# Low regularity ill-posedness and shock formation
for 3D ideal compressible MHD

Xinliang An  Department of Mathematics, National University of Singapore
Singapore<EMAIL_ADDRESS>,  Haoyang Chen  Department of Mathematics,
National University of Singapore
Singapore<EMAIL_ADDRESS>and  Silu Yin  Department of Mathematics,
Hangzhou Normal University
Hangzhou, China<EMAIL_ADDRESS>
###### Abstract.

The study of magnetohydrodynamics (MHD) significantly boosts the understanding
and development of solar physics, planetary dynamics and controlled nuclear
fusion. Dynamical properties of the MHD system involve nonlinear interactions
of waves with multiple travelling speeds (the fast and slow magnetosonic
waves, the Alfvén wave and the entropy wave). One intriguing topic is the
shock phenomena accompanied by the magnetic field, which have been affirmed by
astronomical observations. However, permitting the residence of all above
multi-speed waves, mathematically, whether one can prove shock formation for
3D MHD is still open. The multiple-speed nature of the MHD system makes it
fascinating and challenging. In this paper, we give an affirmative answer to
the above question. For 3D ideal compressible MHD, we construct examples of
shock formation allowing the presence of all characteristic waves with
multiple wave speeds. Building on our construction, we further prove that the
Cauchy problem for 3D ideal MHD is $H^{2}$ ill-posed. And this is caused by
the instantaneous shock formation. In particular, when the magnetic field is
absent, we also provide a desired low-regularity ill-posedness result for the
3D compressible Euler equations, and it is sharp with respect to the
regularity of the fluid velocity. Our proof for 3D MHD is based on a coalition
of a carefully designed algebraic approach and a geometric approach. To trace
the nonlinear interactions of various waves, we algebraically decompose the 3D
ideal MHD equations into a $7\times 7$ non-strictly hyperbolic system. Via
detailed calculations, we reveal its hidden subtle structures. With them we
give a complete description of MHD dynamics up to the earliest singular event,
when a shock forms.

###### Contents

  1. 1 Introduction
    1. 1.1 Main theorems
    2. 1.2 Difficulties and Strategies
    3. 1.3 Main steps in the proof
    4. 1.4 Other related works
    5. 1.5 Acknowledgements
  2. 2 Decomposition of waves
  3. 3 Proof of Theorem 1.1
  4. 4 Proof of shock formation (Theorem 1.2)
    1. 4.1 A priori estimates
    2. 4.2 $L^{\infty}$-estimates and bootstrap argument
    3. 4.3 Shock formation
  5. 5 Proof of $H^{2}$ ill-posedness (Theorem 1.3)
    1. 5.1 Initial data designed for proving ill-posedness
    2. 5.2 Estimate for $\partial_{z_{1}}\rho_{1}$
    3. 5.3 Ill-posedness mechanism
  6. 6 Case of $H_{1}=0$
  7. 7 Proof of $H^{2}$ ill-posedness for 3D compressible Euler equations (Theorem 1.4)
  8. A Proof of Proposition 3.1

##  1\. Introduction

Magnetohydrodynamics (MHD) plays a crucial role in astrophysics, planetary
magnetism and controlled fusion reactor. It operates on every scales, from
vast on stellar winds and solar magnetic activity to the small on controlled
fusion plasmas.

Mathematically, the equations of MHD system describe the dynamics and magnetic
properties of electrically conducting fluid or plasma. To study planetary
dynamics, liquid metals and electrolyte, vibrant mathematical explorations of
incompressible MHD equations are conducted, ranging from well-posedness theory
to singularity formation mechanism and covering topics from Cauchy problems to
free boundary problems.

Compared with the incompressible counterpart, there is less mathematical
literature investigating the compressible MHD. On the contrary, the physical
phenomena governed by compressible MHD are extremely rich, which include the
jets from accretion disks at a galactic scale, stellar magnetism describing
solar activity, and controlled nuclear fusions in a designed tokamak.

Figure 1. Schematic of an accretion disk. From [14] by Davidson.

Figure 2. The structure of the sun. Convection zone is the source of MHD
activity, including the initiation of coronal flux tubes and sunspots. From
[14] by Davidson.

Similarly to other compressible fluids with zero viscosity, shock waves are
anticipated to play a significant role in the context of 3D ideal compressible
MHD with negligible resistivity/viscosity. For instance, a shock wave is
expected to form when the solar wind interacts with the earth’s magnetic
field.

Figure 3. The interaction of the earth’s magnetic field with the solar wind.
From [14] by Davidson.

Shock-wave-related phenomena of MHD raise broad interests among the physical
and the numerical communities. The evolution of compressible MHD involves
interactions of waves with multiple speeds: the fast and slow magnetosonic
waves, the Alfvén wave and the entropy wave. Mathematically, an open question
is to give a rigorous proof of shock formation allowing the residence of all
above multi-speed waves.

In this paper, we provide a confirmative answer to the above question for the
3D ideal compressible MHD. We give a constructive proof including detailed
analysis of the solutions’ behaviours up to the first (shock) singularity. And
the shock forms due to the compression of characteristics. Furthermore, based
on shock formation, we prove that the Cauchy problems of the 3D ideal
compressible MHD equations are ill-posed in $H^{2}(\mathbb{R}^{3})$. We show
that the underlying mechanism of this ill-posedness is the instantaneous shock
formation. To the best of our knowledge, this is also the first nonlinear
result of low regularity ill-posedness for the 3D ideal compressible MHD. In
the absence of the magnetic field, we also provide a desired ill-posedness
result for the 3D compressible Euler equations.

Our research toward low-regularity ill-posedness for MHD is inspired by
remarkable works on wave equations. Under planar symmetry, Lindblad gave sharp
counterexamples to the low-regularity local well-posedness for semilinear and
certain quasilinear wave equations in [29, 30, 31]. In [16], Ettinger-Lindblad
generalized the above results to the Einstein’s equations and constructed a
sharp counterexample for local well-posedness of Einstein vacuum equations in
wave coordinates. An exploration by Granowski [17] later showed that
Lindblad’s $H^{s}$ ill-posedness in [31] is stable under general
perturbations. Without any symmetry assumption, Alinhac [1, 2] firstly proved
shock formation for solutions to quasilinear wave equations in more than one
spatial dimension. He employed a Nash-Moser iteration scheme to overcome the
loss of derivatives in energy estimates. This approach relies heavily on a
non-degeneracy condition posed on initial data and fails to reveal information
beyond the first blow-up point. In 2009, a breakthrough [10] was made by
Christodoulou. Based on a fully geometric approach, he sharpened Alinhac’s
results by giving a detailed understanding and a complete description of shock
formation for 3D irrotational Euler equations. With the full use of the
geometry therein, Chrsitodoulou derived sharper energy estimates which avoid
the potential loss of derivatives. This geometric approach was later extended
to a larger class of wave equations and Euler equations with or without
vorticity. See Christodoulou-Miao [11], Luk-Speck [35], Miao-Yu [37], Speck
[39, 40], Speck-Holzegel-Luk-Wong [41]. Recently, applying a different
approach via self-similar ansatz, Buckmaster-Shkoller-Vicol[5, 6] constructed
shock formation for compressible Euler equations in 2D and 3D allowing the
presence of non-trivial vorticity.

Compared with the single wave equation, fewer results are known for
quasilinear wave systems. Quasilinear wave systems with physical background
(e.g. elasticity and MHD) typically describe phenomena involving interactions
of waves with multiple travelling speeds. An important and challenging task is
to study singularity formation for these multi-wave-speed hyperbolic systems,
including elastic waves, compressible MHD, crystal optics, self-gravitating
relativistic fluids, etc. In [3], we generalized Lindblad’s work on a scalar
wave equation in [31] by showing that the Cauchy problems for 3D elastic
waves, a physical system with multiple wave speeds, are ill-posed in
$H^{3}(\mathbb{R}^{3})$. We also quantitatively demonstrated that the ill-
posedness is driven by shock formation. In this paper, we further generalize
our method and result in [3] to an important quasilinear hyperbolic system
with multiple wave-speed: the 3D ideal compressible MHD system. Compared with
the elastic waves, the MHD system possesses more subtle and more complex
structures, which imposes many new challenges to establish ill-posedness and
shock formation. See Subsection 1.2 for details.

###  1.1. Main theorems

We start from inducing the equations. The 3D ideal compressible MHD, composed
of the Euler equations of fluid dynamics and the coupled Maxwell’s equations
for magnetic field, forms a quasilinear hyperbolic system:

$\left\\{\begin{split}&\partial_{t}\varrho+\nabla\cdot(\varrho u)=0,\\\
&\varrho\\{\partial_{t}+(u\cdot\nabla)\\}u+\nabla
p+\mu_{0}H\times\text{rot}H=0,\\\ &\partial_{t}H-\text{rot}(u\times H)=0,\\\
&\partial_{t}S+(u\cdot\nabla)S=0,\\\ &\nabla\cdot H=0.\end{split}\right.$
(1.1)

Here, $\mu_{0}$ is the magnetic permeability constant, $\varrho$ is the fluid
density111In our analysis, the value of $\varrho$ is close to $1$.,
$u=(u_{1},u_{2},u_{3})\in\mathbb{R}^{3}$ is the fluid velocity,
$H=(H_{1},H_{2},H_{3})\in\mathbb{R}^{3}$ is the magnetic field intensity, $S$
is the entropy and $p$ is the pressure satisfying the equation of state
$p=p(\varrho,S).$ To describe the solar wind, following [14] we use the
polytropic equation of state

$p=Ae^{S}\varrho^{\gamma}$ (1.2)

with $A$ being a positive constant and $\gamma>1$ being the adiabatic gas
constant.

Following Lindblad’s approach toward proving ill-posedness, we study (1.1)
under planar symmetry. By Gauss’s Law, $H_{1}$ is always a constant. See
Section 2. Therefore, we actually work with the following system:

$\partial_{t}\Phi(x,t)+A(\Phi)\partial_{x}\Phi(x,t)=0,$ (1.3)

where $\Phi=(u_{1},u_{2},u_{3},\varrho-1,H_{2},H_{3},S)^{T}$ and
$A(\Phi)\in\mathbb{R}^{7\times 7}$ is a $7\times 7$ matrix. For notational
simplicity, we denote $x_{1}$ by $x$ in this paper. Our proof relies on a
carefully designed algebraic wave decomposition of this $7\times 7$ system.
See Subsection 1.3 and Section 2 for details. The delicate structures of the
decomposed system play a crucial role in our proof. In the following theorem,
these subtle structures are revealed for the first time.

