{2}{q}\right)}\left\|\Delta_{j}B_{0}\right\|_{L^{2}}+c^{1-\frac{2}{q}-\frac{2}{\tilde{q}}}2^{j\left(\frac{3}{2}-\frac{3}{\tilde{r}}\right)}\left\|\Delta_{j}PG\right\|_{L^{\tilde{q}^{\prime}}([0,T);L^{\tilde{r}^{\prime}})}$

and

$\displaystyle 2^{-j\left(\frac{3}{2}-\frac{3}{r}-\frac{2}{q}\right)}$
$\displaystyle\left\|\Delta_{j}B\right\|_{L^{q}([0,T);L^{r})}$
$\displaystyle\lesssim
c^{-1}2^{j}\left\|\Delta_{j}PE_{0}\right\|_{L^{2}}+\left\|\Delta_{j}B_{0}\right\|_{L^{2}}+2^{j\left(\frac{5}{2}-\frac{3}{\tilde{r}}-\frac{2}{\tilde{q}}\right)}\left\|\Delta_{j}PG\right\|_{L^{\tilde{q}^{\prime}}([0,T);L^{\tilde{r}^{\prime}})},$

for all $j\in\mathbb{Z}$ with $2^{j}\leq\sigma c$.

######  Lemma 3.3.

[3, Corollary 2.14] Consider a solution
$(E,B):[0,T)\times\mathbb{R}^{3}\to\mathbb{R}^{6}$ of the damped Maxwell
system (3.13), for some initial data $(E,B)(0,x)=(E_{0},B_{0})(x)$, where
$\sigma>0$ and $c>0$.

For any $\chi\in C^{\infty}_{c}(\mathbb{R}^{d})$ and $s\in\mathbb{R}$, one has
the low-frequency estimates

$\displaystyle\left\|\chi(c^{-1}D)PE\right\|_{L^{m}_{t}([0,T);\dot{B}^{s+\frac{2}{m}}_{2,q})}$
$\displaystyle\lesssim
c^{-\frac{2}{m}}\left\|PE_{0}\right\|_{\dot{B}^{s+\frac{2}{m}}_{2,q}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{s+1}_{2,m}}$
$\displaystyle\quad+c^{-1+\frac{2}{r}-\frac{2}{m}}\left\|PG\right\|_{L_{t}^{r}([0,T);\dot{B}_{2,q}^{s+\frac{2}{m}})},$

for any $1<r\leq m<\infty$ and $1\leq q\leq\infty$, as well as

$\left\|\chi(c^{-1}D)B\right\|_{L^{m}_{t}([0,T);\dot{B}^{s+\frac{2}{m}}_{2,1})}\lesssim
c^{-1}\left\|PE_{0}\right\|_{\dot{B}^{s+1}_{2,m}}+\left\|B_{0}\right\|_{\dot{B}^{s}_{2,m}}+\left\|PG\right\|_{L_{t}^{r}([0,T);\dot{B}_{2,\infty}^{s-1+\frac{2}{r}})},$

for any $1<r<m<\infty$, and

$\left\|\chi(c^{-1}D)B\right\|_{L^{m}_{t}([0,T);\dot{B}^{s+\frac{2}{m}}_{2,q})}\lesssim
c^{-1}\left\|PE_{0}\right\|_{\dot{B}^{s+1}_{2,m}}+\left\|B_{0}\right\|_{\dot{B}^{s}_{2,m}}+\left\|PG\right\|_{L_{t}^{m}([0,T);\dot{B}_{2,q}^{s-1+\frac{2}{m}})},$

for any $1<m<\infty$ and $1\leq q\leq\infty$.

Let us now be more precise about the source term $G$ which will be used in the
application of the preceding two lemmas. Specifically, we will consider the
Maxwell system

(3.14) $\begin{cases}\begin{aligned} \frac{1}{c}\partial_{t}E-\nabla\times
B+\sigma cE&=-\sigma P(u\times B),&\operatorname{div}E&=0,\\\
\frac{1}{c}\partial_{t}B+\nabla\times E&=0,&\operatorname{div}B&=0,\\\
\operatorname{div}u&=0.\end{aligned}\end{cases}$

Furthermore, in order to exploit the dichotomy between high and low
frequencies featured in the estimates from the above lemmas, we consider the
variants of Besov semi-norms

$\left\|f\right\|_{\dot{B}^{s}_{p,q,<}}\overset{\rm
def}{=}\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}\\\ 2^{k}<\sigma
c\end{subarray}}2^{ksq}\left\|\Delta_{k}f\right\|_{L^{p}}^{q}\right)^{\frac{1}{q}}\quad\text{and}\quad\left\|f\right\|_{\dot{B}^{s}_{p,q,>}}\overset{\rm
def}{=}\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}\\\ 2^{k}\geq\sigma
c\end{subarray}}2^{ksq}\left\|\Delta_{k}f\right\|_{L^{p}}^{q}\right)^{\frac{1}{q}},$

as well as the corresponding variants of Chemin–Lerner semi-norms

$\left\|f\right\|_{\widetilde{L}^{r}_{t}\dot{B}^{s}_{p,q,<}}\overset{\rm
def}{=}\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}\\\ 2^{k}<\sigma
c\end{subarray}}2^{ksq}\left\|\Delta_{k}f\right\|_{L^{r}_{t}L^{p}_{x}}^{q}\right)^{\frac{1}{q}}\quad\text{and}\quad\left\|f\right\|_{\widetilde{L}^{r}_{t}\dot{B}^{s}_{p,q,>}}\overset{\rm
def}{=}\left(\sum_{\begin{subarray}{c}k\in\mathbb{Z}\\\ 2^{k}\geq\sigma
c\end{subarray}}2^{ksq}\left\|\Delta_{k}f\right\|_{L^{r}_{t}L^{p}_{x}}^{q}\right)^{\frac{1}{q}},$

for any $s\in\mathbb{R}$ and $0<p,q,r\leq\infty$ (with obvious modifications
if $q$ is infinite). These families of semi-norms have been introduced in [3].
We will utilize them extensively throughout the upcoming sections of our work.

####  3.2.1. Control of high-frequency electromagnetic waves

Here, we establish key bounds on high frequencies of electromagnetic fields.
Lemma 3.4 below combines the damped Strichartz estimates for high
electromagnetic frequencies from Lemma 3.2 with the paradifferential product
laws on axisymmetric vector fields from Lemma 2.3.

The method behind the proof of this lemma is similar to the one used in [3]
(see Lemma 3.8, therein). However, here, we further refine the method by
introducing an additional high–low frequency decomposition of the source term
$P(u\times B)$ in (3.4). This will allow us to obtain stronger estimates (see
(3.15), below).

######  Lemma 3.4.

Let $T\in(0,\infty]$ and $(E,B)$ be a smooth axisymmetric solution to (3.14),
defined on $[0,T)$, for some initial data $(E_{0},B_{0})$ and some
axisymmetric divergence-free vector field $u$. Assume further that $E$ and $u$
are both without swirl and that $B$ has pure swirl.

Then, for any $s\in(-\frac{5}{2},\frac{5}{2})$, $n\in[1,\infty]$ and any
$q\in[\frac{4}{3},\infty]$, it holds that

$\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{s}_{2,n,>}}\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{s}_{2,n,>}}+c^{\frac{1}{2}-\frac{2}{q}}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}\left\|B\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{s}_{2,n}}.$

Moreover, at the endpoint $s=\frac{5}{2}$, we have that

$\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}+c^{-\frac{1}{2}+\frac{2}{p}-\frac{2}{q}}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}\left\|B\right\|_{\widetilde{L}^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}},$

as soon as $2\leq p\leq q\leq\infty$. Furthermore, in the case $p=2$, it holds
that

(3.15)
$\displaystyle\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\quad+c^{\frac{1}{2}-\frac{2}{q}}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}\left(\left\|B\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}+\left\|B\right\|_{L^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,<}}\right),$

for all $q\in[2,\infty]$.

