# Non-uniqueness of Leray weak solutions of the forced MHD equations ††thanks:
Wang was supported by National Key R$\&$D Program of China (No.
2022YFA1005601), National Natural Science Foundation of China (No. 12371114)
and Outstanding Young foundation of Jiangsu Province (No. BK20200042). Xu was
supported by the Postdoctoral Science Foundation of China (2023M731381). Zhang
was supported by National Natural Science Foundation of China (No. 12301133),
the Postdoctoral Science Foundation of China (No. 2023M741441) and Jiangsu
Education Department (No. 23KJB110007).

Jun Wang, Fei Xu , Yong Zhang School of Mathematical Sciences, Jiangsu
University, Zhenjiang, 212013, P.R. China
E-mail<EMAIL_ADDRESS>of Mathematical Sciences, Jiangsu
University, Zhenjiang, 212013, P.R. China
E-mail<EMAIL_ADDRESS>author.
School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, P.R.
China
E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we exhibit non-uniqueness of Leray weak solutions of the forced
magnetohydrodynamic (MHD for short) equations. Similar to the solutions
constructed in [12], we first find a special steady solution of ideal MHD
equations whose linear unstability was proved in [21]. It is possible to
perturb the unstable scenario of ideal MHD to 3D viscous and resistive MHD
equations, which can be regarded as the first unstable ”background” solution.
Our perturbation argument is based on the spectral theoretic approach [35].
The second solution we would construct is a trajectory on the unstable
manifold associated to the unstable steady solution. It is worth noting that
these solutions live precisely on the borderline of the known well-posedness
theory.

_Keywords:_ MHD equations, Leray weak solution, non-uniqueness, megneto-
rotational instability(MRI), constraction mapping principle

_AMS Subjection Classification(2020):_ 76B15, 47J15, 76B03.

##  1 Introduction and main results

Consider the three-dimensional magnetohydrodynamic (MHD for short) system on
$\mathbb{R}^{3}$

$\left\\{\begin{array}[]{llll}\partial_{t}v+v\cdot\nabla v-\Delta v+\nabla
p=H\cdot\nabla H+f_{1}\\\ \partial_{t}H+v\cdot\nabla H-\Delta H=H\cdot\nabla
v+f_{2},\\\ \text{div}v=\text{div}H=0,\end{array}\right.$ (1.1)

where $v(t,x):(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$,
$H(t,x):(0,T)\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$,
$p(t,x)\in\mathbb{R}$ correspond to the velocity field, magnetic field and
pressure of the fluid, respectively, and $f=(f_{1},f_{2})$ is the given body
force. We impose the initial condition

$(v,H)(0,x)=(v^{0},H^{0})(x),\quad x\in\mathbb{R}^{3}.$ (1.2)

Among various hydrodynamic models, the viscous and resistive MHD system is a
canonical macroscopic model to describe the motion of conductive fluid, such
as plasma or liquid metals, under a complicated interaction between the
electromagnetic phenomenon and fluid dynamical phenomenon (see [1]). We refer
the reader to [2, 3, 4, 5] for more physical interpretations of the MHD
system. In particular, in the case without magnetic fields, the system (1.1)
would reduce to the classical incompressible Navier-Stokes equation (NSE for
short). When ignoring viscous and resistive effects, the system (1.1) would
become the ideal MHD system, namely,

$\left\\{\begin{array}[]{llll}\partial_{t}v+v\cdot\nabla v+\nabla
p=H\cdot\nabla H\\\ \partial_{t}H+v\cdot\nabla H=H\cdot\nabla v,\\\
\text{div}v=\text{div}H=0,\end{array}\right.$ (1.3)

with the initial condition

$(v,H)(0,x)=(v^{0},H^{0})(x),\quad x\in\mathbb{R}^{3}$ (1.4)

The incompressible ideal MHD system (1.3) is the classical macroscopic model
coupling the Maxwell equations to the evolution of an electrically conducting
incompressible fluid [3, 4]. In the case $H=0$, it’s obvious that (1.3)
reduces to the Euler equation.

The well-posedness problem of NSE and MHD has been extensively studied in the
literature. For the initial data with finite energy, the existence of global
weak solution $u$ to NSE was first proved by Leray [6] in 1934 and later by
Holf [7] in 1951 in bounded domains, which satisfies
$u\in\mathcal{C}_{weak}([0,+\infty);L^{2}({\Omega}))\cap
L^{2}([0,+\infty);\dot{H}^{1}({\Omega}))$ and obeys the following energy
inequality

$\|u(t)\|_{L^{2}}^{2}+2\nu\int_{t_{0}}^{t}\|\nabla
u(s)\|_{L^{2}}^{2}ds\leq\|u(t_{0})\|_{L^{2}}^{2}$ (1.5)

for any $t>0$ and a.e. $t_{0}\geq 0$. Similar to the Navier-Stokes equation, a
global weak solution (in the sense of Definition 1.1) and local strong
solution to (1.1) with the initial boundary value condition were constructed
by Duvant and Lions [8]. Later, the results were extended to the Cauchy
problem by Sermange and Terman [5], where their main tools are regularity
theory of the Stokes operator and the energy method.

Now let us first recall the notion of Leray weak solution of MHD system (1.1)
for each divergence-free initial value $(v^{0},H^{0})(x)$ and body force
$f_{i}\in L_{t}^{1}L_{x}^{2}(i=1,2)$. Denote that
$L_{\sigma}^{2}(\mathbb{R}^{3})$ is the space of divergence-free vector fields
in $L^{2}(\mathbb{R}^{3})$.

######  Definition 1.1.

The pair $(v,H)\in L^{\infty}(0,T;L_{\sigma}^{2}(\mathbb{R}^{3}))\cap
L^{2}(0,T;W^{1,2}(\mathbb{R}^{3}))$ is called as Leray weak solution in
$[0,T)\times\mathbb{R}^{3}$ if there holds that
(1) The pair $(v,H)$ solves (1.1) in the distribution sense

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{3}}v\cdot\partial_{t}\varphi-v\cdot\nabla
v\cdot\varphi+H\cdot\nabla H\cdot\varphi-\nabla u\cdot\nabla\varphi dxdt$
$\displaystyle=-\int_{\mathbb{R}^{3}}v^{0}\cdot\varphi(\cdot,0)dx-\int_{0}^{T}\int_{\mathbb{R}^{3}}f_{1}\varphi
dtdx$
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{3}}H\cdot\partial_{t}\phi-v\cdot\nabla
H\cdot\phi+H\cdot\nabla v\cdot\phi-\nabla H\cdot\nabla\varphi dxdt$
$\displaystyle=-\int_{\mathbb{R}^{3}}H^{0}\cdot\phi(\cdot,0)dx-\int_{0}^{T}\int_{\mathbb{R}^{3}}f_{2}\phi
dtdx$ (1.6)

for all $\varphi,\phi\in C_{0}^{\infty}([0,T)\times\mathbb{R}^{3})$ and the
initial data $v^{0},H^{0}\in L_{\sigma}^{2}(\mathbb{R}^{3})$.
(2) Such solution pair $(v,H)$ satisfies the energy inequality

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{3}}|v(t)|^{2}+|H(t)|^{2}dx+\int_{0}^{T}\int_{\mathbb{R}^{3}}|\nabla
v(t)|^{2}+|\nabla H(t)|^{2}dxdt$
$\displaystyle\leq\int_{\mathbb{R}^{3}}|v^{0}|^{2}+|H^{0}|^{2}dx+\int_{0}^{T}\int_{\mathbb{R}^{3}}f_{1}v+f_{2}Hdxdt$
(1.7)

It’s known that Leray weak solutions of MHD system (1.1) enjoy several nice
properties, including the partial regularity and weak-strong uniqueness.
However, the uniqueness of the weak solutions still remains one of the most
challenging problems. In this paper, we will answer the uniqueness question in
the negative. Our main result can be stated as follows.

######  Theorem 1.2.

There exist two distinct Leray weak solutions $(v_{1},H_{1})$, $(v_{2},H_{2})$
to the forced viscous and resistive MHD system (1.1) on
$(0,T)\times\mathbb{R}^{3}$ with body force
$(f_{1},f_{2})\in(L_{t}^{1}L_{x}^{2})^{2}$.

Recall the scaling property of MHD equation, if $(v(x,t),H(x,t),p(x,t))$ is a
solution of (1.1) with force $f_{i}(x,t)(i=1,2)$, then for any $\lambda>0$,

$v^{\lambda}(x,t)=\lambda v(\lambda
x,\lambda^{2}t),~{}~{}~{}H^{\lambda}(x,t)=\lambda H(\lambda
x,\lambda^{2}t),~{}~{}~{}p^{\lambda}(x,t)=\lambda^{2}p(\lambda
x,\lambda^{2}t)$ (1.8)

is also a solution with force $f_{i}(x,t)=\lambda^{3}f_{i}(\lambda
x,\lambda^{2}t)$. A particular class of solution are the $self$-$similar$
solutions, that is, the solutions of MHD equations on
$\mathbb{R}^{3}\times\mathbb{R}$ invariant under the scaling symmetry. We will
finish the proof of Theorem 1.2 by following the similar techniques developed
in recent work of Albritton-Brué-Colombo [12]. However, it is worth noting
that the work is not just a parallel extension. The addition of magnetic field
would bring some new difficulties. Especially, it is vital to construct a
smooth and decaying unstable steady state of the forced MHD equations in three
dimension. Considering the similarity variables

$\xi=\frac{x}{\sqrt{t}},~{}~{}~{}\tau=logt,$ (1.9)

the solutions can be expressed as follows in similarity variables by

$\begin{split}&v(x,t)=\frac{1}{\sqrt{t}}V(\xi,\tau),~{}~{}H(x,t)=\frac{1}{\sqrt{t}}W(\xi,\tau),\\\
&f(x,t)=\frac{1}{t^{\frac{3}{2}}}F(\xi,\tau),~{}~{}p(x,t)=\frac{1}{t}P(\xi,\tau)\end{split}$

If $(v,H,p)$ satisfies (1.1) with force $f_{i}(x,t)(i=1,2)$, then profile
$(V,W,P)$ satisfies the time-dependent Leray equations

$\left\\{\begin{array}[]{llll}\partial_{\tau}V-~{}\frac{1}{2}(1+\xi\cdot\nabla_{\xi})V+V\cdot\nabla
V-W\cdot\nabla W-\Delta_{\xi}V+\nabla P=F_{1},\\\
\partial_{\tau}W-~{}\frac{1}{2}(1+\xi\cdot\nabla_{\xi})W+V\cdot\nabla
W-W\cdot\nabla V-\Delta_{\xi}W=F_{2}.\end{array}\right.$ (1.10)