######  Theorem 1.1 (Structures of MHD equations).

For the MHD system (1.3):
$\partial_{t}\Phi(x,t)+A(\Phi)\partial_{x}\Phi(x,t)=0$, one can rewrite it
into the following form:

$\begin{cases}\partial_{s_{i}}\rho_{i}=c_{ii}^{i}v_{i}+\Big{(}\sum\limits_{m\neq
i}c_{im}^{i}w_{m}\Big{)}\rho_{i},\\\
\partial_{s_{i}}w_{i}=-c_{ii}^{i}w_{i}^{2}+\Big{(}\sum\limits_{m\neq
i}(-c_{im}^{i}+\gamma_{im}^{i})w_{m}\Big{)}w_{i}+\sum\limits_{m\neq i,k\neq
i\atop m\neq k}\gamma_{km}^{i}w_{k}w_{m},\\\
\partial_{s_{i}}v_{i}=\Big{(}\sum\limits_{m\neq
i}\gamma_{im}^{i}w_{m}\Big{)}v_{i}+\sum\limits_{m\neq i,k\neq i\atop m\neq
k}\gamma_{km}^{i}w_{k}w_{m}\rho_{i}.\end{cases}$ (1.4)

Here, $\partial_{s_{i}}=\lambda_{i}\partial_{x}+\partial_{t}$ with
$\lambda_{i}$ being the $i^{\text{th}}$ eigenvalue of the $7\times 7$ matrix
$A(\Phi)$. And $\rho_{i}:=\partial_{z_{i}}X_{i}$ is the inverse density of the
characteristics222We use $(X_{i}(z_{i},s_{i}),s_{i})$ to denote the trajectory
of the characteristics with $(z_{i},s_{i})$ being the characteristic
coordinates. See (2.14) for detailed definitions.. We further define
$w_{i}:=l_{i}\partial_{x}\Phi$ and $v_{i}:=\rho_{i}w_{i}$, where $l_{i}$ is
the $i^{\text{th}}$ left eigenvector of $A(\Phi)$. With an 18-page
calculation, we prove that the coefficients
$c^{i}_{ii},c^{i}_{im},\gamma^{i}_{im},\gamma^{i}_{km}$ being non-trivial
functions of $\Phi$ are all of order $1$. (See Section 3 and Appendix A for
the details333These coefficients could potentially be singular due to the
singular factors present in the eigenvectors..) This enables us to reveal the
following structures for the system (1.4). For $\rho_{i}$, we have

$\left\\{\begin{split}\partial_{s_{1}}\rho_{1}\thicksim&v_{1}+\rho_{1}\big{(}\sum\limits_{i\neq
1}w_{i}\big{)},\\\
\partial_{s_{2}}\rho_{2}\thicksim&\cancel{v_{2}}+\boxed{\rho_{2}w_{3}}+\rho_{2}\big{(}\sum\limits_{i\neq
2,3}w_{i}\big{)},\\\
\partial_{s_{3}}\rho_{3}\thicksim&v_{3}+\cancel{\rho_{3}w_{2}}+\rho_{3}\big{(}\sum\limits_{i\neq
2,3}w_{i}\big{)},\\\
\partial_{s_{4}}\rho_{4}\thicksim&\cancel{v_{4}}+\rho_{4}\big{(}\sum\limits_{i\neq
4}w_{i}\big{)},\\\
\partial_{s_{5}}\rho_{5}\thicksim&v_{5}+\boxed{\rho_{5}w_{6}}+\rho_{5}\big{(}\sum\limits_{i\neq
5,6}w_{i}\big{)},\\\
\partial_{s_{6}}\rho_{6}\thicksim&\cancel{v_{6}}+\cancel{\rho_{6}w_{5}}+\rho_{6}\big{(}\sum\limits_{i\neq
5,6}w_{i}\big{)},\\\
\partial_{s_{7}}\rho_{7}\thicksim&v_{7}+\rho_{7}\big{(}\sum\limits_{i\neq
7}w_{i}\big{)}.\\\ \end{split}\right.$ (1.5)

Here, the notation $\cancel{v_{2}}$ denotes the absence of the term $v_{2}$,
$\boxed{\rho_{2}w_{3}}$ denotes the strong interaction term. For $w_{i}$, we
have

$\left\\{\begin{split}\partial_{s_{1}}w_{1}\thicksim&(w_{1})^{2}+\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+{\sum_{m\neq
1,k\neq 1,m\neq k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k}},\\\
\partial_{s_{2}}w_{2}\thicksim&\cancel{(w_{2})^{2}}+\boxed{w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
2,k\neq 2,m\neq k\atop(m,k)\neq(5,6)}w_{m}w_{k},\\\
\partial_{s_{3}}w_{3}\thicksim&(w_{3})^{2}+\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
3,k\neq 3,m\neq k\atop(m,k)\neq(5,6)}w_{m}w_{k},\\\
\partial_{s_{4}}w_{4}\thicksim&\cancel{(w_{4})^{2}}+\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
4,k\neq 4,m\neq k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k},\\\
\partial_{s_{5}}w_{5}\thicksim&(w_{5})^{2}+\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
5,k\neq 5,m\neq k\atop(m,k)\neq(5,6)}w_{m}w_{k},\\\
\partial_{s_{6}}w_{6}\thicksim&\cancel{(w_{6})^{2}}+\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\boxed{w_{5}w_{6}}+\sum_{m\neq
6,k\neq 6,m\neq k\atop(m,k)\neq(2,3)}w_{m}w_{k},\\\
\partial_{s_{7}}w_{7}\thicksim&(w_{7})^{2}+\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
7,k\neq 7,m\neq k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k}.\\\
\end{split}\right.$ (1.6)

Here, the notation $\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}$ denotes the
weak interaction term. For $v_{i}$, we have

$\left\\{\begin{split}\partial_{s_{1}}v_{1}\thicksim&v_{1}\big{(}\sum\limits_{i\neq
1}w_{i}\big{)}+\rho_{1}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
1,k\neq 1,m\neq k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k}\bigg{)},\\\
\partial_{s_{2}}v_{2}\thicksim&\dbox{(\lambda_{2}-\lambda_{3})v_{2}w_{3}}+\big{(}\sum\limits_{i\neq
2,3}w_{i}\big{)}v_{2}+\rho_{2}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
2,k\neq 2,m\neq k\atop(m,k)\neq(5,6)}w_{m}w_{k}\bigg{)},\\\
\partial_{s_{3}}v_{3}\thicksim&\dbox{(\lambda_{2}-\lambda_{3})v_{3}w_{2}}+\big{(}\sum\limits_{i\neq
2,3}w_{i}\big{)}v_{3}+\rho_{3}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
3,k\neq 3,m\neq k\atop(m,k)\neq(5,6)}w_{m}w_{k}\bigg{)},\\\
\partial_{s_{4}}v_{4}\thicksim&v_{4}\big{(}\sum\limits_{i\neq
4}w_{i}\big{)}+\rho_{4}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
4,k\neq 4,m\neq k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k}\bigg{)},\\\
\partial_{s_{5}}v_{5}\thicksim&\dbox{(\lambda_{5}-\lambda_{6})v_{5}w_{6}}+\big{(}\sum\limits_{i\neq
5,6}w_{i}\big{)}v_{5}+\rho_{5}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
5,k\neq 5,m\neq k\atop(m,k)\neq(2,3)}w_{m}w_{k}\bigg{)},\\\
\partial_{s_{6}}v_{6}\thicksim&\dbox{(\lambda_{5}-\lambda_{6})v_{6}w_{5}}+\big{(}\sum\limits_{i\neq
5,6}w_{i}\big{)}v_{6}+\rho_{6}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
6,k\neq 6,m\neq k\atop(m,k)\neq(2,3)}w_{m}w_{k}\bigg{)},\\\
\partial_{s_{7}}v_{7}\thicksim&v_{7}\big{(}\sum\limits_{i\neq
7}w_{i}\big{)}+\rho_{7}\bigg{(}\dbox{(\lambda_{2}-\lambda_{3})w_{2}w_{3}}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}+\sum_{m\neq
7,k\neq 7,m\neq k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k}\bigg{)}.\\\
\end{split}\right.$ (1.7)

######  Remark 1.1 (Potentially harmful terms).

In (1.6)-(1.7), some potentially harmful terms are present. We list them on
the right hand side of the following equations

$\displaystyle\partial_{s_{2}}w_{2}=\boxed{(\gamma_{23}^{2}-c_{23}^{2})w_{2}w_{3}}+\cdots,$
(1.8)
$\displaystyle\partial_{s_{6}}w_{6}=\boxed{(\gamma_{65}^{6}-c_{65}^{6})w_{5}w_{6}}+\cdots,$
(1.9)
$\displaystyle\partial_{s_{2}}\rho_{2}=\boxed{c_{23}^{2}\rho_{2}w_{3}}+\cdots,$
(1.10)
$\displaystyle\partial_{s_{6}}\rho_{6}=\boxed{c_{65}^{6}\rho_{6}w_{5}}+\cdots.$
(1.11)

These potentially harmful terms could result in strong nonlinear interactions
between the two families of waves ($w_{2}$ and $w_{3}$, $w_{5}$ and $w_{6}$)
with almost-the-same speeds. To handle these nonlinear terms is the main task
of Section 4. Compared with elastic waves in [3], there the coefficients of
these potentially harmful nonlinear terms have additional smallness, which
makes the interactions weaker (and thus easier to control) than those in MHD.

######  Remark 1.2.

In Theorem 1.1, our conclusion that all
$c^{i}_{ii}(\Phi),c^{i}_{im}(\Phi),\gamma^{i}_{im}(\Phi),\gamma^{i}_{km}(\Phi)$
are of order $1$ is highly non-trivial. This is true because we carefully
design the wave decomposition in this paper and all the potentially singular
factors in
$c^{i}_{ii}(\Phi),c^{i}_{im}(\Phi),\gamma^{i}_{im}(\Phi),\gamma^{i}_{km}(\Phi)$
are cancelled. To prove this, we conduct an 18-page (mainly in the appendix)
calculation for 299 coefficients. See Proposition 3.1 in Section 3 and
Appendix A for the crucial discussions.

We are now in a good position to state the following result: we construct
examples of shock formation for 3D ideal compressible MHD equations allowing
the presence of nonlinear interactions of all multi-speed waves.

######  Theorem 1.2 (Shock formation for MHD).

For the 3D ideal compressible MHD, there exists a large class of compactly
supported smooth Cauchy initial data that lead to finite-time shock
formations. Let $\varepsilon\in(0,\frac{1}{100})$ be a fixed parameter. Then
there is a constant $\theta_{0}>0$ such that for any $0<\theta<\theta_{0}$,
the following statements hold for the solution to system (1.4) arising from
the initial datum.

  * i)

(Shock formation along characteristics) A shock forms at time $T^{*}$, i.e.,

$\lim_{t\rightarrow
T^{*}}\rho_{1}(z_{0},t)=0\quad\text{and}\quad\lim_{t\rightarrow
T^{*}}w_{1}(z_{0},t)=\infty,$

where at $z_{0}$ the initial datum $w_{1}(z_{1},0)$ attains its maximum. In
particular, the blow-up of $w_{1}$ implies $\lim_{t\rightarrow
T^{*}}|\partial_{x}\Phi|(z_{0},t)=\infty$. The lifespan $T^{*}$ of $w_{1}$
satisfies

$\frac{1}{(1+\varepsilon)^{3}|c_{11}^{1}(0)|w_{1}(z_{0},0)}\leq
T^{*}\leq\frac{1}{(1-\varepsilon)^{3}|c_{11}^{1}(0)|w_{1}(z_{0},0)}.$ (1.12)

  * ii)

(Lower bound of other $\rho_{i}$) For any $i\in\\{2,\cdots,7\\}$, we have
$w_{i}(z_{i},t)=O(\theta)$ and $\rho_{i}(z_{i},t)>C$ with $C$ being a uniform
positive constant for all $z_{i}\in\mathbb{R}$ and any $t<T^{*}$.

  * iii)

(Boundedness of $\Phi$) The original variable $\Phi$ is uniformly bounded. We
have $\Phi(z_{i},t)=O(\theta)$ for all $z_{i}\in\mathbb{R}$ and any $t<T^{*}$.

  * iv)

(Boundedness of $\rho_{i}$ and $v_{i}$) For any $i\in\\{1,\cdots,7\\}$, we
have $\rho_{i}(z_{i},t)=O(1)$ and $v_{i}(z_{i},t)=O(\theta)$ for all
$z_{i}\in\mathbb{R}$ and any $t<T^{*}$.