###### Proof.

Applying Lemma 3.2 yields that

$\displaystyle\left\|\Delta_{j}(E,B)\right\|_{L^{q}_{t}L^{2}}\lesssim
c^{-\frac{2}{q}}\left\|\Delta_{j}(E_{0},B_{0})\right\|_{L^{2}}+c^{-1+2(\frac{1}{p}-\frac{1}{q})}\left\|\Delta_{j}P\big{(}u\times
B\big{)}\right\|_{L_{t}^{p}L^{2}},$

for all $j\in\mathbb{Z}$, with $2^{j}\geq\sigma c$, and any $1\leq p\leq
q\leq\infty$. It then follows, for any $s\in\mathbb{R}$ and any
$n\in[1,\infty]$, that

(3.16)
$\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{s}_{2,n,>}}\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{s}_{2,n,>}}+c^{-1+2(\frac{1}{p}-\frac{1}{q})}\left\|P\big{(}u\times
B\big{)}\right\|_{\widetilde{L}_{t}^{p}\dot{B}^{s}_{2,n}}.$

Thus, choosing $p=\frac{4}{3}$ and utilizing (2.4) we find, for all
$s\in(-\frac{3}{2},\frac{5}{2})$, $n\in[1,\infty]$ and any
$q\in[\frac{4}{3},\infty]$, that

$\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{s}_{2,n,>}}\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{s}_{2,n,>}}+c^{\frac{1}{2}-\frac{2}{q}}\left\|u\right\|_{L^{4}_{t}L^{\infty}\cap\widetilde{L}^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,\infty}}\left\|B\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{s}_{2,n}}.$

Hence, we conclude the proof of the first claim in the lemma by employing the
embedding

$L^{\infty}_{t}\dot{H}^{1}\cap L^{2}_{t}\dot{H}^{2}\hookrightarrow
L^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,1}\hookrightarrow
L^{4}_{t}L^{\infty}\cap\widetilde{L}^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,\infty}.$

We now turn to the endpoint case $(s,n)=(\frac{5}{2},1)$, which corresponds to
the second bound in the statement of the lemma. The natural attempt to
estimate the product $P(u\times B)$ in that case would be by applying (2.5),
which is the corresponding extension of (2.4). Doing so would lead to the
control

(3.17)
$\displaystyle\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}+c^{-1+2\left(\frac{1}{4}+\frac{1}{p}-\frac{1}{q}\right)}\left\|P\big{(}u\times
B\big{)}\right\|_{\widetilde{L}_{t}^{\left(\frac{1}{4}+\frac{1}{p}\right)^{-1}}\dot{B}^{\frac{5}{2}}_{2,1}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}+c^{-\frac{1}{2}-\frac{2}{q}+\frac{2}{p}}\left\|u\right\|_{\widetilde{L}^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B\right\|_{\widetilde{L}^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}},$

as soon as

$1\leq\frac{4p}{p+4}\leq q\leq\infty.$

Observe then that, in order to conclude, the preceding estimate would require
a stronger control on the velocity field $u$, for even though one has that

$\widetilde{L}^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}\hookrightarrow\widetilde{L}^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,1},$

it is unclear whether the embedding

$L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}\hookrightarrow\widetilde{L}^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,1}$

holds or not. Accordingly, (3.17) does not seem to be useful.

Instead, we have to prove the desired estimate by “hand” (that is, by cooking
up a suitable interpolation argument). To that end, we first introduce the
decomposition

$u=u_{\ell}+u_{h}\overset{\rm def}{=}\left(\mathds{1}_{|D|<\frac{\sigma
c}{2}}+\mathds{1}_{|D|\geq\frac{\sigma c}{2}}\right)u.$

Then, by splitting the source term $P(u\times B)$ according to the latter
decomposition of $u$ and by applying (3.16) to each Maxwell system
corresponding to the source terms $P(u_{\ell}\times B)$ and $P(u_{h}\times
B)$, with different values of $p$, one obtains that

(3.18)
$\displaystyle\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{s}_{2,1,>}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{s}_{2,1,>}}+c^{-1+\frac{2}{p}-\frac{2}{q}}\left\|P\big{(}u_{\ell}\times
B\big{)}\right\|_{\widetilde{L}_{t}^{p}\dot{B}^{s}_{2,1,>}}$
$\displaystyle\quad+c^{\frac{2}{p}-\frac{2}{q}}\left\|P\big{(}u_{h}\times
B\big{)}\right\|_{\widetilde{L}_{t}^{\left(\frac{1}{2}+\frac{1}{p}\right)^{-1}}\dot{B}^{s}_{2,1,>}},$

for all $2\leq p\leq q$ and all $s\in\mathbb{R}$. Therefore, applying the
product law estimate (2.5), we infer that

$\displaystyle\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\quad+c^{-1+\frac{2}{p}-\frac{2}{q}}\left\|u_{\ell}\right\|_{\widetilde{L}^{\infty}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B\right\|_{\widetilde{L}^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}}+c^{\frac{2}{p}-\frac{2}{q}}\left\|u_{h}\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B\right\|_{\widetilde{L}^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}}.$

Now, observing that

(3.19)
$\left\|u_{\ell}\right\|_{\widetilde{L}^{\infty}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\lesssim
c^{\frac{1}{2}}\left\|u_{\ell}\right\|_{{L}^{\infty}_{t}\dot{B}^{1}_{2,\infty}}\lesssim
c^{\frac{1}{2}}\left\|u\right\|_{L_{t}^{\infty}\dot{H}^{1}}$

and

$\left\|u_{h}\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\lesssim
c^{-\frac{1}{2}}\left\|u\right\|_{L_{t}^{2}\dot{H}^{2}}$

leads to the desired control

$\displaystyle\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}+c^{-\frac{1}{2}-\frac{2}{q}+\frac{2}{p}}\left\|u\right\|_{L_{t}^{\infty}\dot{H}^{1}\cap
L_{t}^{2}\dot{H}^{2}}\left\|B\right\|_{\widetilde{L}^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}},$

for any $2\leq p\leq q$. This takes care of the second bound in the statement
of the lemma.

Finally, in order for us to justify the last estimate of the lemma (which is
an improvement of the preceding bound in the case $p=2$), we need to further
decompose $B$ in the source term $P(u_{\ell}\times B)$. To that end, we write

$B=B_{\ell}+B_{h}\overset{\rm def}{=}\left(\mathds{1}_{|D|<\frac{\sigma
c}{2}}+\mathds{1}_{|D|\geq\frac{\sigma c}{2}}\right)B,$

which allows to deduce from (3.18) that

(3.20)
$\displaystyle\left\|(E,B)\right\|_{\widetilde{L}^{q}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1,>}}+c^{-\frac{2}{q}}\left\|P\big{(}u_{\ell}\times
B_{\ell}\big{)}\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\quad+c^{-\frac{2}{q}}\left\|P\big{(}u_{\ell}\times
B_{h}\big{)}\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1}}+c^{1-\frac{2}{q}}\left\|P\big{(}u_{h}\times
B\big{)}\right\|_{L_{t}^{1}\dot{B}^{\frac{5}{2}}_{2,1}},$

for all $q\geq 2$, where we have used that

$\widetilde{L}_{t}^{1}\dot{B}^{\frac{5}{2}}_{2,1}=L_{t}^{1}\dot{B}^{\frac{5}{2}}_{2,1}.$

Now, observe that

$\operatorname{supp}\left(\widehat{u_{\ell}\times
B_{\ell}}\right)\subset\left\\{\xi:|\xi|<\sigma c\right\\},$

which implies that

$\Delta_{j}\left(u_{\ell}\times B_{\ell}\right)\equiv 0,$

whenever $2^{j}\geq 2\sigma c$. Consequently, it follows that

$\left\|P\big{(}u_{\ell}\times
B_{\ell}\big{)}\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}=2^{\frac{5}{2}j}\left\|P\Delta_{j}\big{(}u_{\ell}\times
B_{\ell}\big{)}\right\|_{{L}_{t}^{2}L^{2}}=\left\|P\big{(}u_{\ell}\times
B_{\ell}\big{)}\right\|_{L_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}},$

where $j$ is the only integer value which satisfies $\sigma c\leq
2^{j}<2\sigma c$.