A special self-similar solutions of (1.1) correspond to steady state of
(1.10). We will find the first weak solution
$(\overline{V}(\xi),\overline{W}(\xi))$ of (1.10), which is linearly unstable
steady solution. That is to say, the following linearized MHD equations around
the steady state $(\overline{V},\overline{W})$

$\left\\{\begin{array}[]{llll}\partial_{\tau}V-(\frac{1}{2}+\frac{1}{2}\xi\cdot\nabla_{\xi}+\Delta_{\xi})V+\mathbb{P}(\overline{V}\cdot\nabla
V+V\cdot\nabla\overline{V}-\overline{W}\cdot\nabla
W-W\cdot\nabla\overline{W})=0,\\\
\partial_{\tau}W-(\frac{1}{2}+\frac{1}{2}\xi\cdot\nabla_{\xi}+\Delta_{\xi})W+\mathbb{P}(\overline{V}\cdot\nabla
W+V\cdot\nabla\overline{W}-\overline{W}\cdot\nabla
V-W\cdot\nabla\overline{V})=0\end{array}\right.$ (1.11)

have a nontrivial solution of the form
$(V(\xi,\tau),W(\xi,\tau))=(e^{\lambda\tau}\widetilde{V}(\xi),e^{\lambda\tau}\widetilde{W}(\xi))$
with $\lambda>0$. In addition, it follows from Section 3 that we can rewrite
(1.11) as

$\partial_{\tau}\Xi-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})\Xi-\Delta_{\xi}\Xi+\mathbb{P}(\mathfrak{B}(\overline{\Xi},\Xi)+\mathfrak{B}(\Xi,\overline{\Xi}))=0,$
(1.12)

where $\Xi=(V(\xi,\tau),W(\xi,\tau))$ and
$\overline{\Xi}=(\overline{V}(\xi),\overline{W}(\xi)).$ We can also say
$\overline{\Xi}$ is linearly unstable for the dynamics of (1.12) if there
exists an unstable eigenvalue for the linearized operator

$-L_{ss}\Xi=-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})\Xi-\Delta_{\xi}\Xi+\mathbb{P}(\mathfrak{B}(\overline{\Xi},\Xi)+\mathfrak{B}(\Xi,\overline{\Xi})).$

The second solution of (1.10) is a trajectory on the unstable manifold
associated to the most unstable eigenvalue which will be constructed in
Section 5.

Before making a comment on our result in more detail, let us review the
literature on some significant progress towards the non-uniqueness of the
Euler equations and Navier-Stokes equations. In recent two papers [9] and
[10], Vishik answered the non-uniqueness of Euler equation by constructing two
Leray-Holf weak solutions. One solution is an explicit unstable steady
vortices for the Euler dynamics in similarity variables and another is a
trajectory on the unstable manifold associated to the unstable steady state,
which lives precisely on the borderline of the known well-posedness theory.
Later, Albritton et.[11] followed the strategy of Vishik and made some
improvement. Motivated by the Vishik’s work, Albritton et.[12] then
constructed a vortex ring which ’lifts’ Vishik’s unstable vortex to three
dimensions, proving the nonuniqueness of Navier-Stokes equations in the same
way. In addition, we also would like to mention two of particular important
work. The first contribution was made by Jia [13, 14, 15], who developed a
program towards non-uniqueness of Navier-stokes equation without external
force. Compared to Vishik’s approach, the self-similar solutions in [14] are
far from explicit. Therefore, the spectral condition therein seems difficult
to verify analytically, although it has been checked with non-rigorous
numerics in [14]. Second, Buckmaster and Vicol constructed non-unique
distributional solutions of the Navier-Stokes equations in [16] (see also
[17]) and the ideal MHD [18] equations with finite kinetic energy via the
powerful method of convex integration. Recently, the author in [19, 20] proved
the sharp non-uniqueness of weak solutions to 3D viscous and resistive MHD
with finite kinetic energy via method of convex integration. However, these
results mentioned above using convex integration schemes are far from reaching
the regularity $\nabla u\in L_{t,x}^{2}$.

In this paper, we will establish the non-uniqueness of MHD equaitons (1.1)
based on the Leray equations (1.10). For better proceeding, we allow a force
in (1.1), which gives us the freedom to search for a more explicit unstable
profile. The rest of this paper is arranged as follows: In section 2, we
mainly review the linear unstable of the axisymmetric ideal MHD equation
around a rotating flow $(v_{0},H_{0})$ in [21], which contributes to
constructing a unstable steady state profile of (1.10) by choosing a suitable
force. In section 3, we will show that the linearly unstable properties of
axisymmetric case in Theorem 2.1 can be extended to the more general case. In
section 4, we perturb this ideal MHD unstable scenario to 3D viscous and
resistive MHD equations based on the spectral theoretic approach [35]. In
other words, we will establish the linear instability of MHD equations. In
section 5, we use the linear instability proved in Theorem 4.1 to construct
the second Leray weak solution of the forced MHD equations.

##  2  Preliminaries

Firstly, let us pay attention to one simple axisymmetric steady solution
$(v_{0},H_{0})$ among explicit solutions of the incompressible ideal MHD
equations (1.3) (see [22, 23]), where $(v_{0},H_{0})$ is a rotating flow with
a vertical magnetic field, that is,

$\left\\{\begin{array}[]{llll}v_{0}(x)=v_{0}(r)e_{\theta}=r\omega(r)e_{\theta},\\\
H_{0}(x)=\epsilon b(r)e_{z},\end{array}\right.$ (2.1)

where $\epsilon\neq 0$ is a constant, $(r,\theta,z)$ are the cylindrical
coordinates with $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$, $z=x_{3}$,
$(e_{r},e_{\theta},e_{z})$ are unit vectors along $r,\theta,z$ directions,
$\omega(r)\in C^{3}(R_{1},R_{2})$ is the angular velocity of the rotating
fluid, the magnetic profile $b(r)\in C^{3}(R_{1},R_{2})(0\leq
R_{1}<R_{2}=+\infty)$ has a positive lower bound. We will require that
$(\omega(r),b(r))$ has a extra decay at infinity, which guarantee the finite
energy. In [21], they give a rigorous proof of the sharp linear instability
criteria of rotating flows (2.1) with magnetic fields. This smooth and
decaying unstable scenario (2.1) can be regarded as the unstable ”background”
solution of (1.1) in similarity variables by choosing a non-standard force. In
addition, the linear unstable properties of the rotating flow (2.1) play an
important role in constructing non-unique energy weak solutions of (1.1).

The stability criterion for rotating flows with a magnetic field is generally
different from the classical Rayleigh criterion for rotating flows without a
magnetic field. The influence of a vertical and uniform magnetic field
(i.e.,$b(r)$ =constant) on the stability of the rotating flows was first
studied by Velikhov [27] and Chandrasekhar [28], who derived a sufficient
condition for linear stability of a rotating flow in the limit of vanishing
magnetic fields that the square of the angular velocity increases outwards,
i.e.,

$\partial_{r}(\omega^{2})>0,~{}~{}\text{for~{}all}~{}r\in(R_{1},R_{2}).$ (2.2)

If the stability condition (2.2) fails, it was suggested in [27] and [28] that
there is linear instability with small magnetic fields and they also showed
that the unstable eigenvalues are necessarily real. Such instability of
rotating flows induced by small magnetic fields is called magneto-rotational
instability (MRI) in the literature, which has wide application in
astrophysics, particularly to the turbulence and enhanced angular momemtum
transport in astrophysical accretion disks. We refer to the reviews [29, 30,
31, 32] for the history and results of this important topic.

In [21], they answered three natural questions for MRI: Firstly, they give a
sharp instability criterion for general vertical magnetic fields and angular
velocities. Secondly, they show that MRI is due to discrete unstable spectrum,
which is finite. Thirdly, they also prove that the sharp stability or
instability criteria can imply nonlinear stability or instability
respectively. The main proof is based on a local dispersion analysis and a
framework of separable Hamiltonian systems which we will sketch below. The
authors [21] considered the axisymmetric solution of the system (1.3) in the
cylinder
${\Omega}:=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}|R_{1}\leq\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}\leq
R_{2},x_{3}\in T_{2\pi}\\},0\leq R_{1}<R_{2}\leq\infty$. In our paper, we will
consider the case in $\mathbb{R}^{3}$, that is, $0=R_{1}<R_{2}=\infty$. In the
cylindrical coordinates, we denote

$H(t,r,z)=H_{r}(t,r,z)e_{r}+H_{\theta}(t,r,z)e_{\theta}+H_{z}(t,r,z)e_{z}$

and

$v(t,r,z)=v_{r}(t,r,z)e_{r}+v_{\theta}(t,r,z)e_{\theta}+v_{z}(t,r,z)e_{z}$

Due to $\text{div}H=0$, we can define magnetic potential $\psi(t,r,z)$ of
$H(t,r,z)$ by

$\left\\{\begin{array}[]{llll}H_{r}(t,r,z)=\frac{\partial_{z}\psi}{r},\\\
H_{z}(t,r,z)=-\frac{\partial_{r}\psi}{r},\\\
-\frac{1}{r}\partial_{r}(\frac{1}{r}\partial_{r}\psi)-\frac{1}{r^{2}}\partial_{z}^{2}\psi=\frac{1}{r}\partial_{r}H_{z}-\frac{1}{r}\partial_{z}H_{r}.\end{array}\right.$

