# On illposedness of the Hall and electron magnetohydrodynamic equations
without resistivity on the whole space

In-Jee Jeong  Department of Mathematical Sciences and RIM, Seoul National
University, and School of Mathematics, Korea Institute for Advanced Study,
Seoul, Republic of Korea<EMAIL_ADDRESS>and  Sung-Jin Oh  Department of
Mathematics, UC Berkeley, Berkeley (CA), USA and School of Mathematics, Korea
Institute for Advanced Study, Seoul, Republic of Korea.
<EMAIL_ADDRESS>
###### Abstract.

It has been shown in our previous work [19] that the incompressible and
irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are
illposed on flat domains $M=\mathbb{R}^{k}\times\mathbb{T}^{3-k}$ for $0\leq
k\leq 2$. The data and solutions therein were assumed to be independent of one
coordinate, which not only significantly simplifies the systems but also
allows for a large class of steady states. In this work, we remove the
assumption of independence and conclude strong illposedness for compactly
supported data in $\mathbb{R}^{3}$. This is achieved by constructing
degenerating wave packets for linearized systems around time-dependent
axisymmetric magnetic fields. A few main additional ingredients are: a more
systematic application of the generalized energy estimate, use of the
Bogovskiǐ operator, and a priori estimates for axisymmetric solutions to the
Hall- and electron-MHD systems.

##  1\. Introduction

###  1.1. The systems

In this paper, we are concerned with the Cauchy problem for the incompressible
_Hall-magnetohydrodynamics_ (Hall-MHD for short) equations, which describe the
motion of a plasma whose electron speed is significantly faster than the ion
speed. Still, the equations give a single-fluid description of the plasma with
${\bf u}(t):M\rightarrow\mathbb{R}^{3}$ as the so-called bulk plasma velocity.
Together with the magnetic field ${\bf B}(t):M\rightarrow\mathbb{R}^{3}$, the
equations are given by

(Hall-MHD) $\left\\{\begin{aligned} &\partial_{t}{\bf u}+{\bf
u}\cdot\nabla{\bf u}+\nabla{\bf p}-\nu\Delta{\bf u}=(\nabla\times{\bf
B})\times{\bf B},\\\ &\partial_{t}{\bf B}-\nabla\times({\bf u}\times{\bf
B})+\nabla\times((\nabla\times{\bf B})\times{\bf B})=0,\\\ &\nabla\cdot{\bf
u}=\nabla\cdot{\bf B}=0.\end{aligned}\right.$

Here, ${\bf p}(t):M\rightarrow\mathbb{R}$ is the plasma pressure and $\nu\geq
0$ is the plasma viscosity. We assume that the domain is given by a flat
manifold $M=\mathbb{T}^{k}\times\mathbb{R}^{3-k}$ ($0\leq k\leq 3$). Note that
there is no dissipation term for the magnetic field (i.e. no magnetic
resistivity), which will be assumed throughout the paper. The special case
$\nu=0$ is called the _ideal Hall-MHD equation_. This is just the usual ideal
MHD equation with an extra term $\nabla\times((\nabla\times{\bf B})\times{\bf
B})$ on the left hand side, which is referred to as the Hall current term.
This modification over the usual MHD system was suggested by Lighthill [24] to
take into account the disparity between the electron speed and the ion speed,
and is known to be directly responsible for various physical phenomena
including collisionless magnetic reconnection, planetary magnetospheres, and
magnetic field dynamics in neutron stars ([3, 28, 17, 11, 32, 29, 27, 10, 25,
15, 22, 16, 30]). We shall also consider the simpler variant obtained by
formally taking ${\bf u}=0$, called the electron-MHD (E-MHD for short)
equations:

(E-MHD) $\left\\{\begin{aligned} &\partial_{t}{\bf
B}+\nabla\times((\nabla\times{\bf B})\times{\bf B})=0,\\\ &\nabla\cdot{\bf
B}=0.\end{aligned}\right.$

It turns out that, as long as the issue of local wellposedness is concerned,
the reduced system (E-MHD) serves as a good approximation for the more
realistic (Hall-MHD). Due to the potential loss of derivatives present in the
Hall current term, the basic question of local wellposedness had been open in
the irresistive case. In [19] we have established that the Cauchy problem for
the equations (Hall-MHD) and (E-MHD) are strongly illposed in arbitrarily high
Sobolev spaces for smooth initial data, under the simplifying assumption that
the data is independent of one coordinate (this will be always assumed to be
the $z$-direction). This in particular forced the domain $M$ to have at least
one component of $\mathbb{T}$, for the solutions to have finite energy. The
resulting simplified equations are commonly referred to as $2+\frac{1}{2}$
dimensional systems, and have been extensively investigated both in physical
and mathematical literature ([2, 26, 31, 8, 12, 9, 19]).

The goal of the current paper is to remove the assumption of $z$-independence
and conclude illposedness of the Cauchy problem for (Hall-MHD) and (E-MHD) in
the remaining case of $M=\mathbb{R}^{3}$. A simple version of our main result
is as follows:

######  Theorem 1.1 (Nonlinear illposedness, nonexistence, simple version).

  1. (1)

For any $s>\frac{7}{2}$ and $\epsilon>0$, there exists ${\bf B}_{0}\in
H^{s}(\mathbb{R}^{3})$ satisfying $\nabla\cdot{\bf B}_{0}=0$ and $\|{{\bf
B}_{0}}\|_{H^{s}}<\epsilon$ such that there is no associated solution to
(E-MHD) in $L^{\infty}_{t}([0,\delta];H^{s})$ with initial data ${\bf
B}(t=0)={\bf B}_{0}$ for any $\delta>0$.

  2. (2)

For any $s>\frac{7}{2}$ and $\epsilon>0$, there exists ${\bf B}_{0}\in
H^{s}(\mathbb{R}^{3})$ satisfying $\nabla\cdot{\bf B}_{0}=0$ and $\|{{\bf
B}_{0}}\|_{H^{s}}<\epsilon$ such that there is no solution to (Hall-MHD) in
$L^{\infty}_{t}([0,\delta];H^{s})$ with initial data $({\bf u},{\bf
B})(t=0)=(0,{\bf B}_{0})$ for any $\delta>0$.

The precise statements of the main theorems may be found in Section 1.5 below.
We remark that in the case of (E-MHD), we may even take ${\bf B}_{0}$ to be
compactly supported. Our nonlinear illposedness results (Theorems B* and C*)
are based on quantitative illposedness around time-dependent degenerate
axisymmetric magnetic fields (Theorem A*).

The remainder of the introduction is structured as follows. After reviewing
the setup and results from [19] in Sections 1.2 and 1.3, we introduce the
aforementioned class of time-dependent degenerate axisymmetric solutions to
(E-MHD) and (Hall-MHD) in Section 1.4. Then in Section 1.5, we state the
precise versions of the main theorems, and in Section 1.6, we discuss the main
ideas of the proof. Finally, in Section 1.7, we collect some of the notation
and conventions used throughout the paper and conclude with an outline of the
paper.

###  1.2. Energy identities, stationary solutions and linearization

A fundamental property satisfied by (Hall-MHD) and (E-MHD) is the energy
identity. In the following proposition, it will be assumed that
$M=\mathbb{T}^{k}\times\mathbb{R}^{3-k}$ and the solutions have sufficient
smoothness and decay at spatial infinity.

######  Proposition 1.2.

For a solution $({\bf u},{\bf B})$ to (Hall-MHD), we have

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{M}(|{{\bf
u}}|^{2}+|{{\bf
B}}|^{2})(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)=-\nu\int_{M}|{\nabla{\bf
u}}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

Similarly, for a solution ${\bf B}$ to (E-MHD), we have

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{M}|{{\bf
B}}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)=0.$

The proofs of the above identities are straightforward; one simply multiply
the equations for ${\bf u}$ and ${\bf B}$ by ${\bf u}$ and ${\bf B}$
respectively, integrate over the domain $M$, and use integration by parts.
Instead of supplying a proof, we just note that the Hall current term drops
out by the symmetric nature of $\nabla\times$:

$\int_{M}{\bf B}\cdot(\nabla\times((\nabla\times{\bf B})\times{\bf
B})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=\int_{M}(\nabla\times{\bf
B})\cdot((\nabla\times{\bf B})\times{\bf
B})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=0.$

This symmetry in general allows us to gain one derivative back from the Hall
current term; if one attempts to control $\|{\partial^{(N)}{\bf
B}(t)}\|_{L^{2}}$ for a solution ${\bf B}$ to (E-MHD) ($\partial^{(N)}$ refers
to an $N$-th order spatial derivative), we have from the Hall term a
contribution of the form

(1.1) $\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{M}|{\partial^{(N)}{\bf
B}}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=-\int_{M}(\nabla\times\partial^{(N)}{\bf
B})\cdot((\nabla\times{\bf B})\times\partial^{(N)}{\bf
B})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z+\cdots$

where the other terms only involve up to $N$ derivatives of ${\bf B}$. It is
not clear how to handle this integral on the right hand side, and the main
results of this paper and [19] confirms that there are situations where this
expression is an actual loss of a derivative. (In contrast to equations for
which the Cauchy–Kovalevskaya theorem is applicable, the situation is not
better at all in analytic or even in Gevrey regularity classes; see [19,
Section 6].)

While the systems (Hall-MHD) and (E-MHD) possess a large set of stationary
solutions, a special class of them played a crucial role in [19], namely
planar stationary magnetic fields with an additional symmetry. We first recall
the definition:

  * •

(Stationary magnetic field) The solution is of the form ${\bf
B}=\mathring{{\bf B}}$ for (E-MHD), and $({\bf u},{\bf B})=(0,\mathring{{\bf
B}})$ for (Hall-MHD), where $\mathring{{\bf B}}$ is a $t$-independent vector
field on $\mathbb{R}^{3}$ such that $\nabla\cdot\mathring{{\bf B}}=0$
(divergence-free) and $(\nabla\times\mathring{{\bf B}})\times\mathring{{\bf
B}}$ is a pure gradient.

  * •

(Planarity) $\mathring{{\bf B}}$ is independent of the $z$-coordinate and
$\mathring{{\bf B}}^{z}$ = 0.

  * •

(Additional symmetry) $\mathring{{\bf B}}=\mathring{{\bf
B}}^{x}\partial_{x}+\mathring{{\bf B}}^{y}\partial_{y}$, viewed as a vector
field on $\mathbb{R}^{2}_{x,y}$, is invariant under a one-parameter family of
isometries of $\mathbb{R}^{2}_{x,y}$.

The following classification result can be shown:

######  Proposition 1.3 ([19, Section 2.2]).

A smooth planar stationary magnetic field with an additional symmetry is, up
to symmetries, one of the following forms: ($f,g$ are smooth and
$c_{0},c_{1},d\in\mathbb{R}$)

$\mathring{{\bf
B}}=f(y)\partial_{x},\quad(c_{1}y+c_{0})\partial_{x}+d\partial_{y},\quad
g(x^{2}+y^{2})(x\partial_{y}-y\partial_{x}).$

####  1.2.1. Linearized systems around stationary solutions

Given a stationary solution to (Hall-MHD) of the form $(0,\mathring{{\bf
B}})$, let us consider perturbations of the form $({\bf u},{\bf
B})=(u,\mathring{{\bf B}}+b)$. The linearized equation satisfied by $(u,b)$
(i.e., the linearization of (Hall-MHD) around $\mathring{{\bf B}}$) is:

(1.2) $\left\\{\begin{aligned} &\partial_{t}u-\nu\Delta
u=\mathbb{P}((\nabla\times\mathring{{\bf B}})\times b+(\nabla\times
b)\times\mathring{{\bf B}})\\\
&\partial_{t}b+\nabla\times(u\times\mathring{{\bf
B}})+\nabla\times((\nabla\times b)\times\mathring{{\bf
B}})+\nabla\times((\nabla\times\mathring{{\bf B}})\times b)=0,\\\ &\nabla\cdot
u=\nabla\cdot b=0,\end{aligned}\right.$

where $\mathbb{P}$ is the Leray projection operator onto divergence-free
vector fields. In the case of (E-MHD), the linearization is simply given by

(1.3) $\left\\{\begin{aligned} &\partial_{t}b+\nabla\times((\nabla\times
b)\times\mathring{{\bf B}})+\nabla\times((\nabla\times\mathring{{\bf
B}})\times b)=0,\\\ &\nabla\cdot b=0.\end{aligned}\right.$

The $L^{2}$-norm of the perturbation is (formally) under control, for the
linearized systems.

######  Proposition 1.4.

For a sufficiently regular and decaying solution $(u,b)$ to (1.2), we have

(1.4)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{M}|{u}|^{2}(t)+|{b}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)+\nu\int_{M}|{\nabla
u}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle=\int_{M}((b\cdot\nabla)\mathring{{\bf
B}}_{j})u^{j}-((u\cdot\nabla)\mathring{{\bf
B}}_{j})b^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z+\int_{M}((b\cdot\nabla)(\nabla\times\mathring{{\bf
B}})_{j})b^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

Similarly, for a sufficiently regular and decaying solution $b$ to (1.3), we
have

(1.5)
$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{M}|{b}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)=\int_{M}((b\cdot\nabla)(\nabla\times\mathring{{\bf
B}})_{j})b^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

####  1.2.2. Notion of an $L^{2}$-solution.

The above energy identities stated above suggest that any “reasonable”
solution to the linearized systems would enjoy good local-in-time
$L^{2}$-bounds. This motivates the following definition for an
$L^{2}$-solution: on an interval of time $I$, an $L^{2}$-solution means:

  * •

(linearized (Hall-MHD) with $\nu>0$) a pair of vector fields $(u,b)$ such that
$u\in C_{w}(I;L^{2})\cap L^{2}_{t}(I;\dot{H}^{1})$ and $b\in C_{w}(I;L^{2})$
that satisfies (1.2) in the sense of distributions;

  * •

(linearized (Hall-MHD) with $\nu=0$) a pair of vector fields $(u,b)\in
C_{w}(I;L^{2})$ that satisfies (1.2) with $\nu=0$ in the sense of
distributions; or

  * •

(linearized (E-MHD)) a vector field $b\in C_{w}(I;L^{2})$ that satisfies (1.3)
in the sense of distributions.

In the case $M=\mathbb{T}^{3}$, we assume that

(1.6) $\int_{M}u(t)=\int_{M}b(t)=0\quad\hbox{ for all }t\in I,$

where we ignore the condition for $u$ in the case of (E-MHD). Given an initial
data $(u_{0},b_{0})\in L^{2}$, it is not difficult to show existence of an
$L^{2}$-solution $(u,b)$ for (1.2) defined in the sense above (similarly, an
$L^{2}$-solution $b$ given $b_{0}\in L^{2}$ for (1.3)), see [19, Appendix A].
We can further demand that the solution satisfies

$\begin{split}\frac{1}{2}\left(\|{u(t)}\|_{L^{2}(M)}^{2}+\|{b(t)}\|_{L^{2}(M)}^{2}\right)+\nu\|{u}\|_{L^{2}([0,t];\dot{H}^{1})}^{2}\leq\frac{1}{2}\left(\|{u_{0}}\|_{L^{2}(M)}^{2}+\|{b_{0}}\|_{L^{2}(M)}^{2}\right)e^{Ct\|{\nabla\mathring{{\bf
B}}}\|_{C^{1}(M)}}\end{split}$

for all $t>0$ and

$\begin{split}\frac{1}{2}\|{b(t)}\|_{L^{2}(M)}^{2}\leq\frac{1}{2}\|{b_{0}}\|_{L^{2}(M)}^{2}e^{Ct\|{\nabla^{2}\mathring{{\bf
B}}}\|_{L^{\infty}(M)}}\end{split}$

for (1.2) and (1.3), respectively. These estimates are natural in view of
Proposition 1.4. For the simplicity of presentation we shall assume that our
$L^{2}$-solution satisfies these additional estimates.

###  1.3. Illposedness results under $z$-independence

Let us review the linear illposedness results under the assumptions of
$z$-independence. We shall take our domain to be periodic in $z$ and assume
that all the functions are independent of $z$. In particular, $H^{m}$ and
$\nabla$ are identified with $H^{m}_{x,y}$ and $\nabla_{x,y}$, respectively.

####  1.3.1. Assumptions on the background magnetic fields.

Two distinct classes of stationary magnetic fields were considered in [19],
translation symmetric and axisymmetric ones. In the former, we take the domain
$M_{2}=(\mathbb{T},\mathbb{R})_{x}\times(\mathbb{T},\mathbb{R})_{y}$111The
notation $(\mathbb{T},\mathbb{R})_{x}$ means that both $\mathbb{T}_{x}$ and
$\mathbb{R}_{x}$ are allowed.  and set $\mathring{{\bf B}}=f(y)\partial_{x}.$
In the latter, we take $M_{2}=\mathbb{R}_{x,y}$ and $\mathring{{\bf
B}}=f(r)\partial_{\theta}=f(\sqrt{x^{2}+y^{2}})(x\partial_{y}-y\partial_{x}).$
In both cases, it was assumed that $f$ and its derivatives are uniformly
bounded and $f$ is linearly degenerate in the sense that there exists
$r_{0}\neq 0$ such that $f(r_{0})=0$, $f^{\prime}(r_{0})\neq 0$ (in the
axisymmetric case).

####  1.3.2. Linear illposedness: Growth of $L^{2}$-solutions in higher
Sobolev spaces.

We are now in a position to recall the main linear result from [19]:

######  Theorem A (Sharp norm growth).

Take the stationary, planar, and linearly degenerate magnetic field
$\mathring{{\bf B}}$ on $M=M_{2}\times\mathbb{T}_{z}$ as described in the
above. Let $s\geq 0,p\geq 2$ satisfy $s+1/p>1/2$. Then the following
statements hold.

  1. (1)

Consider the linearized (Hall-MHD) with $\nu\geq 0$ around $\mathring{{\bf
B}}$ on a time interval $I\ni 0$. For each $\lambda\in\mathbb{N}$, there
exists an initial data set of the form

     * •

(Case (a): translationally-symmetric background)

$u_{0}=0,\quad b_{0}=\operatorname{Re}(e^{i(\lambda
x+\lambda\mathfrak{G}(y))})\mathfrak{b}(x,y)$

where $G(y)\in C^{\infty}((\mathbb{T},\mathbb{R})_{y})$ and
$\mathfrak{b}(x,y)\in\mathcal{S}((\mathbb{T},\mathbb{R})_{x}\times(\mathbb{T},\mathbb{R})_{y})$
with compact support in $y$ and either compact support in $x$ or real-
analyticity in $x$; or

     * •

(Case (b): axisymmetric background)

$u_{0}=0,\quad
b_{0}=\operatorname{Re}(e^{i(\lambda\theta+\lambda\mathfrak{G}(r))})\mathfrak{b}(r)$

where $G(r)\in C^{\infty}((0,\infty))$ and
$\mathfrak{b}(r)\in\mathcal{C}^{\infty}((0,\infty))$ with compact support in
$r$,

and $\delta>0$ depending only on $\mathring{{\bf B}}$ such that any
corresponding $z$-independent $L^{2}$-solution $(u,b)$ exhibits norm growth of
the form

$\|{b(t)}\|_{W^{s,p}(M)}\geq c_{s,p,\mathring{{\bf B}}}e^{c_{0}(\mathring{{\bf
B}})\cdot(s+\frac{1}{2}-\frac{1}{p})\lambda t}\|{b_{0}}\|_{W^{s,p}(M)}$

for $t\in I$ satisfying $0\leq t<\delta$.

  2. (2)

Consider the linearized (E-MHD) around the stationary solution $\mathring{{\bf
B}}$ on a time interval $I\ni 0$. For each $\lambda\in\mathbb{N}$, there
exists an initial data set of the form

     * •

(Case (a): translationally-symmetric background)

$\quad b_{0}=\operatorname{Re}(e^{i(\lambda
x+\lambda\mathfrak{G}(y)})\mathfrak{b}$

where $G(y)$ and $\mathfrak{b}(x,y)$ are as in part (1); or

     * •

(Case (b): axisymmetric background)

$b_{0}=\operatorname{Re}(e^{i(\lambda\theta+\lambda\mathfrak{G}(r))})\mathfrak{b}(r)$

where $G(r)$ and $\mathfrak{b}(r)$ are as in part (1),

and $\delta>0$ depending only on $\mathring{{\bf B}}$ such that any
corresponding $z$-independent $L^{2}$-solution $b$ on $I$ exhibits norm growth
of the form

$\|{b(t)}\|_{W^{s,p}(M)}\geq c_{s,p,\mathring{{\bf B}}}e^{c_{0}(\mathring{{\bf
B}})\cdot(s+\frac{1}{2}-\frac{1}{p})\lambda t}\|{b_{0}}\|_{W^{s,p}(M)}$

for $t\in I$ satisfying $0\leq t<\delta$.