Hence, by applying (2.5) for classical Besov-space-valued Lebesgue spaces, we
obtain that

$\displaystyle\left\|P\big{(}u_{\ell}\times
B_{\ell}\big{)}\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle=\left\|P\big{(}u_{\ell}\times
B_{\ell}\big{)}\right\|_{L_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}$
$\displaystyle\lesssim\left\|u_{\ell}\right\|_{L_{t}^{\infty}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B_{\ell}\right\|_{L_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1}}\lesssim
c^{\frac{1}{2}}\left\|u\right\|_{L_{t}^{\infty}\dot{H}^{1}}\left\|B\right\|_{L_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,<}},$

where we used (3.19). Similarly, we find, by employing (2.5) and (3.19),
again, that

$\displaystyle\left\|P\big{(}u_{\ell}\times
B_{h}\big{)}\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1}}$
$\displaystyle\lesssim\left\|u_{\ell}\right\|_{\widetilde{L}_{t}^{\infty}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B_{h}\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1}}$
$\displaystyle\lesssim\left\|u_{\ell}\right\|_{\widetilde{L}_{t}^{\infty}\dot{B}^{\frac{3}{2}}_{2,1}}\left(\left\|B\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}+\sum_{\frac{\sigma
c}{4}<2^{j}<\sigma
c}2^{\frac{5}{2}j}\left\|\Delta_{j}B\right\|_{L^{2}_{t}L^{2}}\right)$
$\displaystyle\lesssim
c^{\frac{1}{2}}\left\|u\right\|_{L_{t}^{\infty}\dot{H}^{1}}\left(\left\|B\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}+\left\|B\right\|_{L^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,<}}\right).$

As for the last term in (3.20), applying (2.5), again, in classical Besov-
space-valued Lebesgue spaces, entails that

$\displaystyle\left\|P\big{(}u_{h}\times
B\big{)}\right\|_{L_{t}^{1}\dot{B}^{\frac{5}{2}}_{2,1}}$
$\displaystyle\lesssim\left\|u_{h}\right\|_{L_{t}^{2}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B\right\|_{L_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1}}$
$\displaystyle\lesssim
c^{-\frac{1}{2}}\left\|u\right\|_{L_{t}^{2}\dot{H}^{2}}\left(\left\|B\right\|_{\widetilde{L}_{t}^{2}\dot{B}^{\frac{5}{2}}_{2,1,>}}+\left\|B\right\|_{L^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,<}}\right).$

All in all, gathering the previous estimates and incorporating them into
(3.20) yields the control (3.15), which completes the proof of the lemma. ∎

####  3.2.2. Control of low-frequency electromagnetic waves

It is clear from Lemmas 3.2 and 3.3 that solutions to the damped Maxwell
system enjoy various types of bounds in different regions of low and high
frequencies.

Here, we intend to shed light on the low-frequency control of electromagnetic
fields solving (3.14). In broad terms, the low-frequency bounds on the
electric field $E$ are similar to the same bounds in the high-frequency
regime. However, the low-frequency part of the magnetic field $B$ enjoys
parabolic-type estimates, which are consistent with the limiting system (MHD)
as $c$ goes to infinity. A more precise formulation of that principle is given
in the next lemma.

######  Lemma 3.5.

Let $T\in(0,\infty]$ and $(E,B)$ be a smooth axisymmetric solution to (3.14)
on $[0,T)$, for some initial data $(E_{0},B_{0})$ and some axisymmetric
divergence-free vector field $u$. Assume further that $E$ and $u$ are both
without swirl and that $B$ has pure swirl.

Then, for any $\alpha\in[0,1]$, $n\in[1,\infty]$ and $(m,p)\in[2,\infty]^{2}$,
with

$\alpha+\frac{2}{m}\leq\frac{3}{2},$

and for any $s<\frac{5}{2}$, with $s+\alpha+\frac{2}{m}>0$, one has the
following low-frequency estimates

(3.21)
$\displaystyle\left\|E\right\|_{L^{q}_{t}\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n,<}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n,<}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{1}{2}-\frac{2}{q}}_{2,q,<}}$
$\displaystyle\quad+c^{2\left(\frac{1}{p}+\frac{1}{m}-\frac{1}{q}\right)-1}\left\|u\right\|_{L^{m}_{t}\dot{B}^{\alpha+\frac{2}{m}}_{2,1}}\left\|B\right\|_{L^{p}_{t}\dot{B}^{s}_{2,n}},$

as soon as $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}\leq q<\infty,$ as well
as

(3.22)
$\displaystyle\left\|B\right\|_{L^{q}_{t}\dot{B}^{s+\alpha-\frac{1}{2}+\frac{2}{q}-\frac{2}{p}}_{2,1,<}}$
$\displaystyle\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,q,<}}$
$\displaystyle\quad+\left\|u\right\|_{L^{m}_{t}\dot{B}^{\alpha+\frac{2}{m}}_{2,1}}\left\|B\right\|_{L^{p}_{t}\dot{B}^{s}_{2,\infty}},$

whenever $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}<q<\infty.$

Moreover, in the case where

$1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}=q<\infty,$

it holds that

(3.23)
$\displaystyle\left\|B\right\|_{L^{q}_{t}\dot{B}^{s+\alpha-\frac{1}{2}+\frac{2}{q}-\frac{2}{p}}_{2,n,<}}$
$\displaystyle\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|u\right\|_{L^{m}_{t}\dot{B}^{\alpha+\frac{2}{m}}_{2,1}}\left\|B\right\|_{L^{p}_{t}\dot{B}^{s}_{2,n}}.$

At last, at the endpoint $s=\frac{5}{2}$, we have

(3.24)
$\displaystyle\left\|E\right\|_{L^{q}_{t}\dot{B}^{1+\alpha+\frac{2}{m}}_{2,n,<}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|E_{0}\right\|_{\dot{B}^{1+\alpha+\frac{2}{m}}_{2,n,<}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{2+\alpha+\frac{2}{m}-\frac{2}{q}}_{2,q,<}}$
$\displaystyle\quad+c^{2\left(\frac{1}{p}+\frac{1}{m}-\frac{1}{q}\right)-1}\left\|u\right\|_{L^{m}_{t}\dot{B}^{\alpha+\frac{2}{m}}_{2,1}}\left\|B\right\|_{L^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}},$
$\displaystyle\left\|B\right\|_{L^{q}_{t}\dot{B}^{2+\alpha+\frac{2}{q}-\frac{2}{p}}_{2,n,<}}$
$\displaystyle\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{3+\alpha-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{2+\alpha-\frac{2}{p}}_{2,q,<}}+\left\|u\right\|_{L^{m}_{t}\dot{B}^{\alpha+\frac{2}{m}}_{2,1}}\left\|B\right\|_{L^{p}_{t}\dot{B}^{\frac{5}{2}}_{2,1}},$

as long as $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}\leq q<\infty.$

###### Proof.

Similarly to [3, Lemma 3.9], the proof hinges upon the combination of Lemmas
2.3, 3.2 and 3.3. Thus, on the one hand, applying the low-frequency estimate
from Lemma 3.2, for parameter values such that

$r=\tilde{r}=2$

and

$\tilde{q}^{\prime}=\left(\frac{1}{m}+\frac{1}{p}\right)^{-1},$

yields, as soon as $1\leq\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}\leq
q\leq\infty$, that

$\displaystyle
2^{j\left(s+\alpha+\frac{2}{m}-\frac{3}{2}\right)}\left\|\Delta_{j}E\right\|_{L^{q}_{t}L^{2}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,\infty,<}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{1}{2}-\frac{2}{q}}_{2,\infty,<}}$
$\displaystyle\quad+c^{2\left(\frac{1}{p}+\frac{1}{m}-\frac{1}{q}\right)-1}\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,\infty}\right)}$

and

(3.25) $\displaystyle
2^{j\left(s+\alpha-\frac{1}{2}-\frac{2}{p}+\frac{2}{q}\right)}\left\|\Delta_{j}B\right\|_{L^{q}_{t}L^{2}}$
$\displaystyle\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,\infty,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,\infty,<}}$
$\displaystyle\quad+\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,\infty}\right)},$

for all $j\in\mathbb{Z}$ with $\frac{\sigma c}{2}\leq 2^{j}<\sigma c$.