The system (1.3) can be rewritten in the cylindrical coordinates as

$\left\\{\begin{array}[]{llll}\partial_{t}v_{r}+\partial_{r}p=\frac{\partial_{z}\psi}{r}\partial_{r}(\frac{\partial_{z}\psi}{r})-\frac{\partial_{r}\psi}{r}\partial_{r}(\frac{\partial_{z}\psi}{r})-\frac{H_{\theta}H_{\theta}}{r}-v_{r}\partial_{r}v_{r}-v_{z}\partial_{z}v_{r}+\frac{v_{\theta}v_{\theta}}{r},\\\
\partial_{t}v_{\theta}=\frac{\partial_{z}\psi}{r}\cdot\partial_{r}H_{\theta}-\frac{\partial_{r}\psi}{r}\partial_{z}H_{\theta}+\frac{H_{\theta}\frac{\partial_{z}\psi}{r}}{r}-v_{r}\partial_{r}v_{\theta}-v_{z}\partial_{z}v_{\theta}+\frac{v_{\theta}v_{r}}{r},\\\
\partial_{t}v_{z}+\partial_{z}p=\frac{\partial_{z}\psi}{r}\partial_{r}(-\frac{\partial_{r}\psi}{r})-\frac{\partial_{r}\psi}{r}\partial_{r}(-\frac{\partial_{r}\psi}{r})-v_{r}\partial_{r}v_{z}-v_{z}\partial_{z}v_{z},\\\
\partial_{t}\psi=-v_{r}\partial_{r}\psi-v_{z}\partial_{z}\psi,\\\
\partial_{t}H_{\theta}=\frac{\partial_{z}\psi}{r}\partial_{r}(v_{\theta})-\frac{\partial_{r}\psi}{r}\partial_{r}(v_{\theta})-\frac{H_{\theta}v_{r}}{r}-v_{r}\partial_{r}H_{\theta}-v_{z}\partial_{z}H_{\theta}-\frac{v_{\theta}\frac{\partial_{z}\psi}{r}}{r},\\\
\frac{1}{r}\partial_{r}(ru_{r})+\partial_{z}u_{z}=0.\end{array}\right.$ (2.3)

For steady state, we can take

$\psi_{0}(r)=-\epsilon\int_{0}^{r}sb(s)ds.$

Now let the perturbations be

$\displaystyle u(t,x)=v(t,x)-v_{0}(x);$ $\displaystyle
B_{\theta}(t,x)=H_{\theta}(t,x);$
$\displaystyle\mathcal{P}(t,x)=p(t,x)-p_{0}(x);$
$\displaystyle\varphi(t,r,z)=\psi-\psi_{0}.$

The linearized MHD system around a given steady state
$(v_{0}(x),H_{0}(x),p_{0}(x))$ in the cylindrical coordinates can be reduced
to the following system

$\left\\{\begin{array}[]{llll}\partial_{t}u_{r}=\varepsilon
b(r)\partial_{z}(\frac{\partial_{z}\varphi}{r})+2\omega(r)v_{\theta}-\partial_{r}\mathcal{P},\\\
\partial_{t}u_{\theta}=\varepsilon
b(r)\partial_{z}(B_{\theta})-\frac{u_{r}}{r}\partial_{r}(r^{2}\omega^{2}(r)),\\\
\partial_{t}u_{z}=\varepsilon
b(r)\partial_{z}(-\frac{\partial_{r}\varphi}{r})-\partial_{z}\mathcal{P}+\frac{\varepsilon\partial_{r}b(r)}{r}\partial_{z}\varphi,\\\
\partial_{t}\varphi=\varepsilon rb(r)u_{r},\\\
\partial_{t}B_{\theta}=\varepsilon
rb(r)\partial_{z}u_{\theta}+\partial_{r}\omega(r)\partial_{z}\varphi,\\\
\frac{1}{r}\partial_{r}(ru_{r})+\partial_{z}u_{z}=0.\end{array}\right.$ (2.4)

We impose the system (2.4) with conditions

$\left\\{\begin{array}[]{llll}(u_{r},u_{\theta},u_{z},\varphi,B_{\theta})(t,r,z)|_{t=0}=(u_{r}^{0},u_{\theta}^{0},u_{z}^{0},\varphi^{0},B_{\theta}^{0})(r,z),\\\
(u_{r},u_{\theta},u_{z},\varphi,B_{\theta})(t,r,z)\rightarrow(0,0,0,0,0),~{}~{}~{}\text{as}~{}~{}r\rightarrow\infty,\\\
(u_{r},u_{\theta},u_{z},\varphi,B_{\theta})(t,r,z)=(u_{r},u_{\theta},u_{z},\varphi,B_{\theta})(t,r,z+2\pi).\end{array}\right.$
(2.5)

It is obvious that the linearized axisymmetric MHD equations (2.4) can be
written in a Hamiltonian form

$\begin{matrix}\frac{d}{dt}\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}=\mathbf{J}\mathbf{L}\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}.\end{matrix}$ (2.6)

where $u_{1}=(u_{\theta}+\frac{\partial_{r}\omega(r)}{\varepsilon
b(r)}\varphi,\varphi)$, $u_{2}=(\mbox{\boldmath$u$},B_{\theta})$ with
$\mbox{\boldmath$u$}=(u_{r},u_{z})$. Consider the energy spaces
$\mathbf{X}=X\times Y$ with $X=L^{2}(\mathbb{R}^{3})\times
Z,Y=L_{\sigma}^{2}(\mathbb{R}^{3})\times L^{2}(\mathbb{R}^{3})$, where
$L^{2}(\mathbb{R}^{3})$ is the cylindrically symmetric $L^{2}$ space on
$\mathbb{R}^{3}$,

$L_{\sigma}^{2}(\mathbb{R}^{3}):=\\{\mbox{\boldmath$u$}=u_{r}(r,z)e_{r}+u_{z}(r,z)e_{z}\in
L^{2}(\mathbb{R}^{3})~{}|~{}\text{div}u=0\\}.$

and

$Z=\\{\varphi(r,z)\in
H_{mag}^{1}(\mathbb{R}^{3})|~{}\|\varphi\|_{H_{mag}^{1}(\mathbb{R}^{3})}<\infty\\}$

with

$\|\varphi\|_{H_{mag}^{1}(\mathbb{R}^{3})}=\left(\int_{\Omega}\frac{1}{r^{2}}|\partial_{z}\varphi|^{2}+\frac{1}{r^{2}}|\partial_{r}\varphi|^{2}dx\right)^{\frac{1}{2}}.$

The off-diagonal anti-self-dual operator $\mathbf{J}$ and diagonal self-dual
operator $\mathbf{L}$ are defined by

$\begin{matrix}\mathbf{J}=\begin{pmatrix}0&\mathbb{B}\\\
-\mathbb{B^{\prime}}&0\end{pmatrix}:X^{\ast}\rightarrow
X,~{}~{}~{}\mathbf{L}=\begin{pmatrix}\mathbb{L}&0\\\
0&A\end{pmatrix}:X\rightarrow X^{\ast},\end{matrix}$ (2.7)

in which

$\displaystyle\mathbb{B}\begin{pmatrix}\mbox{\boldmath$u$}\\\
B_{\theta}\end{pmatrix}=\begin{pmatrix}-2\omega(r)u_{r}+\varepsilon
b(r)\partial_{z}B_{\theta}\\\ \varepsilon
rb(r)u_{r}\end{pmatrix}:Y^{\ast}\rightarrow X,$
$\displaystyle\mathbb{B^{\prime}}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}=\begin{pmatrix}\mathbb{P}\begin{pmatrix}-2\omega(r)f_{1}+r\varepsilon
b(r)f_{2}\\\ 0\end{pmatrix}\\\ -\varepsilon
b(r)\partial_{z}f_{1}\end{pmatrix}:X^{\ast}\rightarrow Y,$
$\displaystyle\mathbb{L}=\begin{pmatrix}Id_{1}&0\\\
0&L\end{pmatrix}:X\rightarrow X^{\ast},~{}~{}~{}~{}A=Id_{2}:Y\rightarrow
Y^{\ast},$

with

$L:=-\frac{1}{r}\partial_{r}(\frac{1}{r}\partial_{r}\cdot)-\frac{1}{r^{2}}\partial_{z}^{2}+\mathfrak{F}(r):Z\rightarrow
Z^{\ast},$

and

$\mathfrak{F}(r):=\frac{\partial_{r}\omega^{2}}{\epsilon^{2}b(r)^{2}r}+(\frac{\partial_{r}^{2}b(r)}{r^{2}b(r)}-\frac{\partial_{r}b(r)}{r^{3}b(r)}),$
(2.8)

where $Id_{1}:L^{2}(\mathbb{R}^{3})\rightarrow(L^{2}(\mathbb{R}^{3})^{\ast}$
and $Id_{2}:Y\rightarrow Y^{\ast}$ are the isomorphisms, the operator
$\mathbb{P}$ is the Leray projection from $L^{2}(\Omega)$ to
$L^{2}_{\sigma}(\Omega)$. As proved in [21, Theorem 2.1], when
($\mathbb{L},A,\mathbb{B}$) satisfies the conditions (G1)-(G4) of general
seperable Hamiltonian PDEs, the unstable spectra of (2.6) are all real
discrete and the dimension of the unstable subspace corresponding to positive
(negative) eigenvalues can be determined.

A complete description of the instability spectra and semigroup growth of the
linear axi-symmetric MHD equations (2.4) can be stated as follows:

######  Theorem 2.1.

(refer to [21]) Assume that the steady state $(v_{0},H_{0})(x)$ is given by
(2.1), in which $\omega(r)\in C^{3}(R_{1},R_{2})$, $b(r)\in
C^{3}(R_{1},R_{2})$ with a positive lower bound.
(1) If $R_{1}=0$, $\partial_{r}(\omega^{2})=O(r^{\beta-3})$,
$\partial_{r}b=O(r^{\beta-1})$ for some constant $\beta>0$, as $r\rightarrow
0$.
(2) If $R_{2}=\infty$, $\partial_{r}(\omega^{2})=O(r^{-3-2\alpha})$,
$\partial_{r}b=O(r^{-1-2\alpha})$, for some constant $\alpha>0$, as
$r\rightarrow\infty$.
The operator $\mathbf{JL}$ defined by (2.6) generates a $C^{0}$ group
$e^{t\mathbf{JL}}$ of bounded linear operators on $\mathbf{X}=X\times Y$ and
there exists a decomposition

$\mathbf{X}=E^{u}\oplus E^{c}\oplus E^{s}$

of closed subspace $E^{u,s,c}$ with the following properties:
(i)$E^{c}$, $E^{u}$ and $E^{s}$ are invariant under $e^{t\mathbf{JL}}$.
(ii)$E^{u}(E^{s})$ only consists of eigenvector corresponding to positive
(negative) eigenvalues of $\mathbf{JL}$ and

$\dim E^{u}=\dim E^{s}=n^{-}(\mathbb{L}|_{\overline{R(\mathbb{B})}}),$

where the unstable eigenvalues of the linearized operator
$\mathbf{J}\mathbf{L}$ are all discrete and the number of unstable mode equals
$0<n^{-}(\mathbb{L}|_{\overline{R(\mathbb{B})}})<\infty$, that is, the number
of negative direction of $<\mathbb{L}\cdot,\cdot>$ restricted to
$\overline{R(\mathbb{B})}$ which is shown to be