The choice of profiles $\mathfrak{b}$ and $\mathfrak{G}$ is explicit and will
be explained in the next section. Note that in the above statements,
$z$-independence was assumed for the $L^{2}$-solution. This extra assumption
can be removed as a simple consequence of the results from this paper.

####  1.3.3. Nonlinear illposedness results under $z$-independence

We now describe streamlined versions of main nonlinear illposedness results
obtained in the work [19]. There are two main statements. The first shows
unboundedness of the solution operator, assuming that the solution is well-
defined. This in particular shows failure of the continuity of the solution
map. The second result shows nonexistence for the Cauchy problem, which may be
considered as the strongest notion of illposedness.

We shall need some additional notations to describe the results: let us denote
a function-space ball of radius $\epsilon$ with respect to a norm
$\|{\cdot}\|_{X}$ centered at ${\bf x}$ by

$\mathcal{B}_{\epsilon}({\bf x};X)=\\{{\bf y}\in{\bf x}+X:\|{{\bf y}-{\bf
x}}\|_{X}<\epsilon\\},$

and its restriction to compactly supported functions by

$\mathcal{B}_{\epsilon}({\bf x};X_{comp})=\\{{\bf
y}\in\mathcal{B}_{\epsilon}({\bf x};X):{\bf y}-{\bf x}\hbox{ has compact
support in $M$}\\}.$

######  Theorem B (Unboundedness of the solution map).

Let
$M=(\mathbb{T},\mathbb{R})_{x}\times(\mathbb{T},\mathbb{R})_{y}\times\mathbb{T}_{z}$
and the stationary magnetic field $\mathring{{\bf B}}$ is given either by
$f(y)\partial_{x}$ or $f(r)\partial_{\theta}$ as in Theorem A. Assume that for
some $\epsilon,\delta,r,s,s_{0}>0$, the solution map for (Hall-MHD) (resp.
(E-MHD)) exists as a map

$\displaystyle\mathcal{B}_{\epsilon}((0,\mathring{{\bf B}});H^{r}_{comp}\times
H^{s}_{comp})\to L^{\infty}_{t}([0,\delta];H^{s_{0}-1})\times
L^{\infty}_{t}([0,\delta];H^{s_{0}})$ $\displaystyle\left(\hbox{resp.
}\mathcal{B}_{\epsilon}(\mathring{{\bf B}};H^{s}_{comp})\to
L^{\infty}_{t}([0,\delta];H^{s_{0}})\right).$

Then this solution map is unbounded for $s_{0}\geq 3$.

Moreover, since $\mathring{{\bf B}}$ can be taken arbitrarily close to 0 in
$H^{s}_{comp}$, the same statement holds with $\mathring{{\bf B}}\equiv 0$;
that is, the solution map, if well-defined, is unbounded near the trivial
solution for $s_{0}\geq 3$.

We now state the nonexistence statement.

######  Theorem C (Nonexistence near $0$).

Let $s>7/2$ and
$M=(\mathbb{T},\mathbb{R})_{x}\times\mathbb{R}_{y}\times\mathbb{T}_{z}$ in the
case of (Hall-MHD) and
$M=(\mathbb{T},\mathbb{R})_{x}\times(\mathbb{T},\mathbb{R})_{y}\times\mathbb{T}_{z}$
for (E-MHD). Given any $\epsilon>0$, there exist initial data
$(\mathbf{u}_{0},\mathbf{B}_{0})\in H^{s-1}_{comp}\times H^{s}_{comp}$ for
(Hall-MHD) satisfying
$\|{\mathbf{u}_{0}}\|_{H^{s-1}}+\|{\mathbf{B}_{0}}\|_{H^{s}}<\epsilon$ (resp.
$\mathbf{B}_{0}\in H^{s}_{comp}$ for (E-MHD) satisfying
$\|{\mathbf{B}_{0}}\|_{H^{s}}<\epsilon$) such that for any $\delta>0$, there
is no corresponding $L^{\infty}_{t}([0,\delta];H^{s-1}\times H^{s})$ solution
to (Hall-MHD) (resp. $L^{\infty}_{t}([0,\delta];H^{s})$ solution to (E-MHD)).

###  1.4. Time-dependent axisymmetric magnetic fields and linearized equations

In this work, we focus on axisymmetric background magnetic fields (rather than
translationally symmetric ones), as they are directly responsible for
illposedness in $\mathbb{R}^{3}$. In the axisymmetric case, removing the
assumption of $z$-independence forces one to work with time-dependent magnetic
fields.

To this end, take an axisymmetric vector field of the form ${\bf
B}=\Pi(r,z)\partial_{\theta}=r\Pi(r,z)e_{\theta}$. Then using that
$\nabla\times(\Pi\partial_{\theta})=-r\partial_{z}\Pi\partial_{r}+r^{-1}\partial_{r}(r^{2}\Pi)\partial_{z}$,
we compute

$\begin{split}({\bf B}\cdot\nabla)(\nabla\times{\bf B})&=-r\Pi\partial_{z}\Pi
e_{\theta}=-\Pi\partial_{z}\Pi\partial_{\theta}\end{split}$

and

$\begin{split}((\nabla\times{\bf B})\cdot\nabla){\bf
B}&=(-\partial_{z}\Pi\partial_{r}(r\Pi)+r^{-1}\partial_{r}(r^{2}\Pi)\partial_{z}\Pi)\partial_{\theta}=\Pi\partial_{z}\Pi\partial_{\theta},\end{split}$

so that ${\bf B}(t)=\Pi(t,r,z)\partial_{\theta}$ satisfies (E-MHD) if $\Pi$
solves the inviscid Burgers’ equation:

(1.7) $\begin{split}\partial_{t}\Pi-2\Pi\partial_{z}\Pi=0.\end{split}$

Let us move on to the Hall-MHD case. Unlike the translationally symmetric
case, now that $(\nabla\times{\bf B})\times{\bf B}$ is not a gradient, we
cannot keep ${\bf u}=0$. However, it is still possible to propagate that
velocity is axisymmetric and swirl-free, i.e. ${\bf u}\cdot e^{\theta}=0$. In
this case, it is convenient to introduce $\boldsymbol{\omega}=\nabla\times{\bf
u}=:\Omega(r,z)\partial_{\theta}$; we obtain that $(\Omega,\Pi)$ is a solution
to Hall-MHD system if

(1.8) $\left\\{\begin{aligned}
&\partial_{t}\Omega+(V^{r}\partial_{r}+V^{z}\partial_{z})\Omega+2\Pi\partial_{z}\Pi=\nu(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2})\Omega\\\
&\partial_{t}\Pi+(V^{r}\partial_{r}+V^{z}\partial_{z})\Pi-2\Pi\partial_{z}\Pi=0.\end{aligned}\right.$

This system (as well as (1.7)) seems to have appeared for the first time in
[7], from where we have taken the notation $(\Omega,\Pi)$. It is to be
supplemented with the following relation between $\Omega$ and $V^{r},V^{z}$:

(1.9) $\left\\{\begin{aligned}
&\partial_{z}V^{r}-\partial_{r}V^{z}=r\Omega,\\\
&\partial_{r}V^{r}+\frac{1}{r}V^{r}+\partial_{z}V^{z}=0.\end{aligned}\right.$

As in [7, 23], it will be convenient to introduce a stream function $\Phi$,
such that

$\begin{split}-(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2})\Phi=\Omega,\quad
V^{r}=-r\partial_{z}\Phi,\quad
V^{z}=\frac{1}{r}\partial_{r}(r^{2}\Phi).\end{split}$

Existence, uniqueness, and regularity of $\Phi$ satisfying the above can be
seen by observing that the second-order operator on the left hand side is
simply $-\Delta$ in $\mathbb{R}^{5}$ (see [7]). With some abuse of notation,
from now on we shall refer to a solution of (1.7) by
$\mathbf{\Pi}=\Pi\partial_{\theta}$. Similarly, given a solution
$(\Omega,\Pi)$ of (1.8), the corresponding axisymmetric solution of (Hall-MHD)
will be written as $({\bf V},\mathbf{\Pi})$, with ${\bf
V}=V^{r}\partial_{r}+V^{z}\partial_{z}.$

######  Remark 1.5.

In the translationally symmetric case, a parallel magnetic field ${\bf
B}=F(y,z)\partial_{x}$ defines a stationary solution to (E-MHD). Similarly,
$({\bf u},{\bf B})=(0,F\partial_{x})$ defines a stationary solution to (Hall-
MHD). While one may repeat the analysis in the current paper to the case of
${\bf B}=F(y,z)\partial_{x}$, we omit this case entirely for the sake of
brevity.

####  1.4.1. Linearized equations around time-dependent backgrounds

Given a stationary solution of the form $(0,\mathring{{\bf B}})$
($\mathring{{\bf B}}$, resp.) for (Hall-MHD) ((E-MHD), resp.), the
corresponding linearization and energy identity were given in (1.2) and (1.4)
((1.3) and (1.5), resp.). In the case of the time-dependent axisymmetric
background $({\bf V},{\mathbf{\Pi}})$ for (Hall-MHD), the linearized systems
are more complicated. With perturbations of the form $({\bf u},{\bf B})=({\bf
V}+u,{\mathbf{\Pi}}+b)$, the system is given by

(1.10) $\left\\{\begin{aligned}
&\partial_{t}b+{\mathbf{\Pi}}\cdot\nabla(\nabla\times b)-(\nabla\times
b)\cdot\nabla{\mathbf{\Pi}}-(\nabla\times{\mathbf{\Pi}})\cdot\nabla b+{\bf
V}\cdot\nabla b-{\mathbf{\Pi}}\cdot\nabla u\\\
&\quad=-b\cdot\nabla(\nabla\times{\mathbf{\Pi}})+b\cdot\nabla{\bf
V}-u\cdot\nabla{\mathbf{\Pi}},\\\ &\partial_{t}u+{\bf V}\cdot\nabla u+\nabla
p-\nu\Delta u-(\nabla\times b)\times{\mathbf{\Pi}}=-u\cdot\nabla{\bf
V}+(\nabla\times{\mathbf{\Pi}})\times b,\\\ &\nabla\cdot b=\nabla\cdot
u=0.\end{aligned}\right.$

We then obtain formally that

(1.11)
$\begin{split}&\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{M}|{u}|^{2}(t)+|{b}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)+\nu\int_{M}|{\nabla
u}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\\\
&=\int_{M}((b\cdot\nabla){\mathbf{\Pi}}_{j})u^{j}-((u\cdot\nabla){\mathbf{\Pi}}_{j})b^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z+\int_{M}((b\cdot\nabla)(\nabla\times{\mathbf{\Pi}})_{j})b^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\\\
&\phantom{=}+\int_{M}((b\cdot\nabla){\bf V}_{j})b^{j}-((u\cdot\nabla){\bf
V}_{j})u^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\end{split}$

for a sufficiently smooth and decaying solution $(u,b)$ to (1.10). In the case
of (E-MHD), the corresponding system is simply given by (1.3) where
$\mathring{{\bf B}}$ is replaced by ${\mathbf{\Pi}}(t)$. Similarly, the energy
identity (1.3) holds by replacing ${\bf B}$ with ${\mathbf{\Pi}}(t)$, for a
sufficiently regular solution $b$. For completeness, we write

(1.12) $\left\\{\begin{aligned} &\partial_{t}b+\nabla\times((\nabla\times
b)\times{\mathbf{\Pi}})+\nabla\times((\nabla\times{\mathbf{\Pi}})\times
b)=0,\\\ &\nabla\cdot b=0.\end{aligned}\right.$

and

(1.13)
$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{M}|{b}|^{2}(t)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\right)=\int_{M}((b\cdot\nabla)(\nabla\times{\mathbf{\Pi}})_{j})b^{j}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

We now naturally extend the notion of $L^{2}$-solution to this setting of
time-dependent background: on an interval of time $I$, we demand that

  * •

(linearized (Hall-MHD) with $\nu>0$) a pair of vector fields $(u,b)$ such that
$u\in C_{w}(I;L^{2})\cap L^{2}_{t}(I;\dot{H}^{1})$ and $b\in C_{w}(I;L^{2})$
that satisfies (1.10) in the sense of distributions;

  * •

(linearized (Hall-MHD) with $\nu=0$) a pair of vector fields $(u,b)\in
C_{w}(I;L^{2})$ that satisfies (1.10) with $\nu=0$ in the sense of
distributions; or

  * •

(linearized (E-MHD)) a vector field $b\in C_{w}(I;L^{2})$ that satisfies
(1.12) in the sense of distributions.

Again, in the case $M=\mathbb{T}^{3}$, we assume (1.6). where we ignore the
condition for $u$ in the case of (E-MHD). Given an initial data
$(u_{0},b_{0})\in L^{2}$, it is not difficult to show existence of an
$L^{2}$-solution $(u,b)$ for (1.10) defined in the sense above (similarly, an
$L^{2}$-solution $b$ given $b_{0}\in L^{2}$ for (1.12)), using an argument
parallel to the one given in [19, Appendix A]. In view of the identities
(1.11) and (1.13), it is reasonable to demand further that

(1.14)
$\begin{split}&\frac{1}{2}\left(\|{u(t)}\|_{L^{2}(M)}^{2}+\|{b(t)}\|_{L^{2}(M)}^{2}\right)+\nu\|{u}\|_{L^{2}([0,t];\dot{H}^{1})}^{2}\leq\frac{1}{2}\left(\|{u_{0}}\|_{L^{2}(M)}^{2}+\|{b_{0}}\|_{L^{2}(M)}^{2}\right)e^{Ct(\|{\nabla{\mathbf{\Pi}}}\|_{L^{\infty}_{I}W^{1,\infty}}+\|{\nabla{\bf
V}}\|_{L^{\infty}_{I}L^{\infty}})}\end{split}$

for all $t>0$ and

(1.15)
$\begin{split}\frac{1}{2}\|{b(t)}\|_{L^{2}(M)}^{2}\leq\frac{1}{2}\|{b_{0}}\|_{L^{2}(M)}^{2}e^{Ct\|{\nabla^{2}{\mathbf{\Pi}}}\|_{L^{\infty}_{I}L^{\infty}(M)}}\end{split}$

for (1.10) and (1.12), respectively. We shall assume these estimates for an
$L^{2}$-solution for simplicity.

###  1.5. Main results

We are now ready to state the main results in this work, which are natural
extensions of Theorems A–C to the case of $\mathbb{R}^{3}$.

####  1.5.1. Linear illposedness

Let $b_{0}=b_{0}^{(\lambda)}$ and $h_{0}=h_{0}^{(\lambda)}$ be sequences of
functions parameterized by $\lambda$. For simplicity, let us write

$\begin{split}&b_{0}=h_{0}(1+O(\lambda^{-1}))\end{split}$

to denote that the support of $b_{0}$ is contained in that of $h_{0}$, and
$\|{b_{0}-h_{0}}\|_{W^{s,p}}\lesssim_{s,p}\lambda^{-1}\|{h_{0}}\|_{W^{s,p}}$
for any $s\geq 0$ and $1<p<\infty$.

######  Theorem A* (Norm growth for the linearized (E-MHD) and (Hall-MHD)).

Take $M=\mathbb{R}^{3}$ and let $s\geq 0,p\geq 2$ satisfy $s+1/p>1/2$. Fix
some smooth cutoff function $\chi(z)$ satisfying $\chi(z)=1$ for $|z|\leq 1$.

  1. (1)

Case of (E-MHD): Take a time interval $I\ni 0$ and a time-dependent
axisymmetric solution ${\mathbf{\Pi}}$ to (E-MHD) defined on $I$, such that
the initial data takes the form
${\mathbf{\Pi}}_{0}(r,z)=f(r)\partial_{\theta}$ on $|z|\leq 1$ where $f(r)$
satisfies the assumptions from 1.3.1. We consider the linearized (E-MHD)
(1.12) on around ${\mathbf{\Pi}}$.

Then, there exist $\delta>0$ and a compactly supported smooth function $\chi$
depending only on ${\mathbf{\Pi}}_{0}$ such that with $\mathfrak{G}(r)$ and
$\mathfrak{b}(r)$ as in Theorem A, for $\lambda\in\mathbb{N}$, there exists an
initial data set of the form

(1.16) $\quad
b_{0}^{(\lambda)}=\operatorname{Re}(e^{i(\lambda\theta+\lambda\mathfrak{G}(r))})\mathfrak{b}(r)\chi(z)(1+O(\lambda^{-1}))$

such that any corresponding $L^{2}$-solution $b^{(\lambda)}$ on $I$ exhibits
norm growth of the form

$\|{b^{(\lambda)}(t)}\|_{W^{s,p}(M)}\geq
c_{s,p,{\mathbf{\Pi}}_{0}}e^{c_{0}({\mathbf{\Pi}}_{0})\cdot(s+\frac{1}{2}-\frac{1}{p})\lambda
t}\|{b_{0}^{(\lambda)}}\|_{W^{s,p}(M)}\quad\mbox{for all}\quad t\in
I\cap[0,\delta].$

  2. (2)

Case of (Hall-MHD): Take a time-dependent axisymmetric solution $({\bf
V},{\mathbf{\Pi}})$ of (Hall-MHD) defined on $I$, such that the initial data
is of the form $(0,{\mathbf{\Pi}}_{0})$ where ${\mathbf{\Pi}}_{0}$ is as in
Case (1). Then, for each $\lambda\in\mathbb{N}$, any $L^{2}$-solution
$(u^{(\lambda)},b^{(\lambda)})$ corresponding to the initial data
$(0,b_{0}^{(\lambda)})$ ($b_{0}^{(\lambda)}$ as in Case (1)) satisfies

$\begin{split}&\|{b^{(\lambda)}(t)}\|_{W^{s,p}(M)}\geq
c_{s,p,{\mathbf{\Pi}}_{0}}e^{c_{0}(\mathring{{\bf
B}})\cdot(s+\frac{1}{2}-\frac{1}{p})\lambda
t}\|{b_{0}^{(\lambda)}}\|_{W^{s,p}(M)}\quad\mbox{for all}\quad t\in
I\cap[0,\delta_{1}],\end{split}$

where $\delta_{1}=c_{1}\lambda^{-1}\ln\lambda$ for some $c_{1}>0$ depending
only on ${\mathbf{\Pi}}_{0}$ and $\nu\geq 0$.

######  Remark 1.6.

Unfortunately, in the Hall-MHD case, the growth of $b(t)$ is guaranteed only
in the time interval of length $O(\lambda^{-1}\ln\lambda)$, see 1.6.5. This is
still sufficient to obtain unbounded growth as $\lambda\to\infty$.

####  1.5.2. Nonlinear illposedness

We now present our nonlinear illposedness results on $\mathbb{R}^{3}$.

######  Theorem B* (Unboundedness of the solution map).

Let $M=\mathbb{R}^{3}$ and the axisymmetric solution $({\bf
V},{\mathbf{\Pi}})$ (resp. ${\mathbf{\Pi}}$) is given as in the case (2)
(resp. case (1)) of Theorem A*. Assume that for some
$\epsilon,\delta,s,r,s_{0}>0$, the solution map for (Hall-MHD) (resp. (E-MHD))
exists as a map

$\displaystyle\mathcal{B}_{\epsilon}((0,{\mathbf{\Pi}}_{0});H^{r}_{comp}\times
H^{s}_{comp})\to L^{\infty}_{t}([0,\delta];H^{s_{0}-1})\times
L^{\infty}_{t}([0,\delta];H^{s_{0}})$ $\displaystyle\left(\hbox{resp.
}\mathcal{B}_{\epsilon}({\mathbf{\Pi}}_{0};H^{s}_{comp})\to
L^{\infty}_{t}([0,\delta];H^{s_{0}})\right).$

In the (Hall-MHD) case, assume further that $s<(1+\alpha_{0})s_{0}$ where
$\alpha_{0}>0$ is an absolute constant. Then this solution map is unbounded
for $s_{0}>7/2$, even if we restrict the initial data to be
$C^{\infty}_{comp}$. Moreover, the same statement holds with
${\mathbf{\Pi}}_{0}\equiv 0$; that is, the solution map cannot be bounded near
the trivial solution for $s_{0}>7/2$.

######  Remark 1.7 (Norm inflation).

One may rephrase the above statement as follows: in the (E-MHD) case, assuming
that the solution operator is well-defined from $H^{s}$ to
$L^{\infty}([0,\delta];H^{s_{0}})$, for any
$\varepsilon,M,\delta>\delta^{\prime}>0$, there exists an initial data ${\bf
B}_{0}\in C^{\infty}_{comp}(\mathbb{R}^{3})$ with corresponding solution ${\bf
B}(t)$ defined on $[0,\delta]$ such that

$\begin{split}\|{{\bf B}_{0}}\|_{H^{s}}<\varepsilon,\quad\|{{\bf
B}(\delta^{\prime})}\|_{H^{s_{0}}}>M.\end{split}$

Note that $s$ can be arbitrarily large independently of $s_{0}$, as long as
$s_{0}>7/2$.