On the other hand, employing the first and the second estimates from Lemma 3.3
entails

$\displaystyle\left\|\mathds{1}_{\\{2^{j}<\frac{\sigma
c}{2}\\}}2^{j\left(s+\alpha+\frac{2}{m}-\frac{3}{2}\right)}\left\|\Delta_{j}E\right\|_{L^{2}_{x}}\right\|_{L^{q}_{t}\ell^{n}_{j}}$
$\displaystyle\lesssim
c^{-\frac{2}{q}}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n,<}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{1}{2}-\frac{2}{q}}_{2,q,<}}$
$\displaystyle\quad+c^{2\left(\frac{1}{p}+\frac{1}{m}-\frac{1}{q}\right)-1}\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n}\right)},$

for any $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}\leq q<\infty$ and $1\leq
n\leq\infty$, as well as

$\displaystyle\left\|\mathds{1}_{\\{2^{j}<\frac{\sigma
c}{2}\\}}2^{j\left(s+\alpha-\frac{1}{2}+\frac{2}{q}-\frac{2}{p}\right)}\left\|\Delta_{j}B\right\|_{L^{2}_{x}}\right\|_{L^{q}_{t}\ell^{1}_{j}}$
$\displaystyle\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,q,<}}$
$\displaystyle\quad+\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,\infty}\right)},$

for any $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}<q<\infty$.

All in all, by combining the preceding estimates, we arrive at the conclusion
that

$\displaystyle\left\|E\right\|_{L^{q}_{t}\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n,<}}$
$\displaystyle\lesssim$ $\displaystyle
c^{-\frac{2}{q}}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n,<}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{s+\alpha+\frac{2}{m}-\frac{1}{2}-\frac{2}{q}}_{2,q,<}}$
$\displaystyle+c^{2\left(\frac{1}{p}+\frac{1}{m}-\frac{1}{q}\right)-1}\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n}},$

for any $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}\leq q<\infty$ and $1\leq
n\leq\infty$, as well as

$\left\|B\right\|_{L^{q}_{t}\dot{B}^{s+\alpha-\frac{1}{2}+\frac{2}{q}-\frac{2}{p}}_{2,1,<}}\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,\infty}\right)},$

for any $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}<q<\infty$. Therefore,
applying the product estimates from Lemma 2.3 concludes the proof of (3.21)
and (3.22).

As for the case $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}=q<\infty,$ we
apply the third estimate from Lemma 3.3 instead of the second one to infer
that

$\displaystyle\left\|\mathds{1}_{\\{2^{j}<\frac{\sigma
c}{2}\\}}2^{j\left(s+\alpha-\frac{1}{2}+\frac{2}{q}-\frac{2}{p}\right)}\left\|\Delta_{j}B\right\|_{L^{2}_{x}}\right\|_{L^{q}_{t}\ell^{n}_{j}}$
$\displaystyle\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,q,<}}$
$\displaystyle\quad+\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n}\right)}.$

Hence, combining the latter estimate with (3.25), we obtain that

$\left\|B\right\|_{L^{q}_{t}\dot{B}^{s+\alpha-\frac{1}{2}+\frac{2}{q}-\frac{2}{p}}_{2,n,<}}\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{s+\alpha+\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{s+\alpha-\frac{1}{2}-\frac{2}{p}}_{2,q,<}}+\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{s+\alpha+\frac{2}{m}-\frac{3}{2}}_{2,n}\right)}.$

Our claim (3.23) follows then by employing Lemma 2.3, again.

At last, for the endpoint case $s=\frac{5}{2}$, we deduce from the previous
cases above, for any $n\in[1,\infty]$, that

$\displaystyle\left\|E\right\|_{L^{q}_{t}\dot{B}^{1+\alpha+\frac{2}{m}}_{2,n,<}}$
$\displaystyle\lesssim$ $\displaystyle
c^{-\frac{2}{q}}\left\|E_{0}\right\|_{\dot{B}^{1+\alpha+\frac{2}{m}}_{2,n,<}}+c^{-1}\left\|B_{0}\right\|_{\dot{B}^{2+\alpha+\frac{2}{m}-\frac{2}{q}}_{2,q,<}}$
$\displaystyle+c^{2\left(\frac{1}{p}+\frac{1}{m}-\frac{1}{q}\right)-1}\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\dot{B}^{1+\alpha+\frac{2}{m}}_{2,n}},$

and

$\left\|B\right\|_{L^{q}_{t}\dot{B}^{2+\alpha+\frac{2}{q}-\frac{2}{p}}_{2,n,<}}\lesssim
c^{-1}\left\|E_{0}\right\|_{\dot{B}^{3+\alpha-\frac{2}{p}}_{2,q,<}}+\left\|B_{0}\right\|_{\dot{B}^{2+\alpha-\frac{2}{p}}_{2,q,<}}+\left\|P(u\times
B)\right\|_{L_{t}^{\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}}\left(\dot{B}^{1+\alpha+\frac{2}{m}}_{2,n}\right)},$

as long as $1<\left(\frac{1}{m}+\frac{1}{p}\right)^{-1}\leq q<\infty$.
Therefore, applying Lemma 2.3, again, to estimate the source term concludes
the proof. ∎

######  Remark.

Observe that combining (3.15) and (3.24), with the choice of parameters

$q=p=2,\quad m=4,\quad\alpha=n=1,$

and employing the interpolation inequality

$\left\|u\right\|_{L^{4}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\lesssim\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}$

produces the useful bound

(3.26) $\displaystyle\left\|E\right\|_{L^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1}}$
$\displaystyle\lesssim
c^{-1}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1}}+c^{-\frac{1}{2}}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}\left(\left\|B\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}+\left\|B\right\|_{L^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,<}}\right).$

The latter control of the electric field will come in handy, later on.

####  3.2.3. Persistence of regularity

Observe that the case $q=\infty$ is missing in the bounds from Lemma 3.5,
above. Although it is possible to extend the results from Lemma 3.5 to cover
that case (at the cost of more restrictive assumptions on the third
summability index of Besov norms of the initial data), it is simpler to
establish these missing bounds by utilizing an elementary energy estimate for
Maxwell’s equations, which is the content of the next lemma.

######  Lemma 3.6.

Let $T\in(0,\infty]$ and $(E,B)$ be a smooth axisymmetric solution to (3.14)
on $[0,T)$, for some divergence-free initial data $(E_{0},B_{0})$ and vector
field $u$. Assume further that $E$ and $u$ are both without swirl and that $B$
is a vector field with pure swirl.

Then, for any $s\in(-\frac{3}{2},\frac{5}{2})$, $\epsilon>0$ with
$s+\epsilon<\frac{5}{2}$ and for all $p,q\in[2,\infty]$ with
$\frac{1}{2}=\frac{1}{p}+\frac{1}{q}$, it holds that

$\left\|(E,B)\right\|_{L^{\infty}_{t}\dot{H}^{s}}+c\left\|E\right\|_{L^{2}_{t}\dot{H}^{s}}\lesssim\left\|(E_{0},B_{0})\right\|_{\dot{H}^{s}}+\left\|u\right\|_{L^{q}_{t}\dot{H}^{\frac{3}{2}-\epsilon}}\left\|B\right\|_{L^{p}_{t}\dot{H}^{s+\epsilon}},$

on the time interval $[0,T)$. Moreover, in the endpoint case $\epsilon=0$, we
have that

$\left\|(E,B)\right\|_{L^{\infty}_{t}\dot{H}^{s}}+c\left\|E\right\|_{L^{2}_{t}\dot{H}^{s}}\lesssim\left\|(E_{0},B_{0})\right\|_{\dot{H}^{s}}+\left\|u\right\|_{L^{q}_{t}\dot{B}^{\frac{3}{2}}_{2,1}}\left\|B\right\|_{L^{p}_{t}\dot{H}^{s}},$

on the time interval $[0,T)$.