$\overline{R(\mathbb{B})}=\\{(g_{1},g_{2})\in
X|g_{j}(r,z)=\sum_{k=1}^{\infty}e^{ikz}\widetilde{\varphi}_{k,j}(r),j=1,2\\}.$

It follows that
$n^{-}(\mathbb{L}|_{\overline{R(\mathbb{B})}})=\Sigma_{k=1}^{\infty}n^{-}(\mathbb{L}_{k})$,
where the operator $\mathbb{L}_{k}:H_{mag}^{r}\rightarrow(H_{mag}^{r})^{*}$ is
defined by

$\mathbb{L}_{k}:=-\frac{1}{r}\partial_{r}(\frac{1}{r}\partial_{r}\cdot)+\frac{k^{2}}{r^{2}}+\mathfrak{F}(r)$
(2.9)

for any $k\geq 0$, with $\mathfrak{F}(r)$ defined in $(\ref{e2.8})$.
$n^{-}(\mathbb{L}_{k})$ denotes the number of negative directions of
$<\mathbb{L}_{k}\cdot,\cdot>$. In particular, $n^{-}(\mathbb{L}_{k})=0$ when
$k$ is large enough.
(3) If there exists $r_{0}\in(R_{1},R_{2})$ such that
$\partial_{r}(\omega^{2})|_{r=r_{0}}<0$, then for $\epsilon^{2}$ small enough
the steady state $(v_{0},H_{0})(x)$ in (2.1) is linearly unstable to axi-
symmetric perturbations.
(iii) The sharp exponential growth estimate for the linearized MHD equation
(2.6) along the most unstable modes

$\|e^{t\mathbf{JL}}(u_{1},u_{2})(0)\|_{\mathbf{X}}\lesssim e^{\Lambda
t}\|(u_{1},u_{2})(0)\|_{\mathbf{X}}$ (2.10)

where $\Lambda>0$ determined the maximal growth rate.

######  Remark 2.2.

(i) In this paper, we will consider the case $R_{2}=\infty$ and study the
nonlinear instability of (1.1) to construct non-uniqueness Leray weak
solution, which is located in finite energy class
$L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{R}^{3}))$.
Indeed, here we need to require a stronger condition $\omega=O(r^{-1-\alpha})$
for $\alpha>0$ as $r\rightarrow\infty$, which ensure that
$\nabla(v_{0},H_{0})\in L^{2}(\mathbb{R}^{3})$.

(ii) It follows from [21] that any order derivative of $(u_{1},u_{2})$ can be
written in a Hamiltonian form

$\begin{matrix}\frac{d}{dt}\begin{pmatrix}\partial_{z}^{\alpha}u_{1}\\\
\partial_{z}^{\alpha}u_{2}\end{pmatrix}=\mathbf{J}\mathbf{L}\begin{pmatrix}\partial_{z}^{\alpha}u_{1}\\\
\partial_{z}^{\alpha}u_{2}\end{pmatrix}.\end{matrix}$ (2.11)

where $u_{1}=(u_{\theta}+\frac{\partial_{r}\omega(r)}{\varepsilon
b(r)}\varphi,\varphi)$, $u_{2}=(\mbox{\boldmath$u$},B_{\theta})$ with
$\mbox{\boldmath$u$}=(u_{r},u_{z})$. Together with the fact
$\|(u,B)(t)\|_{L^{2}}\sim\|(u_{1},u_{2})(t)\|_{X}$ and
$\|\partial_{z}^{\alpha}(u,B)\|_{L^{2}}\sim\|\partial_{z}^{\alpha}(u,B)\|_{X}$.
One can get

$\|(u,B)(t)\|_{L^{2}}\lesssim e^{\Lambda t}\|(u,B)(0)\|_{L^{2}}$ (2.12)

and

$\|\partial_{z}^{\alpha}(u,B)(t)\|_{L^{2}}\lesssim e^{\Lambda
t}\|\partial_{z}^{\alpha}(u,B)(0)\|_{L^{2}},~{}~{}|\alpha|\leq s~{}(s\geq 0).$
(2.13)

It also follows [21] that $\|\partial_{r}^{\alpha}(u,B)(t)\|_{L^{2}}\lesssim
e^{\Lambda t}\|(u,B)(0)\|_{H^{s}}$ for $|\alpha|=s$.

##  3 The linear instability and semigroup estimates of ideal MHD system

In this subsection, we are going to show that the linearly unstable properties
of axisymmetric case in Theorem 2.1 can be extended to the more general case.
To this end, we first linearize the ideal MHD equations (1.3) around the
axisymmetric steady solution $(v_{0},H_{0})(x)$ to obtain the following system
(3.3). Then we will obtain the linear instability of (3.3) as in Theorem 3.1.

Assume axisymmetric steady solution $(v_{0},H_{0})(r,z)$ is perturbed by a
small disturbance

$\displaystyle v(t,r,z)=v_{0}(r,z)+\varepsilon u(t,r,z),\quad
H(t,r,z)=H_{0}(r,z)+\varepsilon B(t,r,z),$ $\displaystyle
p(t,r,z)=p_{0}(r,z)+\varepsilon\mathcal{P}(t,r,z).$ (3.1)

We shall rewrite the ideal MHD system (1.3) in following perturbation form:

$\left\\{\begin{array}[]{llll}\varepsilon\partial_{t}u+\varepsilon
v_{0}\cdot\nabla u+\varepsilon u\cdot\nabla v_{0}-\varepsilon H_{0}\cdot\nabla
B-\varepsilon B\cdot\nabla
H_{0}+\varepsilon\nabla\mathcal{P}=\varepsilon^{2}B\cdot\nabla
B-\varepsilon^{2}u\cdot\nabla u,\\\ \varepsilon\partial_{t}B+\varepsilon
v_{0}\cdot\nabla B+\varepsilon u\cdot\nabla H_{0}-\varepsilon H_{0}\cdot\nabla
u-\varepsilon B\cdot\nabla v_{0}=\varepsilon^{2}B\cdot\nabla
u-\varepsilon^{2}u\cdot\nabla B,\\\ \text{div}\varepsilon
u=\text{div}\varepsilon B=0,\end{array}\right.$ (3.2)

Then we obtain the following linearized system of the MHD system (1.3) around
steady solution $(v_{0},H_{0},p_{0})(x)$ by estimating (3.2) at order
$\varepsilon$

$\left\\{\begin{array}[]{llll}\partial_{t}u+v_{0}\cdot\nabla u+u\cdot\nabla
v_{0}-H_{0}\cdot\nabla B-B\cdot\nabla H_{0}+\nabla\mathcal{P}=0,\\\
\partial_{t}B+v_{0}\cdot\nabla B+u\cdot\nabla H_{0}-H_{0}\cdot\nabla
u-B\cdot\nabla v_{0}=0,\\\ \text{div}u=\text{div}B=0.\end{array}\right.$ (3.3)

Moreover, (3.3) can be rewritten as (2.4) in cylindrical coordinates, which is
equivalent to the linearized system (2.6). If we define

$\displaystyle(u,B,\mathcal{P})(x,t)=(u,B,\mathcal{P})(x)e^{\Lambda_{0}t}=(u,B,\mathcal{P})(r,z)e^{\Lambda_{0}t}$
(3.4)

with

$\displaystyle u_{r}=\widetilde{u_{r}}(r)cos(kz)\cdot
e^{\Lambda_{0}t},~{}~{}u_{\theta}=\widetilde{u_{\theta}}(r)cos(kz)\cdot
e^{\Lambda_{0}t},~{}~{}u_{z}=\widetilde{u_{z}}(r)sin(kz)\cdot
e^{\Lambda_{0}t},$ $\displaystyle\varphi=\widetilde{\varphi}(r)cos(kz)\cdot
e^{\Lambda_{0}t},~{}~{}B_{\theta}=\widetilde{B_{\theta}}(r)sin(kz)\cdot
e^{\Lambda_{0}t},~{}~{}\mathcal{P}=\widetilde{\mathcal{P}}(r)cos(kz)\cdot
e^{\Lambda_{0}t},$ (3.5)

where $\Lambda_{0}$ is one of the unstable eigenvalues of (2.6) and
$(u,B,\mathcal{P})(x,t)$ will satisfy equation (2.4). Naturally, the
nontrivial (3.4) with $\Lambda_{0}>0$ is a linear unstable solution of (3.3).
We call the pair $(u,B)$ solve (3.3) in the distribution sense if

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{3}}u\cdot\partial_{t}\varphi-
v_{0}\cdot\nabla u\cdot\varphi-u\cdot\nabla v_{0}\cdot\varphi+H_{0}\cdot\nabla
B\cdot\varphi+B\cdot\nabla H_{0}\cdot\varphi dxdt$
$\displaystyle=-\int_{\mathbb{R}^{3}}u^{0}\cdot\varphi(\cdot,0)dx,$
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{3}}B\cdot\partial_{t}\phi-
v_{0}\cdot\nabla B\cdot\phi-u\cdot\nabla H_{0}\cdot\phi+H_{0}\cdot\nabla
u\cdot\phi+B\cdot\nabla v_{0}\cdot\phi dxdt$
$\displaystyle=-\int_{\mathbb{R}^{3}}B^{0}\cdot\phi(\cdot,0)dx$ (3.6)

for initial data $u^{0},B^{0}\in L_{\sigma}^{2}(\mathbb{R}^{3})$ and test
function $\varphi,\phi\in\mathcal{D_{T}}$, where
$\mathcal{D_{T}}:=\\{\varphi\in
C_{0}^{\infty}((0,T);\mathbb{R}^{3}),\text{div}\varphi=0\\}$.