######  Theorem C* (Nonexistence).

Let $s>7/2$ and $M=\mathbb{R}^{3}$. Given any $\epsilon>0$, there exists an
initial data $\mathbf{B}_{0}\in H^{s}_{comp}$ satisfying $\nabla\cdot{\bf
B}_{0}=0$ and $\|{\mathbf{B}_{0}}\|_{H^{s}}<\epsilon$ such that for any
$\delta>0$, there is _no_ associated solution to (E-MHD) belonging to
$L^{\infty}_{t}([0,\delta];H^{s})$. In the (Hall-MHD) case, there is
$\mathbf{B}_{0}\in H^{s}$ satisfying $\nabla\cdot{\bf B}_{0}=0$ and
$\|{\mathbf{B}_{0}}\|_{H^{s}}<\epsilon$ such that for any $\delta>0$, there is
_no_ solution to (Hall-MHD) belonging to
${L^{\infty}_{t}([0,\delta];H^{s-1}\times H^{s})}$ corresponding to the
initial data $(\mathbf{u}_{0}\equiv 0,\mathbf{B}_{0})$.

Some remarks about these theorems are in order.

######  Remark 1.8 (Degenerate dispersion).

As in [19], the main illposedness mechanism exploited in this paper is
_degenerate dispersion_ , i.e., the rapid growth of frequencies of solutions
that travel towards a degeneracy of the principal term. This mechanism may be
found in other classes of quasilinear dispersive equations as well; see [5,
20].

######  Remark 1.9 (Wellposedness in the nondegenerate case).

In [21], the following local well(!)posedness theorem for (E-MHD) on
$\mathbb{R}^{3}$ was proved:

> Let $s>\frac{7}{2}$ and consider a vector field ${\bf
> B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfying $\nabla\cdot{\bf
> B}_{0}=0$. Assume furthermore that ${\bf B}_{0}$ satisfies the following
> properties:
>
>   1. (1)
>
> Nondegeneracy. ${\bf B}_{0}(x)\neq 0$ at every point $x\in\mathbb{R}^{3}$,
>
>   2. (2)
>
> Asymptotic uniformity. $\|{{\bf B}_{0}-{\bf
> e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<+\infty$,
>
>   3. (3)
>
> Nontrapping. Every nonconstant solution $(X,\Xi)(t)$ to the Hamiltonian
> system associated with ${\bf p}_{{\bf B}_{0}}(x,\xi)={\bf
> B}_{0}(x)\cdot\xi|{\xi}|$ escapes to $x^{3}=\pm\infty$, i.e.,
> $X^{3}(t)\to\infty$ or $X^{3}(t)\to-\infty$ as $t\to\infty$.
>
>

>
> Then the Cauchy problem for (E-MHD) with ${\bf B}(t=0)={\bf B}_{0}$ is
> locally wellposed.

We remark that the techniques in [21] can be used to prove a similar theorem
for (Hall-MHD) as well; see [21, Remark 1.3]. The theorems in this paper
complement this result by exhibiting a strong nonlinear illposedness
phenomenon in a neighborhood of certain _degenerate_ initial data (e.g.,
$\mathring{{\bf B}}=0$).

Observe that the above theorem from [21] requires ${\bf B}_{0}$ to be
asymptotic to the uniform magnetic field ${\bf e}_{3}$. Another natural
setting where local wellposedness may hold is when ${\bf B}_{0}$ close to a
_nondegenerate_ axisymmetric magnetic field. Such a result would justify the
axisymmetric ansatz under a nondegeneracy assumption, which is interesting in
view of the finite time blow-up result of Chae–Weng [7] for the axisymmetric
(Hall-MHD). However, the proof in [21] fails due to the difficulty of
establishing local smoothing effect in this setting.

######  Remark 1.10.

The nonlinear illposedness results can be similarly stated and proved in the
scale of $C^{k,\alpha}$-spaces as long as $k+\alpha\geq 2$ for ${\bf B}$ in
the (E-MHD) case. In the (Hall-MHD) case, it is natural to put $({\bf u},{\bf
B})\in C^{k-1,\alpha}\times C^{k,\alpha}$ with $k+\alpha\geq 2$.

###  1.6. Ideas of the proof

In this section, we discuss key ideas of the proof, mainly emphasizing the
additional difficulties arising from having compact $z$-support. Main ideas in
the $z$-independent case are given in [19, Sections 1.6, 1.7].

####  1.6.1. Analysis at the level of bicharacteristics.

Here we give a heuristic explanation as to why the same illposedness results
are expected in the $z$-dependent case as well. This is most easily seen under
the (formal) framework of bicharacteristics. As argued in [19], our basic
viewpoint towards (Hall-MHD) is to regard it as a quasilinear dispersive
system, and the basic mechanism for the illposedness is given by the existence
bicharacteristics quickly converging to the degeneracy of the background
magnetic field. Let us recall the notion of bicharacteristics: Linearizing
around a stationary magnetic field $\mathring{{\bf B}}$ and neglecting the
equation for the velocity, we may write the resulting equation in the form

$\begin{split}\partial_{t}b+(\mathring{{\bf B}}\cdot\nabla)\nabla\times
b=l.o.t.,\end{split}$

where the right hand side contains either first or zeroth order terms in $b$.
We may then diagonalize the matrix-valued symbol ${\bf p}=-(\mathring{{\bf
B}}\cdot\xi)\xi\times$ on the subspace
$\\{\zeta\in\mathbb{R}^{3}:\xi\cdot\zeta=0\\}$ with eigenvalues $\pm p=\pm
i\mathring{{\bf B}}\cdot\xi|\xi|$. Then neglecting the lower order terms, the
above vector equation splits into two scalar dispersive PDE of the form

$\begin{split}\partial_{t}b_{\pm}\pm ip(i^{-1}\nabla)b_{\pm}=0.\end{split}$

Then the associated ODE system with the Hamiltonian vector field
$(\nabla_{\xi}p,-\nabla_{\mathbf{x}}p)$ on $T^{*}M$

$\left\\{\begin{aligned} \dot{X}&=\nabla_{\xi}p(X,\Xi),\\\
\dot{\Xi}&=-\nabla_{\mathbf{x}}p(X,\Xi)\end{aligned}\right.$

is referred to as the Hamiltonian ODE, and its solution $(X,\Xi)(t)$ a
bicharacteristic. This is a natural generalization of the notion of the group
velocity from constant coefficient dispersive PDEs. Given a Gaussian-like
initial data with physical and frequency centers $(X_{0},\Xi_{0})$, the
associated bicharacteristics $(X,\Xi)(t)$ describes the evolution of the
(approximate) centers, at least for some period of time.

We now take the concrete choice $\mathring{{\bf B}}=y\chi(z)\partial_{x}$
where $\chi(\cdot)\geq 0$ is a smooth cutoff of $O(1)$ length scale. The
associated Hamiltonian is $y(X)\chi(z(X))\Xi_{x}|\Xi|$ and since $\Xi_{x}$ is
conserved along bicharacteristics (from $x$-invariance of $\mathring{{\bf
B}}$), we conclude that $y(X)\chi(z(X))|\Xi|$ is conserved as well. Hence as
in the case $\chi\equiv 1$, $y\rightarrow 0$ will imply
$|\Xi|\rightarrow+\infty$.

Towards this goal we first note that, as long as $|z(X)|\lesssim 1$, the
Hamiltonian simply coincides with the one from the $z$-independent case. Take
initial data of the form $X(0)=(0,1,z(0))$ ($x(0)=0$ can be taken without loss
of generality) and $\Xi(0)=(\lambda,-\lambda,\Xi_{z}(0))$ where
$|z(0)|,|\Xi_{z}(0)|\lesssim 1$. As long as $|z(X)|\lesssim 1$, we have

$\left\\{\begin{aligned} \dot{z}&=\lambda y\frac{\Xi_{z}}{|\Xi|},\\\
\dot{\Xi_{z}}&=0,\end{aligned}\right.$

and hence we see that if we have $|y(X)|\lesssim 1$ and
$|\Xi_{z}|/|\Xi|\lesssim\lambda^{-1}$ for some interval of time, we can
guarantee that $|z(X)|\lesssim 1$ as well. But then this means that the
behavior of the bicharacteristics is the same as in the $z$-independent case;
in particular,

$\begin{split}|\Xi(t)|\simeq\lambda e^{\lambda t},\quad y(t)\simeq e^{-\lambda
t}\end{split}$

which in turn guarantees $|y(X)|\ll 1$ and $|\Xi_{z}|/|\Xi|\ll\lambda^{-1}$!
(This circular argument can be made rigorous with a continuity argument.) This
shows that the bicharacteristics from the $z$-independent case still provides
a good approximation within a time interval which is independent of $\lambda$,
as long as the initial $z$-frequency is kept at $O(1)$. Essentially the same
analysis can be repeated for the axisymmetric magnetic field of the form
$f(r)\chi(z)\partial_{\theta}$ with the caveat that it actually does not
define a stationary solution to (Hall-MHD); see 1.6.5 below. This heuristic
argument suggests that given a $z$-independent linear approximate solution, a
simple cutoff in the $z$-direction still gives an approximate solution. Of
course, this procedure gives rise to various technical difficulties, which
shall be discussed in the following.

####  1.6.2. Generalized energy identities.

Our main technical tool in the illposedness arguments (both linear and
nonlinear) is the so-called generalized energy identity, which is a
deceptively simple consequence of the (usual) energy identity. Let us briefly
explain the ideas. Consider a linear equation of the form

(1.17) $\begin{split}\partial_{t}b+\mathcal{L}[b]=\mathcal{B}[b],\end{split}$

where $\mathcal{L}$ is antisymmetric and $\mathcal{B}$ is a zeroth order
operator;

$\begin{split}\langle{\mathcal{L}[f],g}\rangle=-\langle{f,\mathcal{L}[g]}\rangle,\quad\|{\mathcal{B}[f]}\|_{L^{2}}\lesssim\|{f}\|_{L^{2}}.\end{split}$

Then one can see that not only the $L^{2}$-norm of a solution $b$ is bounded
in terms of the initial data, but also does the $L^{2}$-inner product of two
solutions: for $b$ and $\tilde{b}$ satisfying (1.17), we have

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\langle{b,\tilde{b}}\rangle=\langle{\mathcal{B}[b],\tilde{b}}\rangle-\langle{\mathcal{L}[b],\tilde{b}}\rangle+\langle{b,\mathcal{B}[\tilde{b}]}\rangle-\langle{b,\mathcal{L}[\tilde{b}]}\rangle\end{split}$

so that after cancellations,

(1.18)
$\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}t}\langle{b,\tilde{b}}\rangle\right|\lesssim\|{b}\|_{L^{2}}\|{\tilde{b}}\|_{L^{2}}\end{split}$

which implies (after integrating the above in time and then using that the
$L^{2}$-norm is stable under (1.17))

$\begin{split}\left|\langle{b(t),\tilde{b}(t)}\rangle-\langle{b_{0},\tilde{b}_{0}}\rangle\right|\lesssim
t\|{b_{0}}\|_{L^{2}}\|{\tilde{b}_{0}}\|_{L^{2}}.\end{split}$

Finally, if the “angle” between $b_{0}$ and $\tilde{b}_{0}$ is small; that is,
$\langle{b_{0},\tilde{b}_{0}}\rangle\gtrsim\|{b_{0}}\|_{L^{2}}\|{\tilde{b}_{0}}\|_{L^{2}}$,
we conclude that for some interval $I$ of time,

(1.19) $\begin{split}\inf_{t\in I}\,\langle{b(t),\tilde{b}(t)}\rangle\geq
c\|{b_{0}}\|_{L^{2}}\|{\tilde{b}_{0}}\|_{L^{2}}\end{split}$

for some $c>0$. In the following we shall frequently refer to (1.18) (or its
simple consequence (1.19)) as generalized energy estimate. The merit in the
generalized energy estimate lies in the flexibility involved in its
derivation; say we cannot solve (1.17) directly, but somehow are given an
approximate solution $\tilde{b}$, satisfying instead

$\begin{split}\partial_{t}\tilde{b}+\mathcal{L}[\tilde{b}]=\mathcal{B}^{\prime}[\tilde{b}]\end{split}$

where $\mathcal{B}^{\prime}$ is some bounded operator in $L^{2}$ which does
not need to be related in any way with $\mathcal{B}$. But note that in this
case, the estimates (1.18) and (1.19) still hold under the same assumptions.
Furthermore, one may observe that we actually do not need
$\mathcal{B}^{\prime}$ and $\mathcal{B}$ to be bounded operators in $L^{2}$,
but just need that
$\|{\mathcal{B}^{*}[\tilde{b}]}\|_{L^{2}},\|{\mathcal{B}^{\prime}[\tilde{b}]}\|_{L^{2}}\lesssim\|{\tilde{b}}\|_{L^{2}}$.
In conclusion, as far as one is concerned with deriving the generalized energy
estimate, one may regard a priori and a posteriori bounded terms as errors.

Having a generalized energy estimate in hand, the task of showing growth of
$b$ in high-regularity Sobolev spaces reduces to that of showing decay of an
approximate solution $\tilde{b}$ in low-regularity Sobolev spaces. As a simple
example, if we are able to show $\|{\tilde{b}(t)}\|_{L^{1}}\lesssim
e^{-t}\|{\tilde{b}_{0}}\|_{L^{2}}$, we conclude at once from (1.19) that
$\|{b(t)}\|_{L^{\infty}}\gtrsim e^{t}\|{b_{0}}\|_{L^{2}}$. Of course in
general showing degeneration in low regularity spaces could be a difficult
task.

So far we have demonstrated that showing growth of a solution to (1.17) can be
reduces to construction and analysis of an appropriate approximate solution.
There is a general such procedure, commonly called as WKB analysis, which we
briefly illustrate in 1.6.3. Then we are in a position to discuss additional
difficulties in proving the generalized energy estimate in the $z$-dependent
case, over the $z$-independent case.

####  1.6.3. WKB analysis.

Keeping the notation above, say we are interested in constructing an
approximate solution $\tilde{b}$ for (1.17). The discussion in the above
suggest that we can completely neglect the a priori bounded terms from the
beginning and consider $z$-independent solutions; say the latter independence
assumption simplifies the operator $\mathcal{L}$ into $\mathcal{L}^{\prime}$;
now we want to approximately solve

(1.20)
$\begin{split}\partial_{t}\tilde{b}+\mathcal{L}^{\prime}\tilde{b}=0.\end{split}$

We use the standard WKB analysis, which can be viewed as an expansion of the
solution in frequency. Take $\lambda\gg 1$ and prepare an ansatz of the
following form

$\begin{split}\tilde{b}=e^{i\lambda\Phi}(h^{(0)}+\lambda^{-1}h^{(1)}+\cdots).\end{split}$

We have a freedom of choice for the initial data of $\Phi,h^{(0)}$, and so on,
which we can use to our advantage. Formally plugging in the ansatz into (1.20)
and organizing the terms by the powers of $\lambda$, one obtains a hierarchy
of evolution equations; the first one involving only $\Phi$, the second
describing the evolution of $h^{(0)}$ with coefficients depending on $\Phi$,
and so on. While this inductively determines each function in the expansion,
it turns out that in our concrete cases, $e^{i\lambda\Phi}h^{(0)}$ is already
provides a good approximate solution, in the $L^{2}$-sense.

####  1.6.4. Bogovskiǐ operator and broken antisymmetry.

As we have mentioned in the above, we would like to declare that
$\chi\tilde{b}$ still provides a good approximation, where $\chi$ is a simple
cutoff function in $z$, and $\tilde{b}$ is $z$-independent. An immediate
problem one faces is that $\chi\tilde{b}$ cannot be divergence free, a
property which is essential for the principal operator $\mathcal{L}$ to be
anti-symmetric to begin with. Moreover, the initial data $b_{0}$ should be
divergence-free, close to $\chi\tilde{b}$ at least in the $L^{2}$-norm, and
have compact support. It is tempting to just apply the Leray projector on
$\chi\tilde{b}$ but in terms of the compact support property, a much better
alternative is provided by the Bogovskiǐ operator introduced in [4]. This is
also used in an essential way in the (Hall-MHD) case; $\chi\tilde{u}$ is not
divergence-free, so to remove the pressure term, we need to test $u$ against
the divergence-free part (defined by the Bogovskiǐ operator) of
$\chi\tilde{u}$.

We still need to explain how the first difficulty is handled. The principal
operator in the Hall current term is of second order and involves
$\partial_{zx}$ and $\partial_{zy}$ derivatives. While going through the
generalized energy identity, there appear terms with $\partial_{x}$ and
$\partial_{y}$ on $\tilde{b}$ when a $z$-derivatives falls on the cutoff
(anti-symmetry is broken). However all such terms altogether cancel out to
give rise an $L^{2}$-bounded term after some juggling of derivatives with the
divergence-free condition on $b$. Of course this was anticipated from 1.6.1,
or can be alternatively seen by repeating the WKB analysis in the
$z$-dependent case.

####  1.6.5. A priori estimates for the axisymmetric Hall-MHD

Utilizing the axisymmetric background magnetic fields (over the
translationally symmetric ones) is the key to obtain illposedness for compact
data in $\mathbb{R}^{3}$. However, compared to the setting of [19], one
immediately sees a problem: there are no non-trivial axisymmetric _stationary_
magnetic fields for (E-MHD) with compact support in $\mathbb{R}^{3}$. This
forces us to work with time-dependent background solutions, which sounds like
a nonsense because in the end, we have proven nonexistence of a solution to
the initial value problem. Interestingly enough it is known that under the
assumption of axisymmetry, existence and uniqueness of local-in-time smooth
axisymmetric solutions can be established ([7, 18]).222This statement should
be interpreted carefully, as uniqueness is not guaranteed for (E-MHD) and
(Hall-MHD), so that there is no reason to believe that the solution is
axisymmetric even if the initial data is so. This point is emphasized in [18].
We give a proof of a priori estimates both for the (E-MHD) and (Hall-MHD)
cases, although some cases have been already covered in [18]. In the case of
(E-MHD), the resulting equation for the axisymmetric magnetic field is exactly
the (inviscid) Burgers equation in the $z$-direction, which is locally
wellposed but definitely blows up in finite time for compact support initial
data. This is not an issue for linear illposedness since the growth happens
essentially instantaneously. Even though the magnetic field is time-dependent,
all we need is that this background is constant in the $z$-direction for some
intervals in $t$ and $z$, which holds for solutions to the Burgers equation.
In view of this, serious difficulties arise in the (Hall-MHD) case. Even if
the background velocity is initially zero, the velocity becomes immediately
nonzero by the axisymmetric magnetic field for $t>0$, which in turn breaks
$z$-invariance in any small interval of $z$. This is handled by quantifying
smallness of the $z$-variation of the magnetic field for small $t>0$. A much
more serious problem is that, in general, the velocity field will immediately
move the point (a hyperplane in $\mathbb{R}^{3}$, strictly speaking) of
degeneracy for the magnetic field. Recall that the illposedness mechanism is
the convergence of wave packet solutions towards the degenerate point. The
fact that the degenerate point is moving with time destroys the estimates for
the wavepackets which were valid for an $O(1)$-interval of time; in the worst
possible case, the estimates are only valid for a time interval of size
$O(\lambda^{-1}\ln\lambda)$, where $\lambda\gg 1$ is the frequency of the wave
packet at the initial time. Even showing this requires proving smallness-in-
time estimates for various quantities involving the background axisymmetric
solution.

######  Remark 1.11.

It is desirable (and could be of independent interest) to have a situation
where the degenerate point is forced to be fixed, even under the axisymmetric
Hall-MHD evolution. This is the case for axis-degenerate magnetic fields,
which are simply steady magnetic fields which are degenerate on the axis
$\\{r=0\\}$. One may perform the analysis of wave packets in this case, which
gives admissible error estimates for $O(\lambda^{\alpha})$-timescales where
$\alpha>0$ is a factor depending on the order of axis degeneracy.