######  Remark 1.3.

For the decomposed system (1.4), initial data are compactly supported on
$[-2\eta,2\eta]$ with $\eta>0$. Then, along different families of
characteristics, there are seven strips $\mathcal{R}_{i}$, $i=1\cdots 7$
issuing from the support $[-2\eta,2\eta]$. To conduct our analysis, we use the
following picture. The shock is constructed to happen at $t=T_{\eta}^{*}$ in
the first characteristic strip $\mathcal{R}_{1}$ with the fastest propagation
wave speed. For $t<T_{\eta}^{*}$, there is no other singular point in the
spacetime region and the solution is smooth. See Figure 4.

$2\eta$$-2\eta$$t=0$$t=t_{0}^{(\eta)}$$t=T_{\eta}^{*}$$z_{0}$$(z_{0},T_{\eta}^{*})$$\mathcal{R}_{1}$$\mathcal{R}_{2}\cup\mathcal{R}_{3}$$\mathcal{R}_{4}$$\mathcal{R}_{5}\cup\mathcal{R}_{6}$$\mathcal{R}_{7}$
Figure 4. First shock forms at $(z_{0},T_{\eta}^{*})$ in the characteristic
strip $\mathcal{R}_{1}$. In this picture, the domain of our consideration is
divided into three regions: (a) The grey region is called the characteristic
strips. (b) The light grey region denotes the disjoint domain between two
separate strips. (c) The dark grey region denotes the domain where the strips
overlap with each other.

######  Remark 1.4.

We design the above strips
$\\{\mathcal{R}_{1},\mathcal{R}_{2}\cup\mathcal{R}_{3},\mathcal{R}_{4},\mathcal{R}_{5}\cup\mathcal{R}_{6},\mathcal{R}_{7}\\}$
and regions with colors to conduct our analysis. We remark that the solution
is not necessarily being zero outside of the characteristic strips. In
particular, the solutions’ dynamics in the light-grey region of Figure 4 is
far from being trivial. For our proof, it is crucial to investigate the
different behaviours of the solutions in and out of these characteristic
strips. We will derive a priori estimates for solutions in these colored
regions in Subsection 4.1.

Moreover, based on our constructed shock formation, we further provide a
counterexample to $H^{2}$ local well-posedness. It is caused by the
instantaneous shock formation. The ill-posedness theorem is stated as below.

######  Theorem 1.3 ($H^{2}$ ill-posedness for MHD).

The Cauchy problems of the 3D ideal compressible MHD equations (1.1) are ill-
posed in $H^{2}(\mathbb{R}^{3})$ in the following sense:

There exists a family of compactly supported, smooth initial data
$\Phi_{0}^{(\eta)}$ satisfying

${\|\Phi_{0}^{(\eta)}\|}_{{\dot{H}}^{2}(\mathbb{R}^{3})}\to
0\quad\text{as}\quad\eta\to 0,$

where $\eta>0$ is a small parameter and it identifies the datum in this
family. For each $\eta$, the Cauchy problem of the 3D ideal MHD (1.3) admits a
solution that ceases to be regular in finite time. Let $T_{\eta}^{*}$ be the
largest time such that the solution $\Phi_{\eta}\in
C^{\infty}\big{(}\mathbb{R}^{3}\times[0,T_{\eta}^{*})\big{)}$. The following
statements hold:

  * i)

(Instantaneous shock formation) Evolving from each initial datum
$\Phi_{0}^{(\eta)}$, a shock forms at $T_{\eta}^{*}$. And as $\eta\to 0$, we
have $T_{\eta}^{*}\to 0$. This means that the constructed initial data
$\Phi_{0}^{(\eta)}$ give rise to an instantaneous shock formation.

  * ii)

(Blow-up of $H^{1}$-norm) For each solution $\Phi_{\eta}$, with
$\Omega_{T_{\eta}^{*}}$ being a spatial neighborhood of the first (shock)
singularity. the $H^{1}$-norm $\|\Phi_{\eta}\|_{H^{1}(\Omega_{T_{\eta}^{*}})}$
blows up at $T^{*}_{\eta}$. In particular,

$\|\partial_{x}u_{1}^{(\eta)}(\cdot,T_{\eta}^{*})\|_{L^{2}(\Omega_{T_{\eta}^{*}})}=+\infty,\
\|\partial_{x}\varrho^{(\eta)}(\cdot,T_{\eta}^{*})\|_{L^{2}(\Omega_{T_{\eta}^{*}})}=+\infty.$
(1.13)

######  Remark 1.5.

Noted that the $L^{\infty}$ blow-up of the solutions’ first derivatives in
Theorem 1.2 is not adequate to imply the blow-up of the $H^{1}$ norm as in
(1.13). To prove (1.13), we crucially employ the vanishing of the inverse
density $\rho$, which is the geometric description of the shock formation. Our
exploration of the characteristics’ geometric behaviours via $\rho$ bridges
the gap between the shock formation (pointwise blow-up of the first
derivatives) and the blow-up of $H^{1}$-norms. See Subsection 5.3 for the
details.

######  Remark 1.6.

For 3D ideal compressible MHD (1.1), both our examples of the shock formation
allowing the presence of waves with multiple travelling speeds and the
associated low-regularity ill-posedness are new. With the subtle structures we
found, for our constructed solutions, we give a complete description of the
MHD dynamics up to the time $T^{*}_{\eta}$ when the first (shock) singularity
happens.

If the magnetic field vanishes, i.e., $H_{1}=H_{2}=H_{3}\equiv 0$, our system
(1.1) reduces to the 3D compressible Euler equations. Our results thus give

######  Theorem 1.4 ($H^{2}$ ill-posedness for Euler equations).

Consider the following 3D compressible Euler equations allowing non-trivial
entropy and vorticity

$\left\\{\begin{split}&\partial_{t}\varrho+\nabla\cdot(\varrho u)=0,\\\
&\varrho\\{\partial_{t}+(u\cdot\nabla)\\}u+\nabla p=0,\\\
&\partial_{t}S+(u\cdot\nabla)S=0.\end{split}\right.$ (1.14)

Then the Cauchy problem of (1.14) is $H^{2}$ ill-posed with respect to the
fluid velocity and density. More precisely, there exits a family of compactly
supported smooth initial data for (1.14) satisfying

${\|\varrho_{0}^{(\eta)}\|}_{{\dot{H}}^{2}(\mathbb{R}^{3})}+{\|u_{0}^{(\eta)}\|}_{{\dot{H}}^{2}(\mathbb{R}^{3})}+{\|S_{0}^{(\eta)}\|}_{{\dot{H}}^{2}(\mathbb{R}^{3})}\to
0\quad\text{as}\quad\eta\to 0.$

After a finite time $T_{\eta}^{*}$ a shock forms and the solution ceases to be
smooth. Furthermore, tied to this family of initial data the shock formation
is instantaneous, i.e., $T_{\eta}^{*}\to 0$ as $\eta\to 0$. Moreover, the
$H^{1}$-norm of the velocity $u_{1}$ and density $\varrho$ blow up at the
shock formation time $T^{*}_{\eta}$:

$\|u_{1}^{(\eta)}(\cdot,T_{\eta}^{*})\|_{H^{1}(\Omega_{T_{\eta}^{*}})}=+\infty,\quad\|\varrho^{(\eta)}(\cdot,T_{\eta}^{*})\|_{H^{1}(\Omega_{T_{\eta}^{*}})}=+\infty$

where $\Omega_{T_{\eta}^{*}}$ is a spatial region around the first (shock)
singularity.

######  Remark 1.7.

The above ill-posedness result for the 3D compressible Euler equations is
sharp with respect to the regularity of the fluid velocity $u$ and density
$\varrho$. The classical local well-posedness (LWP) result for the 3D
compressible Euler equations holds in $H^{s}$ with $s>\frac{5}{2}$ for initial
data of $u$ and $\varrho$. The regularity for local well-posedness is lowered
to $s>2$ for fluid velocity and density by Disconzi-Luo-Mazzone-Speck [13],
Wang [46] and Zhang-Andersson [52]. Our ill-posedness result thus acts as a
sharp counterpart of these local well-posedness results. As a special scenario
of 3D compressible Euler equations, for the incompressible case, Bourgain-Li
[4] proved the sharp low-regularity ill-posedness result with respect to the
regularity of the vorticity $\omega$. Their initial data require fluid
velocity $u$ and vorticity $\omega$ satisfying $(u,\omega)\in
H^{\frac{5}{2}}\times H^{\frac{3}{2}}$. With the picture below, one can see
that our Theorem 1.4 serves as a sharp companion with respect to the fluid
velocity $u$ and density $\varrho$. See Section 7 for more detailed
discussions.

$s$: regularity for the velocity$s^{\prime}$: regularity for the
vorticity2$\frac{5}{2}$$\frac{3}{2}$2LWP proved by Wang [46](See also
Disconzi-Luo-Mazzone-Speck [13], Zhang-Andersson [52])LWP by the Conjecture in
[46]Bourgain-Li’s ill-posedness [4]An-Chen-Yin’s ill-posedness

###  1.2. Difficulties and Strategies

To prove the ill-posedness result, as in [3, 16, 30, 31] we work under planar
symmetry, the MHD system then obeys

$\left\\{\begin{split}&\varrho\partial_{t}u_{1}+\varrho
u_{1}\partial_{x}u_{1}+c^{2}\partial_{x}\varrho+\partial_{S}p\partial_{x}S+\mu_{0}H_{2}\partial_{x}H_{2}+\mu_{0}H_{3}\partial_{x}H_{3}=0,\\\
&\varrho\partial_{t}u_{2}+\varrho
u_{1}\partial_{x}u_{2}-\mu_{0}H_{1}\partial_{x}H_{2}=0,\\\
&\varrho\partial_{t}u_{3}+\varrho
u_{1}\partial_{x}u_{3}-\mu_{0}H_{1}\partial_{x}H_{3}=0,\\\
&\partial_{t}\varrho+u_{1}\partial_{x}\varrho+\varrho\partial_{x}u_{1}=0,\\\
&\partial_{t}H_{2}+u_{1}\partial_{x}H_{2}-H_{1}\partial_{x}u_{2}+H_{2}\partial_{x}u_{1}=0,\\\
&\partial_{t}H_{3}+u_{1}\partial_{x}H_{3}-H_{1}\partial_{x}u_{3}+H_{3}\partial_{x}u_{1}=0,\\\
&\partial_{t}S+u_{1}\partial_{x}S=0,\end{split}\right.$ (1.15)

where $c=\sqrt{\partial_{\varrho}p}$ denotes the sound speed. For (1.15), by
Gauss’s Law, $H_{1}$ is a constant and hence (1.15) can be viewed as a
$7\times 7$ first-order hyperbolic system:

$\partial_{t}\Phi+A(\Phi)\partial_{x}\Phi=0,$ (1.16)

where $\Phi=(u_{1},u_{2},u_{3},\varrho-1,H_{2},H_{3},S)^{T}$ and $A(\Phi)$ is
a $7\times 7$ matrix. We then explore the structures of (1.16) and proceed to
do analysis, but several difficulties arise:

$\bullet$ Invalidity of Riemann invariants for a $7\times 7$ system: For
hyperbolic system with different travelling speeds, to prove shock formation,
a key ingredient is to understand the nonlinear interaction of waves
propagating in different speeds. For a $2\times 2$ first-order hyperbolic
system, there exist two Riemann invariants, which can be constructed
explicitly and verify two transport equations. These two Riemann invariants
carry crucial information about the nonlinear structure of the $2\times 2$
first-order hyperbolic system. And with them other geometric quantities can be
solved along the characteristics. However, for a large system, such as our
$7\times 7$ system, it is almost impossible to find proper Riemann invariants
with explicit formulas. Here we adopt a different approach. We appeal to
John’s classic method [25], i.e., decomposition of waves. This approach was
used by us in [3] to study the elastic waves equations, and it will further be
extended to the $7\times 7$ MHD system in this paper. Applying the following
formula of decomposition of waves

$\partial_{x}\Phi=\sum_{k=1}^{7}w_{k}r_{k},$

we rewrite (1.16) as a diagonal Riccati-type system:

$(\partial_{t}+\lambda_{i}\partial_{x})w_{i}=-c_{ii}^{i}(w_{i})^{2}+\Big{(}\sum_{m\neq
i}(-c_{im}^{i}+\gamma_{im}^{i})w_{m}\Big{)}w_{i}+\sum_{m\neq i,k\neq i\atop
m\neq k}\gamma_{km}^{i}w_{k}w_{m},$ (1.17)

where $w_{i}:=l_{i}(\Phi)\partial_{x}\Phi$ and $l_{i}(\Phi)$ are the left
eigenvectors of $A(\Phi)$. Here the coefficients $c_{ii}^{i}$, $c_{im}^{i}$,
$\gamma_{km}^{i}$, $\gamma_{im}^{i}$ are functions of the unknowns $\Phi$. A
fundamental technical point in our work is to bound these coefficients. This
is closely relevant to the choice of right eigenvectors $r_{i}$.