In order to deal with nonlinear term conveniently, we will introduce a
trilinear form $\mathfrak{B_{0}}$. First, we define a trilinear form by
setting

$b(u,v,\omega)=\sum_{i,j=1}^{3}\int_{\Omega}u_{i}\partial_{i}v_{j}\omega_{j}dx=\int_{\Omega}u\cdot\nabla
v\cdot\omega dx.$ (3.7)

We can easily get by a direct calculation

$b(u,v,\omega)=-b(u,\omega,v).$

In order to write (3.4) as a simpler form, we define a trilinear form
$\mathfrak{B_{0}}$ as

$\mathfrak{B_{0}}(\Phi^{1},\Phi^{2},\Phi^{3})=b(u,v,\omega)-b(\mathbb{U},\mathbb{V},\omega)+b(u,\mathbb{V},\mathbb{W})-b(\mathbb{U},v,\mathbb{W}),$

where
$\Phi^{1}=(u,\mathbb{U}),\Phi^{2}=(v,\mathbb{V}),\Phi^{3}=(\omega,\mathbb{W})$.
Due to the continuous $b$, one derives that $\mathfrak{B_{0}}$ is trilinearly
continuous. This let us give a continuous bilinear operator $\mathfrak{B}$ as

$\langle\mathfrak{B}(\Phi^{1},\Phi^{2}),\Phi^{3}\rangle=\mathfrak{B_{0}}(\Phi^{1},\Phi^{2},\Phi^{3}).$

If we choose $\varphi,\phi\in C_{0,\sigma}^{\infty}(\mathbb{R}^{3})$, one can
rewrite (3.4) as

$\displaystyle\partial_{t}(u,\varphi)+b(v_{0},u,\varphi)+b(u,v_{0},\varphi)-b(H_{0},B,\varphi)-b(B,H_{0},\varphi)=0,$
$\displaystyle\partial_{t}(B,\phi)+b(v_{0},B,\phi)+b(u,H_{0},\phi)-b(H_{0},u,\phi)-b(B,v_{0},\phi)=0,$

with $(f,g)=\int_{R^{3}}f\cdot gdx$. Then this system can be equivalent to the
following formula

$\partial_{t}(\Gamma,\Psi)+\mathfrak{B_{0}}(\Gamma_{0},\Gamma,\Psi)+\mathfrak{B_{0}}(\Gamma,\Gamma_{0},\Psi)=0,$
(3.8)

where $\Gamma=(u,B)$, $\Gamma_{0}=(v_{0},H_{0})$. Using the operator
$\mathfrak{B}$ previously defined, (3.2) is equivalent to the following
formula

$\partial_{t}\Gamma+\mathfrak{B}(\Gamma_{0},\Gamma)+\mathfrak{B}(\Gamma,\Gamma_{0})=0.$
(3.9)

It is obvious that (3.8) is a weak formulation of the problem (3.9). In the
sense of distribution, the system (3.3) can be expressed as (3.9).

Suppose $\Gamma_{0}$ is linearly unstable for the dynamics of (3.3), there
exists an unstable eigenvalue for the linearized operator

$-A\Gamma=\mathfrak{B}(\Gamma_{0},\Gamma)+\mathfrak{B}(\Gamma,\Gamma_{0})$
(3.10)

whose domain

$\mathcal{D}(A):=\\{\Gamma\in
L_{\sigma}^{2}(\mathbb{R}^{3}):\Gamma_{0}\cdot\nabla\Gamma\in
L^{2}(\mathbb{R}^{3})\\}.$

in which $\Gamma=(u,B)$, $\Gamma_{0}=(v_{0},H_{0})$. It is easy to see that
$A$ at least has a unstable eigenvalue $\Lambda_{0}>0$ from (3.3) and (3.4).
The main results can be stated as follows:

######  Theorem 3.1.

Assume that the steady state $(v_{0},H_{0})(x)$ is given by (2.1), in which
$\omega(r)\in C^{3}(R_{1},R_{2})$ and $b(r)\in C^{3}(R_{1},R_{2})$ satisfy the
conditions (1), (2) and (3) in Theorem 2.1, then the linearized operator $A$
defined by (3.10) generates a $C^{0}$ group $e^{At}$ of bounded linear
operator on $\mathcal{D}(A)\subset L^{2}_{\sigma}(\mathbb{R}^{3})\rightarrow
L^{2}_{\sigma}(\mathbb{R}^{3})$. And there exists a decomposition

$L^{2}(\Omega)=E^{u}\oplus E^{c}\oplus E^{s}$

of closed subspaces $E^{u,s,c}$ with the following properties:

i) $E^{u},E^{c},E^{s}$ are invariant under $e^{At}$.

ii)The operator $A$ defined by (3.10) is linear instability and the unstable
spectra are all real and discrete. $E^{u}(E^{s})$ only consists of
eigenvectors corresponding to positive (negative) eigenvalues of $A$ and
dimension is finite.

(iii)The sharp exponential growth estimate for the $C^{0}$ group $e^{At}$
along the most unstable modes

$\|e^{At}\Gamma_{0}\|_{L^{2}}\lesssim e^{\Lambda t}\|\Gamma_{0}\|_{L^{2}}$

where $\Lambda>0$ is the maximum instability eigenvalue.

###### Proof.

We mainly prove the operator $A$ can inherit the instability nature of the
operator $\mathbf{J}\mathbf{L}$. Note that in the cylindrical coordinates, the
linearized system (3.3) can be rewritten as (2.4), which is equivalent to the
linearized system (2.6) with
$b_{r}=\frac{\partial_{z}\varphi}{r},b_{z}=\frac{\partial_{r}\varphi}{r}$.
According to Theorem 2.1, the unstable spectra of (2.6) are all discrete and
finite, we can denote the maximum instability eigenvalue of the operator
$\mathbf{J}\mathbf{L}$ as $a>0$, where
$a^{2}=\max\\{\sigma(-\mathbb{B^{\prime}}\mathbb{L}\mathbb{B}A)\\}$. We can
construct a maximal growing normal mode of the linear equation (2.4) with
$\Lambda_{0}=a$ as (3.4). It’s known that $(u,B,\mathcal{P})$ so defined is
also the solution of (3.3). It can be proved that the operator $A$ has maximum
unstable eigenvalue $a>0$. Then, there holds

$\|e^{At}\Gamma_{0}\|_{L^{2}}\lesssim e^{at}\|\Gamma_{0}\|_{L^{2}}.$ (3.11)

where $a=\sup\\{\lambda:\lambda\in\sigma(A)\\}$ is the spectral bound of
$e^{At}$ equaling to growth bound
$\omega_{0}(e^{At}):=\inf\\{\omega\in\mathbb{R}:\|e^{At}\|_{L^{2}\rightarrow
L^{2}}\leq M(\omega)e^{\omega t}\\}$.

∎

##  4 Linear instability: from the ideal MHD equation to MHD equation

In this section, we concern the linear instability of the following Leray
equation around steady state $(\beta V_{0}(\xi),\beta W_{0}(\xi))$ ($\beta\gg
1$)

$\left\\{\begin{array}[]{ll}\partial_{\tau}U-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})U+\beta\mathbb{P}(V_{0}\cdot\nabla
U+U\cdot\nabla V_{0}-W_{0}\cdot\nabla W-W\cdot\nabla
W_{0})-\Delta_{\xi}U=0,\\\
\partial_{\tau}W-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})W+\beta\mathbb{P}(V_{0}\cdot\nabla
W+U\cdot\nabla W_{0}-W_{0}\cdot\nabla U-W\cdot\nabla
V_{0})-\Delta_{\xi}W=0,\end{array}\right.$ (4.1)

in which $(V_{0}(\xi),W_{0}(\xi))$ is the axisymmetric velocity profile and
magnetic profile associated with $(v_{0}(x),H_{0}(x))$. That is to say, they
are equal under the similarity variables (1.9) as $t=1$. Using the same method
introduced in Section 3, we can rewrite (4.1) as

$\partial_{\tau}\Xi-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})\Xi-\Delta_{\xi}\Xi+\beta\mathbb{P}(\mathfrak{B}(\Xi_{0},\Xi)+\mathfrak{B}(\Xi,\Xi_{0}))=0,$
(4.2)

where $\Xi=(V(\xi,\tau),W(\xi,\tau))$ and $\Xi_{0}=(V_{0}(\xi),W_{0}(\xi)).$
In this section, we concern ourselves with instability for the operator

$-L_{ss}^{\beta}\Xi=-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})\Xi-\Delta_{\xi}\Xi+\beta\mathbb{P}(\mathfrak{B}(\Xi_{0},\Xi)+\mathfrak{B}(\Xi,\Xi_{0}))$
(4.3)

whose domain is

$\mathcal{D}(L_{ss}^{\beta}):=\\{\Xi\in L_{\sigma}^{2}(\Omega):\Xi\in
H^{2}(\Omega),\xi\cdot\nabla_{\xi}\Xi\in L^{2}(\Omega\\}.$

Indeed, we $claim$ that $\beta\Xi_{0}$ is linearly unstable for the dynamics
of (4.3), namely, there exists an unstable eigenvalue
$\widetilde{\lambda_{\beta}}>0$ for the linearized operator $L_{ss}^{\beta}$.

It is difficult to study the unstable eigenvalue for the linearized operator
$L_{ss}^{\beta}$ directly. Thus, multiplying both sides of (4.3) by
$\frac{1}{\beta}$, we can obtain that

$-T_{\beta}\Xi:=-\frac{1}{\beta}[(\frac{1}{2}+\frac{1}{2}\xi\cdot\nabla_{\xi})+\Delta_{\xi}]\Xi+\mathbb{P}(\mathfrak{B}(\Xi_{0},\Xi)+\mathfrak{B}(\Xi,\Xi_{0}))$

The main terms in the operator $-T_{\beta}$ are arising from the nonlinearity
of the equation; the extra term, including the Laplacian, can be considered as
perturbations. In the following, we will prove the existence of the unstable
eigenvalue $\lambda_{\beta}$ of $T_{\beta}$. Then
$\widetilde{\lambda_{\beta}}=\beta\lambda_{\beta}$ would be taken as an
unstable eigenvalue of the linearized operator $L_{ss}^{\beta}$.

Let us firstly define

$-T_{\infty}\Xi:=\mathfrak{B}(\Xi_{0},\Xi)+\mathfrak{B}(\Xi,\Xi_{0}).$ (4.4)

It follows from Theorem 3.1 that the operator $T_{\infty}$ is linearly
unstable and exist unstable eigenvalue. Now let us state our main result in
this section as follows.

######  Theorem 4.1.

(the instability of self-similar MHD) Let $\Lambda_{0}$ be an unstable
eigenvalue of $T_{\infty}$ with $\Lambda_{0}>0$. For any $\varepsilon>0$,
there exists $\beta_{0}>0$ such that, for all $\beta>\beta_{0}$,
$T_{\beta}|_{\mathbf{X}_{sub}}$ has an unstable eigenvalue $\lambda_{\beta}>0$
satisfying $|\lambda_{\beta}-\Lambda_{0}|<\varepsilon$. We can conclude that
$L_{ss}^{\beta}$ has unstable eigenvalue $\widetilde{\lambda_{\beta}}$ with
$\widetilde{\lambda_{\beta}}=\beta\lambda_{\beta}$ and the corresponding
unstable modes belongs to $L^{2}(\mathbb{R}^{3})$.