####  1.6.6. Nonlinear illposedness

There is no general method for passing from a linear illposedness result to a
corresponding nonlinear result; in certain cases a nonlinear system can be
wellposed while it has a linearization which is illposed ([14]). In our case,
we use the (generalized) energy estimate in a crucial way in deducing a
nonlinear illposedness result from a linear one. Using a similar notation from
the above, the equation for the perturbation takes the form

$\begin{split}\partial_{t}b+\mathcal{L}[b]=\mathcal{B}[b]+Q[b,b]\end{split}$

where we write $Q[b,b]$ for the nonlinear terms. In practice, we have in $Q$
terms of the form $b\nabla^{2}b$ and $(\nabla b)^{2}$. Note that upon assuming
towards contradiction that $b$ belongs to some $H^{s}$ with $s$ large, we may
bound the contribution from $Q$ in deriving a generalized energy estimate for
$b$ using Sobolev inequalities:

$\begin{split}\left|\langle{Q[b,b],\tilde{b}}\rangle\right|\lesssim\|{b}\|_{H^{s}}\|{b}\|_{L^{2}}\|{\tilde{b}}\|_{L^{2}}.\end{split}$

This allows us to again deduce (1.19), which gives a contradiction to the
boundedness of $b$ upon taking the initial frequency support of $b$ to be very
large. This simple estimate makes precise the general belief that the
nonlinear evolution follows approximately the linear one for a short time
interval. In the three-dimensional case, one will see a slight difference in
the minimum threshold for $s$ (compared to the $z$-independent case) since one
needs to use three-dimensional embedding results rather than two-dimensional
ones.

The above gives an outline of how the unboundedness of the solution operator
(assuming existence) can be proved. As we have done in [19], nonexistence of
the solution operator can be actually shown; the simple idea is to prepare a
background solution with countably many points of degeneracy. Then for each
degenerate point one puts a small (in some fixed $H^{s}$ with $s$ large
enough) perturbation with increasing frequency. If the same argument as in the
above can be repeated independently near each degenerate point, it will show
that at some positive time moment (where the solution is assumed to exist),
the $H^{s}$-norm should be greater than any large number, which is a
contradiction.

Other than a few minor issues (related with ensuring the divergence-free
conditions for some vector functions), there is not much additional
difficulties in proving nonlinear illposedness in the $z$-dependent case over
the $z$-independent case. On the contrary, having an unbounded direction
$\mathbb{R}_{z}$ can be used to our advantage in the nonexistence argument, as
we can place countably many points of degeneracy along the $z$-direction which
was definitely not possible in our previous work.

###  1.7. Notation, conventions and some useful vector calculus identities

Here, we collect some notation, conventions and vector calculus identities
that will be used freely in the remainder of the paper.

#### Notation and conventions

By $A\lesssim B$, we mean that there exists some positive constant $C>0$ such
that $|{A}|\leq CB$. The dependency of the implicit constant $C$ is specified
by subscripts, e.g. $A\lesssim_{E}B$. By $A\simeq B$, we mean $A\lesssim B$
and $A\gtrsim B$.

We denote by $\mathbb{R}$ the real line, $\mathbb{Z}$ the set of integers,
$\mathbb{T}=\mathbb{R}/{2\pi\mathbb{Z}}$ the torus with length $2\pi$,
$\mathbb{N}_{0}=\\{0,1,2,\ldots\\}$ the set of nonnegative integers and
$\mathbb{N}=\\{1,2,\ldots\\}$ the set of positive integers.

We write $M$ for the $3$-dimensional domain of the form
$\mathbb{T}^{k}\times\mathbb{R}^{3-k}$ ($0\leq k\leq 3$) equipped with the
rectangular coordinates $(x,y,z)$, and $M^{2}=M^{2}_{x,y}$ for the two-
dimensional projection of $M$ along the $z$-axis. We use the notation
$\langle{u,v}\rangle$ and $\langle{u,v}\rangle_{M^{2}}$ for the standard
$L^{2}$-inner product for vector fields on $M$ and $M^{2}$, respectively;
i.e.,

$\langle{{\bf u},{\bf v}}\rangle=\int_{M}{\bf u}\cdot{\bf
v}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z,\quad\langle{u,v}\rangle_{M^{2}}=\int_{M^{2}}u\cdot
v\,\mathrm{d}x\mathrm{d}y.$

Given a vector $u$ on $M^{2}$, we define its perpendicular $u^{\perp}$ by
$(-u^{y},u^{x})^{\top}$ and the perpendicular gradient operator
$\nabla^{\perp}$ by $(-\partial_{y},\partial_{x})$. We use the usual notation
$W^{s,p}$ for the $L^{p}$-based Sobolev space of regularity $s$; when $p=2$,
we write $H^{s}=W^{s,2}$. The mixed Lebesgue norm $L^{p}_{x}L^{q}_{y}$ is
defined as

$\|{u}\|_{L^{p}_{x}L^{q}_{y}}=\|{\|{u(x,y)}\|_{L^{q}_{y}}}\|_{L^{p}_{x}}.$

The norm $L^{p}_{t}H^{s}$ is defined similarly.

Given any space $X$ of functions on $M$, we denote by $X_{comp}(M)$ the
subspace of compactly supported elements of $X$, and by $X_{loc}(M)$ the space
of functions $u$ such that $\chi u\in X_{loc}(M)$ for any smooth compactly
supported function $\chi$ on $M$.

#### Vector calculus identities

We recall some useful vector calculus identities:

(1.21) $\displaystyle{\bf U}\times({\bf V}\times{\bf W})$ $\displaystyle={\bf
V}({\bf U}\cdot{\bf W})-{\bf W}({\bf U}\cdot{\bf V}),$ (1.22)
$\displaystyle\nabla\times({\bf U}\times{\bf V})$ $\displaystyle=({\bf
V}\cdot\nabla){\bf U}+{\bf U}(\nabla\cdot{\bf V})-({\bf U}\cdot\nabla){\bf
V}-{\bf V}(\nabla\cdot{\bf U}),$ (1.23) $\displaystyle(\nabla\times{\bf
U})\times{\bf V}$ $\displaystyle=({\bf V}\cdot\nabla){\bf U}-{\bf
V}_{j}\nabla{\bf U}^{j},$ (1.24) $\displaystyle\nabla\times(\nabla\times{\bf
U})$ $\displaystyle=-\Delta{\bf U}+\nabla(\nabla\cdot{\bf U}).$

#### Vector calculus in cylindrical coordinates

The cylindrical coordinates $(r,\theta,z)$ are defined by

$r=\sqrt{x^{2}+y^{2}},\quad\theta=\tan^{-1}\frac{y}{x}.$

As in [19], we use the _coordinate derivative basis_
$(\partial_{r},\partial_{\theta},\partial_{z})$ to decompose vectors into
components, i.e., given a vector ${\bf U}$ on $M$, we define its components
${\bf U}^{r},{\bf U}^{\theta},{\bf U}^{z}$ by

${\bf U}={\bf U}^{r}\partial_{r}+{\bf U}^{\theta}\partial_{\theta}+{\bf
U}^{z}\partial_{z}.$

(Another common choice is
$(e_{r},e_{\theta},e_{z})=(\partial_{r},r^{-1}\partial_{\theta},\partial_{z})$.)
With this convention, the inner product is given by

(1.25) ${\bf U}\cdot{\bf V}={\bf U}^{r}{\bf V}^{r}+r^{2}{\bf U}^{\theta}{\bf
V}^{\theta}+{\bf U}^{z}{\bf V}^{z},$

where ${\bf V}={\bf V}^{r}+{\bf V}^{\theta}\partial_{\theta}+{\bf
V}^{z}\partial_{z}$. For a scalar-valued function $f$, the gradient and
Laplacian in the cylindrical coordinates are

(1.26) $\nabla
f=(\partial_{r}f)\partial_{r}+(r^{-2}\partial_{\theta}f)\partial_{\theta}+(\partial_{z}f)\partial_{z},$
(1.27) $\begin{split}\Delta
f=r^{-1}\partial_{r}(r\partial_{r}f)+r^{-2}\partial_{\theta\theta}f+\partial_{zz}f.\end{split}$

When $f$ is independent of $z$, the perpendicular gradient takes the form

(1.28)
$\nabla^{\perp}f=-(r^{-1}\partial_{\theta}f)\partial_{r}+(r^{-1}\partial_{r}f)\partial_{\theta}.$

Next, the curl and divergence in the cylindrical coordinates are

(1.29) $\displaystyle\nabla\times({\bf U}^{r}{\partial_{r}}+{\bf
U}^{\theta}{\partial_{\theta}}+{\bf U}^{z}{\partial_{z}})=$
$\displaystyle(r^{-1}\partial_{\theta}{\bf U}^{z}-r\partial_{z}{\bf
U}^{\theta}){\partial_{r}}+(r^{-1}\partial_{z}{\bf
U}^{r}-r^{-1}\partial_{r}{\bf U}^{z}){\partial_{\theta}}$
$\displaystyle+r^{-1}(\partial_{r}(r^{2}{\bf
U}^{\theta})-\partial_{\theta}{\bf U}^{r}){\partial_{z}},$ (1.30)
$\nabla\cdot({\bf U}^{r}{\partial_{r}}+{\bf
U}^{\theta}{\partial_{\theta}}+{\bf
U}^{z}{\partial_{z}})=r^{-1}\partial_{r}(r{\bf U}^{r})+\partial_{\theta}{\bf
U}^{\theta}+\partial_{z}{\bf U}^{z}.$

Finally, the material derivative is given by

(1.31) $\begin{split}({\bf U}\cdot\nabla){\bf V}&=({\bf U}^{r}\partial_{r}{\bf
V}^{r}+{\bf U}^{\theta}\partial_{\theta}{\bf V}^{r}+{\bf
U}^{z}\partial_{z}{\bf B}^{r}-r{\bf U}^{\theta}{\bf
V}^{\theta})\partial_{r}\\\ &\qquad+(r^{-1}{\bf U}^{r}\partial_{r}(r{\bf
V}^{\theta})+{\bf U}^{\theta}\partial_{\theta}{\bf V}^{\theta}+{\bf
U}^{z}\partial_{z}{\bf V}^{\theta}+r^{-1}{\bf U}^{\theta}{\bf
V}^{r})\partial_{\theta}\\\ &\qquad+({\bf U}^{r}\partial_{r}{\bf V}^{z}+{\bf
U}^{\theta}\partial_{\theta}{\bf V}^{z}+{\bf U}^{z}\partial_{z}{\bf
V}^{z})\partial_{z}.\end{split}$

### Organization of the paper

The rest of this paper is organized as follows. In Section 2, we recall the
Bogovskiǐ operator and derive useful a priori estimates for axisymmetric
solutions to (E-MHD) and (Hall-MHD) respectively in Sections 2.1 and 2.2. Then
the construction of degenerating wavepackets from [19] is reviewed in Sections
2.3–2.4. Section 3 is dedicated to the proofs of linear illposedness results.
In Sections 4 and 5 we complete the proofs of unboundedness and nonexistence
of the solution operator, respectively.

### Acknowledgements

I.-J. Jeong has been supported by the National Research Foundation of Korea
(NRF) grant No. 2022R1C1C1011051. S.-J. Oh was partially supported the Sloan
Research Fellowship and the National Science Foundation CAREER Grant under
NSF-DMS-1945615.

##  2\. Preliminaries

###  2.1. Bogovskiǐ operator

In later sections, it will be important to be able to systematically “invert”
the divergence operator to ensure divergence-free condition on the initial
perturbations. One may simply apply the Leray projector to a vector field, but
then the compact support property will be immediately lost. To avoid this
problem we use the Bogovskiǐ operator introduced in [4]. Detailed information
regarding the properties of this operator can be found in several textbooks;
see for instance [1, 13]. Given a bounded open set $U\subset\mathbb{R}^{3}$,
we write $d(U)=\sup_{\mathbf{x},\mathbf{y}\in U}|\mathbf{x}-\mathbf{y}|$ and
$i(U)=\sup_{\exists\mathbf{x}\in{U},B(\mathbf{x},R)\subset{U}}R$.

######  Proposition 2.1 ([13, Lemma 3.1]).

Fix some smooth function $w\in C^{\infty}$ supported in the unit ball in
$\mathbb{R}^{3}$ and satisfying $\int w=1$, and let $U\subset\mathbb{R}^{3}$
be a bounded open set with Lipschitz boundary. Take $\mathbf{x}_{0}\in U$ such
that $B(\mathbf{x}_{0},i(U))\subset\overline{U}$ and $\overline{U}$ is star-
shaped with respect to $B(\mathbf{x}_{0},i(U))$ (i.e., for every ${\bf
x}\in\overline{U}$ and ${\bf x}^{\prime}\in B(\mathbf{x}_{0},i(U))$, the line
segment from ${\bf x}$ to ${\bf x}^{\prime}$ is contained in $\overline{U}$).
Rescale $w$ to define
$w_{U}(\mathbf{x})=i(U)^{-3}w(i(U)^{-1}(\mathbf{x}-\mathbf{x}_{0}))$. We
define the kernel

(2.1)
$\begin{split}G_{U}(\mathbf{x},\mathbf{y})=\int_{0}^{1}\frac{\mathbf{x}-\mathbf{y}}{s}w\left(\mathbf{y}+\frac{\mathbf{x}-\mathbf{y}}{s}\right)\frac{ds}{s^{3}}.\end{split}$

Assume that $g\in L^{2}(\mathbb{R}^{3})$ is supported in $U$ and satisfies
$\int_{U}g=0$. Then, we have that $h$ defined by

$\begin{split}&h(\mathbf{x})=\int_{\mathbb{R}^{3}}G_{U}(\mathbf{x},\mathbf{y})g(\mathbf{y})\mathrm{d}\mathbf{y}\end{split}$

is supported in $U$ and solves $\mathrm{div}\,h=g$, with the estimates

(2.2)
$\begin{split}\|{h}\|_{L^{2}}\leq\frac{C_{n}}{d({U})}\left(\frac{d({U})}{i({U})}\right)^{n}\left(1+\frac{d({U})}{i({U})}\right)\|{g}\|_{L^{2}},\qquad\|{\nabla
h}\|_{L^{2}}\leq
C\left(\frac{d({U})}{i({U})}\right)^{3}\left(1+\frac{d({U})}{i({U})}\right)\|{g}\|_{L^{2}},\end{split}$

for a universal constant $C>0$.

For our applications, we shall take ${\bf x}_{0}$ to lie on the $z$-axis and
$U$ to be an (axisymmetric) cylinder of the form
$U=\\{(x,y,z):|{(x,y)}|<\ell,\,|{z}|<\ell\\}$ for some $\ell>0$. Abusing the
terminology, we will refer to such cylinder $U$ as a _cube_ with side-length
$\ell$. Choosing $i(U)=c\ell$ for some universal small constant $c>0$, we
simply have

(2.3) $\begin{split}\|{\nabla^{m+1}h}\|_{L^{2}}\leq
C_{m}\|{\nabla^{m}g}\|_{L^{2}}\end{split}$

for all $m\geq 0$. Indeed, observe that $d(U)$ is comparable to $\ell$ (and
hence to $i(U)$, and that the star-shape condition follows from the convexity
of $U$.

Moreover, we shall replace the kernel $G_{U}$ in (2.1) by

$\begin{split}\mathring{G}_{U}(r,\theta,z,r^{\prime},\theta^{\prime},z^{\prime}):=\int_{0}^{2\pi}G_{U}(r,\theta+\theta_{0},z,r^{\prime},\theta^{\prime}+\theta_{0},z^{\prime})\,\mathrm{d}\theta_{0}\end{split}$

and define

$\mathrm{div}^{-1}_{U}g({\bf x}):=\int_{\mathbb{R}^{3}}\mathring{G}_{U}({\bf
x},{\bf y})g({\bf y})\,\mathrm{d}{\bf y}.$

With this choice, we keep the right-inverse and support-preserving properties
of the operator in Proposition 2.1, and have the additional convenient
property that $[\partial_{\theta},\mathrm{div}^{-1}_{U}]=0$. In what follows,
if the choice of $U$ is clear from the context, we will simply write
$\mathrm{div}^{-1}$.

###  2.2. A priori estimates for axisymmetric solutions

The purpose of subsection is to derive a few elementary a priori estimates for
axisymmetric solutions to (E-MHD) and (Hall-MHD), which will serve as
background solutions in our proof of instability/illposedness.

We begin by recalling that if ${\bf U}$ be of the form ${\bf
U}=U(r,z)\partial_{\theta}$, then $(-\Delta){\bf
U}=-\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}\right)U\partial_{\theta}$.
We shall use the notation

$-\not{\Delta}U=-\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}\right)U.$

In what follows, we write ${\mathbf{\Pi}}$ instead ${\bf B}$ for an
axisymmetric solution to (E-MHD). The system (E-MHD) for solutions of the form
${\mathbf{\Pi}}=\Pi(r,z)\partial_{\theta}$ may be rewritten as

(2.4) $\partial_{t}\Pi-2\Pi\partial_{z}\Pi=0.$

For an axisymmetric solution to (Hall-MHD), we write $({\bf
V},{\mathbf{\Pi}})$ instead $({\bf u},{\bf B})$. The system (Hall-MHD) for
solutions of the form $({\bf
V},{\mathbf{\Pi}})=(V^{r}(r,z)\partial_{r}+V^{z}(r,z)\partial_{z},\Pi(r,z)\partial_{\theta})$
may be rewritten as

(2.5) $\left\\{\begin{aligned}
\partial_{t}\Omega+(V^{r}\partial_{r}+V^{z}\partial_{z})\Omega+2\Pi\partial_{z}\Pi&=\nu\not{\Delta}\Omega,\\\
\partial_{t}\Pi+(V^{r}\partial_{r}+V^{z}\partial_{z})\Pi-2\Pi\partial_{z}\Pi&=0,\\\
\partial_{r}V^{r}+\partial_{z}V^{z}&=0,\end{aligned}\right.$

where $\nabla\times{\bf V}=\Omega(r,z)\partial_{\theta}$. Conversely, under a
mild decay assumption at spatial infinity (which will always be satisfied in
our application), ${\bf V}$ may be recovered from $\Omega$ by introducing the
stream function $\Phi(r,z)$ that solves $-\not{\Delta}\Phi=\Omega,$ with
respect to which we have

$V^{r}=-r\partial_{z}\Phi,\quad V^{z}=\frac{1}{r}\partial_{r}(r^{2}\Phi).$

####  2.2.1. $H^{m}$-estimates and local wellposedness

Unlike the general non-axisymmetric case, (2.4) and (2.5) are locally
wellposed in (sufficiently high regularity) Sobolev spaces. Here we derive
$H^{m}$ a priori estimates and corresponding local wellposedness statements
for (2.4) and (2.5).

We start with a lemma concerning the relationship between the standard
$H^{m}$-norm for a vector field formulated with respect to the standard basis
$(e_{x},e_{y},e_{z})$ and those for components with respect to
$(\partial_{r},\partial_{\theta},\partial_{z})$ in the axisymmetric case.

######  Lemma 2.2.

Let ${\bf U},{\bf V}\in\mathcal{S}(\mathbb{R}^{3})$ be of the form ${\bf
U}=U^{\theta}(r,z)\partial_{\theta}$ and ${\bf
V}=V^{r}(r,z)\partial_{r}+V^{z}(r,z)\partial_{z}$.