$\bullet$ Non-strict hyperbolicity: For the hyperbolic system (1.16), via
algebraic calculations, we notice that the MHD system (1.15) is non-strictly
hyperbolic444For more discussions on non-strictly hyperbolic system, we refer
to Liu [33], Liu-Xin [34] and Zhou [54]., with eigenvalues of the coefficient
matrix $(A(\Phi))_{7\times 7}$ satisfying

$\displaystyle\begin{split}\lambda_{7}(\Phi)<\lambda_{6}(\Phi)\leq\lambda_{5}(\Phi)<\lambda_{4}(\Phi)<\lambda_{3}(\Phi)\leq\lambda_{2}(\Phi)<\lambda_{1}(\Phi).\end{split}$

When $\Phi$ is sufficiently small, two pairs of characteristics have almost-
the-same travelling speed $\lambda_{2}(\Phi)\thickapprox\lambda_{3}(\Phi),\
\lambda_{5}(\Phi)\thickapprox\lambda_{6}(\Phi).$ This yields the long-time
overlapping of the corresponding characteristic strips
$\mathcal{R}_{2},\mathcal{R}_{3}$ (or $\mathcal{R}_{5},\mathcal{R}_{6}$)
mentioned in Remark 1.3. Substantial nonlinear interactions will be carried
out in the overlapped regions and they may effect other strips. We overcome
this obstacle by taking unions of the overlapped strips and considering the
wave propagations in and out of the following five strips:

$\\{\mathcal{R}_{1},\mathcal{R}_{2}\cup\mathcal{R}_{3},\mathcal{R}_{4},\mathcal{R}_{5}\cup\mathcal{R}_{6},\mathcal{R}_{7}\\}.$
(1.18)

Tracing the dynamics of MHD (a $7\times 7$ non-strictly hyperbolic system)
with analyzing the solutions’ behaviours in and out of these five
characteristic strips
$\\{\mathcal{R}_{1},\mathcal{R}_{2}\cup\mathcal{R}_{3},\mathcal{R}_{4},\mathcal{R}_{5}\cup\mathcal{R}_{6},\mathcal{R}_{7}\\}$
is new. These five strips will completely separate from each other when
$t>t_{0}^{(\eta)}$, where $t_{0}^{(\eta)}$ can be precisely calculated. See
the following picture sketching the dynamics:

$2\eta$$-2\eta$$t=0$$t=t_{0}^{(\eta)}$$\mathcal{R}_{1}$$\mathcal{R}_{2}\cup\mathcal{R}_{3}$$\mathcal{R}_{4}$$\mathcal{R}_{5}\cup\mathcal{R}_{6}$$\mathcal{R}_{7}$
Figure 5. Separation of five characteristic strips.

Estimates before $t_{0}^{(\eta)}$ can be carried out similarly as in [12] by
Christodoulou-Perez, where they studied the propagation of electromagnetic
waves in nonlinear crystals and solved a first-order strictly hyperbolic
system. However, after $t_{0}^{(\eta)}$, the non-strict hyperbolicity
dramatically amplifies the nonlinear interactions. $\mathcal{R}_{2}$ and
$\mathcal{R}_{3}$ (or $\mathcal{R}_{5}$ and $\mathcal{R}_{6}$) could overlap
with each other for a long time. This overlapping feature results in strong
nonlinear interaction. Here we illustrate the intuition with the interactions
in $\mathcal{R}_{2}$ and $\mathcal{R}_{3}$. In strictly hyperbolic case as in
[12], since $\mathcal{R}_{2}$ and $\mathcal{R}_{3}$ will separate after
$t_{0}^{(\eta)}$, the interaction of corresponding characteristic waves is
weak. However, for our non-strictly hyperbolic system, due to the overlapping
feature, the characteristics $\mathcal{C}_{2},\mathcal{C}_{3}$ may still
intersect in $\mathcal{R}_{2}$ (or $\mathcal{R}_{3}$) after a long time. This
forces us to estimate in $\mathcal{R}_{2}\cup\mathcal{R}_{3}$ where the
interaction is conceivably stronger. Nonetheless, we show that the strong
interaction will not jeopardize the desired blow-up mechanism. To guarantee
that the interaction terms are still controllable, we need to employ the
subtle structures of the equations. This will be discussed in the following
paragraph.

$\bullet$ Subtle structures in MHD system have to be employed: The subtleties
lying in the structures of (1.17) are of paramount importance in our proof. A
thorough analysis of the structures is essential for tracing the geometric
behaviours of the characteristics in and out of the above five strips (1.18)
up to shock formation time $T^{*}_{\eta}$. We list some crucial structures for
$w_{i}$ here:

$\left\\{\begin{split}(\partial_{t}+\lambda_{1}\partial_{x})w_{1}\thicksim&(w_{1})^{2}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{5}-\lambda_{6})w_{5}w_{6}+\cdots,\\\
(\partial_{t}+\lambda_{2}\partial_{x})w_{2}\thicksim&\cancel{(w_{2})^{2}}+\boxed{w_{2}w_{3}}+(\lambda_{5}-\lambda_{6})w_{5}w_{6}+\cdots,\\\
(\partial_{t}+\lambda_{3}\partial_{x})w_{3}\thicksim&(w_{3})^{2}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{5}-\lambda_{6})w_{5}w_{6}+\cdots,\\\
(\partial_{t}+\lambda_{4}\partial_{x})w_{4}\thicksim&\cancel{(w_{4})^{2}}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{5}-\lambda_{6})w_{5}w_{6}+\cdots,\\\
(\partial_{t}+\lambda_{5}\partial_{x})w_{5}\thicksim&(w_{5})^{2}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{5}-\lambda_{6})w_{5}w_{6}+\cdots,\\\
(\partial_{t}+\lambda_{6}\partial_{x})w_{6}\thicksim&\cancel{(w_{6})^{2}}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+\boxed{w_{5}w_{6}}+\cdots,\\\
(\partial_{t}+\lambda_{7}\partial_{x})w_{7}\thicksim&(w_{7})^{2}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{5}-\lambda_{6})w_{5}w_{6}+\cdots.\\\
\end{split}\right.$ (1.19)

The coefficients of the deleted terms in (1.19) are proved to be zero. In
other words, there is no Riccati-type terms appearing in the equations of
$\\{w_{i}\\}_{i=2,4,6}$. The vanishing of these terms indicates the linear
degeneracy of the corresponding characteristics. In contrast, the non-
vanishing of Riccati-type terms such as
${(w_{1})}^{2},{(w_{3})}^{2},{(w_{5})}^{2},{(w_{7})}^{2}$ demonstrate the
genuine nonlinearity of the corresponding characteristics and they may drive
the blow-ups of the solutions.

Now we compare with the circumstance for the elastic waves in [3]. The system
of $w_{i}$ in [3] reads

$\left\\{\begin{split}(\partial_{t}+\lambda_{1}\partial_{x})w_{1}\thicksim&(w_{1})^{2}+\cancel{w_{2}w_{3}}+\cancel{w_{4}w_{5}}+\cdots,\\\
(\partial_{t}+\lambda_{2}\partial_{x})w_{2}\thicksim&\cancel{(w_{2})^{2}}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{4}-\lambda_{5})w_{4}w_{5}+\cdots,\\\
(\partial_{t}+\lambda_{3}\partial_{x})w_{3}\thicksim&(w_{3})^{2}+\cancel{w_{2}w_{3}}+\cancel{w_{4}w_{5}}+\cdots,\\\
(\partial_{t}+\lambda_{4}\partial_{x})w_{4}\thicksim&(w_{4})^{2}+\cancel{w_{2}w_{3}}+\cancel{w_{4}w_{5}}+\cdots,\\\
(\partial_{t}+\lambda_{5}\partial_{x})w_{5}\thicksim&\cancel{(w_{5})^{2}}+(\lambda_{2}-\lambda_{3})w_{2}w_{3}+(\lambda_{4}-\lambda_{5})w_{4}w_{5}+\cdots,\\\
(\partial_{t}+\lambda_{6}\partial_{x})w_{6}\thicksim&(w_{6})^{2}+\cancel{w_{2}w_{3}}+\cancel{w_{4}w_{5}}+\cdots\end{split}\right.$
(1.20)

where $\lambda_{2}\thickapprox\lambda_{3}$ and
$\lambda_{5}\thickapprox\lambda_{6}$.

In the equations for $w_{2}$ and $w_{5}$ in (1.20), observe that the
coefficients of $w_{2}w_{3}$ and $w_{4}w_{5}$ contain small factors
$\lambda_{2}-\lambda_{3}$ and $\lambda_{4}-\lambda_{5}$ owing to the non-
strict hyperbolicity. This suggests that the corresponding related
interactions are weak. We were able to employ the bi-characteristic
coordinates transformation
$(x,t)=\big{(}X_{j}(y_{j},t^{\prime}(y_{i},y_{j})),t^{\prime}(y_{i},y_{j})\big{)}$
to control these terms. For the equation of $w_{2}$, we integrate along
$\mathcal{C}_{2}$ inside $\mathcal{R}_{3}$ to deduce

$\begin{split}&\int_{\mathcal{C}_{2}}\big{|}(\lambda_{2}-\lambda_{3})w_{2}\big{(}X_{2}(z_{2},t^{\prime}),t^{\prime}\big{)}w_{3}\big{(}X_{2}(z_{2},t^{\prime}),t^{\prime}\big{)}\big{|}dt^{\prime}\\\
\leq&\int_{y_{3}\in[\eta,2\eta]}\Big{|}w_{2}w_{3}\big{(}y_{3},t^{\prime}(y_{2},y_{3})\big{)}\cancel{(\lambda_{2}-\lambda_{3})}\frac{\rho_{3}\big{(}y_{3},t^{\prime}(y_{2},y_{3})\big{)}}{\cancel{\lambda_{2}-\lambda_{3}}}\Big{|}dy_{3}.\end{split}$

The only singular factor in the above integral is
$\frac{1}{\lambda_{2}-\lambda_{3}}$, which comes from the coordinate
transformation. As we can see, it can be exactly cancelled by the small factor
$\lambda_{2}-\lambda_{3}$ in the coefficient.