Before showing the proof of Theorem 4.1, we first give a spectral perturbation
Lemma due to Kato [35].

######  Lemma 4.2.

Consider the operator $T$ in the finite dimensional space, assume $T(\tau)$ be
continuous at $\tau=0$, then the eigenvalues of $T(\tau)$ are continuous at
$\tau=0$.

Now let us prove Theorem 4.1.

###### Proof.

We define the following operator

$-T_{\tau}\Xi:=-\tau[\frac{1}{2}+\frac{1}{2}\xi\cdot\nabla_{\xi}+\Delta_{\xi}]\Xi+\mathbb{P}(\mathfrak{B}(\Xi_{0},\Xi)+\mathfrak{B}(\Xi,\Xi_{0}))$

with domain

$\mathcal{D}:=\\{\Xi\in L_{\sigma}^{2}(\mathbb{R}^{3}):\Xi\in
H^{2},\xi\cdot\nabla_{\xi}\Xi\in L^{2}(\mathbb{R}^{3})\\}.$

For any $\tau_{1},\tau_{2}\in\mathbb{R}$, $\Xi\in\mathcal{D}$, it’s not
difficult to check that

$\|T_{\tau_{1}}\Xi-
T_{\tau_{2}}\Xi\|_{L^{2}}\leq|\tau_{1}-\tau_{2}|\|\Xi\|_{\mathcal{D}},$ (4.5)

which gives the continuity of $T(\tau)$ with respect to $\tau$.

For our perturbation argument, we will consider a new finite dimensional
subspace

$\mathbf{X}_{sub}=\mathcal{D}\cap E^{u}$

by taking $\tau=\frac{1}{\beta}$, where $E^{u}$ defined in Theorem 3.1 only
consists of eigenvectors corresponding to positive eigenvalues of
$T_{\infty}$. Since $\Lambda_{0}$ is an unstable eigenvalue of $T_{\infty}$
with $\Lambda_{0}>0$, then we use Kato’s perturbation Lemma 4.2 on
$\mathbf{X}_{sub}$ to deduce that there is an unstable eigenvalue
$\lambda_{\beta}$ of $T_{\beta}$. So
$\widetilde{\lambda_{\beta}}=\beta\lambda_{\beta}$ would be an unstable
eigenvalue of the linearized operator $L_{ss}^{\beta}$. ∎

##  5 Nonlinear instability

In this section, we demonstrate how to use the linear instability established
in Theorem 4.1 to construct non-unique Leray weak solution to the forced MHD
equations. The refined non-uniqueness Theorem can be stated as follows:

######  Theorem 5.1.

There exists a smooth decaying unstable velocity and magnetic profile
$\beta\Xi_{0}=(\beta V_{0}(\xi),\beta W_{0}(\xi))$ of (1.10) with force
profile

$F(\xi,\tau)=-\frac{\beta}{2}(1+\xi\cdot\nabla_{\xi})\Xi_{0}-\beta\Delta_{\xi}\Xi_{0}+\beta^{2}\mathbb{P}(\mathfrak{B}(\Xi_{0},\Xi_{0})+\mathfrak{B}(\Xi_{0},\Xi_{0}))$

satisfying the following properties:
(A) The linearized operator $L_{ss}^{\beta}$ defined in (4.3) ($\beta$
sufficiently large) has real discrete unstable eigenvalue $\lambda_{\beta}$.
Then $a$ ($a>0)$ can be chosen to be the maximally unstable, the corresponding
non-trivial smooth eigenfunction $\eta$ belonging to $H^{k}(\mathbb{R}^{3})$
for all $k\geq 0$:

$L_{ss}^{\beta}\eta=a\eta.$ (5.1)

We can look for another solution $\Xi$ to (1.10) that vanishes as
$\tau\rightarrow-\infty$ with the ansatz

$\Xi=\beta\Xi_{0}+\Xi^{lim}+\Xi^{per},$

where

$\Xi^{lim}=e^{a\tau}\eta$ (5.2)

solves the linear equation

$\partial_{\tau}\Xi^{lim}=L_{ss}^{\beta}\Xi^{lim}.$ (5.3)

(B) Substituting this ansatz into (1.10), there exists $T\in\mathbb{R}$ and a
velocity field and magnetic profile $\Xi^{per}$ satisfying (5.35) and

$\|\Xi^{per}\|_{H^{k}}\leq
Ce^{2a\tau}~{}~{}~{}~{}~{}\forall~{}~{}\tau\in(-\infty,T)$ (5.4)

for all $k\geq 0$. Correspondingly, in the similarity variable (1.9), we
construct the first Leray weak solution of the equation (1.1) is

$(v_{1},H_{1})(x,t)=(\frac{\beta}{\sqrt{t}}V_{0}(\xi),\frac{\beta}{\sqrt{t}}W_{0}(\xi))$

with force $f_{i}(x,t)=\frac{1}{t^{\frac{3}{2}}}F_{i}(\xi,\tau)$ for $(i=1,2)$
on a time interval $[0,e^{T}]$. Based on this, the second weak solution of the
MHD equations (1.1) constructed is

$(v_{2}(x,t),H_{2}(x,t))=\frac{1}{\sqrt{t}}\Xi(\xi,\tau)$

on $\mathbb{R}^{3}\times(0,e^{T})$ with zero initial data and same forcing
term $f_{i}(x,t)$ for $(i=1,2)$.

The solutions constructed above live at critical regularity. One may easily
verify that for any $p\in[2,+\infty]$, $j,k\geq 0$ and $t\in(0,e^{T})$, we
have

$\displaystyle
t^{\frac{k}{2}}\|\nabla^{k}\Gamma_{0}(\cdot,t)\|_{L^{p}}+t^{\frac{k}{2}}\|\nabla^{k}\Gamma(\cdot,t)\|_{L^{p}}\lesssim
t^{\frac{1}{2}(\frac{3}{p}-1)},$ $\displaystyle
t^{j+\frac{k}{2}}\|\partial_{t}^{j}\nabla^{k}f(\cdot,t)\|_{L^{p}}\lesssim
t^{\frac{1}{2}(\frac{3}{p}-3)}$

This is enough to bootstrap $\Gamma_{0}$, $\Gamma$ to smoothness in
$\mathbb{R}^{3}\times(0,e^{T})$. As mentioned above a second solution to (MHD)
is sought as a trajectory on the unstable manifold of $\beta\Xi_{0}$
associated to the most unstable eigenvalue $a$ of $L_{ss}^{\beta}$.

###  5.1  The semigroup generated by $L^{\beta}_{ss}$

In this subsection, we introduce the $C^{0}$ semigroup $e^{\tau
L^{\beta}_{ss}}$ generated by $L_{ss}^{\beta}$. We combine Theorem 3.1 and
Theorem 4.1 to prove some results for the spectrum and the semigroup
estimation of $L^{\beta}_{ss}$. It follows from [14, Lemma 2.1] that the
spectrum of $L^{0}_{ss}$ satisfies
$\sigma(L^{0}_{ss})\subset\\{\lambda\in\mathbb{C}:Re(\lambda)\leq-\frac{1}{4}\\}.$
According to Theorem 3.1, it is known that $L^{\beta}_{ss}-L^{0}_{ss}$ has
finite unstable discrete spectrum $\lambda>0$ and the unstable subspace is
finite-dimensional.

We refer the reader to [34] for definition of the $spectral~{}bound$ of
$L_{ss}^{\beta}$ as

$s(L_{ss}^{\beta})=:\sup\\{Re(\lambda):\lambda\in\sigma(L_{ss}^{\beta})\\},$

which is bounded by the $growth~{}bound$

$\omega_{0}(L^{\beta}_{ss}):=\inf\\{\omega\in R:\|e^{\tau
L^{\beta}_{ss}}\|_{L_{\sigma}^{2}\rightarrow L_{\sigma}^{2}}\leq
M(\omega)e^{\tau\omega}\\}$

of semigroup. Then we can obtain the following result.

######  Lemma 5.2.

$L_{ss}^{\beta}$ is the generator of a strongly continuous semigroup $e^{\tau
L_{ss}^{\beta}}$ on $L^{2}(\mathbb{R}^{3})$.
$\sigma(L^{\beta}_{ss})\cap\\{\lambda:Re(\lambda)>0\\}$ consists of only
finitely many eigenvalues with finite multiplicity, then the growth bound
$\omega_{0}(L_{ss}^{\beta})$ of $e^{\tau L_{ss}^{\beta}}$ equals

$s(L_{ss}^{\beta}):=\sup\\{z_{0}:z_{0}\in\sigma(L_{ss}^{\beta})<\infty\\}.$

In other words, assume $a=s(L_{ss}^{\beta})>0$, then $a<\infty$, and there
exist $\lambda\in\sigma(L_{ss}^{\beta})$ and $\eta\in D(L_{ss}^{\beta})$ and
$L_{ss}^{\beta}\eta=\lambda\eta$. Moreover, for every $\delta>0$, there is a
constant $M(\delta)$ with the property that

$\|e^{\tau L_{ss}^{\beta}}\Xi(0,\cdot)\|_{L^{2}(\mathbb{R}^{3})}\leq
M(\delta)e^{(a+\delta)\tau}\|\Xi(0,\cdot)\|_{L^{2}(\mathbb{R}^{3})},~{}~{}\forall~{}\tau\geq
0,\Xi\in L^{2}.$ (5.5)

######  Lemma 5.3.

By (5.2), we have the following energy estimate

$\|\Xi^{lim}\|_{H^{k}}\leq C(k,\eta)e^{a_{0}\tau}$ (5.6)

We consider the following system, the linearized system of the homogeneous MHD
system (1.4) around the axisymmetric steady solution $(\beta v_{0},\beta
H_{0},\beta^{2}p_{0})(x)$ in (2.1) as