###### Proof.

We begin with localizing (3.14) in frequencies by applying $\Delta_{j}$, for
$j\in\mathbb{Z}$. Then, by an $L^{2}$ energy estimate, we find, for any
$j\in\mathbb{Z}$, that

$\displaystyle\frac{1}{2c}\left\|\Delta_{j}(E,B)(t)\right\|_{L^{2}}^{2}+\sigma
c\left\|\Delta_{j}E\right\|_{L^{2}_{t}L^{2}}^{2}$
$\displaystyle\leq\frac{1}{2c}\left\|\Delta_{j}(E_{0},B_{0})\right\|_{L^{2}}^{2}+\sigma\left\|\Delta_{j}P(u\times
B)\right\|_{L^{2}_{t}L^{2}}\left\|\Delta_{j}E\right\|_{L^{2}_{t}L^{2}}$
$\displaystyle\leq\frac{1}{2c}\left\|\Delta_{j}(E_{0},B_{0})\right\|_{L^{2}}^{2}+\frac{\sigma}{2c}\left\|\Delta_{j}P(u\times
B)\right\|_{L^{2}_{t}L^{2}}^{2}$ $\displaystyle\quad+\frac{\sigma
c}{2}\left\|\Delta_{j}E\right\|_{L^{2}_{t}L^{2}}^{2}.$

Therefore, it follows that

$\left\|(E,B)(t)\right\|_{\dot{H}^{s}}^{2}+\sigma
c^{2}\left\|E\right\|_{L^{2}_{t}\dot{H}^{s}}^{2}\leq\left\|(E_{0},B_{0})\right\|_{\dot{H}^{s}}^{2}+\sigma\left\|P(u\times
B)\right\|_{L^{2}_{t}\dot{H}^{s}}^{2},$

for any $s\in(-\frac{3}{2},\frac{5}{2})$.

Finally, we conclude the proof by employing Lemma 2.3 to control the source
term in the right-hand side, above. ∎

##  4\. Asymptotic analysis of electromagnetic fields

In this section, we are going to make (3.11) and (3.12) precise. In
particular, we are going to study the convergence

$\frac{1}{c}\partial_{t}E\to 0,$

as $c\to\infty$.

For simplicity, we will drop $\sigma$ from the equations (1.1) by fixing its
value $\sigma=1$, and we emphasize that the analysis we perform here holds for
any non-negative value of that parameter.

###  4.1. Asymptotic analysis of Ampère’s equation

From (1.1), observe, at least formally, that $E$ vanishes when $c$ goes to
infinity. Moreover, Ampère’s equation allows us to obtain

$j\rightarrow\nabla\times B,$

as $c\to\infty$, in some suitable weak sense.

In the next lemma, we establish a more precise description of the preceding
convergence in adequate functional spaces. This step is crucial in the proof
of Theorem 1.1 and will come in handy in Section 5.

######  Proposition 4.1.

Let $T\in(0,\infty]$ and $(u,E,B)$ be smooth solution to the
Navier–Stokes–Maxwell equations (1.1), defined on $[0,T)$, where $u$ and $E$
have no swirl and $B$ has pure swirl.

Then, on the time interval $[0,T)$, for all $s\in[0,\frac{1}{2}]$,
$p\in[2,\infty]$ and for any $c>0$, it holds that

$\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{p}_{t}\dot{H}^{s}}=\left\|j-\nabla\times
B\right\|_{L^{p}_{t}\dot{H}^{s}}\lesssim c^{-\frac{2}{p}}\left\|\nabla\times
B_{0}-j_{0}\right\|_{\dot{H}^{s}}+c^{-\frac{2}{p}}\left\|E_{0}\right\|_{\dot{H}^{s+1}}+c^{-\left(\frac{2}{p}+1\right)}\mathcal{A}_{s}(u,E,B),$

where

$\displaystyle\mathcal{A}_{s}(u,E,B)$ $\displaystyle\overset{\rm
def}{=}\Big{(}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}+\mathcal{E}_{0}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}\Big{)}\Big{(}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{3}{2}}}+c\left\|E\right\|_{L^{2}_{t}\dot{H}^{s+\frac{3}{2}}}\Big{)}$
$\displaystyle\quad+\left\|u\right\|_{L^{\infty}_{t}L^{2}}^{\frac{1}{2}-s}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}}^{s+\frac{1}{2}}\left\|u\right\|_{L^{2}_{t}\dot{H}^{2}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}.$

###### Proof.

The proof relies on a key idea from [4, Proposition 3.3], which we adapt to
the three-dimensional setting.

We begin by applying a time derivative to Ampère’s equation to obtain the
following damped wave equation for $E$

$\frac{1}{c}\partial_{tt}E-c\Delta E+c\,\partial_{t}E=-\partial_{t}P(u\times
B),$

where we have used Faraday’s equation and Ohm’s law, as well.

Then, we localize in frequencies by applying $\Delta_{j}$, for
$j\in\mathbb{Z}$, and we perform an $L^{2}$ energy estimate followed by
Hölder’s inequality to find, for all $t\in[0,T)$, that

$\displaystyle\frac{1}{2}\frac{d}{dt}\Bigg{(}\frac{1}{c^{4}}\left\|\Delta_{j}\partial_{t}E(t)\right\|_{L^{2}}^{2}$
$\displaystyle+\frac{1}{c^{2}}\left\|\Delta_{j}\nabla
E(t)\right\|_{L^{2}}^{2}\Bigg{)}+\frac{1}{c^{2}}\left\|\Delta_{j}\partial_{t}E(t)\right\|_{L^{2}}^{2}$
$\displaystyle\qquad\qquad\leq\frac{1}{c^{3}}\left|\int_{\mathbb{R}^{3}}\Delta_{j}\partial_{t}P(u\times
B)\Delta_{j}\partial_{t}E(t,x)dx\right|$
$\displaystyle\qquad\qquad\leq\frac{1}{c^{3}}\left\|\Delta_{j}\partial_{t}P(u\times
B)(t)\right\|_{L^{2}}\left\|\Delta_{j}\partial_{t}E(t)\right\|_{L^{2}}$
$\displaystyle\qquad\qquad\leq\frac{1}{2c^{4}}\left\|\Delta_{j}\partial_{t}P(u\times
B)(t)\right\|_{L^{2}}^{2}+\frac{1}{2c^{2}}\left\|\Delta_{j}\partial_{t}E(t)\right\|_{L^{2}}^{2}.$

Therefore, we find, for any $s\in\mathbb{R}$, that

$\frac{d}{dt}\left(\frac{1}{c^{4}}\left\|\partial_{t}E(t)\right\|_{\dot{H}^{s}}^{2}+\frac{1}{c^{2}}\left\|E(t)\right\|_{\dot{H}^{s+1}}^{2}\right)+\frac{1}{c^{2}}\left\|\partial_{t}E(t)\right\|_{\dot{H}^{s}}^{2}\leq\frac{1}{c^{4}}\left\|\partial_{t}P(u\times
B)(t)\right\|_{\dot{H}^{s}}^{2}.$

Consequently, employing Hölder’s inequality followed by Young’s inequality for
products, we obtain that

(4.1)
$\displaystyle\frac{1}{c^{2-\frac{2}{p}}}\left\|\partial_{t}E\right\|_{L^{p}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim\frac{1}{c^{2}}\left\|\partial_{t}E\right\|_{L^{\infty}_{t}\dot{H}^{s}}+\frac{1}{c}\left\|\partial_{t}E\right\|_{L^{2}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim\frac{1}{c}\Big{(}\left\|\nabla\times
B_{0}-j_{0}\right\|_{\dot{H}^{s}}+\left\|E_{0}\right\|_{\dot{H}^{s+1}}\Big{)}+\frac{1}{c^{2}}\left\|\partial_{t}P(u\times
B)\right\|_{L^{2}_{t}\dot{H}^{s}},$

for every $p\in[2,\infty]$, where we have used Ampère’s equation to express
the initial data of $\frac{1}{c}\partial_{t}E$ in terms of $B_{0}$ and
$j_{0}$.