However, for the ideal compressible MHD, equations of $w_{2}$ and $w_{6}$ in
(1.19) do not include small factor $\lambda_{2}-\lambda_{3}$ or
$\lambda_{5}-\lambda_{6}$ in front of $w_{2}w_{3}$ or $w_{5}w_{6}$, which
renders our approach in [3] via the bi-characteristic coordinates to fail. A
more delicate choice of the initial data for $w_{2},w_{3},w_{5},w_{6}$ and
additional a priori estimates for

$\check{W}(t):=\max\Big{\\{}\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{w_{2},w_{3}\\},\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq t}\\{w_{5},w_{6}\\}\Big{\\}}.$

are employed to handle these terms.

Moreover, for (1.19), the interaction terms $w_{2}w_{3}$ and $w_{5}w_{6}$ also
appear in the equations of $w_{1}$, $w_{3}$, $w_{4}$, $w_{5}$, $w_{7}$. While
the corresponding terms vanish in (1.20) since their coefficients are all
zero. Luckily, in (1.19) we can prove these nonlinear interactions are weak
and negligible compared with the ones in the equations of $w_{2}$ and $w_{6}$.
In particular, the interaction terms $w_{2}w_{3},w_{5}w_{6}$ are proved to be
small compared with the dominant Riccati terms since they have small
coefficients $\lambda_{2}-\lambda_{3},\lambda_{5}-\lambda_{6}$. And the
smallness here enables us to use bi-characteristic coordinates to control
$w_{2}w_{3},w_{5}w_{6}$ with desired bounds.

$\bullet$ Delicate choice of right eigenvectors: To explore the subtle
structures of our system, we need to control the following non-trivial
coefficients $c_{im}^{i}$, $\gamma_{im}^{i}$ and $\gamma_{km}^{i}$ in the
decomposed system (1.4), where

$\displaystyle c_{im}^{i}:=\nabla_{\Phi}\lambda_{i}\cdot r_{m},$
$\displaystyle\gamma_{im}^{i}:=-(\lambda_{i}-\lambda_{m})l_{i}\cdot(\nabla_{\Phi}r_{i}\cdot
r_{m}-\nabla_{\Phi}r_{m}\cdot r_{i}),\quad m\neq i,$
$\displaystyle\gamma_{km}^{i}:=-(\lambda_{k}-\lambda_{m})l_{i}\cdot(\nabla_{\Phi}r_{k}\cdot
r_{m}),\qquad\qquad\qquad k\neq i,\ m\neq i.$

Here, $\Phi=(u_{1},u_{2},u_{3},\varrho-1,H_{2},H_{3},S)^{T}$. And
$\lambda_{i}$, $l_{i}$, $r_{i}$ are the eigenvalues, left and right
eigenvectors of $A(\Phi)$, respectively. If all these coefficients are of
$O(1)$, the system will then take the schematic form presented in (1.19) and
in Theorem 1.1. As we mentioned before, the realization of these estimates for
$c_{im}^{i}$, $\gamma_{im}^{i}$, $\gamma_{km}^{i}$ relies heavily on carefully
designed right eigenvectors $r_{i}$. For example, to evaluate
$\gamma_{64}^{2}$, one attempt is to take the following right eigenvectors

$\begin{split}&\tilde{r}_{2}=\left(\begin{array}[]{cc}0\\\
\frac{C_{a}H_{3}}{H_{1}H_{2}}\\\ -\frac{C_{a}}{H_{1}}\\\ 0\\\
-\frac{H_{3}}{H_{2}}\\\ 1\\\ 0\end{array}\right),\
\tilde{r}_{4}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\
-\frac{\varrho}{\gamma}\\\ 0\\\ 0\\\ 1\end{array}\right),\
\tilde{r}_{6}=\left(\begin{array}[]{cc}0\\\ \frac{C_{a}}{H_{1}}\\\
-\frac{C_{a}H_{2}}{H_{1}H_{3}}\\\ 0\\\ 1\\\ -\frac{H_{2}}{H_{3}}\\\
0\end{array}\right),\end{split}$ (1.21)

where $C_{a}$ is the Alfvén wave speed (See Section 2), and the left
eigenvector

$\tilde{l}_{2}=(0,\frac{H_{1}H_{2}H_{3}}{2C_{a}(H_{2}^{2}+H_{3}^{2})},-\frac{H_{2}^{2}H_{1}}{2C_{a}(H_{2}^{2}+H_{3}^{2})},0,-\frac{H_{2}H_{3}}{2(H_{2}^{2}+H_{3}^{2})},\frac{H_{2}^{2}}{2(H_{2}^{2}+H_{3}^{2})},0).$
(1.22)

Here $H_{1}\neq 0$ and the elements in $\tilde{l}_{2}$ are of order $1$. And
it follows

$\gamma_{64}^{2}=-(\lambda_{6}-\lambda_{4})\tilde{l}_{2}\cdot(\nabla_{\Phi}\tilde{r}_{6}\cdot\tilde{r}_{4})=-\frac{(\lambda_{6}-\lambda_{4})H_{2}}{4\gamma
H_{3}}.$ (1.23)

However, since both $H_{2}$ and $H_{3}$ are unknowns with values close to $0$,
the term $\frac{H_{2}}{H_{3}}$ is very problematic and may exhibit unexpected
singularity. It may render the failure of the whole argument. To overcome this
problem, we renormalize and design the right eigenvectors and left
eigenvectors in the following form

$\begin{split}&r_{2}=\left(\begin{array}[]{cc}0\\\ \frac{C_{a}H_{3}}{H_{1}}\\\
-\frac{C_{a}H_{2}}{H_{1}}\\\ 0\\\ -H_{3}\\\ H_{2}\\\ 0\end{array}\right),\
r_{4}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\ -\frac{\varrho}{\gamma}\\\ 0\\\
0\\\ 1\end{array}\right),\ r_{6}=\left(\begin{array}[]{cc}0\\\
\frac{C_{a}H_{3}}{H_{1}}\\\ -\frac{C_{a}H_{2}}{H_{1}}\\\ 0\\\ H_{3}\\\
-H_{2}\\\ 0\end{array}\right),\end{split}$
$l_{2}=(0,\frac{H_{1}H_{3}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},-\frac{H_{2}H_{1}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0,-\frac{H_{3}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},\frac{H_{2}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0).$

Note that the right eigenvectors are more regular, but the left eigenvectors
are more singular, since they always satisfy $l_{i}r_{j}=\delta_{ij}$ and
$\tilde{l}_{i}\tilde{r}_{j}=\delta_{ij}$. Miraculously, with the new set of
$r_{2},r_{4},r_{6}$ and $l_{2}$, we have

$\displaystyle\gamma_{64}^{2}$ $\displaystyle=$
$\displaystyle-(\lambda_{6}-\lambda_{4})l_{2}\cdot(\nabla_{\Phi}r_{6}\cdot
r_{4})$ $\displaystyle=$
$\displaystyle-(\lambda_{6}-\lambda_{4})l_{2}\cdot\Big{(}0,\frac{C_{a}H_{3}}{2H_{1}\gamma},-\frac{C_{a}H_{2}}{2H_{1}\gamma},0,0,0,0\Big{)}^{\top}$
$\displaystyle=$
$\displaystyle-(\lambda_{6}-\lambda_{4})\Big{(}\frac{H_{3}^{2}}{4\gamma(H_{2}^{2}+H_{3}^{2})}+\frac{H_{2}^{2}}{4\gamma(H_{2}^{2}+H_{3}^{2})}\Big{)}$
$\displaystyle=$ $\displaystyle-\frac{\lambda_{6}-\lambda_{4}}{4\gamma}=O(1).$

Compared with (1.23), the singular term is cancelled! Note that in $l_{2}$, a
potentially singular factor $\frac{1}{H_{2}^{2}+H_{3}^{2}}$ appears since
$H_{2}$ and $H_{3}$ are close to zero. But with a detailed matrix
multiplication, the singular denominator is cancelled! Not only for
$\gamma^{2}_{64}$, we have checked 299 terms with 18 pages calculations and
find that with our designed right and left eigenvectors in (2.9) and (2.11),
all these coefficients $c^{i}_{im}$, $\gamma^{i}_{im}$ and $\gamma^{i}_{km}$
are uniformly bounded.

$\bullet$ Control of nonlinear interactions of multiple waves: Due to the
multiple-wave-speed trait of the MHD, the decomposed system (1.4) exhibits
nonlinear interactions among different characteristic strips. The high
complexity of the coupled nonlinear terms poses a major technical difficulty,
especially after the separating time $t_{0}^{(\eta)}$. To disclose how
different waves interact with each other, we must distinguish its influence to
other characteristic waves in its own strips and in the exterior. Firstly,
outside of the strips, we estimate the following quantities

$\displaystyle
V_{i}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{i},\atop 0\leq
t^{\prime}\leq t}|w_{i}(x^{\prime},t^{\prime})|,\quad\text{for}\quad i=1,4,7,$
$\displaystyle
V_{\bar{2}}(t):=\max\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{|w_{2}(x^{\prime},t^{\prime})|,|w_{3}(x^{\prime},t^{\prime})|\\},$
$\displaystyle
V_{\bar{5}}(t):=\max\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq
t}\\{|w_{5}(x^{\prime},t^{\prime})|,|w_{6}(x^{\prime},t^{\prime})|\\},$
$\displaystyle V(t):=\max_{i}V_{i}(t)\quad\text{for}\
i=1,\bar{2},4,\bar{5},7.$

Note that $V_{\bar{2}}$ and $V_{\bar{5}}$ are defined in a form different from
that of $V_{1}$, $V_{4}$, $V_{7}$. This is because, as we emphasized before,
for our non-strictly hyperbolic system, the almost-repeated characteristic
waves $w_{2},w_{3}$ (or $w_{5},w_{6}$) are treated together in
$\mathcal{R}_{2}\cup\mathcal{R}_{3}$ (or
$\mathcal{R}_{5}\cup\mathcal{R}_{6}$). We then prescribe initial data such
that only $w_{1}$ will blow up. This is feasible since the equation of $w_{1}$
is of Riccati type. Yet it is still hard to solve the Riccati equation
directly in the presence of other nonlinear terms. To overcome this
difficulty, we introduce the geometric quantities—the inverse densities
$\rho_{i}$ to characterize the compression of the $i^{\text{th}}$
characteristics. To control $w_{i}$, we appeal to investigating dynamics of
$\rho_{i}$ and $v_{i}:=\rho_{i}w_{i}$ instead by estimating

$\displaystyle S_{i}(t):=$
$\displaystyle\sup_{(z^{\prime}_{i},s^{\prime}_{i})\atop
z^{\prime}_{i}\in[-2\eta,2\eta],\ 0\leq s^{\prime}_{i}\leq
t}\rho_{i}(z^{\prime}_{i},s^{\prime}_{i}),$ $\displaystyle J_{i}(t):=$
$\displaystyle\sup_{(z^{\prime}_{i},s^{\prime}_{i})\atop
z^{\prime}_{i}\in[-2\eta,2\eta]\ 0\leq s^{\prime}_{i}\leq
t}|v_{i}(z^{\prime}_{i},s^{\prime}_{i})|.$

For the purpose of demonstration, we use $\rho_{1}$ as an example. To bound
$\rho_{1}$, we estimate $S_{1}$ via integrating (2.25)

$\frac{\partial\rho_{1}}{\partial s_{1}}=O(J_{1}+VS_{1}).$

The first step then is to estimate $J_{1}$. We integrate (2.27)

$\frac{\partial v_{1}}{\partial s_{1}}=O(VJ_{1}+V^{2}S_{1}).$

Here, it is important to analyze the behaviours of the nonlinear terms on the
right. The next step is to investigate $V_{i}$. Since a priori we do not have
the estimates for the solution’s life span $T$, we employ the bi-
characteristic coordinates to control $V_{i}$. Note that to control $\rho_{1}$
in $\mathcal{R}_{1}$, we carefully estimate $J_{1}$ and all $V_{i}$
simultaneously. Compared with controlling $\rho_{1}$ in $\mathcal{R}_{1}$, the
situation in the overlapped strips is more intricate. We refer the readers to
Section 4.1 for more details.