  1. (1)

For $U^{\theta}$ and $r^{-1}V^{r}$, we have the formulae

(2.6) $\displaystyle U^{\theta}(r,z)$
$\displaystyle=\frac{1}{r^{2}}\int_{0}^{r}(-\partial_{y}{\bf
U}^{x}+\partial_{x}{\bf
U}^{y})(r^{\prime},z)r^{\prime}\,\mathrm{d}r^{\prime},$ (2.7)
$\displaystyle\frac{1}{r}V^{r}(r,z)$
$\displaystyle=\frac{1}{r^{2}}\int_{0}^{r}(\partial_{x}{\bf
V}^{x}+\partial_{y}{\bf
V}^{y})(r^{\prime},z)r^{\prime}\,\mathrm{d}r^{\prime}.$

  2. (2)

For any nonnegative integer $m$, we have

(2.8) $\displaystyle\|{{\bf U}}\|_{H^{m}}^{2}$
$\displaystyle\simeq_{m}\|{rU^{\theta}}\|_{H^{m}(r\mathrm{d}r\mathrm{d}z)}^{2}+\|{U^{\theta}}\|_{H^{m-1}(r\mathrm{d}r\mathrm{d}z)}^{2},$
(2.9) $\displaystyle\|{{\bf V}}\|_{H^{m}}^{2}$
$\displaystyle\simeq_{m}\|{V^{r}}\|_{H^{m}(r\mathrm{d}r\mathrm{d}z)}+\|{r^{-1}V^{r}}\|_{H^{m-1}(r\mathrm{d}r\mathrm{d}z)}^{2}+\|{V^{z}}\|_{H^{m}(r\mathrm{d}r\mathrm{d}z)}^{2},$

where in the case $m=0$, the terms involving $H^{m-1}$ are ignored. Moreover,
we have

(2.10) $\displaystyle\|{\nabla(U^{\theta}\partial_{\theta})}\|_{L^{2}}^{2}$
$\displaystyle\simeq\|{r\partial_{r}U^{\theta}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}^{2}+\|{r\partial_{z}U^{\theta}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}^{2}+\|{U^{\theta}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}^{2},$
(2.11) $\displaystyle\|{\nabla(V^{r}\partial_{r})}\|_{L^{2}}^{2}$
$\displaystyle\simeq\|{\partial_{r}V^{r}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}^{2}+\|{\partial_{z}V^{r}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}^{2}+\|{r^{-1}V^{r}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}^{2}.$

######  Remark 2.3.

For $V^{z}$, by Hardy’s inequality on $\mathbb{R}^{3}$, we simply have

$\|{(r^{2}+z^{2})^{-\frac{1}{2}}V^{z}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}\lesssim\|{\nabla
V^{z}}\|_{L^{2}(r\mathrm{d}r\mathrm{d}z)}\simeq\|{\nabla(V^{z}\partial_{z})}\|_{L^{2}}.$

###### Proof.

The inequalities $\lesssim_{m}$ in (2.8)–(2.9) are straightforward
consequences of the relation
$(\partial_{r},\partial_{\theta},\partial_{z})=(r^{-1}xe_{x}+r^{-1}ye_{y},xe_{y}-ye_{x},e_{z})$.
To prove the reverse inequality, as well as (2.6)–(2.7) and (2.10)–(2.11), we
introduce the two-dimensional scaling vector field
$S=r\partial_{r}=x\partial_{x}+y\partial_{y}$ and note the algebraic
identities

$\displaystyle(S+2)U^{\theta}$
$\displaystyle=-\partial_{y}(-yU^{\theta})+\partial_{x}(xU^{\theta})=-\partial_{y}{\bf
U}^{x}+\partial_{x}{\bf U}^{y},$ $\displaystyle(S+2)(r^{-1}V^{r})$
$\displaystyle=\partial_{x}(xr^{-1}V^{r})+\partial_{y}(yr^{-1}V^{r})=\partial_{x}{\bf
V}^{x}+\partial_{y}{\bf V}^{y}.$

Due to the similarity of these identities, the proofs for ${\bf U}$ and ${\bf
V}$ are similar after this point; henceforth, we only focus on the case of
${\bf U}$. Noting that $(S+2)U^{\theta}=r^{-1}\partial_{r}(r^{2}U^{\theta})$,
(2.6) follows from the fundamental theorem of calculus and the vanishing of
$r^{2}U^{\theta}$ at $r=0$. Furthermore, for any multiindex $\beta$ of order
$\ell-1$, we have the commutator identity

$(S+\ell+1)\nabla_{x,y}^{\beta}U^{\theta}=\nabla_{x,y}^{\beta}(S+2)U^{\theta}=\nabla_{x,y}^{\beta}(-\partial_{y}{\bf
U}^{x}+\partial_{x}{\bf U}^{y}).$

Hence,

$\|{(S+\ell+1)\nabla_{x,y}^{\beta}U^{\theta}}\|_{L^{2}}\lesssim\|{\nabla_{x,y}^{\ell}{\bf
U}}\|_{L^{2}}.$

We claim that, for any $u=u(r)$ that vanishes sufficiently fast as
$r\to\infty$, the following Hardy’s inequality holds:

$\|{(r\partial_{r}+\ell+1)u}\|_{L^{2}(r\mathrm{d}r)}^{2}\geq\ell^{2}\|{u}\|_{L^{2}(r\mathrm{d}r)}^{2}.$

Assuming the claim, (2.10) and the inequality $\gtrsim_{m}$ in (2.8) readily
follow. To prove the claim, we follow the classical proof of Hardy’s
inequality and argue as follows:

$\displaystyle\int((r\partial_{r}+\ell+1)u)^{2}\,r\mathrm{d}r$
$\displaystyle=\int\left(((r\partial_{r}+1)u)^{2}+2\ell
u(r\partial_{r}+1)u+\ell^{2}u^{2}\right)r\,\mathrm{d}r$
$\displaystyle=\int((r\partial_{r}+1)u)^{2}r+\ell\partial_{r}u^{2}r^{2}+\ell(\ell+2)u^{2}r\,\mathrm{d}r$
$\displaystyle\geq\int((r\partial_{r}+1)u)^{2}r+\ell^{2}u^{2}r\,\mathrm{d}r,$

where we used an integration by parts and the vanishing of $u(r)$ as
$r\to\infty$ on the last line. Since the first term on the last line is
nonnegative, the claim follows. ∎

We now prove an $H^{m}$ a priori estimate for the axisymmetric (E-MHD) system
(2.4). Let ${\mathbf{\Pi}}=\Pi(r,z)\partial_{\theta}\in C_{t}(I;H^{\infty})$
and fix $m\geq 4$. For any multiindex $\alpha$ of order at most $m$, we
compute

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|{\nabla^{\alpha}{\mathbf{\Pi}}}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|{\nabla^{\alpha}(\Pi\partial_{\theta})}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=2\int\nabla^{\alpha}(\Pi\partial_{z}\Pi\,\partial_{\theta})\cdot\nabla^{\alpha}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle\quad=2\int\Pi\partial_{z}\nabla^{\alpha}(\Pi\,\partial_{\theta})\cdot\nabla^{\alpha}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z+2\int\\{\nabla^{\alpha}(\Pi\partial_{z}(\Pi\,\partial_{\theta}))-\Pi\partial_{z}\nabla^{\alpha}(\Pi\,\partial_{\theta})\\}\cdot\nabla^{\alpha}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

For the first term, we perform an integration by parts in $z$ and estimate as
follows:

$\displaystyle\left|{\int\Pi\partial_{z}\nabla^{\alpha}(\Pi\,\partial_{\theta})\cdot\nabla^{\alpha}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z}\right|$
$\displaystyle=\left|{\int(\partial_{z}\Pi)\nabla^{\alpha}(\Pi\,\partial_{\theta})\cdot\nabla^{\alpha}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z}\right|\lesssim\|{\partial_{z}\Pi}\|_{L^{\infty}}\|{{\mathbf{\Pi}}}\|_{H^{m}}^{2}\lesssim\|{{\mathbf{\Pi}}}\|_{H^{m}}^{3},$

where we used the Sobolev inequality,
$\|{\partial_{z}\Pi}\|_{H^{2}}\lesssim\|{{\mathbf{\Pi}}}\|_{H^{4}}$ and $m\geq
4$ for the last inequality. For the second term, we use the fact that no
factor of $\Pi$ is differentiated $m+1$-times. Using $\partial_{\theta}=x{\bf
e}_{y}-y{\bf e}_{x}$ and Lemma 2.2, we have

$\displaystyle\left|{\int\\{\nabla^{\alpha}(\Pi\partial_{z}(\Pi\,\partial_{\theta}))-\Pi\partial_{z}\nabla^{\alpha}(\Pi\,\partial_{\theta})\\}\cdot\nabla^{\alpha}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z}\right|$
$\displaystyle\lesssim\sum_{\beta:|{\beta}|\geq
1,\,\beta\leq\alpha}\|{\nabla^{\beta}\Pi\nabla^{\alpha-\beta}\partial_{z}{\mathbf{\Pi}}}\|_{L^{2}}\|{{\mathbf{\Pi}}}\|_{H^{m}}\lesssim\|{{\mathbf{\Pi}}}\|_{H^{m}}^{3},$

where we used $m\geq 4$ on the last line. Integrating the resulting
differential inequality in time, we obtain the a priori bound

(2.12)
$\sup_{t\in[0,T_{0}]}\|{{\mathbf{\Pi}}(t)}\|_{H^{m}}\lesssim\|{{\mathbf{\Pi}}(0)}\|_{H^{m}},\qquad
T_{0}=T_{0}(\|{{\mathbf{\Pi}}(0)}\|_{H^{m}})>0.$

Moreover, we have the following local wellposedness result:

######  Proposition 2.4.

Given initial data ${\mathbf{\Pi}}_{0}=\Pi_{0}(r,z)\partial_{\theta}\in H^{m}$
with $m\geq 4$, there exists a unique solution ${\mathbf{\Pi}}\in
C_{t}([0,T_{0}];H^{m})$ to the system (2.4) with
${\mathbf{\Pi}}(0)={\mathbf{\Pi}}_{0}$, where $T_{0}$ depends only on
$\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}$. Moreover, the solution satisfies the a
priori estimate (2.12) on $[0,T_{0}]$.

Uniqueness can be proved along the similar lines as in the proof of the
$H^{m}$ a priori estimate.333We note that the uniqueness statement says that
the solution to (E-MHD) under the axisymmetric assumption is unique. Of
course, without a uniqueness statement for the original (E-MHD) system, one
cannot exclude the possibility that an axisymmetric initial data evolving into
a non-axisymmetric solution as well. For existence, one may use a vanishing
viscosity method (as sketched below for the more complicated case of (2.5)).
We omit the straightforward details.

######  Remark 2.5.

Using fractional Sobolev spaces, the regularity requirement may be optimized
to $m>7/2$.

Next, we turn to the axisymmetric (Hall-MHD) system, i.e., (2.5). Let
$(\boldsymbol{\omega},{\mathbf{\Pi}})=(\Omega(r,z)\partial_{\theta},\Pi(r,z)\partial_{\theta})\in
C_{t}(I;H^{\infty})$ and fix $m\geq 4$. Let ${\bf
V}=V^{r}(r,z)\partial_{r}+V^{\theta}(r,z)$ be given from $\boldsymbol{\omega}$
by the Biot–Savart law; hence

(2.13) $\|{\nabla{\bf
V}}\|_{H^{m-1}}\lesssim\|{\boldsymbol{\omega}}\|_{H^{m-1}}.$

By Lemma 2.2, it follows that $\|{\nabla
V^{r}}\|_{H^{m-1}}+\|{r^{-1}V^{r}}\|_{L^{2}}+\|{\nabla
V^{z}}\|_{H^{m-1}}\lesssim\|{\nabla{\bf V}}\|_{H^{m-1}}.$ For any multiindex
$\alpha$ of order at most $m-1$, we use the $\Omega$-equation in (2.5) to
compute

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|{\nabla^{\alpha}\boldsymbol{\omega}}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=\int\nabla^{\alpha}(\partial_{t}\Omega\partial_{\theta})\cdot\nabla^{\alpha}(\Omega\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle=-\int(V^{r}\partial_{r}+V^{z}\partial_{z})\nabla^{\alpha}(\Omega\partial_{\theta})\cdot\nabla^{\alpha}(\Omega\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle\mathrel{\phantom{=}}-\int\\{\nabla^{\alpha}((V^{r}\partial_{r}+V^{z}\partial_{z})\Omega\partial_{\theta})-(V^{r}\partial_{r}+V^{z}\partial_{z})\nabla^{\alpha}(\Omega\partial_{\theta})\\}\cdot\nabla^{\alpha}(\Omega\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle\mathrel{\phantom{=}}-2\int\nabla^{\alpha}(\Pi\partial_{z}\Pi\,\partial_{\theta})\cdot\nabla^{\alpha}(\Omega\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z-\nu\int|{\nabla\nabla^{\alpha}(\Omega\,\partial_{\theta})}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

The first term vanishes after an integration by parts (since ${\bf V}$ is
divergence-free) and the last term is nonpositive. Estimating the remaining
terms as in the previous proof of (2.4), we obtain

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|{\nabla^{\alpha}\boldsymbol{\omega}}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle\lesssim\|{\boldsymbol{\omega}}\|_{H^{m-1}}^{3}+\|{{\mathbf{\Pi}}}\|_{H^{m}}^{2}\|{\boldsymbol{\omega}}\|_{H^{m-1}},$

provided that $m\geq 4$. Next, for any multiindex $\alpha^{\prime}$ of order
at most $m$, we use the $\Pi$-equation in (2.5) to compute

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|{\nabla^{\alpha^{\prime}}{\mathbf{\Pi}}}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z=\int\nabla^{\alpha^{\prime}}(\partial_{t}\Pi\partial_{\theta})\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle=-\int\nabla^{\alpha^{\prime}}((V^{r}\partial_{r}+V^{z}\partial_{z})\Pi\partial_{\theta})\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z+2\int\nabla^{\alpha^{\prime}}(\Pi\partial_{z}\Pi\,\partial_{\theta})\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle=-\int(V^{r}\partial_{r}+V^{z}\partial_{z})\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle\mathrel{\phantom{=}}-\int\\{\nabla^{\alpha^{\prime}}((V^{r}\partial_{r}+V^{z}\partial_{z})\Pi\partial_{\theta})-(V^{r}\partial_{r}+V^{z}\partial_{z})\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\\}\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$
$\displaystyle\mathrel{\phantom{=}}+2\int\Pi\partial_{z}\nabla^{\alpha^{\prime}}(\Pi\,\partial_{\theta})\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z+2\int\\{\nabla^{\alpha^{\prime}}(\Pi\partial_{z}(\Pi\,\partial_{\theta}))-\Pi\partial_{z}\nabla^{\alpha^{\prime}}(\Pi\,\partial_{\theta})\\}\cdot\nabla^{\alpha^{\prime}}(\Pi\partial_{\theta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}z.$

The first term vanishes after an integration by parts (since ${\bf V}$ is
divergence-free). Estimating the remaining terms as in the proof of (2.4), we
obtain

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|{\nabla^{\alpha^{\prime}}{\mathbf{\Pi}}}|^{2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z\lesssim(\|{\boldsymbol{\omega}}\|_{H^{m-1}}+\|{{\mathbf{\Pi}}}\|_{H^{m}})\|{{\mathbf{\Pi}}}\|_{H^{m}}^{2}.$

We now sum up these differential inequalities and integrate in time. As a
result, for $m\geq 4$, we obtain the a priori bound

(2.14)
$\displaystyle\sup_{t\in[0,T_{0}]}\left(\|{(\boldsymbol{\omega}(t),{\mathbf{\Pi}}(t))}\|_{H^{m-1}\times
H^{m}}\right)+\left(\nu\int_{0}^{T_{0}}\|{\boldsymbol{\omega}(t)}\|_{H^{m}}^{2}\,\mathrm{d}t\right)^{\frac{1}{2}}\lesssim_{m}\|{(\boldsymbol{\omega}(0),{\mathbf{\Pi}}(0))}\|_{H^{m-1}\times
H^{m}}$

on a time interval $[0,T_{0}]$, for $T_{0}$ sufficiently small depending on
$\|{(\boldsymbol{\omega}(0),{\mathbf{\Pi}}(0))}\|_{H^{m-1}\times H^{m}}$.
Moreover, we have the following local wellposedness result:

######  Proposition 2.6.

Given initial data
$(\boldsymbol{\omega}_{0}=\Omega(r,z)\partial_{\theta},{\mathbf{\Pi}}_{0}=\Pi_{0}(r,z)\partial_{\theta})\in
H^{m-1}\times H^{m}$ with $m\geq 4$, there exists a unique solution
$(\boldsymbol{\omega},{\mathbf{\Pi}})\in C_{t}([0,T_{0}];H^{m-1}\times H^{m})$
to (2.5) with $({\bf V}(0),{\mathbf{\Pi}}(0))=({\bf
V}_{0},{\mathbf{\Pi}}_{0})$ such that the LHS of (2.14) is finite, where
$T_{0}$ depends only on $\|{({\bf V}_{0},{\mathbf{\Pi}}_{0})}\|_{H^{m-1}\times
H^{m}}$. Moreover, the solution satisfies the a priori estimate (2.14) on
$[0,T_{0}]$ with an implicit constant that only depends on $m$.

As before, uniqueness can be proved along the similar lines as in the proof of
the $H^{m}$ a priori estimate. For the proof of existence, we may add an
artificial viscosity term $\kappa\Delta\Pi$ ($\kappa>0$) on the RHS of the
$\Pi$-equation in (2.5), for which local wellposedness is proved in [6]. For
this viscous system, an argument similar to the proof of the $H^{m}$ a priori
estimate leads to a $\kappa$-independent $H^{m}$ a priori estimate of the form
(2.14) on a short time interval $[0,T_{0}]$, with $T_{0}$ only depending on
$\|{({\bf V}_{0},{\mathbf{\Pi}}_{0})}\|_{H^{m-1}\times H^{m}}$. Then by taking
the limit $\kappa\to 0$, we obtain a solution to (2.5) in the desired function
space. We omit the straightforward details.

Remark 2.5 apply to the case of (Hall-MHD) as well.

####  2.2.2. Further properties

We now discuss further properties of the axisymmetric solutions that will be
used later. We begin with the case of (E-MHD). Observe that (2.4) with
$\eta=0$ is simply the inviscid Burgers’ equation in $(t,z)$. As a result, it
obeys the following finite speed of propagation property:

######  Lemma 2.7.

For any $C^{1}$ solution $\Pi$ to (2.4), the value of $\Pi(t,r,z)$ is uniquely
determined by $\Pi_{0}$ on
$\\{(r,z^{\prime})\in\\{r\\}\times\mathbb{R}:|{z-z^{\prime}}|\leq
t\|{\Pi_{0}}\|_{L^{\infty}}\\}$.

In general, we have the following estimate near the initial time:

######  Lemma 2.8.

Let $m\geq 10$. Consider an initial data set
${\mathbf{\Pi}}_{0}=\Pi_{0}(r,z)\partial_{\theta}\in H^{m}$ and the
corresponding solution ${\mathbf{\Pi}}$ to (2.4) on $[0,T_{0}]$ given by
Proposition 2.4. Then for $t\in[0,T_{0}]$, we have

(2.15)
$\begin{split}\left\|{\Pi-\left(\Pi_{0}+\Pi_{0}\partial_{z}\Pi_{0}\right)}\right\|_{H^{m-3}}\lesssim_{m}t^{2}\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}^{3}.\end{split}$

We remark that the factor $t^{2}$ allows us to treat the expression on the LHS
as an acceptable error in the proof of Sobolev instability/illposedness of
(E-MHD) outside axisymmetry below.

###### Proof.

In this proof, we suppress the dependence of implicit constants on $m$. By
Taylor expansion in $t$, we have

(2.16)
$\|{{\mathbf{\Pi}}(t)-{\mathbf{\Pi}}(0)-t\partial_{t}{\mathbf{\Pi}}(0)}\|_{H^{m-2}}\lesssim
t^{2}\sup_{t^{\prime}\in[0,t]}\|{\partial_{t}^{2}{\mathbf{\Pi}}(t^{\prime})}\|_{H^{m-2}}.$

By (2.4) and Lemma 2.2, the LHS of (2.16) dominates the LHS of (2.15). Hence,
it only remains to bound the RHS of (2.16). By (2.4), for each fixed
$t\in[0,T_{0}]$, we have

$\displaystyle\|{\partial_{t}{\mathbf{\Pi}}}\|_{H^{m-1}}$
$\displaystyle\lesssim\|{{\mathbf{\Pi}}}\|_{H^{m}}^{2},\qquad\|{\partial_{t}^{2}{\mathbf{\Pi}}}\|_{H^{m-2}}\lesssim\|{{\mathbf{\Pi}}}\|_{H^{m-1}}\|{\partial_{t}{\mathbf{\Pi}}}\|_{H^{m-1}}\lesssim\|{{\mathbf{\Pi}}}\|_{H^{m}}^{3}.$

Then by the a priori estimate (2.12), the desired bound follows. ∎

In the case of (Hall-MHD), we consider initial data of the form
$(\boldsymbol{\omega}_{0},{\mathbf{\Pi}}_{0})=(0,\Pi(r,z)\partial_{\theta})$.
For the corresponding solution, the following comparison with the solution to
(2.4) with the same initial data ${\mathbf{\Pi}}_{0}=\Pi_{0}\partial_{\theta}$
holds near the initial time:

######  Lemma 2.9.

Let $\nu\geq 0$ and $m\geq 10$. Consider an initial data set
$(\boldsymbol{\omega}_{0},{\mathbf{\Pi}}_{0})=(0,\Pi(r,z)\partial_{\theta})$,
where ${\mathbf{\Pi}}_{0}\in H^{m}$, and let
$(\boldsymbol{\omega},{\mathbf{\Pi}})$ be the corresponding solution to (2.5)
on $[0,T_{0}]$ given by Proposition 2.6. Moreover, let
${\mathbf{\Pi}}^{e}=\Pi^{e}(r,z)\partial_{\theta}$ be the solution to (2.4)
with ${\mathbf{\Pi}}^{e}(t=0)=\Pi_{0}(r,z)\partial_{\theta}$. Then for
$t\in[0,T_{0}]$, we have

(2.17) $\displaystyle\|{\nabla{\bf V}}\|_{H^{m-3}}$
$\displaystyle\lesssim_{m}t(\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}^{2}+\nu\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}),$
(2.18) $\displaystyle\|{\Pi-\Pi^{e}}\|_{H^{m-3}}$
$\displaystyle\lesssim_{m}t^{2}\exp(C\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}t)(\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}^{3}+\nu\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}^{2}).$

We note that the factor $t^{2}$ in (2.18) is again crucial to treat
$\Pi-\Pi^{e}$ as an acceptable error in the proof of Sobolev
instability/illposedness of (Hall-MHD) outside axisymmetry below.