Hence, we are now left to control $\partial_{t}P(u\times B)$. To this end, we
shall first transform the time derivative into spatial derivatives by using
the momentum and Faraday’s equations from (1.1). Accordingly, we obtain that

$\partial_{t}P(u\times B)=P\left(\partial_{t}u\times
B\right)+P\left(u\times\partial_{t}B\right)=-\sum_{i=1}^{4}\mathcal{I}_{i},$

where

$\displaystyle\mathcal{I}_{1}$ $\displaystyle\overset{\rm
def}{=}P\Big{(}P\left(u\cdot\nabla u\right)\times B\Big{)},$
$\displaystyle\mathcal{I}_{2}$ $\displaystyle\overset{\rm
def}{=}-P\Big{(}\Delta u\times B\Big{)},$ $\displaystyle\mathcal{I}_{3}$
$\displaystyle\overset{\rm def}{=}-P\Big{(}P\left(j\times B\right)\times
B\Big{)},$ $\displaystyle\mathcal{I}_{4}$ $\displaystyle\overset{\rm
def}{=}cP\Big{(}u\times(\nabla\times E)\Big{)}.$

We are now going to utilize (2.3) from Lemma 2.3 to estimate each term
separately.

For $\mathcal{I}_{1}$, we find that

$\displaystyle\left\|\mathcal{I}_{1}\right\|_{L^{2}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim\left\|u\cdot\nabla
u\right\|_{L^{2}_{t}\dot{H}^{s}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}$
$\displaystyle\lesssim\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{1}{2}}}\left\|\nabla
u\right\|_{L^{2}_{t}\dot{H}^{1}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}.$

Therefore, as long as $s\in[0,\frac{1}{2}]$, we obtain, by interpolation, that

$\displaystyle\left\|\mathcal{I}_{1}\right\|_{L^{2}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim\left\|u\right\|_{L^{\infty}_{t}L^{2}}^{\frac{1}{2}-s}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}}^{s+\frac{1}{2}}\left\|u\right\|_{L^{2}_{t}\dot{H}^{2}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}.$

As for $\mathcal{I}_{2}$, we employ Lemma 2.3, again, to find, for any
$s\in[0,\frac{1}{2}]$, that

$\displaystyle\left\|\mathcal{I}_{2}\right\|_{L^{2}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim\left\|\Delta
u\right\|_{L^{2}_{t}L^{2}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{3}{2}}}$
$\displaystyle\lesssim\left\|u\right\|_{L^{2}_{t}\dot{H}^{2}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{3}{2}}}.$

In order to estimate $\mathcal{I}_{3}$, we utilize Lemma 2.3 twice to obtain,
for any $s\in[0,\frac{1}{2}]$, that

$\displaystyle\left\|\mathcal{I}_{3}\right\|_{L^{2}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim\left\|P\big{(}j\times
B\big{)}\right\|_{L^{2}_{t}L^{2}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{3}{2}}}$
$\displaystyle\lesssim\left\|j\right\|_{L^{2}_{t}L^{2}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{3}{2}}}$
$\displaystyle\lesssim\mathcal{E}_{0}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{\frac{3}{2}}}\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{s+\frac{3}{2}}},$

where we have used the energy estimate (3.1) in the last inequality. Finally,
the control of the last term $\mathcal{I}_{4}$ is achieved by classical
product laws, which give that

$\displaystyle\left\|\mathcal{I}_{4}\right\|_{L^{2}_{t}\dot{H}^{s}}\lesssim\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}}\left\|cE\right\|_{L^{2}_{t}\dot{H}^{s+\frac{3}{2}}}.$

All in all, gathering the previous bounds yields

(4.2) $\left\|\partial_{t}(u\times
B)\right\|_{L^{2}_{t}\dot{H}^{s}}\lesssim\mathcal{A}_{s}(u,E,B),$

where $\mathcal{A}_{s}(u,E,B)$ is defined in the statement of the proposition,
above. The proof is then concluded by incorporating (4.2) into (4.1). ∎

###  4.2. Almost-parabolic estimates on the magnetic field

When considering the limiting system (MHD), one can show, with standard energy
estimates, that the quantities $B$ and $\Gamma=\frac{B_{\theta}}{r}$ are
globally controlled in $L^{2}_{t}\dot{H}^{1}$ and $L^{\infty}_{t}L^{p}\cap
L^{2}_{t}\dot{H}^{1}$, for all $p\in[1,\infty]$, respectively.

Here, we establish an asymptotic version of these bounds for (1.1), thereby
justifying (3.12). The key observation in the proof below consists in
considering the term $\frac{1}{c^{2}}\partial_{tt}\Gamma$ in the equation

(4.3)
$\frac{1}{c^{2}}\partial_{tt}\Gamma+\partial_{t}\Gamma+u\cdot\nabla\Gamma-\big{(}\Delta+\frac{\partial_{r}}{r}\big{)}\Gamma=0$

as an error, for large values of $c$. Accordingly, one should treat (4.3) as a
parabolic equation with a vanishing source term, as $c\to\infty$. A more
precise statement of that is given in the next proposition

######  Proposition 4.2.

Let $T\in\mathbb{R}^{+}\cup\\{\infty\\}$ and $(u,E,B)$ be a smooth
axisymmetric solution to (1.1), defined on $[0,T)$, where $u$ and $E$ have no
swirl and $B$ has pure swirl.

Then, it holds that

(4.4)
$\left\|B\right\|_{L^{2}_{t}\dot{H}^{1}}\leq\mathcal{E}_{0}+\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}L^{2}}.$

Moreover, $\Gamma$ enjoys the bounds

(4.5)
$\left\|\Gamma\right\|_{L^{2}_{t}\dot{H}^{1}}\lesssim\left\|\Gamma_{0}\right\|_{L^{2}}+c^{-1}\left\|E_{0}\right\|_{\dot{H}^{2}}+c^{-1}\left(\left\|B\right\|_{L^{\infty}_{t}\dot{H}^{2}}+c\left\|E\right\|_{L^{2}_{t}\dot{H}^{2}}\right),$

and, for any $p\in[2,\infty)$,

(4.6)
$\left\|\Gamma\right\|_{L^{\infty}_{t}L^{p}}\lesssim\left\|\Gamma_{0}\right\|_{L^{p}}+c^{-1}\left\|E_{0}\right\|_{\dot{W}^{2,p}}+\left\|E\right\|_{L^{2}_{t}\dot{W}^{2,p}},$

where all the time norms above are taken over the whole interval $[0,T).$

######  Remark.

Due to Lemma 3.6 and Proposition 4.1, observe that the last terms in the
right-hand side of (4.4) and (4.5) can be seen as errors for large values of
$c$.

Likewise, by virtue of (3.26) and the embedding

$\dot{B}^{\frac{5}{2}}_{2,1}\hookrightarrow\dot{W}^{2,3}(\mathbb{R}^{3}),$

note that (4.6) implies, for $p=3$, that

$\left\|\Gamma\right\|_{L^{\infty}_{t}L^{3}}\lesssim\left\|\Gamma_{0}\right\|_{L^{3}}+c^{-1}\left\|(E_{0},B_{0})\right\|_{\dot{B}^{\frac{5}{2}}_{2,1}}+c^{-\frac{1}{2}}\left\|u\right\|_{L^{\infty}_{t}\dot{H}^{1}\cap
L^{2}_{t}\dot{H}^{2}}\left(\left\|B\right\|_{\widetilde{L}^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,>}}+\left\|B\right\|_{L^{2}_{t}\dot{B}^{\frac{5}{2}}_{2,1,<}}\right).$

This bound will come in handy, later on.