$\bullet$ Meaning of ill-posedness: For ideal compressible MHD (1.1), we
further prove the $H^{2}$ ill-posedness in 3D and we also demonstrate that the
ill-posedness is driven by shock formation. In particular, the $H^{2}$-norm of
the solutions in 3D blows up at the shock formation time $T^{*}_{\eta}$ with
$T^{*}_{\eta}\to 0$ as $\eta\to 0$.

Based on our decomposition of waves, we trace the evolution of inverse
densities $\rho_{i}$ ($i=1,\cdots,7$). In $\mathcal{R}_{1}$, we prove that
$\rho_{1}(z_{0},t)\to 0$ as $t\to T_{\eta}^{*}$ which renders $T_{\eta}^{*}$
to be the first shock formation time. And $\rho_{1}$ stays positive outside
$\mathcal{R}_{1}$. Then, with a uniform bound of $\partial_{z_{1}}\rho_{1}$
obtained in Subsection 5.2, in a constructed spatial region
$\Omega_{T^{*}_{\eta}}$, we prove that

$\begin{split}\|\Phi_{\eta}(\cdot,T^{*}_{\eta})\|_{H^{1}(\Omega_{T^{*}_{\eta}})}^{2}&\geq
C_{\eta}{\int_{z_{0}}^{z_{0}^{*}}\frac{1}{\rho_{1}(z,T_{\eta}^{*})}dz}\overset{(*)}{=}C_{\eta}{\int_{z_{0}}^{z_{0}^{*}}\frac{1}{\rho_{1}(z,T_{\eta}^{*})-\rho_{1}(z_{0},T_{\eta}^{*})}dz}\\\
&\geq
C_{\eta}\int_{z_{0}}^{z_{0}^{*}}\frac{1}{(z-z_{0})\sup|\partial_{z}\rho_{1}|}dz\\\
&\geq C_{\eta}\int_{z_{0}}^{z_{0}^{*}}\frac{1}{z-z_{0}}dz=+\infty.\end{split}$

As we mentioned in Remark 1.5, one can see from equality $(*)$ that the
vanishing of $\rho_{1}$ crucially drives the above blow-up. Moreover, for our
choice of initial data, we have $T_{\eta}^{*}\to 0$ as $\eta\to 0$. These
conclusions reveal a hidden mechanism: the instantaneous shock formation
underlies the $H^{2}$ ill-posedness.

###  1.3. Main steps in the proof

We outline our proof of Theorem 1.2-1.1 in this subsection.

Step 1: Decomposition of waves (Section 2). Our first step is to algebraically
decompose the non-strictly hyperbolic $7\times 7$ system (1.16). In this
paper, we will use both characteristic coordinates $(z_{i},s_{i})$ and bi-
characteristic coordinates $(y_{i},y_{j})$. Let $\mathcal{C}_{i}(z_{i})$ be
the $i^{\text{th}}$ characteristic with propagation speed $\lambda_{i}$
starting at $z_{i}$. Denote the corresponding $i^{\text{th}}$ characteristic
strip to be $\mathcal{R}_{i}:=\cup_{z_{i}\in I_{0}}\mathcal{C}_{i}(z_{i})$,
where $I_{0}$ is the support of initial data. Define the inverse density of
the $i^{\text{th}}$ characteristics

$\rho_{i}:=\partial_{z_{i}}X_{i}.$

For fixed $i$, let

$w_{i}:=l_{i}\partial_{x}\Phi,\quad\text{and}\quad v_{i}:=\rho_{i}w_{i}.$

Then, with the left and right eigenvectors satisfying
$l_{i}r_{j}=\delta_{ij}$, one can decompose the original system in the
following way

$\partial_{x}\Phi=\sum_{k}w_{k}r_{k}.$

One can verify that these quantities satisfy

$\displaystyle\partial_{s_{i}}\rho_{i}=$ $\displaystyle
c_{ii}^{i}v_{i}+\Big{(}\sum_{m\neq i}c_{im}^{i}w_{m}\Big{)}\rho_{i},$ (1.24)
$\displaystyle\partial_{s_{i}}w_{i}=$ $\displaystyle-
c_{ii}^{i}(w_{i})^{2}+\Big{(}\sum_{m\neq
i}(-c_{im}^{i}+\gamma_{im}^{i})w_{m}\Big{)}w_{i}+\sum_{m\neq i,k\neq i\atop
m\neq k}\gamma_{km}^{i}w_{k}w_{m},$ (1.25)
$\displaystyle\partial_{s_{i}}v_{i}=$ $\displaystyle\Big{(}\sum_{m\neq
i}\gamma_{im}^{i}w_{m}\Big{)}v_{i}+\sum_{m\neq i,k\neq i\atop m\neq
k}\gamma_{km}^{i}w_{k}w_{m}\rho_{i}.$ (1.26)

Step 2: Analysis of structures (Section 3). We explore the subtle structures
of the above decomposed system (1.24)-(1.26) by carrying out a thorough
calculation of all the coefficients
$c^{i}_{ii},c^{i}_{im},\gamma^{i}_{im},\gamma^{i}_{km}$. Firstly, recall that
$c_{ii}^{i}$ being non-vanishing corresponds to the genuine non-linearity. We
show that $c_{11}^{1}<0$ in Lemma 3.1. Hence the genuinely nonlinear condition
is verified and $w_{1}$ satisfies a Riccati-type equation. In fact, the blow-
up of $w_{1}$ builds on the genuine non-linearity. Then, we provide an
explicit detailed computation of all coefficients. Based on our calculation,
the decomposed system admits a schematic structure (1.5)-(1.7). We present the
key ideas in Section 3. The Appendix A, a 15-page concrete calculation of the
system’s structure, supplies all ingredients to complete the proof of Theorem
1.1.

Step 3: A priori estimates (Subsection 4.1). For the decomposed system (1.4),
characteristic waves in different $\mathcal{R}_{i}$ interact with each other.
In particular, two pairs of characteristic strips ($\mathcal{R}_{2}$ and
$\mathcal{R}_{3}$, $\mathcal{R}_{5}$ and $\mathcal{R}_{6}$) may overlap for a
long time which yields strong nonlinear interactions. Our strategy is to study
the dynamics of waves in the following five strips:

$\\{\mathcal{R}_{1},\mathcal{R}_{2}\cup\mathcal{R}_{3},\mathcal{R}_{4},\mathcal{R}_{5}\cup\mathcal{R}_{6},\mathcal{R}_{7}\\}.$

Then we will prove that a shock happens (and only happens) in
$\mathcal{R}_{1}$. Initially, we prepare compactly supported smooth data
$\\{w_{i}^{(\eta)}(z_{i},0)\\}$ satisfying (4.5). Observe that after a finite
time $t_{0}^{(\eta)}$ (see (4.3)) these five strips are separated. Our a
priori estimates are thus divided into two parts: before and after
$t_{0}^{(\eta)}$.

  * •

Estimates for $\mathbf{t\in[0,t_{0}^{(\eta)}]}$: Before the separating time
$t_{0}^{(\eta)}$, all the characteristic strips $\\{\mathcal{R}_{i}\\}$
overlap with each other. We define

$\displaystyle
W(t):=\max_{i\in\\{1,2,3,4,5,6,7\\}}\sup_{(x^{\prime},t^{\prime})\atop 0\leq
t^{\prime}\leq t}|w_{i}(x^{\prime},t^{\prime})|.$

In Section 4.1, via a Grönwall-type argument, we firstly obtain

$W(t)=O(W_{0}^{(\eta)})$

with $W_{0}^{(\eta)}:=\max_{i}\sup_{z_{i}}|w_{i}(z_{i},0)|$. Next, we
introduce the upper bounds $\rho_{i},v_{i}$ inside the strips

$\displaystyle S_{i}(t):=$
$\displaystyle\sup_{(z^{\prime}_{i},s^{\prime}_{i})\atop
z^{\prime}_{i}\in[-2\eta,2\eta],\ 0\leq s^{\prime}_{i}\leq
t}\rho_{i}(z^{\prime}_{i},s^{\prime}_{i}),$ $\displaystyle S(t):=$
$\displaystyle\max_{i\in\\{1,2,3,4,5,6,7\\}}S_{i}(t),$ $\displaystyle
J_{i}(t):=$ $\displaystyle\sup_{(z^{\prime}_{i},s^{\prime}_{i})\atop
z^{\prime}_{i}\in[-2\eta,2\eta]\ 0\leq s^{\prime}_{i}\leq
t}|v_{i}(z^{\prime}_{i},s^{\prime}_{i})|,$ $\displaystyle J(t):=$
$\displaystyle\max_{i\in\\{1,2,3,4,5,6,7\\}}J_{i}(t).$

With $t_{0}^{(\eta)}$ explicitly given in (4.3), it is straight forward to
estimate these quantities before $t_{0}^{(\eta)}$. Integrating the transport
equations under the characteristic coordinates, we prove that when
$t<t_{0}^{(\eta)}$

$\displaystyle S(t)=O(1),\quad\quad J(t)=$ $\displaystyle O(W_{0}^{(\eta)}).$

  * •

Estimates for $\mathbf{t\in[t_{0}^{(\eta)},T]}$: For $t$ larger than
$t_{0}^{(\eta)}$, $\mathcal{R}_{2}$ and $\mathcal{R}_{3}$ (or
$\mathcal{R}_{5}$ and $\mathcal{R}_{6}$) is not completely separated due to
the non-strict hyperbolicity. Nonetheless, the aforementioned five
characteristic strips
$\mathcal{R}_{1},\mathcal{R}_{2}\cup\mathcal{R}_{3},\mathcal{R}_{4},\mathcal{R}_{5}\cup\mathcal{R}_{6},\mathcal{R}_{7}$
are well separated. Adapted to this peculiarity, to replace $W$ in this part
we introduce the following quantity

$\check{W}(t):=\max\Big{\\{}\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{w_{2},w_{3}\\},\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq t}\\{w_{5},w_{6}\\}\Big{\\}}.$

Furthermore, to give a thorough description of the dynamics after the
separating time $t_{0}^{(\eta)}$, we need to estimate $w_{i}$ outside of the
strips and we define

$\displaystyle
V_{1}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{1},\atop 0\leq
t^{\prime}\leq t}|w_{1}(x^{\prime},t^{\prime})|,\quad
V_{\bar{2}}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{|w_{2}(x^{\prime},t^{\prime})|,|w_{3}(x^{\prime},t^{\prime})|\\},$
$\displaystyle
V_{4}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{4},\atop 0\leq
t^{\prime}\leq t}|w_{4}(x^{\prime},t^{\prime})|,\quad
V_{\bar{5}}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq
t}\\{|w_{5}(x^{\prime},t^{\prime})|,|w_{6}(x^{\prime},t^{\prime})|\\},$
$\displaystyle
V_{7}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{7},\atop 0\leq
t^{\prime}\leq t}|w_{7}(x^{\prime},t^{\prime})|,\quad
V(t):=\max_{i=1,\bar{2},4,\bar{5},7}V_{i}(t).$

Estimates of $V$ before $t_{0}^{(\eta)}$ can be similarly obtained as we just
discussed

$V(t)=O(\eta[W_{0}^{(\eta)}]^{2}).$

After $t_{0}^{(\eta)}$, as we emphasized in Subsection 1.2, we apply the bi-
characteristic coordinates to bound certain quantities. Specifically, we use
this technique to estimate $V_{i}$. In summary, for $t\in[t_{0}^{(\eta)},T]$,
we obtain the following a priori estimates in the characteristic strips:

$\displaystyle S=$ $\displaystyle O(1+tVS+tJ+tS\check{W}),$ $\displaystyle J=$
$\displaystyle O(W_{0}^{(\eta)}+tVJ+tV^{2}S+t\check{W}J+tVS\check{W}),$
$\displaystyle V=$ $\displaystyle O(\eta[W_{0}^{(\eta)}]^{2}+tV^{2}+\eta
VJ+tV\check{W}+t(\check{W})^{2}),$ $\displaystyle\check{W}=$ $\displaystyle
O(\eta[W_{0}^{(\eta)}]^{2}+t\check{W}^{2}+tV\check{W}+tV^{2}).$

Step 4: $L^{\infty}$-estimates and bootstrap argument (Subsection 4.2). We
setup bootstrap assumptions in (4.59)(4.60)(4.61). Based on the above
estimates in Subsection 4.1, via a bootstrap argument we obtain the desired
$L^{\infty}$-estimates of the solution:

$J=O(W_{0}^{(\eta)}),\,S=O(1),\,V=O\Big{(}\eta[W_{0}^{(\eta)}]^{2}\Big{)},\,\check{W}=O(\eta[W_{0}^{(\eta)}]^{2}).$

We depict the detailed behaviour of the solution up to the shock formation
time. See Subsection 4.2 for the details.