$\left\\{\begin{array}[]{llll}\partial_{t}u+\beta\mathbb{P}(v_{0}\cdot\nabla
u+u\cdot\nabla v_{0}-H_{0}\cdot\nabla B-B\cdot\nabla H_{0})-\Delta u=0,\\\
\partial_{t}B+\beta\mathbb{P}(v_{0}\cdot\nabla B+u\cdot\nabla
H_{0}-H_{0}\cdot\nabla u-B\cdot\nabla v_{0})-\Delta B=0,\\\
\text{div}u=\text{div}B=0,\end{array}\right.$ (5.7)

can be rewrite as

$\left\\{\begin{array}[]{llll}\partial_{t}\Gamma+A\Gamma+\beta\mathbb{P}(\mathfrak{B}(\Gamma_{0},\Gamma)+\mathfrak{B}(\Gamma,\Gamma_{0}))=0\\\
\text{div}\Gamma=0,\end{array}\right.$ (5.8)

in the sense of distribution, endowed with the initial condition

$(u,B)(0,x)=(u^{0},B^{0})(x)$ (5.9)

where $\Gamma=(u,B)$, $\Gamma_{0}=(v_{0},H_{0})$, $\Gamma^{0}=(u^{0},B^{0})$.
In this case, the solution to (5.8) is formally given by

$\Gamma=e^{t\Delta}\Gamma^{0}-\int_{0}^{t}e^{(t-s)\Delta}\beta\mathbb{P}(\mathfrak{B}(\Gamma_{0},\Gamma)+\mathfrak{B}(\Gamma,\Gamma_{0}))(s)ds.$
(5.10)

######  Lemma 5.4.

[36] Let $\Omega\subset\mathbb{R}^{3}$ is a smooth bounded domain or
$\mathbb{R}^{3}$, Let $1<r\leq q<\infty$ and
$\sigma=\sigma(q,r)=\frac{3}{2}(\frac{1}{r}-\frac{1}{q})\geq 0$, we have

$\displaystyle\|e^{t\Delta}\mathbb{P}f\|_{q}\leq Ct^{-\sigma}\|f\|_{r},$
$\displaystyle\|\nabla e^{t\Delta}\mathbb{P}f\|_{q}\leq
Ct^{-\sigma-\frac{1}{2}}\|f\|_{r}.$ (5.11)

######  Lemma 5.5.

(Parabolic regularity) Assume $a=s(L_{ss}^{\beta})>0$, for any
$\sigma_{2}\geq\sigma_{1}\geq 0$ and $\delta>0$, it holds

$\|e^{\tau L_{ss}^{\beta}}\Xi(0,\cdot)\|_{H^{\sigma_{2}}}\leq
M(\sigma_{1},\sigma_{2},\delta,\beta)e^{(a+\delta)\tau}\tau^{-\frac{\sigma_{2}-\sigma_{1}}{2}}\|\Xi(0,\cdot)\|_{H^{\sigma_{1}}}$
(5.12)

for any $\Xi(0,\cdot)\in L_{\sigma}^{2}\cap H^{\sigma_{1}}(\mathbb{R}^{3})$.

###### Proof.

Firstly, we prove for any $0\leq m\leq k$, $\Xi(0,\cdot)\in L_{\sigma}^{2}\cap
H^{m}(\mathbb{R}^{3})$, it holds

$\|e^{\tau L_{ss}^{\beta}}\Xi(0,\cdot)\|_{H^{k}}\leq
M(k,m)\tau^{-\frac{k-m}{2}}\|\Xi(0,\cdot)\|_{H^{m}},\tau\in(0,2).$ (5.13)

We study the problem in physical variables: setting

$u(x,t)=\frac{1}{\sqrt{t+1}}U(\frac{x}{\sqrt{t+1}},log^{(t+1)}),~{}~{}B(x,t)=\frac{1}{\sqrt{t+1}}W(\frac{x}{\sqrt{t+1}},log^{(t+1)}),$
$v_{0}(x,t)=\frac{1}{\sqrt{t+1}}V_{0}(\frac{x}{\sqrt{t+1}},log^{(t+1)}),~{}~{}H_{0}(x,t)=\frac{1}{\sqrt{t+1}}W_{0}(\frac{x}{\sqrt{t+1}},log^{(t+1)}),$

satisfying equation (5.7), which can be expressed as (5.8). Since $\Gamma_{0}$
is smooth function in bounded domain, we can easily prove that

$\|\Gamma\|_{2}+t^{\frac{1}{2}}\|\nabla\Gamma\|_{2}\leq
C(\Gamma_{0},\beta)\|\Gamma(0,\cdot)\|_{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t\in(0,10)$
(5.14)

which gives

$\|\Xi\|_{2}+\tau^{\frac{1}{2}}\|\nabla\Xi\|_{2}\leq
C(\Xi_{0},\beta)\|\Xi(0,\cdot)\|_{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau\in(0,2)$
(5.15)

The latter implies (5.12) for $k=1$ and $m=0,1$. The general case follows by
induction studying the equation solved by $\nabla^{k}\Xi$ which has a
structure similar to (5.8) but with forcing and additional lower order terms.
Secondly, we prove for any $\delta>0$ it holds

$\|e^{\tau L_{ss}^{\beta}}\Xi(0,\cdot)\|_{H^{k}}\leq
M(k,\delta)e^{\tau(a+\delta)}\|\Xi(0,\cdot)\|_{L^{2}},~{}~{}\tau\geq 2.$
(5.16)

Using the semigroup property in Step 1 with $m=0$, and (5.6), we have

$\displaystyle\|e^{\tau L_{ss}^{\beta}}\Xi(0,\cdot)\|_{H^{k}}$
$\displaystyle=\|e^{\kappa
L_{ss}^{\beta}}(e^{(\tau-k)L_{ss}^{\beta}}\Xi(0,\cdot))\|_{H^{k}}$
$\displaystyle\leq M(k)k^{-\frac{k}{2}}\|e^{\kappa
L_{ss}^{\beta}}(e^{(\tau-k)L_{ss}^{\beta}}\Xi(0,\cdot))\|_{L^{2}}$
$\displaystyle\leq
M(k,\delta)k^{-\frac{k}{2}}e^{(\tau-k)(a+\delta)}\|\Xi(0,\cdot)\|_{L^{2}}$
(5.17)

The claimed inequality (5.12) follows by choosing $\kappa=1$.
It is immediate to see that the combination of (5.13) and (5.16) gives (5.12)
for integers $\sigma_{2}\geq\sigma_{1}\geq 0$. This completes the proof. ∎

###  5.2  Nonlinear construction

In the conditions of Theorem 3.1, we consider smooth, compactly supported
velocity and magnetic profile $(\beta v_{0},\beta H_{0})$. Since
$(u^{lim},B^{lim})$ satisfy (5.7), we can get $(u^{per},B^{per})$ satisfies
the following nonlinear problem

$\displaystyle\left\\{\begin{array}[]{llll}&\partial_{t}u^{per}+\beta\mathbb{P}(v_{0}\cdot\nabla
u^{per}+u^{per}\cdot\nabla v_{0}-H_{0}\cdot\nabla H^{per}-H^{per}\cdot\nabla
H_{0})-\Delta u^{per}\\\ &=B\cdot\nabla B-u\cdot\nabla u,\\\
&\partial_{t}B^{per}+\beta\mathbb{P}(v_{0}\cdot\nabla
B^{per}+u^{per}\cdot\nabla H_{0}-H_{0}\cdot\nabla u^{per}-B^{per}\cdot\nabla
v_{0})-\Delta B^{per}\\\ &=B\cdot\nabla u-u\cdot\nabla B,\\\
&div~{}u=divB=0,\end{array}\right.$ (5.23)

we can write $f_{1}=B\cdot\nabla B-u\cdot\nabla u$ and $f_{2}=B\cdot\nabla
u-u\cdot\nabla B$, in which $B(x,t)=B^{lim}+B^{per}$ and
$u(x,t)=u^{lim}+u^{per}$. In similarity variable,

$\begin{split}&\xi=\frac{x}{{t}},~{}~{}~{}~{}~{}~{}~{}\tau=logt,~{}~{}~{}~{}~{}~{}~{}~{}~{}f(x,t)=\frac{1}{t^{\frac{3}{2}}}F(\xi,\tau)\\\
&u^{lim}(x,t)=\frac{1}{\sqrt{t}}V^{lim}(\xi,\tau),~{}~{}B^{lim}(x,t)=\frac{1}{\sqrt{t}}W^{per}(\xi,\tau),\\\
&u^{per}(x,t)=\frac{1}{\sqrt{t}}V^{per}(\xi,\tau),~{}~{}B^{per}(x,t)=\frac{1}{\sqrt{t}}W^{per}(\xi,\tau),\end{split}$

then (5.23) can be expressed as

$\displaystyle\left\\{\begin{array}[]{llll}&\partial_{\tau}V^{per}-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})V^{per}-\Delta_{\xi}V^{per}+\beta\mathbb{P}(V_{0}\cdot\nabla
V^{per}+V^{per}\cdot\nabla V_{0}-W_{0}\cdot\nabla W^{per}\\\
&-W^{per}\cdot\nabla W_{0})=F_{1},\\\
&\partial_{\tau}W^{per}-~{}\frac{1}{2}(1+\xi\cdot\nabla_{\xi})W^{per}-\Delta_{\xi}W^{per}+\beta\mathbb{P}(V_{0}\cdot\nabla
W^{per}+V^{per}\cdot\nabla W_{0}-W_{0}\cdot\nabla V^{per}\\\
&-W^{per}\cdot\nabla V_{0})=F_{2}\\\
&\text{div}V^{per}=\text{div}W^{per}=0,\end{array}\right.$ (5.29)

where

$\displaystyle F_{1}$
$\displaystyle=(W^{lim}+W^{per})\cdot\nabla(W^{lim}+W^{per})-(V^{lim}+V^{per})\cdot\nabla(V^{lim}+V^{per})$
$\displaystyle=W^{lim}\cdot\nabla W^{lim}+W^{per}\cdot\nabla
W^{lim}+W^{lim}\cdot\nabla W^{per}+W^{per}\cdot\nabla
W^{per}-V^{lim}\cdot\nabla V^{lim}$ $\displaystyle-V^{per}\cdot\nabla
V^{lim}-V^{lim}\cdot\nabla V^{per}-V^{per}\cdot\nabla V^{per}$ (5.30)

and

$\displaystyle F_{2}$
$\displaystyle=(W^{lim}+W^{per})\cdot\nabla(V^{lim}+V^{per})-(V^{lim}+V^{per})\cdot\nabla(W^{lim}+W^{per})$
$\displaystyle=W^{lim}\cdot\nabla V^{lim}+W^{per}\cdot\nabla
V^{lim}+W^{lim}\cdot\nabla V^{per}+W^{per}\cdot\nabla
V^{per}-V^{lim}\cdot\nabla W^{lim}$ $\displaystyle-V^{per}\cdot\nabla
W^{lim}-V^{lim}\cdot\nabla W^{per}-V^{per}\cdot\nabla W^{per}.$ (5.31)