###### Proof.

It is readily seen that, by virtue of the energy inequality (3.1), Ampère’s
equation from (1.1) entails that

$\left\|B\right\|_{L^{2}_{t}\dot{H}^{1}}\leq\left\|j\right\|_{L^{2}_{t}L^{2}}+\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}L^{2}}\leq\mathcal{E}_{0}+\left\|\frac{1}{c}\partial_{t}E\right\|_{L^{2}_{t}L^{2}},$

thereby establishing (4.4).

We now focus on the proof of (4.5) and (4.6). To that end, multiplying (4.3)
by $\Gamma|\Gamma|^{p-2}$, integrating with respect to space variables and
using the identities

$-\int_{\mathbb{R}^{3}}(\Delta\Gamma)\Gamma|\Gamma|^{p-2}dx-\int_{\mathbb{R}^{3}}\frac{\partial_{r}\Gamma}{r}\Gamma|\Gamma|^{p-2}dx=\underbrace{\frac{4(p-1)}{p^{2}}\left\|\nabla(|\Gamma(t)|^{\frac{p}{2}})\right\|_{L^{2}}^{2}+\frac{2\pi}{p}\int_{\mathbb{R}}|\Gamma(t,r=0,z)|^{p}dz}_{\overset{\rm
def}{=}X_{p}(t)}$

and

$\int_{\mathbb{R}^{3}}(\partial_{tt}\Gamma)\Gamma|\Gamma|^{p-2}dx=\partial_{tt}\Big{(}\underbrace{\frac{1}{p}\left\|\Gamma(t)\right\|_{L^{p}}^{p}}_{\overset{\rm
def}{=}Y_{p}(t)}\Big{)}-\underbrace{(p-1)\int_{\mathbb{R}^{3}}|\partial_{t}\Gamma|^{2}|\Gamma|^{p-2}dx}_{\overset{\rm
def}{=}A_{p}(t)},$

yields, for all $t\in[0,T)$, that

(4.7)
$\frac{1}{c^{2}}Y^{\prime\prime}_{p}(t)+Y^{\prime}_{p}(t)+X_{p}(t)=\frac{1}{c^{2}}A_{p}(t).$

Therefore, integrating in time, we obtain that

(4.8)
$Y_{p}(t)+\int_{0}^{t}X_{p}(\tau)d\tau=Y_{p}(0)+\frac{1}{c^{2}}Y_{p}^{\prime}(0)-\frac{1}{c^{2}}Y_{p}^{\prime}(t)+\frac{1}{c^{2}}\int_{0}^{t}A_{p}(\tau)d\tau.$

Now, we need to take care of the term $-\frac{1}{c^{2}}Y_{p}^{\prime}(t)$,
above. To that end, observing that (4.7) can be rewritten as

$\frac{1}{c^{2}}\frac{d}{dt}\Big{(}Y_{p}^{\prime}(t)e^{c^{2}t}\Big{)}=\Big{(}\frac{1}{c^{2}}A_{p}(t)-X_{p}(t),\Big{)}e^{c^{2}t},$

which we then integrate with respect to the time variable, we find that

$-\frac{1}{c^{2}}Y^{\prime}_{p}(t)=-\frac{1}{c^{2}}Y_{p}^{\prime}(0)e^{-c^{2}t}+\int_{0}^{t}e^{-c^{2}(t-\tau)}\Big{(}X_{p}(\tau)-\frac{1}{c^{2}}A_{p}(\tau)\Big{)}d\tau.$

Hence, plugging the latter identity into (4.8) yields that

(4.9) $\displaystyle Y_{p}(t)+\int_{0}^{t}X_{p}(\tau)d\tau$
$\displaystyle=Y_{p}(0)+\frac{1-e^{-c^{2}t}}{c^{2}}Y_{p}^{\prime}(0)+\int_{0}^{t}e^{-c^{2}(t-\tau)}X_{p}(\tau)d\tau$
$\displaystyle\quad+\frac{1}{c^{2}}\int_{0}^{t}A_{p}(\tau)d\tau-\frac{1}{c^{2}}\int_{0}^{t}e^{-c^{2}(t-\tau)}A_{p}(\tau)d\tau.$

Next, utilizing Faraday’s equation from (1.1), we write, for any $t\in(0,T)$,
that

(4.10)
$\left(\partial_{t}\Gamma\right)=\left(\frac{\partial_{t}B}{r}\right)\cdot
e_{\theta}=-c\left(\frac{\nabla\times E}{r}\right)\cdot e_{\theta},$

whereby

$\left(\partial_{t}\Gamma\right)|_{t=0}=-c\left(\frac{\nabla\times
E_{0}}{r}\right)\cdot e_{\theta}.$

Accordingly, it follows that

$\displaystyle\frac{1}{c^{2}}|Y^{\prime}_{p}(0)|=\frac{1}{c^{2}}\left|\Big{(}\int_{\mathbb{R}^{3}}(\partial_{t}\Gamma)\Gamma|\Gamma|^{p-2}dx\Big{)}_{|_{t=0}}\right|$
$\displaystyle\leq\frac{1}{c}\int_{\mathbb{R}^{3}}\frac{|\nabla\times
E_{0}|}{r}|\Gamma_{0}|^{p-1}dx$
$\displaystyle\quad\leq\frac{1}{c}\left\|\frac{\nabla\times
E_{0}}{r}\right\|_{L^{p}}\left\|\Gamma_{0}\right\|_{L^{p}}^{p-1}.$

Thus, recalling the inequality

(4.11)
$ab\leq\varepsilon^{\alpha}\frac{a^{\alpha}}{\alpha}+\varepsilon^{-\alpha^{\prime}}\frac{b^{\alpha^{\prime}}}{\alpha^{\prime}},$

for any $\varepsilon>0$ and $\alpha\in(1,\infty)$, where $\alpha^{\prime}$
denotes the conjugate of $\alpha$, and by virtue of Lemma 2.2, we end up with

(4.12) $\displaystyle\frac{1}{c^{2}}|Y^{\prime}_{p}(0)|$
$\displaystyle\leq\left(\left(\frac{1}{c}\left\|\frac{\nabla\times
E_{0}}{r}\right\|_{L^{p}}\right)^{\frac{1}{p}}\left\|\Gamma_{0}\right\|_{L^{p}}^{\frac{1}{p^{\prime}}}\right)^{p}$
$\displaystyle\leq\left(\frac{1}{pc}\left\|\frac{\nabla\times
E_{0}}{r}\right\|_{L^{p}}+\frac{1}{p^{\prime}}\left\|\Gamma_{0}\right\|_{L^{p}}\right)^{p}$
$\displaystyle\lesssim\Big{(}\frac{1}{c}\left\|E_{0}\right\|_{\dot{W}^{2,p}}+\left\|\Gamma_{0}\right\|_{L^{p}}\Big{)}^{p}.$

Consequently, for $p=2$, we find that (4.9) entails that

$\left\|\Gamma(t)\right\|_{L^{2}}^{2}\lesssim\left\|\Gamma_{0}\right\|_{L^{2}}^{2}+c^{-2}\left(\left\|E_{0}\right\|_{\dot{H}^{2}}+\left\|\partial_{t}\Gamma\right\|_{L^{2}_{t}L^{2}}\right)^{2}$

and

$\left\|\Gamma\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}\lesssim\left\|\Gamma_{0}\right\|_{L^{2}}^{2}+c^{-2}\left(\left\|E_{0}\right\|_{\dot{H}^{2}}+\left\|\Gamma\right\|_{L^{\infty}_{t}\dot{H}^{1}}+\left\|\partial_{t}\Gamma\right\|_{L^{2}_{t}L^{2}}\right)^{2}.$

Finally, employing (4.10) and Lemma 2.2 in the last bound leads to (4.5), and
(4.6) in the case $p=2$.