Step 5: Shock formation (Subsection 4.3). We shall prove that a shock forms in
$\mathcal{R}_{1}$ (Theorem 1.2). With the estimates obtained in Section 4.2,
we derive the following sharp control of $\rho_{1}$:

$\rho_{1}\sim 1-tW_{0}^{(\eta)}.$

This implies that $\rho_{1}$ would become $0$ (a shock forms) at a finite time
$T_{\eta}^{*}$ satisfying

$T_{\eta}^{*}\sim\frac{1}{W_{0}^{(\eta)}}.$

Moreover, we deduce that $v_{1}$ has a positive lower bound
$(1-\varepsilon)W_{0}^{(\eta)}$. Since $v_{1}=\rho_{1}w_{1}$, it follows that
$w_{1}$ will blow up at $T_{\eta}^{*}$.

Step 6: $H^{2}$ ill-posedness (Section 5). Furthermore, we prove the $H^{2}$
ill-posedness (Theorem 1.3). We first choose a family of appropriate
$H^{2}$-initial data (5.1)(5.4) for $w_{i}^{(\eta)}$ in Subsection 5.1. For
each $\eta$, in Subsection 5.3 we prove that the $H^{1}$-norm of the solution
to (1.1) tends to infinity as $t\to T_{\eta}^{*}$. Here, we apply an upper
bound estimate of $\partial_{z_{1}}\rho_{1}$ obtained in Subsection 5.2.
Moreover, for our solution, as $\eta\to 0$, we have $T_{\eta}^{*}\to 0$. The
underlying reason for ill-posedness, embedded in our proof, is indeed the
instantaneous shock formation. This is the desired ill-posedness result for 3D
ideal compressible MHD in $H^{2}(\mathbb{R}^{3})$.

Finally, in Section 6 and Section 7, we prove Theorem 1.2-1.3 for the case
when $H_{1}=0$ (we treat the case with non-trivial $H_{1}$ in all other
sections) and for the 3D compressible Euler equations (i.e., $H=0$),
respectively.

###  1.4. Other related works

In this subsection, we refer to related works in MHD. For 3D ideal
compressible MHD, Ohno-Shirota [38] proved that a characteristic fixed-
boundary linearized problem is ill-posed in the standard Sobolev spaces
$H^{l}$ with $l\geq 2$. We also refer to [9] by Chen-Young-Zhang for a
conditional555In [9], the authors considered the case of $H_{1}=0$ and imposed
the condition that the fluid density $\varrho$ has global lower and upper
bounds. shock formation result for 1D compressible MHD. Another active topic
for this system is the free-boundary problem. We refer to Chen-Wang [8],
Lindblad-Zhang [32], Trakhinin-Wang [43, 44] and Zhang [53] for related
results. The readers can also read Hu-Wang [22], Jiang-Ju-Li-Xin [24] and Li-
Li [27] for other mathematical topics about compressible MHD.

For incompressible MHD, the global well-posedness of Cauchy problem has been
studied by Cai-Lei [7], Deng-Zhang [15], He-Xu-Yu [20], Hu-Lin [21], Lin-Zhang
[28], Wei-Zhang [48, 49, 50], Xu-Zhang [51]. Jeong-Oh proved ill-posedness
results for the Hall- and electron-MHD in [23]. For free-boundary problems, we
refer to Gu-Luo-Zhang [18], Hao-Luo [19], Luo-Zhang [36], Wang-Xin [47] and
references therein.

###  1.5. Acknowledgements

The authors would like to thank Prof. Zhouping Xin for an enlightening
discussion.

##  2\. Decomposition of waves

In this section, we rewrite the MHD system with the wave decomposition method.
We will compute the eigenvalues and eigenvectors of the coefficient matrix for
the decomposed system. We then define the corresponding characteristics and
introduce the characteristic (and bi-characteristic) coordinates. With them we
will trace the dynamics of characteristic waves of different travelling
speeds.

For the 3D ideal compressible MHD (1.1), we collect the unknowns as a $8\times
1$ vector-valued function $\mathcal{U}$ with variables $x_{1},x_{2},x_{3}$ and
$t$. We let

$\mathcal{U}(x_{1},x_{2},x_{3},t)=(u_{1},u_{2},u_{3},\varrho,H_{1},H_{2},H_{3},S)^{T}(x_{1},x_{2},x_{3},t).$

To prove ill-posedness for (1.1), as in [3, 16, 30, 31], we work under planar
symmetry. And the solution satisfies
$\mathcal{U}(x_{1},x_{2},x_{3},t)=U(x_{1},t).$ For notational simplicity, we
denote $x_{1}$ by $x$ hereafter.

By the fifth equation of (1.1), i.e., Gauss’s Law, we obtain

$\nabla\cdot H=\frac{\partial H_{1}}{\partial x_{1}}+\frac{\partial
H_{2}}{\partial x_{2}}+\frac{\partial H_{3}}{\partial x_{3}}=\frac{\partial
H_{1}}{\partial x}=0,\quad\text{hence}\,\,H_{1}(x,t)=H_{1}(t).$

And from the third equation of (1.1), we further have $\frac{dH_{1}}{dt}=0.$
Thus, $H_{1}(x,t)$ equals a constant during the evolution. We require that the
constant $H_{1}$ is small in the following sense

$H_{1}^{2}\ll\min\\{\mu_{0}^{-1}A\gamma,1\\}.$ (2.1)

We then collect the remaining unknowns
$(u_{1},u_{2},u_{3},\varrho,H_{2},H_{3},S)^{T}$ and they satisfy the following
equations

$\left\\{\begin{split}&\varrho\partial_{t}u_{1}+\varrho
u_{1}\partial_{x}u_{1}+c^{2}\partial_{x}\varrho+\partial_{S}p\partial_{x}S+\mu_{0}H_{2}\partial_{x}H_{2}+\mu_{0}H_{3}\partial_{x}H_{3}=0,\\\
&\varrho\partial_{t}u_{2}+\varrho
u_{1}\partial_{x}u_{2}-\mu_{0}H_{1}\partial_{x}H_{2}=0,\\\
&\varrho\partial_{t}u_{3}+\varrho
u_{1}\partial_{x}u_{3}-\mu_{0}H_{1}\partial_{x}H_{3}=0,\\\
&\partial_{t}\varrho+u_{1}\partial_{x}\varrho+\varrho\partial_{x}u_{1}=0,\\\
&\partial_{t}H_{2}+u_{1}\partial_{x}H_{2}-H_{1}\partial_{x}u_{2}+H_{2}\partial_{x}u_{1}=0,\\\
&\partial_{t}H_{3}+u_{1}\partial_{x}H_{3}-H_{1}\partial_{x}u_{3}+H_{3}\partial_{x}u_{1}=0,\\\
&\partial_{t}S+u_{1}\partial_{x}S=0.\end{split}\right.$ (2.2)

Here $c=\sqrt{\partial_{\varrho}p}$ is the sound speed and the pressure $p$
satisfies the equation of state (1.2).

We start to rewrite (2.2) as a $7\times 7$ hyperbolic system. Let

$\Phi:=(u_{1},u_{2},u_{3},\varrho-1,H_{2},H_{3},S)^{T}(x,t).$

From (2.2), $\Phi$ obeys

$\partial_{t}\Phi+A(\Phi)\partial_{x}\Phi=0$ (2.3)

with the coefficient matrix

$A(\Phi)=\begin{pmatrix}u_{1}&0&0&c^{2}/\varrho&\mu_{0}H_{2}/\varrho&\mu_{0}H_{3}/\varrho&\partial_{S}p/\varrho\\\
0&u_{1}&0&0&-\mu_{0}H_{1}/\varrho&0&0\\\
0&0&u_{1}&0&0&-\mu_{0}H_{1}/\varrho&0\\\ \varrho&0&0&u_{1}&0&0&0\\\
H_{2}&-H_{1}&0&0&u_{1}&0&0\\\ H_{3}&0&-H_{1}&0&0&u_{1}&0\\\
0&0&0&0&0&0&u_{1}\end{pmatrix}.$ (2.4)

The eigenvalues of (2.4) are

$\begin{split}&\lambda_{1}=u_{1}+C_{f},\quad\lambda_{2}=u_{1}+C_{a},\quad\lambda_{3}=u_{1}+C_{s},\quad\lambda_{4}=u_{1},\\\
&\lambda_{5}=u_{1}-C_{s},\quad\lambda_{6}=u_{1}-C_{a},\quad\lambda_{7}=u_{1}-C_{f},\end{split}$
(2.5)

where

$\begin{split}&C_{f}=\Big{\\{}\frac{\mu_{0}}{2\varrho}(H_{1}^{2}+H_{2}^{2}+H_{3}^{2})+\frac{c^{2}}{2}+\frac{1}{2}\sqrt{[\frac{\mu_{0}}{\varrho}(H_{1}^{2}+H_{2}^{2}+H_{3}^{2})+c^{2}]^{2}-\frac{4\mu_{0}}{\varrho}H_{1}^{2}c^{2}}\Big{\\}}^{1/2},\\\
&C_{s}=\Big{\\{}\frac{\mu_{0}}{2\varrho}(H_{1}^{2}+H_{2}^{2}+H_{3}^{2})+\frac{c^{2}}{2}-\frac{1}{2}\sqrt{[\frac{\mu_{0}}{\varrho}(H_{1}^{2}+H_{2}^{2}+H_{3}^{2})+c^{2}]^{2}-\frac{4\mu_{0}}{\varrho}H_{1}^{2}c^{2}}\Big{\\}}^{1/2},\\\
&C_{a}=\sqrt{\frac{\mu_{0}}{\varrho}}H_{1}\end{split}$ (2.6)

are called the fast, the slow and the Alfvén wave speeds, respectively. Here,
the characteristic speed $\lambda_{4}$ corresponds to the propagation speed of
the entropy disturbance that just follows the stream-line of the flow. The
pair $\lambda_{2}$ and $\lambda_{6}$ correspond respectively to the speeds of
the Alfvén disturbances moving forward and backward along the flow. And
$\lambda_{3},\lambda_{5}$ are speeds of forward and backward slow-magneto-
acoustic disturbances, whereas $\lambda_{1},\lambda_{7}$ constitute speeds of
forward and backward fast-magneto-acoustic disturbances. Noting that $H_{1}$
satisfies (2.1), when $|\Phi|$ is small enough, one can verify that

$C_{s}\leq C_{a}<C_{f}.$ (2.7)

The bulk of the paper is devoted to studying the case of non-trivial $H_{1}$.
The case of $H_{1}=0$ will be studied in Section 6. With non-trivial $H_{1}$,
by (2.7), the corresponding eigenvalues obtained in (2.5) satisfy:

$\begin{split}\lambda_{7}<\lambda_{6}\leq\lambda_{5}<\lambda_{4}<\lambda_{3}\leq\lambda_{2}<\lambda_{1}.\end{split}$
(2.8)

In (2.8), the equalities hold if and only if $H_{2}^{2}+H_{3}^{2}=0$. Since
$\lambda_{2},\lambda_{3}$ (or $\lambda_{5},\lambda_{6}$) could take the same
value, the system (2.3) is not strictly hyperbolic.