###### Proof.

In this proof, we suppress the dependence of implicit constants on $m$. By
(2.5) and (2.13), for each $t\in[0,T_{0}]$, we may estimate

$\displaystyle\|{\partial_{t}\nabla{\bf
V}}\|_{H^{m-3}}\lesssim\|{\partial_{t}\boldsymbol{\omega}}\|_{H^{m-3}}\lesssim\|{\boldsymbol{\omega}}\|_{H^{m-1}}^{2}+\|{{\mathbf{\Pi}}}\|_{H^{m}}^{2}+\nu\|{\boldsymbol{\omega}}\|_{H^{m-1}}$

By hypothesis, $\boldsymbol{\omega}_{0}=0$ (hence ${\bf V}(0)=0$). Hence, by
integrating in time and using the a priori estimate (2.14), obtain (2.17).
Next, by subtracting (2.4) from the $\Pi$-equation in (2.5), we obtain

$\displaystyle\partial_{t}(\Pi-\Pi^{e})=-(V^{r}\partial_{r}+V^{z}\partial_{z})\Pi+2(\Pi-\Pi^{e})\partial_{z}\Pi+2\Pi^{e}\partial_{z}(\Pi-\Pi^{e}).$

We use this equation to compute
$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{{\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}}\|_{H^{m-2}}^{2}$.
Proceeding as in the proof of the a priori estimate (2.14), we obtain the
differential inequality

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{{\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}}\|_{H^{m-2}}^{2}\lesssim\|{\nabla{\bf
V}}\|_{H^{m-3}}\|{{\mathbf{\Pi}}}\|_{H^{m-1}}\|{{\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}}\|_{H^{m-2}}+(\|{{\mathbf{\Pi}}}\|_{H^{m-1}}+\|{{\mathbf{\Pi}}^{e}}\|_{H^{m-1}})\|{{\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}}\|_{H^{m-2}}^{2}$

while $({\mathbf{\Pi}}-{\mathbf{\Pi}}^{e})(0)=0$. Cancelling a factor of
$\|{{\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}}\|_{H^{m-1}}$, applying Grönwall’s
inequality and using the a priori estimates (2.14) and (2.17), we obtain

$\|{{\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}}\|_{H^{m-2}}\lesssim
t^{2}\exp(C\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}t)(\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}^{3}+\nu\|{{\mathbf{\Pi}}_{0}}\|_{H^{m}}^{2}).$

We note that the factor $t^{2}$ comes from integrating the factor $t$ from
(2.17) for $\|{\nabla V}\|_{H^{m-3}}$. Then (2.18) follows via Lemma 2.2. ∎

###  2.3. The $(2+\frac{1}{2})$-dimensional reduction

The systems (Hall-MHD) and (E-MHD) significantly simplifies under the
$(2+\frac{1}{2})$-dimensional reduction, by which we man that the solutions
are assumed to be independent of the $z$-coordinate.

####  2.3.1. Linearized equations under the $(2+\frac{1}{2})$-dimensional
reduction.

In this section, we write down the simpler system of equations for (1.2) and
(1.3), under the $(2+\frac{1}{2})$-dimensional assumption.

We take $\mathring{{\bf B}}=f(r)\partial_{\theta}$ and introduce $\psi$ and
$\omega$ satisfying

$\begin{split}(\nabla\times b)^{z}=-\Delta\psi,\qquad(\nabla\times
u)^{z}=\omega.\end{split}$

Since $(\nabla\times\mathring{{\bf B}})^{z}=r^{-1}\partial_{r}(r^{2}f),$ we
obtain that $\psi=(-\Delta)^{-1}((\nabla\times\mathring{{\bf
B}})^{z}-r^{-1}\partial_{r}(r^{2}f)).$ Then, we have that (1.2) and (1.3)
simplify to become

(2.19) $\left\\{\begin{aligned}
&\partial_{t}u^{z}-f(r)\partial_{\theta}b^{z}-\nu\Delta u^{z}=0,\\\
&\partial_{t}\omega-\left(f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)\right)\partial_{\theta}\psi+f(r)\partial_{\theta}\Delta\psi-\nu\Delta\omega=0,\\\
&\partial_{t}b^{z}-f(r)\partial_{\theta}u^{z}+\left(f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)\right)\partial_{\theta}\psi-f(r)\partial_{\theta}\Delta\psi=0,\\\
&\partial_{t}\psi-f(r)\partial_{\theta}(-\Delta)^{-1}\omega+f(r)\partial_{\theta}b^{z}=0.\end{aligned}\right.$

and

(2.20) $\left\\{\begin{aligned}
&\partial_{t}b^{z}-f(r)\partial_{\theta}\Delta\psi+\left(f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)\right)\partial_{\theta}\psi=0,\\\
&\partial_{t}\psi+f(r)\partial_{\theta}b^{z}=0,\end{aligned}\right.$

respectively. Note that $(b^{r},b^{\theta},u^{r},u^{\theta})$ can be recovered
from $\psi$ and $\omega$ by the following relations:

$\begin{split}b^{r}=\frac{1}{r}\partial_{\theta}\psi,\quad
b^{\theta}=-\frac{1}{r}\partial_{r}\psi,\quad
u^{r}=\frac{1}{r}\partial_{\theta}(-\Delta)^{-1}\omega,\quad
u^{\theta}=-\frac{1}{r}\partial_{r}(-\Delta)^{-1}\omega.\end{split}$

Strictly speaking, some justification is necessary for these systems; one
needs to make sure that the Biot–Savart type identities in the above are valid
at the level of $(u,b)\in L^{2}$ and that a $z$-independent $L^{2}$-solution
$(u,b)$ for (1.2) ($b$ for (1.3), resp.) with $\mathring{{\bf
B}}=f(r)\partial_{\theta}$ gives rise to a unique solution
$(u^{z},\omega,b^{z},\psi)$ for (2.19) ($(b^{z},\psi)$ for (2.20), resp.).
These issues are handled in [19, Proposition 2.1].

####  2.3.2. Approximate solutions and generalized energy identities

It will be convenient to consider quadruples
$(\tilde{u}^{z},\tilde{\omega},\tilde{b}^{z},\tilde{\psi})$ (pairs
$(\tilde{b}^{z},\tilde{\psi})$, resp.) which solve (2.19) ((2.20), resp.) up
to an $L^{2}$-error. The tildes were inserted to emphasize that they are not
exact solutions to the linear systems. We follow the notation for the errors
used in [19]: given $\mathring{{\bf B}}=f(r)\partial_{\theta}$ and
$(\tilde{u}^{z},\tilde{\omega},\tilde{b}^{z},\tilde{\psi})$, we define
$(\boldsymbol{\delta}_{\tilde{u}^{z}}^{(\nu)},\boldsymbol{\delta}_{\tilde{\omega}}^{(\nu)},\boldsymbol{\delta}_{\tilde{b}^{z}},\boldsymbol{\delta}_{\tilde{\psi}})$
by

(2.21) $\left\\{\begin{aligned}
&\partial_{t}\tilde{u}^{z}-f(r)\partial_{\theta}\tilde{b}^{z}-\nu\Delta\tilde{u}^{z}=\boldsymbol{\delta}_{\tilde{u}^{z}}^{(\nu)},\\\
&\partial_{t}\tilde{\omega}-\left(f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)\right)\partial_{\theta}\tilde{\psi}+f(r)\partial_{\theta}\Delta\tilde{\psi}-\nu\Delta\tilde{\omega}=\boldsymbol{\delta}_{\tilde{\omega}}^{(\nu)},\\\
&\partial_{t}\tilde{b}^{z}-f(r)\partial_{\theta}\tilde{u}^{z}+\left(f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)\right)\partial_{\theta}\tilde{\psi}-f(r)\partial_{\theta}\Delta\tilde{\psi}=\boldsymbol{\delta}_{\tilde{b}^{z}},\\\
&\partial_{t}\tilde{\psi}-f(r)\partial_{\theta}(-\Delta)^{-1}\tilde{\omega}+f(r)\partial_{\theta}\tilde{b}^{z}=\boldsymbol{\delta}_{\tilde{\psi}},\end{aligned}\right.$

and in the case of (E-MHD), we introduce

(2.22) $\left\\{\begin{aligned}
&\partial_{t}\tilde{b}^{z}-f(r)\partial_{\theta}\Delta\tilde{\psi}+\left(f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)\right)\partial_{\theta}\tilde{\psi}=\boldsymbol{\epsilon}_{\tilde{b}^{z}},\\\
&\partial_{t}\tilde{\psi}+f(r)\partial_{\theta}\tilde{b}^{z}=\boldsymbol{\epsilon}_{\tilde{\psi}}.\end{aligned}\right.$

Moreover, we use the notation
$\boldsymbol{\delta}_{\tilde{b}}=(\boldsymbol{\delta}_{\tilde{b}^{x}},\boldsymbol{\delta}_{\tilde{b}^{y}},\boldsymbol{\delta}_{\tilde{b}^{z}})=(-\nabla^{\perp}\boldsymbol{\delta}_{\tilde{\psi}},\boldsymbol{\delta}_{\tilde{b}}^{z})$.
Similarly $\boldsymbol{\epsilon}_{\tilde{b}}$ and
$\boldsymbol{\delta}_{\tilde{u}}$ are defined;
$\boldsymbol{\epsilon}_{\tilde{b}}=(-\nabla^{\perp}\boldsymbol{\epsilon}_{\tilde{\psi}},\boldsymbol{\epsilon}_{\tilde{b}^{z}})$
and
$\boldsymbol{\delta}_{\tilde{u}}=(-\nabla^{\perp}(-\Delta)^{-1}\boldsymbol{\delta}_{\tilde{\omega}},\boldsymbol{\delta}_{\tilde{u}^{z}})$.
With this notation, given a solution $(u,b)$ to (1.2), a straightforward
(formal) computation shows the following identity:

(2.23)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\langle{\tilde{b},b}\rangle+\langle{\tilde{u},u}\rangle\right)+2\nu\langle{\nabla\tilde{u},\nabla
u}\rangle$ $\displaystyle=$
$\displaystyle-\langle{(rf^{\prime\prime}+3f^{\prime})r^{-1}\partial_{\theta}\tilde{\psi},b^{z}}\rangle-\langle{\tilde{b}^{z},(rf^{\prime\prime}+3f^{\prime})r^{-1}\partial_{\theta}\psi}\rangle-\langle{(rf^{\prime}+2f)\nabla\tilde{\psi},u^{r,\theta}}\rangle-\langle{\tilde{u}^{r,\theta},(rf^{\prime}+2f)\nabla\psi}\rangle$
$\displaystyle+\langle{\nabla^{\perp}\boldsymbol{\delta}_{\tilde{\psi}},\nabla^{\perp}\psi}\rangle+\langle{\nabla^{\perp}\tilde{\psi},\nabla^{\perp}\boldsymbol{\delta}_{\psi}}\rangle+\langle{\boldsymbol{\delta}_{\tilde{b}},b^{z}}\rangle+\langle{\tilde{b}^{z},\boldsymbol{\delta}_{b}}\rangle$
$\displaystyle-\langle{\nabla^{\perp}(-\Delta)^{-1}\boldsymbol{\delta}_{\tilde{\omega}},u^{r,\theta}}\rangle-\langle{\tilde{u}^{r,\theta},\nabla^{\perp}(-\Delta)^{-1}\boldsymbol{\delta}_{\omega}}\rangle+\langle{\boldsymbol{\delta}_{\tilde{u}}^{(\nu)},u^{z}}\rangle+\langle{\tilde{u}^{z},\boldsymbol{\delta}_{u}^{(\nu)}}\rangle.$

Here, we note that $(u,b)$ does not have to be $z$-independent for the
identity to hold. From (2.23), we obtain

(2.24)
$\begin{split}&\left|\frac{\mathrm{d}}{\mathrm{d}t}\left(\langle{\tilde{b},b}\rangle+\langle{\tilde{u},u}\rangle\right)\right|\lesssim\nu\|{\nabla\tilde{u}}\|_{L^{2}}\|{\nabla
u}\|_{L^{2}}+(\|{b}\|_{L^{2}}+\|{u}\|_{L^{2}})(\|{\tilde{b}}\|_{L^{2}}+\|{\tilde{u}}\|_{L^{2}}+\|{\boldsymbol{\delta}_{\tilde{b}}}\|_{L^{2}}+\|{\boldsymbol{\delta}_{\tilde{u}}}\|_{L^{2}}),\end{split}$

where the implicit constant depends on $f$. Similarly, when $b$ is a solution
to (1.3), we have

(2.25)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle{\tilde{b},b}\rangle=$
$\displaystyle-\langle{(rf^{\prime\prime}+3f^{\prime})r^{-1}\partial_{\theta}\tilde{\psi},b^{z}}\rangle-\langle{\tilde{b}^{z},(rf^{\prime\prime}+3f^{\prime})r^{-1}\partial_{\theta}\psi}\rangle$
$\displaystyle+\langle{\nabla^{\perp}\boldsymbol{\epsilon}_{\tilde{\psi}},\nabla^{\perp}\psi}\rangle+\langle{\nabla^{\perp}\tilde{\psi},\nabla^{\perp}\boldsymbol{\epsilon}_{\psi}}\rangle+\langle{\boldsymbol{\epsilon}_{\tilde{b}},b^{z}}\rangle+\langle{\tilde{b}^{z},\boldsymbol{\epsilon}_{b}}\rangle,$

which gives the estimate

(2.26)
$\begin{split}&\left|\frac{\mathrm{d}}{\mathrm{d}t}\langle{\tilde{b},b}\rangle\right|\lesssim\|{b}\|_{L^{2}}(\|{\tilde{b}}\|_{L^{2}}+\|{\boldsymbol{\epsilon}_{\tilde{b}}}\|_{L^{2}}).\end{split}$

This shows that a quadruple
$(\tilde{u}^{z},\tilde{\omega},\tilde{b}^{z},\tilde{\psi})$ (a pair
$(\tilde{b}^{z},\tilde{\psi})$, resp.) is a good approximation to a solution
to (1.2) ((1.3), resp.) if (i) $b$ and $\tilde{b}$ are close in $L^{2}$ at the
initial time, and (ii) the $L^{2}$-norm of the error is under control. For
this reason, as far as the argument involving the generalized energy identity
goes, it is natural and convenient to absorb terms that are bounded either a
priori or a posteriori by the $L^{2}$-norm of $\tilde{b}$ into the error
(similar to the way the $O(1)$-notation works).

Some regularity is necessary to justify the above identities (and the
resulting estimates), which is handled in [19, Proposition 2.3]; here, we
shall just assume the approximate solutions have some additional regularity:

######  Proposition 2.10.

The energy identity (2.23) holds under the following assumptions:
$(u^{z},\omega,b^{z},\psi)$ is derived from an $L^{2}$-solution $(u,b)$, and
$(\tilde{u}^{z},\tilde{\omega},\tilde{b}^{z},\tilde{\psi})$ obeys444Here, by
the assertion $\nabla(-\Delta)^{-1}\tilde{\omega}\in X$, we mean
$\tilde{\omega}$ is of the form $-\Delta w$ where $\nabla w\in X$.

$(\tilde{u}^{z},\nabla(-\Delta)^{-1}\tilde{\omega})\in
C_{t}(I;L^{2}),\quad(\tilde{b}^{z},\nabla\tilde{\psi})\in C_{t}(I;L^{2})\cap
L^{1}_{t}(I;H^{1}),$

and the error obeys
$(\boldsymbol{\delta}_{\tilde{u}}^{(\nu)},\boldsymbol{\delta}_{\tilde{b}})\in
L^{1}_{t}(I;L^{2}),$ and when $\nu>0$, also
$\nabla\tilde{u}^{z},\tilde{\omega}\in L^{2}_{t}(I;L^{2}).$ Analogously,
(2.25) holds when $(b^{z},\psi)$ is derived from an $L^{2}$-solution $b$ and
$(\tilde{b}^{z},\tilde{\psi})$ obeys

$(\tilde{b}^{z},\nabla\tilde{\psi})\in C_{t}(I;L^{2})\cap L^{1}_{t}(I;H^{1})$

and the error obeys $\boldsymbol{\epsilon}_{\tilde{b}}\in L^{1}_{t}(I;L^{2}).$

###  2.4. Degenerating wave packets for the axisymmetric background

In this section, we review the construction of degenerating wave packets for
the axisymmetric background from [19, Sections 3–4], which is the key
ingredient in the proof of illposedness. For simplicity, we shall impose the
following assumptions on the stationary magnetic background.

Assumptions on $\mathring{{\bf B}}=f(r)\partial_{\theta}$. We assume that $f$
is a smooth function of $r$ and satisfies the following.

  * •

There exist some $1\geq\ell=\ell_{f}>0,r_{0}=r_{0,f}$ such that $20\ell\geq
r_{0}\geq 2\ell$ and $f(r_{0})=0$.

  * •

The support of $f$ is contained in $[r_{0}-\ell,r_{0}+2\ell]$.

  * •

For $r_{1}=r_{0}+\ell$, we have

$\begin{split}\frac{1}{2}\leq f^{\prime}(r)\leq 1,\qquad
r\in[r_{0},r_{1}].\end{split}$

  * •

Lastly, we set $C_{M}=\ell^{-1}\max_{1\leq j\leq
M}(\ell^{j}\|{f^{(j)}(r)}\|_{L^{\infty}[r_{0},r_{1}]}).$ We also let $C_{M}$
to depend on $M$.

Basically, we are assuming that $f$ is characterized by a single length scale
$\ell$. (This is actually not a very stringent restriction, since the scaling
property of the linearized equation gives the wave packets for the rescaled
$f$ by a simple rescaling of the time variable.)

For convenience of the reader, we review the construction of degenerating wave
packet, under the above simplifying assumptions for $f(r)$. The starting point
is to take another time derivative on the $\psi$ equation in (2.20):

(2.27)
$\begin{split}\partial_{t}^{2}\psi+f^{2}\partial_{\theta}^{2}\Delta\psi=0,\end{split}$

where we have neglected the last term on the left hand side of the $b^{z}$
equation (which is an allowable error). Once we find a solution $\tilde{\psi}$
to this equation up to an allowable error, we shall take $\tilde{b}^{z}$ by
$-(f\partial_{\theta})^{-1}\partial_{t}\tilde{\psi}$, again up to an allowable
error. Expanding the Laplacian in the cylindrical coordinates, and introducing
$\tau=\lambda t$ for some $\lambda\in\mathbb{N}$, (2.27) becomes

$\partial_{\tau}^{2}\tilde{\psi}+f^{2}(\lambda^{-1}\partial_{\theta})^{2}\partial_{r}^{2}\tilde{\psi}+\frac{f^{2}}{r}(\lambda^{-1}\partial_{\theta})^{2}\partial_{r}\tilde{\psi}+\frac{\lambda^{2}f^{2}}{r^{2}}(\lambda^{-1}\partial_{\theta})^{4}\tilde{\psi}=0.$

In the domain $r\in[r_{0},r_{1}]$, we make a change of variables
$\eta=\eta(r)$, where

$\eta^{\prime}(r)=\frac{1}{f(r)},\qquad\eta(r_{1})=0.$

Note that $\eta\to-\infty$ as $r\to r_{0}^{+}$. Moreover,
$\partial_{\eta}f=f\partial_{r}f$ and $\partial_{\eta}r=f$. Introducing a new
dependent variable

$\varphi(\tau,\eta)=e^{-i\lambda\theta}\left(\frac{r}{f}\right)^{\frac{1}{2}}\tilde{\psi},$

$\varphi$ solves the following remarkably simple equation in the
$(\tau,\eta)$-coordinates: introducing $f_{*}=f/r$,

(2.28)
$\displaystyle\partial_{\tau}^{2}\varphi-\partial_{\eta}^{2}\varphi+\lambda^{2}f_{*}^{2}\varphi=0.$

Note that $f_{*}(\eta)\leq\frac{1}{3}\exp(\eta)\leq\frac{1}{3}$ for $\eta\leq
0$ from the assumption $r_{0}\geq 3\ell$. Furthermore, derivatives decay
exponentially as well as $\eta\to-\infty$:
$|\partial_{\eta}^{(m)}f_{*}(\eta)|\lesssim_{m}\exp(\eta)$. Now, we take the
following ansatz for $\varphi$:

(2.29)
$\varphi(\tau,\eta)=\lambda^{-1}e^{i\lambda\Phi(\tau,\eta)}h(\tau,\eta),$

where $\Phi$ and $h$ are independent of $\lambda$. The prefactor
$\lambda^{-1}$ is simply a convenient normalization. Plugging this ansatz into
(2.28) and organizing the terms with the same order in $\lambda$, we obtain
the Hamilton–Jacobi equation