Note that using the same method as in section 2, we can rewrite equation
(5.29) as

$\partial_{\tau}\Xi^{per}-\frac{1}{2}(1+\xi\cdot\nabla_{\xi})\Xi^{per}-\Delta_{\xi}\Xi^{per}+\beta\mathbb{P}(\mathfrak{B}(\Xi_{0},\Xi^{per})+\mathfrak{B}(\Xi^{per},\Xi_{0}))=F,$
(5.32)

where $\Xi^{per}=(V^{per}(\xi,\tau),W^{per}(\xi,\tau))$,
$\Xi_{0}=(V_{0}(\xi,\tau),W_{0}(\xi,\tau))$, $F=(F_{1},F_{2})$ and
$\mathbb{P}$ is the Leray projector. We define the total energy by

$\|\Xi^{per}\|_{\mathbb{X}}=\|(V^{per},W^{per})\|_{\mathbb{X}}:=\sup_{\tau<T}e^{-(a+\varepsilon_{0})\tau}(\|V^{per}(\cdot,\tau)\|_{H^{N}}+\|W^{per}(\cdot,\tau)\|_{H^{N}}).$

######  Proposition 5.6.

Assume $a=s(L_{ss}^{\beta})>0$ and $N>\frac{5}{2}$ is an integer. Then there
exist $T=T(\Xi_{0},\Xi^{lim})$, $\varepsilon_{0}>0$ and $\Xi^{per}\in
C((-\infty,T];H^{N}(\mathbb{R}^{3};\mathbb{R}^{3}))$, a solution to (5.32),
such that

$\|E^{per}(\cdot,\tau)\|_{H^{N}}\leq
e^{(a+\varepsilon_{0})\tau},~{}~{}~{}~{}for~{}any~{}\tau<T$ (5.33)

###### Proof.

Firstly, we introduce the Banach space

$\mathbb{X}:=\\{\Xi^{per}\in
C((-\infty,T];H^{N}(\mathbb{R}^{3};\mathbb{R}^{3})):\sup_{\tau<T}e^{-(a+\varepsilon_{0})\tau}\|\Xi^{per}(\cdot,\tau)\|_{H^{N}}<\infty\\}$

with the norm

$\|\Xi^{per}\|_{\mathbb{X}}:=\sup_{\tau<T}e^{-(a+\varepsilon_{0})\tau}\|\Xi^{per}(\cdot,\tau)\|_{H^{N}}$

By Duhamel’s formula, (5.32) can be expressed as the following functional

$\mathcal{T}(\Xi^{per})(\cdot,\tau)=-\int_{-\infty}^{\tau}e^{(\tau-s)L_{ss}^{\beta}}\circ\mathbb{P}Fds$
(5.34)

By parabolic regularity theory, any $\Xi^{per}\in\mathbb{X}$ such that
$T(\Xi^{per})=\Xi^{per}$ is a solution to (5.32) satisfying the statement of
the Proposition by the following contraction mapping principle. ∎

######  Proposition 5.7.

Let $B_{\mathbb{X}}$ be the closed unit ball of $\mathbb{X}$. Then for $T$
sufficiently large and negative, and $N>\frac{5}{2}$, $\mathcal{T}$ map
$B_{\mathbb{X}}\rightarrow B_{\mathbb{X}}$ is a contraction.

###### Proof.

According to the definition of total energy, $\mathcal{T}$ splits into three
terms $\mathcal{T}(E^{per})\simeq B(E^{per},E^{per})+L(\cdot,E^{per})+G$,
defined by

$G=\int_{-\infty}^{\tau}e^{(\tau-s)L_{ss}^{\beta}}(W^{lim}\cdot\nabla
E^{lim,\perp}-V^{lim}\cdot\nabla E^{lim})ds=G^{0,1}+G^{0,2}$ $\displaystyle
L(\cdot,E^{per})=\int_{-\infty}^{\tau}e^{(\tau-s)L_{ss}^{\beta}}(W^{lim}\cdot\nabla
E^{per,\perp}+W^{per}\cdot\nabla E^{lim,\perp}-V^{lim}\cdot\nabla
E^{per}-V^{per}\cdot\nabla E^{lim})ds$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=L^{1,1}+L^{1,2}+L^{1,3}+L^{1,4}$

and

$B(E^{per},E^{per})=\int_{-\infty}^{\tau}e^{(\tau-s)L_{ss}^{\beta}}(W^{per}\cdot\nabla
E^{per,\perp}-V^{per}\cdot\nabla E^{per})ds=B^{2,1}+B^{2,2},$

where $E^{\perp}=(W,V)$, $G\in\mathbb{X}$, $L:\mathbb{X}\rightarrow\mathbb{X}$
is a bounded linear operator and
$B:\mathbb{X}\times\mathbb{X}\rightarrow\mathbb{X}$ is a bounded bilinear
form. By a simple computation, we can obtain that

$\|\mathcal{T}(E^{per}_{1}-E^{per}_{2})\|_{\mathbb{X}}\leq(2\|B\|+\|L\|)\|E^{per}_{1}-E^{per}_{2}\|_{\mathbb{X}}.$
(5.35)

To prove the operator $\mathcal{T}$ is contraction, it is sufficient for us to
show $2\|B\|+\|L\|<1$.

Firstly, we apply Lemma 5.5 to get

$\|B^{2,1}\|_{H^{N+\frac{1}{2}}}\leq\int_{-\infty}^{\tau}(\tau-s)^{-\frac{3}{4}}e^{(\tau-s)(a+\delta)}\|(W^{per}\cdot\nabla
E^{per,\perp})\|_{H^{N-1}}(s)ds.$ (5.36)

Since the space $H^{N-1}$ is an algebra, then

$\|W^{per}\cdot\nabla E^{per,\perp}\|_{H^{N-1}}\leq
C(N)\|E^{per}\|_{H^{N}}^{2}\leq
e^{2(a+\varepsilon_{0})s}\|E^{per}\|_{\mathbb{X}}^{2}$ (5.37)

and, using that $a+2\varepsilon_{0}-\delta>0$, we obtain

$\displaystyle\|B^{2,1}\|_{H^{N+\frac{1}{2}}}$ $\displaystyle\leq
C(N,\delta)\|E^{per}\|_{\mathbb{X}}^{2}\int_{-\infty}^{\tau}\frac{e^{(\tau-s)(a+\delta)}e^{2(a+\varepsilon_{0})s}}{(\tau-s)^{\frac{1}{2}}}ds\leq
C(N,\delta)e^{2(a+\varepsilon_{0})\tau}\|E^{per}\|_{\mathbb{X}}^{2}.$ (5.38)

Hence, $\|B^{2,1}\|_{\mathbb{X}}\leq
e^{T(a+\varepsilon_{0})}\|E^{per}\|_{\mathbb{X}}^{2}$. As a consequence of
Lemma 5.5, we get

$\displaystyle\|L^{1,1}\|_{H^{N}}$ $\displaystyle\leq
M(N,\delta)\int_{-\infty}^{\tau}(\tau-s)^{-\frac{1}{2}}e^{(a+\varepsilon_{0})(\tau-s)}\|W^{lim}\cdot\nabla
E^{per}\|_{H^{N-1}}ds$ $\displaystyle\leq
M(N,\delta)\int_{-\infty}^{\tau}(\tau-s)^{-\frac{1}{2}}e^{(a+\varepsilon_{0})(\tau-s)}e^{(2a+\varepsilon_{0})s}\|E^{per}\|_{\mathbb{X}}ds$
(5.39)

By employing Lemma 5.3 and Lemma 5.5, we deduce

$\|L^{1,1}\|_{H^{N}}\leq
C(N,\delta,a)e^{(2a+\varepsilon_{0})\tau}\|E^{per}\|_{\mathbb{X}}$ (5.40)

and

$\displaystyle\|G^{0,1}\|_{H^{N}}$
$\displaystyle\leq\int_{-\infty}^{\tau}e^{(a+\delta)(\tau-s)}\|(W^{lim}\cdot\nabla
E^{lim,\perp})\|_{H^{N}}ds$
$\displaystyle\leq\int_{-\infty}^{\tau}e^{(a+\delta)(\tau-s)}e^{2as}ds\leq
C(N,\delta,a)e^{2a\tau},$ (5.41)

provided $\delta<a$, as a consequence of Lemma 5.3 and Lemma 5.5, which leads
to the estimates

$\|L^{1,1}\|_{\mathbb{X}}\leq
C(N,\delta,a)e^{aT}\|E^{per}\|_{\mathbb{X}},~{}~{}~{}\|G^{0,1}\|_{\mathbb{X}}\leq
e^{(a-\varepsilon_{0})T}.$

Similar estimates can be carried on $B^{2,2}(V^{per},E^{per})$,
$L^{1,2}(W^{per},E^{lim}),L^{1,3}(V^{lim},E^{per})$,
$L^{1,4}(V^{per},E^{lim}),G^{0,2}(V^{lim},E^{lim})$.

Based on these estimates above, it follows that for $T$ sufficiently large and
negative and $\|E^{per}\|_{\mathbb{X}}\leq 1$, we can make $2\|B\|+\|L\|<1$,
which gives $\|\mathcal{T}(E^{per})\|_{\mathbb{X}}\leq 1$. That is to say,
$\mathcal{T}|_{B(\mathbb{X})}$ is contractive. This finish the proof. ∎

In Theorem 5.1, $\beta\Xi_{0}$ is the solution of (1.10) by choosing the force
$\widetilde{F}(\xi,\tau)$. Let $E^{per}\in\mathbb{X}$ be the unique fixed
point of $\mathcal{T}$ guaranteed by Proposition 5.6 and showed that $E^{per}$
decays as $e^{(a+\varepsilon_{0})\tau}$ as $\tau\rightarrow-\infty$. By
induction, we can bootstrap it to $O(e^{2a\tau})$ decay in $H^{N}$ for
$N>\frac{5}{2}$. We can construct $E=\beta\Xi_{0}+E^{lim}+E^{per}$ solving
(1.10) which is not equal to $\beta\Xi_{0}$. Finally, undoing the similarity
variable transform for both $\beta\Xi_{0}$ and $E$ gives a pair of distinct
solutions of (1.1).

## Data Availability Statements

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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