Then, in order to deal with the range $p\in(2,\infty)$, we deduce from (4.9)
that

$\left\|\Gamma\right\|_{L^{\infty}_{t}L^{p}}^{p}\lesssim\left\|\Gamma_{0}\right\|_{L^{\infty}_{t}(L^{p})}^{p}+\frac{1}{c^{2}}|Y^{\prime}_{p}(0)|+\frac{1}{c^{2}}\int_{0}^{t}\left\|\partial_{t}\Gamma(\tau)\right\|_{L^{p}}^{2}\left\|\Gamma(\tau)\right\|_{L^{p}}^{p-2}d\tau.$

Therefore, employing (4.12) to control $Y^{\prime}_{p}(0)$ yields

$\left\|\Gamma\right\|_{L^{\infty}_{t}L^{p}}^{p}\lesssim\left\|\Gamma_{0}\right\|_{L^{\infty}_{t}L^{p}}^{p}+\Big{(}\frac{1}{c}\left\|E_{0}\right\|_{\dot{W}^{2,p}}+\left\|\Gamma_{0}\right\|_{L^{p}}\Big{)}^{p}+\frac{1}{c^{2}}\left\|\partial_{t}\Gamma\right\|_{L^{2}_{t}L^{p}}^{2}\left\|\Gamma\right\|_{L^{\infty}_{t}L^{p}}^{p-2}.$

After that, we utilize (4.11) to obtain that

$\left\|\Gamma\right\|_{L^{\infty}_{t}L^{p}}^{p}\lesssim\left\|\Gamma_{0}\right\|_{L^{\infty}_{t}L^{p}}^{p}+\Big{(}\frac{1}{c}\left\|E_{0}\right\|_{\dot{W}^{2,p}}+\left\|\Gamma_{0}\right\|_{L^{p}}\Big{)}^{p}+\frac{1}{c^{p}}\left\|\partial_{t}\Gamma\right\|_{L^{2}_{t}L^{p}}^{p}.$

At last, by using (4.10) and Lemma 2.2, again, we arrive at the desired bound
(4.6). This completes the proof of the proposition. ∎

##  5\. Closing the estimates and proof of Theorem 1.1

We are now ready to establish the final global bounds which lead to the
existence of global solutions of the Navier–Stokes–Maxwell equations (1.1).
This is going to be done, first, by gathering all the estimates from the
previous sections to produce a nonlinear energy estimate. Then, under adequate
assumptions in the regime $c\to\infty$, the nonlinear energy bound will allow
us to deduce the desired global control, uniformly with respect to the speed
of light.

Throughout this section, for the sake of simplicity and clarity, the subscript
“$t$” that appears in time-Lebesgue spaces $L^{p}_{t}$ should be understood as
the endpoint of the time interval, i.e., for example, we will be using the
notation

$\left\|f\right\|_{L^{p}_{t}\dot{H}^{s}}\overset{\rm
def}{=}\left\|f\right\|_{L^{p}\left([0,t);\dot{H}^{s}(\mathbb{R}^{3})\right)}.$

Accordingly, all the quantities involving time-Lebesgue norms are continuous
functions on the real half-line $[0,\infty)$, for all $p\in[1,\infty]$.

###  5.1. Approximation scheme and compactness

Solutions of (1.1) will be constructed by a standard compactness method based
on the smooth approximation of solutions. Although this is now classical in
the literature, we briefly recall here the principle ideas in that method.
This will allow us to justify all the formal computations in the derivation of
our global a priori bounds, below.

We follow the approach laid out in [3, Section 3.1], for instance. Thus, we
begin with approximating the Navier–Stokes–Maxwell equations (1.1) by a new
system of equations which has a unique smooth solution.

An admissible approximation should preserve the structures satisfied by the
original system such as, in our case, the energy inequality (1.2) and the
axisymmetric properties. A possible choice of that approximation is given, for
$n\in\mathbb{N}$, by

$\begin{cases}\begin{aligned} &\partial_{t}u_{n}+(S_{n}u_{n})\cdot\nabla
u_{n}=\nu\Delta u_{n}-\nabla p_{n}+j_{n}\times
B_{n},&\operatorname{div}u_{n}=0,&\\\
&\frac{1}{c}\partial_{t}E_{n}-\nabla\times
B_{n}=-(S_{n}j_{n}),&\operatorname{div}E_{n}=0,&\\\
&\frac{1}{c}\partial_{t}B_{n}+\nabla\times
E_{n}=0,&\operatorname{div}B_{n}=0,&\\\
&j_{n}=\sigma\big{(}cE_{n}+S_{n}P(u_{n}\times
B_{n})\big{)},&\operatorname{div}j_{n}=0,&\end{aligned}\end{cases}$

supplemented with the initial data $(u_{n},E_{n},B_{n})|_{t=0}\overset{\rm
def}{=}S_{n}(u_{0},E_{0},B_{0})$, where $S_{n}$ is a radial cutoff Fourier
multiplier which restricts the frequencies to the set $\\{|\xi|\leq 2^{n}\\}$
and converges to the identity as $n\to\infty$.

Showing that the approximate system, for any fixed $n\in\mathbb{N}$, has a
unique global solution is a routine procedure based on standard methods.
Moreover, one can show that the corresponding solutions are smooth in time and
space and satisfy the energy inequality

$\left\|\left(u_{n},E_{n},B_{n}\right)(t)\right\|_{L^{2}}^{2}+2\nu\int_{0}^{t}\left\|\nabla
u_{n}(\tau)\right\|_{L^{2}}^{2}d\tau+\frac{2}{\sigma}\int_{0}^{t}\left\|j_{n}(\tau)\right\|_{L^{2}}^{2}d\tau=\left\|S_{n}(u_{0},E_{0},B_{0})\right\|_{L^{2}}^{2}\leq\mathcal{E}_{0}^{2},$

where we recall that

$\mathcal{E}_{0}=\left\|(u_{0},E_{0},B_{0})\right\|_{L^{2}}.$

If, furthermore, the initial data are assumed to be axisymmetric, and the
cutoff operator $S_{n}$ does not alter that structure (which is the case when
$S_{n}$ can be characterized as a convolution with a radial function), then
the approximate smooth solution remains axisymmetric for all times.

Noting, once again, that the energy inequality above is not sufficient to
ensure the stability, as $n\to\infty$, of the nonlinear term

$j_{n}\times B_{n},$

our strategy thus consists in looking for better bounds in higher regularity
spaces, uniformly with respect to the regularizing parameter $n$. In
particular, we will obtain new bounds which will allow us to establish the
strong relative compactness of all vector field $u_{n}$, $E_{n}$, $B_{n}$ and
$j_{n}$ in $L^{2}_{\text{loc},t,x}$ and, then, conclude that the approximate
solutions converge, as $n\to\infty$, to an exact solution of (1.1).

With such strong bounds, the full justification of the stability of the
approximate system follows from standard compactness arguments, which we will
therefore omit. We refer to [3] for some details on similar arguments applied
to construction of global solutions of the Euler–Maxwell system in two
dimensions of space.

Normally, we should prove the a priori estimates on the approximate system
above. However, as usual, since the approximate system enjoys the same
structure as the original one, we will, from now on, assume that the solutions
to (1.1) are smooth and we will perform all estimates directly on (1.1).

###  5.2. Weak–strong uniqueness

The functional spaces used in Theorem 1.1 are sufficient to prove uniqueness
results for (1.1). Although this is not hard to show, we choose to provide in
the following proposition a weak–strong stability result with a self-contained
proof which covers the uniqueness of solutions claimed in Theorem 1.1. See
also [6, Proposition 3.11] for a similar weak–strong principle for the same
system.

######  Proposition 5.1 ($L^{2}$ weak–strong stability).

Let $c>0$ and $(u_{i},E_{i},B_{i})_{i\in\\{1,2\\}}$ be two weak solutions of
(1.1) associated with the same initial data and satisfying the energy
inequality (1.2). Assume moreover that

$u_{2}\in L^{2}_{\mathrm{loc}}(\mathbb{R}^{+};L^{\infty}),\qquad E_{2}\in
L^{2}_{\mathrm{loc}}(\mathbb{R}^{+};L^{3}),\qquad B_{2}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{+};L^{3}).$

Then, the two solutions are equal.

###### Proof.

We define the difference of the two solutions by