(2.30)
$\begin{split}-(\partial_{\tau}\Phi)^{2}+(\partial_{\eta}\Phi)^{2}+f_{*}^{2}=0\end{split}$

and the associated transport equation

(2.31)
$\begin{split}(\partial_{\tau}\Phi\partial_{\tau}-\partial_{\eta}\Phi\partial_{\eta})h=-\frac{1}{2}(\partial_{\tau}^{2}\Phi-\partial_{\eta}^{2}\Phi)h\end{split}$

with some remaining terms, which will be incorporated into the error term. We
take the initial condition $h(0,\eta)$ for (2.31) to be a smooth function
$h_{0}$ with compact support in the region $\\{-1<\eta<0\\}$, and we simply
choose the function

(2.32)
$\begin{split}\Phi(\tau,\eta)=\tau+\eta+\int_{-\infty}^{\eta}\left((1-f_{*}^{2}(\eta^{\prime}))^{\frac{1}{2}}-1\right)\,\mathrm{d}\eta^{\prime}\end{split}$

as the solution of (2.30). The integral is finite thanks to exponential decay
of $f_{*}$. Then, (2.31) becomes

$\begin{split}\partial_{\tau}h-(1-f_{*}^{2})^{\frac{1}{2}}\partial_{\eta}h=-\frac{1}{2}\frac{f_{*}\partial_{\eta}f_{*}}{(1-f_{*}^{2})^{\frac{1}{2}}}h\end{split}$

and using that $f_{*}$ and its $\eta$-derivatives decay exponentially as
$\eta\to-\infty$, we obtain (see [19, Lemmas 3.6, 3.7])

######  Lemma 2.11.

The solution $h(\tau,\eta)$ to (2.31) with (2.32) exists for all $\tau\geq 0$
and satisfies the estimates

$\begin{split}\max_{0\leq k\leq m}\sup_{\tau\geq
0}\|{\partial_{\tau}^{k}\partial_{\eta}^{m-k}h}\|_{L^{p}(\mathbb{R}_{\eta})}\leq
C_{m}\|{h_{0}}\|_{W^{m,p}(\mathbb{R}_{\eta})},\qquad\operatorname{supp}_{\eta}(h(\tau,\cdot))\subset(-1-\tau,-\tau/2).\end{split}$

Since we have specified $\Phi$ and $h$, this gives $\varphi$ via (2.29).
Returning to the $(t,r,\theta)$ coordinates system, we define

(2.33)
$\begin{split}\tilde{\psi}=\operatorname{Re}[f^{\frac{1}{2}}\varphi]=\operatorname{Re}[\lambda^{-1}e^{i\lambda(\theta+\Phi)}\ell
f^{\frac{1}{2}}h],\qquad\tilde{b}^{z}=-\operatorname{Re}[(i\lambda
f)^{-1}\lambda^{-1}\partial_{t}(e^{i\lambda(\theta+\Phi)})\ell
f^{\frac{1}{2}}h].\end{split}$

At the initial time, we set

(2.34) $\begin{split}g_{0}(r)=(f/r)^{\frac{1}{2}}h(0,r(\eta)),\qquad
G(r)=\Phi(0,\eta(r)).\end{split}$

This will be convenient since

$\begin{split}\|{g_{0}}\|_{L^{2}}^{2}\sim\int|g_{0}(r)|^{2}r\mathrm{d}r=\int
fh^{2}\mathrm{d}r=\int
h(0,\eta)^{2}\,\mathrm{d}\eta=\|{h(0,\cdot)}\|_{L^{2}_{\eta}}^{2}.\end{split}$

Similarly, we have that

$\begin{split}\frac{1}{C_{m}}\sum_{j=0}^{m}\ell^{-j}\|{\partial_{\eta}^{(j)}h(0,\eta)}\|_{L^{2}_{\eta}}\leq\|{g_{0}}\|_{H^{m}}\leq
C_{m}\sum_{j=0}^{m}\ell^{-j}\|{\partial_{\eta}^{(j)}h(0,\eta)}\|_{L^{2}_{\eta}}.\end{split}$

Writing (2.33) at $t=0$ using $g_{0}$ gives

$\begin{split}&\tilde{b}^{z}_{(\lambda)}(0)=-f^{-1}\ell\operatorname{Re}[e^{i\lambda(\theta+G(r))}g_{0}],\qquad\tilde{\psi}_{(\lambda)}(0)=\lambda^{-1}\ell\operatorname{Re}[e^{i\lambda(\theta+G(r))}g_{0}].\end{split}$

Note that given a smooth $g_{0}$, we may obtain $h(0,\cdot)$ using (2.34) and
then define the associated wave packet
$(\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)})$. Based on these
considerations, we arrive at the following key propositions, see [19,
Propositions 3.1, 3.4].

######  Proposition 2.12 (Degenerating wave packets for (E-MHD)).

Given a smooth radial function $g_{0}(r)$ satisfying
$\operatorname{supp}g_{0}\subseteq(\tfrac{1}{2}(r_{0}+r_{1}),r_{1})$ and
$\lambda\in\mathbb{N}$, the associated wave packet
$(\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)})[g_{0}]$ constructed in
the above satisfies the following

  * •

(initial data) at $t=0$, we have

$c\|{g_{0}}\|_{L^{2}}-C_{1}(\lambda\ell)^{-1}\|{g_{0}}\|_{H^{1}}\leq\|{\tilde{b}^{z}_{(\lambda)}(0)}\|_{L^{2}}+\|{\nabla\tilde{\psi}_{(\lambda)}(0)}\|_{L^{2}}\leq
C\|{g_{0}}\|_{L^{2}}+C_{1}(\lambda\ell)^{-1}\|{g_{0}}\|_{H^{1}};$

  * •

(regularity estimates) for any $0\leq m<M$ and ${t\geq 0}$,

$\displaystyle\sup_{0\leq j\leq
m}\|{(\lambda^{-2}\partial_{t})^{k}(\lambda^{-1}\partial_{\theta})^{j}(\lambda^{-1}f\partial_{r})^{m-k-j}(\tilde{b}^{z}_{(\lambda)},\nabla\tilde{\psi}_{(\lambda)})(t)}\|_{L^{2}}\leq
C_{M}(1+(\lambda\ell)^{-1})^{m+1}\|{g_{0}}\|_{H^{m+1}};$

  * •

(degeneration) for any $1\leq p\leq 2$ and ${t\geq 0}$,

(2.35)
$\|{\tilde{b}^{z}_{(\lambda)}(t)}\|_{L^{2}_{\theta}L^{p}_{rdr}}+\|{\nabla\tilde{\psi}_{(\lambda)}(t)}\|_{L^{2}_{\theta}L^{p}_{rdr}}\leq
C_{1}e^{-c_{0}(\frac{1}{p}-\frac{1}{2})\lambda t}\|{g_{0}}\|_{H^{1}};$

  * •

(error bounds) for ${t\geq 0}$, we have

(2.36)
$\begin{split}\|{\boldsymbol{\epsilon}_{\tilde{b}}[\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)}](t)}\|_{L^{2}}\leq
C_{2}\|{g_{0}}\|_{H^{2}}.\end{split}$

In the above statements, $c,c_{0},C>0$ are absolute constants.

###### Sketch of the proof.

In the estimates, it is implicitly used that $\ell\leq 1$ and $\lambda\geq 1$.

To observe the initial data estimate, we consider
$\partial_{r}\tilde{\psi}_{(\lambda)}(0)$. The main term is when the
derivative falls on $\Phi$, which gives
$i\partial_{r}\Phi\ell(f/r)^{\frac{1}{2}}he^{i\lambda(\theta+\Phi)}$. Taking
its $L^{2}$ norm gives

$\begin{split}\iint|\partial_{r}\Phi|^{2}\ell^{2}|h|^{2}\frac{f}{r}\,r\mathrm{d}r\mathrm{d}\theta=2\pi\int|\partial_{\eta}\Phi|^{2}|\partial_{\eta}r|^{-2}\ell^{2}|h(0,\eta)|^{2}\,\mathrm{d}\eta\sim\int|h(0,\eta)|^{2}\,\mathrm{d}\eta\sim\|{g_{0}}\|_{L^{2}}^{2}\end{split}$

where we have used that $|\partial_{\eta}r|\sim\ell$ and
$|\partial_{\eta}\Phi|\sim 1$ on the support of $h(0,\cdot)$. ∎

######  Proposition 2.13 (Degenerating wave packets for (Hall-MHD)).

Under the assumptions of Proposition 2.12, in addition to
$(\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)})$, take

(2.37)
$\tilde{u}^{z}_{(\lambda)}[g_{0}]=-\tilde{\psi}_{(\lambda)}[g_{0}],\quad\tilde{\omega}_{(\lambda)}[g_{0}]=-\tilde{b}^{z}_{(\lambda)}[g_{0}].$

Then the following properties hold:

  * •

(smoothing for fluid components) for ${t\geq 0}$, we have

$\displaystyle\|{\tilde{u}^{z}_{(\lambda)}(t)}\|_{L^{2}}+\|{\nabla^{\perp}(-\Delta)^{-1}\tilde{\omega}_{(\lambda)}(t)}\|_{L^{2}}$
$\displaystyle\leq C_{1}\lambda^{-1}\ell\|{g_{0}}\|_{H^{1}},$
$\displaystyle\|{\nabla\tilde{u}^{z}_{(\lambda)}(t)}\|_{L^{2}}+\|{\tilde{\omega}_{(\lambda)}(t)}\|_{L^{2}}$
$\displaystyle\leq C_{1}\|{g_{0}}\|_{H^{1}};$

  * •

(error estimates) for ${t\geq 0}$, we have
$\boldsymbol{\delta}_{\tilde{u}^{z}}^{(\nu)}[\tilde{u}^{z}_{(\lambda)},\tilde{\omega}_{(\lambda)},\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)}]+\nu\Delta\tilde{\psi}=0$
and

$\displaystyle\|{\nabla^{\perp}(-\Delta)^{-1}(\boldsymbol{\delta}_{\tilde{\omega}}^{(\nu)}[\tilde{u}^{z}_{(\lambda)},\tilde{\omega}_{(\lambda)},\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)}]+\nu\Delta\tilde{\omega})(t)}\|_{L^{2}}$
$\displaystyle\leq C_{2}\lambda^{-1}\|{g_{0}}\|_{H^{2}},$
$\displaystyle\|{(\boldsymbol{\delta}_{\tilde{b}^{z}}^{(\nu)},\nabla\boldsymbol{\delta}_{\tilde{\psi}}^{(\nu)})[\tilde{u}^{z}_{(\lambda)},\tilde{\omega}_{(\lambda)},\tilde{b}^{z}_{(\lambda)},\tilde{\psi}_{(\lambda)}](t)}\|_{L^{2}}$
$\displaystyle\leq C_{2}\|{g_{0}}\|_{H^{2}}.$

######  Remark 2.14 (Rescaling wave packets).

If $\tilde{f}$ is a smooth function such that $\tilde{f}=\mu f$ for some
$\mu>0$ with $f$ satisfying the assumptions in the beginning of Section 2.4,
then defining wave packets corresponding to $\tilde{f}$ by simply rescaling
the time variable $t$ of wave packets for $f$ by $\mu t$, the estimates in the
above Propositions hold (with $t$ replaced by $\mu t$). This will be useful in
the nonexistence proof from Section 5.

##  3\. Proof of linear statements

In this section, we prove the linear illposedness result, Theorem A*.

###  3.1. Case of (E-MHD)

We first handle the (E-MHD) case. Let us give an overview of the proof:

  * •

In 3.1.1, we fix our choice of background magnetic fields $\mathring{{\bf B}}$
(compact support in $z$) and $\tilde{{\bf B}}$ (independent in $z$) as well as
initial data $b_{0}$ (compact support in $z$) and $\tilde{b}_{0}$ (independent
in $z$).

  * •

In 3.1.2, we prove that $b$ and $\chi\tilde{b}$ remains close in $L^{2}$ for
some interval of time, where $\chi$ is a cutoff in the $z$-direction adapted
to the support of $\mathring{{\bf B}}$ and $b$.

  * •

Then in 3.1.3, we conclude the proof using Sobolev inequalities.

#### Notations and conventions

Throughout the section, we shall assume that $f(r)$ satisfies the assumptions
from Section 2.4 and furthermore that $\|{f}\|_{L^{\infty}}\leq 1$,
$\ell=\ell_{f}\leq 1$, and $\lambda\geq 1$ is taken sufficiently large with
respect to $\ell$ so that $\lambda\ell\gg 1$. For a given $\ell$, we fix a
smooth cutoff function $\chi=\chi_{\ell}:\mathbb{R}_{z}\to\mathbb{R}_{\geq 0}$
such that $\chi(z)=1$ for $|z|\leq\ell$ and $\chi(z)=0$ for $|z|>2\ell$. In
this section, we use tildes to denote $z$-independent functions, and for
those, we use the notation $\|{\cdot}\|_{L^{2}_{x,y}}$. Note that
$\|{\chi\tilde{b}}\|_{L^{2}}^{2}\sim\ell\|{\tilde{b}}\|_{L^{2}_{x,y}}^{2}$.

####  3.1.1. Choice of initial data

Let $b$ be an $L^{2}$-solution defined on $[0,\delta]$ of the linearized E-MHD
system (1.3), written in the following convenient form:

$\begin{split}&\partial_{t}b+({\bf B}\cdot\nabla)(\nabla\times
b)-((\nabla\times b)\cdot\nabla){\bf B}+(b\cdot\nabla)(\nabla\times{\bf
B})-((\nabla\times{\bf B})\cdot\nabla)b=0.\end{split}$

We shall take ${\bf B}$ to be the time-dependent background

$\begin{split}&{\bf B}(t)=\Pi(t,r,z)\partial_{\theta}\end{split}$

with $\Pi$ being the unique smooth local solution to (1.7) with initial data
$\Pi_{0}(r,z)=f(r)\chi(z/10)$. The lifespan of $\Pi$ is proportional to
$\|{\partial_{z}\Pi_{0}}\|_{L^{\infty}}^{-1}\gtrsim\ell$, and hence as long as
$\ell\gtrsim\delta$, the background is well-defined on the time interval of
$b$. Furthermore, we shall use the following simple observation which states
that $\Pi$ is independent of $z$ in the support of $\chi$ for an interval of
time.

######  Lemma 3.1.

We have $\chi(z)\partial_{z}\Pi(t,r,z)=0$ for $0\leq t\leq 4\ell$.

###### Proof.

Differentiating (1.7) in $z$, we have

$\begin{split}&\partial_{t}(\partial_{z}\Pi)+2\Pi\partial_{z}(\partial_{z}\Pi)=-2(\partial_{z}\Pi)^{2}.\end{split}$

Defining the characteristics by $Z(t,z_{0})=2\Pi(t,Z(t,z_{0}))$ and
$Z(0,z_{0})=z_{0}$, we have from the above that $\partial_{z}\Pi(t)=0$ on the
set $Z(t,[-10\ell,10\ell])$. Since $\chi$ vanishes for $|z|\geq 2\ell$, it
suffices that

$2t=\int_{0}^{t}\|{2\Pi(s)}\|_{L^{\infty}}ds\leq 8\ell$

for $\chi(z)\partial_{z}\Pi(t,r,z)=0$. ∎

In light of the above lemma, we shall (implicitly) assume that $\delta\leq
4\ell$.

We take the degenerating wave packets corresponding to $\widetilde{{\bf
B}}=f(r)\partial_{\theta}$ from the previous section, and introduce the
simplifying notation $\tilde{b}=(-\nabla^{\perp}\tilde{\psi},\tilde{b}^{z})$.
Here, we fix some universal smooth function $h(0,\cdot)$ and use (2.34) to
define $g_{0}$. We shall freely use the estimates
$\|{g_{0}}\|_{H^{m}_{x,y}}\lesssim_{m}\ell^{-m}\|{g_{0}}\|_{L^{2}_{x,y}}$, and
suppress from writing out the dependence of constants in $m,C_{m}$.

The equation for $\tilde{b}$ can be written in the following economical form

$\begin{split}\partial_{t}\tilde{b}+(\tilde{{\bf
B}}\cdot\nabla)(\nabla\times\tilde{b})-((\nabla\times\tilde{b})\cdot\nabla)\tilde{{\bf
B}}=\boldsymbol{\epsilon}_{\tilde{b}},\end{split}$

with

$\sup_{t\in[0,\delta]}\|{\boldsymbol{\epsilon}_{\tilde{b}}}\|_{L^{2}_{x,y}}\lesssim\ell^{-2}\|{g_{0}}\|_{L^{2}_{x,y}}.$

We now fix our choice of initial data $b_{0}$. To take it to be close in
$L^{2}$ to $\chi(z)\tilde{b}_{0}$, under the divergence-free constraint, we
write

$\begin{split}&b_{0}=\chi(z)\tilde{b}_{0}+\zeta\end{split}$

for some $\zeta$. Then, taking the divergence of both sides, we see that
$\zeta$ must satisfy

$\begin{split}&0=\chi^{\prime}(z)\tilde{b}_{0}^{z}+\mathrm{div}\,\zeta,\end{split}$

which suggests the choice
$b_{0}:=\chi(z)\tilde{b}_{0}-\mathrm{div}_{U}^{-1}(\chi^{\prime}\tilde{b}_{0}^{z})$,
where $U$ can be taken to be any cube of side length $20\ell$ (say) containing
the support of $\chi^{\prime}\tilde{b}_{0}^{z}$. Using (2.3), we estimate that

$\begin{split}\|{\partial_{\theta}(\mathrm{div}_{U}^{-1}\partial_{\theta}^{-1}(\chi^{\prime}\tilde{b}_{0}^{z}))}\|_{L^{2}}\lesssim\lambda^{-1}\|{\chi^{\prime}\lambda\partial_{\theta}^{-1}\tilde{b}_{0}^{z}}\|_{L^{2}}\lesssim\lambda^{-1}\ell^{\frac{1}{2}}\|{\tilde{b}_{0}^{z}}\|_{L^{2}_{x,y}}.\end{split}$

Hence we obtain in particular that

(3.1)
$\begin{split}&\langle{b_{0},\chi\tilde{b}_{0}}\rangle=\ell^{\frac{1}{2}}\|{b_{0}}\|_{L^{2}}\|{\tilde{b}_{0}}\|_{L^{2}_{x,y}}(1+O(\lambda^{-1})).\end{split}$

####  3.1.2. Generalized energy estimate

Now that we have specified $b$ and $\tilde{b}$, let us proceed with an
argument involving the general energy identity: we compute

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\langle{b,\chi\tilde{b}}\rangle&=-\langle{\nabla\times((\nabla\times
b)\times\mathring{{\bf
B}}),\chi\tilde{b}}\rangle-\langle{b,\chi\nabla\times((\nabla\times\tilde{b})\times\tilde{{\bf
B}})}\rangle\\\ &\qquad+\langle{((\nabla\times\mathring{{\bf
B}})\cdot\nabla)b,\chi\tilde{b}}\rangle-\langle{(b\cdot\nabla)(\nabla\times\mathring{{\bf
B}}),\chi\tilde{b}}\rangle+\langle{b,\chi\boldsymbol{\epsilon}_{\tilde{b}}}\rangle.\end{split}$

First, we take care of the terms that are trivially bounded in $L^{2}$:

$\begin{split}\left|-\langle{(b\cdot\nabla)(\nabla\times\mathring{{\bf
B}}),\chi\tilde{b}}\rangle+\langle{b,\chi\boldsymbol{\epsilon}_{\tilde{b}}}\rangle\right|&\lesssim\|{\nabla^{2}\mathring{{\bf
B}}}\|_{L^{\infty}}\|{b}\|_{L^{2}}\|{\chi\tilde{b}}\|_{L^{2}}+\|{b}\|_{L^{2}}\|{\chi\boldsymbol{\epsilon}_{\tilde{b}}}\|_{L^{2}}\lesssim\ell^{-4+\frac{1}{2}}\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}\end{split}$

Then, we rewrite the first term as

$\begin{split}-\langle{\nabla\times((\nabla\times b)\times\mathring{{\bf
B}}),\chi\tilde{b}}\rangle&=\langle{b,\nabla\times((\nabla\times(\chi\tilde{b}))\times\mathring{{\bf
B}})}\rangle=\langle{b,(\mathring{{\bf
B}}\cdot\nabla)(\nabla\times(\chi\tilde{b}))}\rangle-\langle{b,(\nabla\times(\chi\tilde{b}))\cdot\nabla\mathring{{\bf
B}}}\rangle\end{split}$

and organize the remaining terms as follows:

$\begin{split}\mathrm{I}=\langle{((\nabla\times\mathring{{\bf
B}})\cdot\nabla)b,\chi\tilde{b}}\rangle,\end{split}$
$\begin{split}\mathrm{II}=-\langle{b,(\nabla\times(\chi\tilde{b}))\cdot\nabla\mathring{{\bf
B}}-\chi(\nabla\times\tilde{b})\cdot\nabla\tilde{{\bf B}}}\rangle,\end{split}$

and

$\begin{split}\mathrm{III}=\langle{b,(\mathring{{\bf
B}}\cdot\nabla)(\nabla\times(\chi\tilde{b}))-\chi(\tilde{{\bf
B}}\cdot\nabla)(\nabla\times\tilde{b})}\rangle.\end{split}$

Estimate for I. Using $\nabla\times\mathring{{\bf
B}}=-r\partial_{z}\Pi\partial_{r}+r^{-1}\partial_{r}(r^{2}\Pi)\partial_{z}=r^{-1}\partial_{r}(r^{2}f)\partial_{z}$
on the support of $\chi$,

$\begin{split}|\mathrm{I}|=|\langle{r^{-1}\partial_{r}(r^{2}\Pi)\partial_{z}b,\chi\tilde{b}}\rangle|&\lesssim\ell^{-\frac{1}{2}}(\|{rf^{\prime}(r)}\|_{L^{\infty}}+\|{f}\|_{L^{\infty}})\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}\lesssim\ell^{-\frac{1}{2}}\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Estimate for II. Note that we obtain cancellations unless a derivative falls
on $\chi$ in the expression $\nabla\times(\chi\tilde{b})$, simply because
$\mathring{{\bf B}}=\tilde{{\bf B}}$ on the support of $\chi$. Then there are
no derivatives falling on $b,\tilde{b}$, so that

$\begin{split}&|\mathrm{II}|\lesssim\ell^{-\frac{1}{2}}\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Estimate for III. We begin with writing
$\nabla\times(\chi\tilde{b})=\chi\nabla\times\tilde{b}+\chi^{\prime}(z)(-r\tilde{b}^{\theta}\partial_{r}+r^{-1}\tilde{b}^{r}\partial_{\theta})$
and

$\begin{split}\mathrm{III}&=-\langle{b^{r},\chi^{\prime}(z)f(r)\partial_{\theta}(r\tilde{b}^{\theta})}\rangle+\langle{rb^{\theta},r\chi^{\prime}(z)f(r)\partial_{\theta}(r^{-1}\tilde{b}^{r})}\rangle\\\
&=\langle{b^{r},\chi^{\prime}(z)f(r)(r\partial_{r}\tilde{b}^{r}+\tilde{b}^{r})}\rangle+\langle{\partial_{r}b^{r}+r^{-1}b^{r}+\partial_{z}b^{z},\chi^{\prime}(z)f(r)r\tilde{b}^{r}}\rangle.\end{split}$

Note that an integration by parts gives

$\begin{split}&\langle{b^{r},\chi^{\prime}(z)f(r)r\partial_{r}\tilde{b}^{r}}\rangle=-\langle{\partial_{r}b^{r},\chi^{\prime}(z)f(r)r\tilde{b}^{r}}\rangle-2\langle{b^{r},\chi^{\prime}(z)f(r)\tilde{b}^{r}}\rangle-\langle{b^{r},\chi^{\prime}(z)f^{\prime}(r)r\tilde{b}^{r}}\rangle,\end{split}$

so that after cancellations,

$\begin{split}|\mathrm{III}|=|\langle{b^{r},\chi^{\prime}(z)f^{\prime}(r)r\tilde{b}^{r}}\rangle-\langle{b^{z},\chi^{\prime\prime}(z)f(r)r\tilde{b}^{r}}\rangle|&\lesssim(\ell^{-\frac{3}{2}}\|{rf}\|_{L^{\infty}}+\ell^{-\frac{1}{2}}\|{rf^{\prime}(r)}\|_{L^{\infty}})\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}\\\
&\lesssim\ell^{-\frac{1}{2}}\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Collecting all the estimates, we conclude that

(3.2)
$\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}t}\langle{b,\chi\tilde{b}}\rangle\right|\lesssim\ell^{-\frac{3}{2}}\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

####  3.1.3. Growth of Sobolev norms

Restricting further to a time interval of length $\epsilon\ell^{2}$ with some
small absolute constant $\epsilon>0$, we may ensure that $\|{b}\|_{L^{2}}\leq
2\|{b_{0}}\|_{L^{2}}$. Within this time interval, we integrate (3.2) in time
and use (3.1) to obtain

$\begin{split}&\langle{b,\chi\tilde{b}}\rangle(t)\geq\frac{1}{2}\ell^{\frac{1}{2}}\|{b_{0}}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}},\end{split}$

by further restricting the interval of time to satisfy
$t\leq\epsilon\ell^{2}$. The rest of the proof is straightforward using the
degeneration estimate from Proposition 2.12 and completely parallel to the
corresponding argument from [19, proof of Theorem A].

In the case $s=0$ and $p>2$, we use the degeneration estimate (2.35) for
$\|{\tilde{b}}\|_{L^{2}_{\theta}L^{p^{\prime}}_{rdr}}$ where $p^{\prime}<2$ is
the dual exponent of $p$, and

$\begin{split}&\|{b(t)}\|_{L^{p}}\|{\chi\tilde{b}(t)}\|_{L^{p^{\prime}}}\geq\langle{b,\chi\tilde{b}}\rangle(t)\geq\frac{1}{2}\ell^{\frac{1}{2}}\|{b_{0}}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}\end{split}$

with

$\begin{split}\|{\chi\tilde{b}(t)}\|_{L^{p^{\prime}}}\lesssim_{\ell}\|{\tilde{b}(t)}\|_{L^{2}_{\theta}L^{p^{\prime}}_{rdr}}\end{split}$

to deduce that, up to some powers of $\ell$,

$\begin{split}&\|{b(t)}\|_{L^{p}}\gtrsim_{\ell}e^{c_{0}(\frac{1}{2}-\frac{1}{p})\lambda
t}\|{b_{0}}\|_{L^{2}},\qquad t\in[0,\epsilon\ell^{2}].\end{split}$

The general case of $s\geq 0$ and $s+\frac{1}{2}-\frac{1}{p}>0$ can be handled
in a similar way. ∎

###  3.2. Case of (Hall-MHD)

In this subsection, we treat the (Hall-MHD) case. While the computations are
significantly more involved in this case, the overall strategy is essentially
the same with the case of (E-MHD), and in particular, the initial data will be
taken to be $u_{0}=0$ and $b_{0}$ the same as before. The key difference lies
in the way a suitable lower bound is obtained in the $L^{2}$-inner product
$\langle{b,\chi\tilde{b}}\rangle$.

For convenience, let us recall the system of equations (1.10) satisfied by an
$L^{2}$-solution $(u,b)$ around a time-dependent axisymmetric background:

(3.3) $\left\\{\begin{aligned}
&\partial_{t}b+{\mathbf{\Pi}}\cdot\nabla(\nabla\times b)-(\nabla\times
b)\cdot\nabla{\mathbf{\Pi}}-(\nabla\times{\mathbf{\Pi}})\cdot\nabla b+{\bf
V}\cdot\nabla b-{\mathbf{\Pi}}\cdot\nabla u\\\
&\quad=-b\cdot\nabla(\nabla\times{\mathbf{\Pi}})+b\cdot\nabla{\bf
V}-u\cdot\nabla{\mathbf{\Pi}},\\\ &\partial_{t}u+{\bf V}\cdot\nabla u+\nabla
p-\nu\Delta u-(\nabla\times b)\times{\mathbf{\Pi}}=-u\cdot\nabla{\bf
V}+(\nabla\times{\mathbf{\Pi}})\times b.\end{aligned}\right.$

Here, we used the notation ${\mathbf{\Pi}}=\Pi\partial_{\theta}$ and ${\bf
V}=\nabla\times(-\Delta)^{-1}(\Omega\partial_{\theta})$, where the pair of
scalar-valued functions $(\Pi,\Omega)$ provides a solution to (1.8).

Furthermore, recall the system for $(\tilde{u},\tilde{b})$ (omitting the
superscript $(\nu)$ for simplicity)

$\left\\{\begin{aligned}
&\partial_{t}\tilde{u}-f\partial_{\theta}\tilde{b}=\boldsymbol{\delta}_{\tilde{u}}\\\
&\partial_{t}\tilde{b}+f\partial_{\theta}(\nabla\times\tilde{b})+r^{-1}f\partial_{\theta}\tilde{b}\partial_{\theta}=\boldsymbol{\delta}_{\tilde{b}}.\end{aligned}\right.$

We shall need the following simple estimate, which is a straightforward
consequence of Proposition 2.12:

(3.4)
$\begin{split}\|{\partial_{r}\tilde{b}}\|_{L^{2}_{x,y}}&\lesssim\lambda\ell^{-2}\|{f^{-1}}\|_{L^{\infty}(\mathrm{supp}(\tilde{b}(t)))}\|{g_{0}}\|_{L^{2}_{x,y}}\lesssim\lambda\ell^{-2}e^{c_{1}\lambda
t}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Similarly, we have

(3.5)
$\begin{split}\|{\partial_{\theta}\tilde{b}}\|_{L^{2}_{x,y}}&\lesssim\lambda\ell^{-2}\|{g_{0}}\|_{L^{2}_{x,y}},\qquad\|{\partial_{\theta}\partial_{r}\tilde{b}}\|_{L^{2}_{x,y}}\lesssim\lambda^{2}\ell^{-3}e^{c_{1}\lambda
t}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

We write out the terms as follows:

$\begin{split}&\frac{\mathrm{d}}{\mathrm{d}t}\left(\langle{u,\chi\tilde{u}-\mathrm{div}^{-1}(\chi^{\prime}\tilde{u}^{z})}\rangle+\langle{b,\chi\tilde{b}}\rangle\right)=\mathrm{I}+\mathrm{II}+\mathrm{III}+\mathrm{IV}+\mathrm{V}+\mathrm{VI},\end{split}$

where

$\begin{split}&\mathrm{I}=\langle{\partial_{t}u+\nabla
p,-\mathrm{div}^{-1}(\chi^{\prime}\tilde{u}^{z})}\rangle+\langle{u,\mathrm{div}^{-1}(\chi^{\prime}\partial_{t}\tilde{u}^{z})}\rangle,\end{split}$
$\begin{split}&\mathrm{II}=\langle{u,\chi\boldsymbol{\delta}_{\tilde{u}}}\rangle+\langle{b,\chi\boldsymbol{\delta}_{\tilde{b}}}\rangle,\end{split}$
$\begin{split}&\mathrm{III}=\langle{-b\cdot\nabla(\nabla\times{\mathbf{\Pi}})+b\cdot\nabla{\bf
V}-u\cdot\nabla{\mathbf{\Pi}},\chi\tilde{u}}\rangle+\langle{-u\cdot\nabla{\bf
V}+(\nabla\times{\mathbf{\Pi}})\times b,\chi\tilde{b}}\rangle,\end{split}$
$\begin{split}&\mathrm{IV}=-\langle{{\bf V}\cdot\nabla
u,\chi\tilde{u}}\rangle-\langle{{\bf V}\cdot\nabla
b,\chi\tilde{b}}\rangle,\end{split}$
$\begin{split}&\mathrm{V}=\langle{\nu\Delta
u,\chi\tilde{u}}\rangle+\langle{(\nabla\times
b)\times{\mathbf{\Pi}},\chi\tilde{u}}\rangle+\langle{(\nabla\times{\mathbf{\Pi}})\cdot\nabla
b,\chi\tilde{b}}\rangle,\end{split}$

and finally further decompose the remaining terms as
$\mathrm{VI}=\mathrm{VI}_{1}+\mathrm{VI}_{2}+\mathrm{VI}_{3}$ where

$\begin{split}\mathrm{VI}_{1}&=\langle{u,\chi
f\partial_{\theta}\tilde{b}}\rangle+\langle{{\mathbf{\Pi}}\cdot\nabla
u,\chi\tilde{b}}\rangle,\\\
\mathrm{VI}_{2}&=-\langle{{\mathbf{\Pi}}\cdot\nabla\nabla\times
b,\chi\tilde{b}}\rangle+\langle{b,-\chi
f\partial_{\theta}\nabla\times\tilde{b}}\rangle,\\\
\mathrm{VI}_{3}&=\langle{\nabla\times
b\cdot\nabla{\mathbf{\Pi}},\chi\tilde{b}}\rangle-\langle{b,\chi
r^{-1}f\partial_{\theta}\tilde{b}\partial_{\theta}}\rangle.\end{split}$

We begin with rewriting $\mathrm{I}$ as

$\begin{split}\mathrm{I}&=\langle{{\bf V}\cdot\nabla u-\nu\Delta
u-(\nabla\times b)\times{\mathbf{\Pi}}+u\cdot\nabla{\bf
V}-(\nabla\times{\mathbf{\Pi}})\times
b,\mathrm{div}^{-1}(\chi^{\prime}\tilde{u}^{z})}\rangle\\\
&\quad+\langle{u,\mathrm{div}^{-1}(\chi^{\prime}f\partial_{\theta}\tilde{b})}\rangle+\langle{u,\mathrm{div}^{-1}(\chi^{\prime}\boldsymbol{\delta}_{\tilde{u}})}\rangle.\end{split}$

Using

$\begin{split}&\|{\nabla\mathrm{div}^{-1}(\chi^{\prime}\tilde{u}^{z})}\|_{L^{2}_{x,y}}\lesssim\|{\chi^{\prime}\tilde{u}^{z}}\|_{L^{2}_{x,y}}\lesssim\lambda^{-1}\ell^{\frac{1}{2}}\|{g_{0}}\|_{L^{2}_{x,y}},\quad\|{\mathrm{div}^{-1}(\chi^{\prime}\tilde{u}^{z})}\|_{L^{2}_{x,y}}\lesssim\ell^{-1}\|{\chi^{\prime}\tilde{u}}\|_{L^{2}_{x,y}}\lesssim\lambda^{-1}\ell^{-\frac{1}{2}}\|{g_{0}}\|_{L^{2}_{x,y}},\end{split}$

we bound

$\begin{split}&\left|\langle{{\bf V}\cdot\nabla u-\nu\Delta u-(\nabla\times
b)\times{\mathbf{\Pi}}+u\cdot\nabla{\bf V}-(\nabla\times{\mathbf{\Pi}})\times
b,\mathrm{div}^{-1}(\chi^{\prime}\tilde{u}^{z})}\rangle\right|\\\
&\quad\lesssim\lambda^{-1}\ell^{\frac{1}{2}}\left((\|{\nabla{\bf
V}}\|_{L^{\infty}}+\ell^{-1}\|{{\bf
V}}\|_{L^{\infty}})\|{u}\|_{L^{2}}+(\|{\nabla{\mathbf{\Pi}}}\|_{L^{\infty}}+\ell^{-1}\|{{\mathbf{\Pi}}}\|_{L^{\infty}})\|{b}\|_{L^{2}}+\nu\|{\nabla
u}\|_{L^{2}}\right)\|{g_{0}}\|_{L^{2}_{x,y}}\end{split}$

and

$\begin{split}&\left|\langle{u,\mathrm{div}^{-1}(\chi^{\prime}f\partial_{\theta}\tilde{b})}\rangle+\langle{u,\mathrm{div}^{-1}(\chi^{\prime}\boldsymbol{\delta}_{\tilde{u}})}\rangle\right|\lesssim\ell^{\frac{1}{2}}(1+\ell^{-1})\|{u}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Terms in $\mathrm{II}$ and $\mathrm{III}$ can be handled in a straightforward
manner: we have

$\begin{split}&|\mathrm{II}|\lesssim\ell^{-\frac{3}{2}}(\|{u}\|_{L^{2}}+\|{b}\|_{L^{2}})\|{g_{0}}\|_{L^{2}_{x,y}}\end{split}$

and

$\begin{split}&|\mathrm{III}|\lesssim\ell^{\frac{1}{2}}\left((\|{\nabla^{2}{\mathbf{\Pi}}}\|_{L^{\infty}}+\|{\nabla{\mathbf{\Pi}}}\|_{L^{\infty}}+\|{\nabla{\bf
V}}\|_{L^{\infty}})\|{b}\|_{L^{2}}+(\|{\nabla{\mathbf{\Pi}}}\|_{L^{\infty}}+\|{\nabla{\bf
V}}\|_{L^{\infty}})\|{u}\|_{L^{2}}\right)\|{g_{0}}\|_{L^{2}_{x,y}},\end{split}$

respectively. We now treat the terms in $\mathrm{IV}$. Noting that

$\begin{split}&\langle{{\bf V}\cdot\nabla
b,\chi\tilde{b}}\rangle=-\langle{b,\chi
V^{r}\partial_{r}\tilde{b}+V^{z}\chi^{\prime}\tilde{b}}\rangle,\quad\langle{{\bf
V}\cdot\nabla u,\chi\tilde{u}}\rangle=-\langle{u,\chi
V^{r}\partial_{r}\tilde{u}+V^{z}\chi^{\prime}\tilde{u}}\rangle\end{split}$

and using (3.4) to bound $\langle{b,\chi V^{r}\partial_{r}\tilde{b}}\rangle$,
we obtain

$\begin{split}&|\mathrm{IV}|\lesssim\ell^{\frac{1}{2}}((1+\ell^{-1})+\lambda
e^{c\lambda t})\|{{\bf
V}}\|_{L^{\infty}}(\|{u}\|_{L^{2}}+\|{b}\|_{L^{2}})\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Next, recalling that

$\begin{split}&\nabla\times(\Pi\partial_{\theta})=(-r\partial_{z}\Pi)\partial_{r}+(r^{-1}\partial_{r}(r^{2}\Pi))\partial_{z},\end{split}$

we write

$\begin{split}&\langle{(\nabla\times{\mathbf{\Pi}})\cdot\nabla
b,\chi\tilde{b}}\rangle=\langle{-r\partial_{z}\Pi\partial_{r}b,\chi\tilde{b}}\rangle+\langle{r^{-1}\partial_{r}(r^{2}\Pi)\partial_{z}b,\chi\tilde{b}}\rangle\end{split}$

and we may bound

$\begin{split}&\left|\langle{-r\partial_{z}\Pi\partial_{r}b,\chi\tilde{b}}\rangle\right|\lesssim\left|\langle{b,\chi\partial_{r}(r\partial_{z}\Pi\tilde{b})}\rangle\right|\lesssim\ell^{\frac{1}{2}}(\ell^{-1}\|{{\mathbf{\Pi}}}\|_{L^{\infty}}+\lambda
e^{c_{1}\lambda
t}\|{\partial_{z}{\mathbf{\Pi}}}\|_{L^{\infty}})\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

The remaining terms from $\mathrm{V}$ are easy to bound; we end up with

$\begin{split}&\left|\mathrm{V}\right|\lesssim\ell^{\frac{1}{2}}\left(\nu\|{\nabla
u}\|_{L^{2}}+(\lambda e^{c_{1}\lambda
t}\|{\partial_{z}{\mathbf{\Pi}}}\|_{L^{\infty}}+\ell^{-1}\|{{\mathbf{\Pi}}}\|_{L^{\infty}})\|{b}\|_{L^{2}}\right)\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

Finally, we write

$\begin{split}&\mathrm{VI}=\mathrm{VI}^{e}+\langle{({\mathbf{\Pi}}-{\mathbf{\Pi}}^{e})\cdot\nabla
u,\chi\tilde{b}}\rangle-\langle{({\mathbf{\Pi}}-{\mathbf{\Pi}}^{e})\cdot\nabla\nabla\times
b,\chi\tilde{b}}\rangle+\langle{\nabla\times
b\cdot\nabla({\mathbf{\Pi}}-{\mathbf{\Pi}}^{e}),\chi\tilde{b}}\rangle\end{split}$

where ${\mathbf{\Pi}}^{e}$ is as in Lemma 2.9 and

$\begin{split}\mathrm{VI}^{e}&:=\langle{u,\chi
f\partial_{\theta}\tilde{b}}\rangle+\langle{{\mathbf{\Pi}}^{e}\cdot\nabla
u,\chi\tilde{b}}\rangle-\langle{{\mathbf{\Pi}}^{e}\cdot\nabla\nabla\times
b,\chi\tilde{b}}\rangle-\langle{b,\chi
f\partial_{\theta}\nabla\times\tilde{b}}\rangle+\langle{\nabla\times
b\cdot\nabla{\mathbf{\Pi}}^{e},\chi\tilde{b}}\rangle-\langle{b,\chi
r^{-1}f\partial_{\theta}\tilde{b}\partial_{\theta}}\rangle.\end{split}$

From the computations in the case of E-MHD in 3.1.2,

$\begin{split}&\left|\mathrm{VI}^{e}\right|\lesssim\ell^{-\frac{3}{2}}\|{b}\|_{L^{2}}\|{g_{0}}\|_{L^{2}_{x,y}}.\end{split}$

For the other terms, we have

$\begin{split}&\left|\langle{({\mathbf{\Pi}}-{\mathbf{\Pi}}^{e})\cdot\nabla