A very crucial step is to take proper right eigenvectors of $A(\Phi)$. By
carefully designing, we take the right eigenvectors of $A(\Phi)$ as follows:

$\begin{split}&r_{1}=\left(\begin{array}[]{cc}\frac{C_{a}^{2}}{C_{f}}-C_{f}\\\
\frac{C_{a}^{2}H_{2}}{C_{f}H_{1}}\\\ \frac{C_{a}^{2}H_{3}}{C_{f}H_{1}}\\\
\varrho(\frac{C_{a}^{2}}{C_{f}^{2}}-1)\\\ -H_{2}\\\ -H_{3}\\\
0\end{array}\right),\ r_{2}=\left(\begin{array}[]{cc}0\\\
\frac{C_{a}H_{3}}{H_{1}}\\\ -\frac{C_{a}H_{2}}{H_{1}}\\\ 0\\\ -H_{3}\\\
H_{2}\\\ 0\end{array}\right),\
r_{3}=\left(\begin{array}[]{cc}\frac{C_{a}^{2}}{C_{s}}-C_{s}\\\
\frac{C_{a}^{2}H_{2}}{C_{s}H_{1}}\\\ \frac{C_{a}^{2}H_{3}}{C_{s}H_{1}}\\\
\varrho(\frac{C_{a}^{2}}{C_{s}^{2}}-1)\\\ -H_{2}\\\ -H_{3}\\\
0\end{array}\right),\ r_{4}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\
-\frac{\varrho}{\gamma}\\\ 0\\\ 0\\\ 1\end{array}\right),\\\
&r_{5}=\left(\begin{array}[]{cc}\frac{C_{a}^{2}}{C_{s}}-C_{s}\\\
\frac{C_{a}^{2}H_{2}}{C_{s}H_{1}}\\\ \frac{C_{a}^{2}H_{3}}{C_{s}H_{1}}\\\
-\varrho(\frac{C_{a}^{2}}{C_{s}^{2}}-1)\\\ H_{2}\\\ H_{3}\\\
0\end{array}\right),\ r_{6}=\left(\begin{array}[]{cc}0\\\
\frac{C_{a}H_{3}}{H_{1}}\\\ -\frac{C_{a}H_{2}}{H_{1}}\\\ 0\\\ H_{3}\\\
-H_{2}\\\ 0\end{array}\right),\
r_{7}=\left(\begin{array}[]{cc}\frac{C_{a}^{2}}{C_{f}}-C_{f}\\\
\frac{C_{a}^{2}H_{2}}{C_{f}H_{1}}\\\ \frac{C_{a}^{2}H_{3}}{C_{f}H_{1}}\\\
-\varrho(\frac{C_{a}^{2}}{C_{f}^{2}}-1)\\\ H_{2}\\\ H_{3}\\\
0\end{array}\right).\end{split}$ (2.9)

As emphasized in Subsection 1.2, this selection of right eigenvectors is a key
point to our proof. The corresponding left eigenvectors $l_{i}$ are set dual
to the right ones, i.e.,

$l_{i}r_{j}=\delta_{ij}\quad\text{for}\ i,j=1,\cdots,7.$ (2.10)

Here $\delta_{ij}$ is the Kronecker symbol. Specifically, we list these
$l_{i}$:

$\displaystyle l_{1}=$
$\displaystyle\bigg{(}-\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})},\frac{H_{1}H_{2}C_{f}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{H_{1}H_{3}C_{f}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{C_{f}^{2}C_{s}^{2}}{2\varrho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$
$\displaystyle-\frac{H_{2}C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{H_{3}C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{)},$ $\displaystyle l_{2}=$
$\displaystyle\bigg{(}0,\frac{H_{1}H_{3}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},-\frac{H_{1}H_{2}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0,-\frac{H_{3}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},\frac{H_{2}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0\bigg{)},$
$\displaystyle l_{3}=$
$\displaystyle\bigg{(}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})},-\frac{H_{1}H_{2}C_{s}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{H_{1}H_{3}C_{s}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{C_{f}^{2}C_{s}^{2}}{2\varrho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$
$\displaystyle\frac{H_{2}C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{H_{3}C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{)},$ $\displaystyle l_{4}=$
$\displaystyle\big{(}0,0,0,0,0,0,1\big{)},$ (2.11) $\displaystyle l_{5}=$
$\displaystyle\bigg{(}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})},-\frac{H_{1}H_{2}C_{s}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{H_{1}H_{3}C_{s}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{C_{f}^{2}C_{s}^{2}}{2\varrho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$
$\displaystyle-\frac{H_{2}C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{H_{3}C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},-\frac{C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{)},$ $\displaystyle l_{6}=$
$\displaystyle\bigg{(}0,\frac{H_{1}H_{3}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},-\frac{H_{1}H_{2}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0,\frac{H_{3}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},-\frac{H_{2}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0\bigg{)},$
$\displaystyle l_{7}=$
$\displaystyle\bigg{(}-\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})},\frac{H_{1}H_{2}C_{f}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{H_{1}H_{3}C_{f}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{C_{f}^{2}C_{s}^{2}}{2\varrho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$
$\displaystyle\frac{H_{2}C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{H_{3}C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\frac{C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{)}$

with $C_{f},C_{s},C_{a}$ defined in (2.6).

######  Remark 2.1.

Note that there is the singular factor $\frac{1}{H_{2}^{2}+H_{3}^{2}}$ present
in the $2^{\text{nd}}$,$3^{\text{rd}}$,$5^{\text{th}}$,$6^{\text{th}}$
components of all $l_{j}$ with $j\neq 4$. Nevertheless, with our carefully
designed right eigenvectors (2.9), all these $\frac{1}{H_{2}^{2}+H_{3}^{2}}$
factors are miraculously cancelled when calculating the coefficients
$\gamma^{i}_{km}$ in Theorem 1.1. The details of these cancellations will be
demonstrated in Section 3 and Appendix A.

We now introduce the characteristics as well as the associated characteristic
and bi-characteristic coordinates. We define the $i^{\text{th}}$
characteristic $\mathcal{C}_{i}(z_{i})$, which originates from $z_{i}$, to be
the image $(X_{i}(z_{i},t),t)$ of solutions to the following ODE:

$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}X_{i}(z_{i},t)=\lambda_{i}\big{(}\Phi(X_{i}(z_{i},t),t)\big{)},\quad
t\in[0,T],\\\ X_{i}(z_{i},0)=z_{i}.\end{array}\right.$ (2.14)

For any $(x,t)\in\mathbb{R}\times[0,T]$, there is a unique
$(z_{i},s_{i})\in\mathbb{R}\times[0,T]$ such that
$(x,t)=\big{(}X_{i}(z_{i},s_{i}),s_{i}\big{)}$ with $X_{i}$ satisfying (2.14).
We define this $(z_{i},s_{i})$ to be the characteristic coordinates of
$(x,t)$. Moreover, we define bi-characteristic coordinates: consider two
transversal characteristics $\mathcal{C}_{i}(z_{i})$ and
$\mathcal{C}_{j}(z_{j})$ with $i\neq j$. Note that the characteristic
$\mathcal{C}_{i}(z_{i})$ can be parameterized by $z_{j}$. Thus given
$(z_{i},z_{j})$ one can uniquely locate a point where the two characteristics
$\mathcal{C}_{i}(z_{i})$ and $\mathcal{C}_{j}(z_{j})$ intersect. To
distinguish with the characteristic coordinates above, we denote
$(z_{i},z_{j})$ by $(y_{i},y_{j})$ in this new coordinate system. Then, given
$(x,t)\in\mathbb{R}\times[0,T]$, the bi-characteristic coordinates of $(x,t)$
is defined to be the unique $(y_{i},y_{j})\in\mathbb{R}^{2}$ such that with
$t=t^{\prime}(y_{i},y_{j})$, we have

$(x,t)=\big{(}X_{i}(y_{i},t^{\prime}(y_{i},y_{j})),t^{\prime}(y_{i},y_{j})\big{)}=\big{(}X_{j}(y_{j},t^{\prime}(y_{i},y_{j})),t^{\prime}(y_{i},y_{j})\big{)}.$
(2.15)

To describe the compression among the $i^{\text{th}}$ characteristics, we
introduce a geometric quantity $\rho_{i}$ (the inverse density of the
$i^{\text{th}}$ characteristics)

$\rho_{i}:=\partial_{z_{i}}X_{i}.$ (2.16)

Following from (2.14), for the initial data we have

$\rho_{i}(z_{i},0)=1,$ (2.17)

The coordinate transformations comply with the following rules:

$\partial_{z_{i}}=\rho_{i}\partial_{x},\quad\partial_{s_{i}}=\lambda_{i}\partial_{x}+\partial_{t},\quad\partial_{y_{i}}t^{\prime}=\frac{\rho_{i}}{\lambda_{j}-\lambda_{i}},\quad\partial_{y_{j}}t^{\prime}=\frac{\rho_{j}}{\lambda_{i}-\lambda_{j}},$
(2.18)
$\partial_{y_{i}}=\frac{\rho_{i}}{\lambda_{j}-\lambda_{i}}\partial_{s_{j}}=\partial_{z_{i}}+\frac{\rho_{i}}{\lambda_{j}-\lambda_{i}}\partial_{s_{i}},$
(2.19)

and

$dx=\rho_{i}dz_{i}+\lambda_{i}ds_{i},\quad dt=ds_{i},\quad dz_{i}=dy_{i},\quad
dz_{j}=dy_{j},$ (2.20)
$dx=\frac{\rho_{i}\lambda_{j}}{\lambda_{j}-\lambda_{i}}dy_{i}+\frac{\rho_{j}\lambda_{i}}{\lambda_{i}-\lambda_{j}}dy_{j},\quad
dt=\frac{\rho_{i}}{\lambda_{j}-\lambda_{i}}dy_{i}+\frac{\rho_{j}}{\lambda_{i}-\lambda_{j}}dy_{j}.$
(2.21)

Now we are ready to decompose system (2.3) via the decomposition of waves. See
[3, 12, 25] for more applications of this method. For fixed $i$, setting

$w_{i}:=l_{i}\partial_{x}\Phi,$ (2.22)

and

$v_{i}:=\rho_{i}w_{i},$ (2.23)

then by (2.10), we have

$\partial_{x}\Phi=\sum_{k=1}^{7}w_{k}r_{k}.$ (2.24)

The above equation serves as a decomposition of $\partial_{x}\Phi$. By
calculations similar to John [25], one can verify that
$\\{\rho_{i},w_{i},v_{i}\\}_{i=1,\cdots,7}$ satisfy the following transport
equations

$\displaystyle\partial_{s_{i}}\rho_{i}=$ $\displaystyle
c_{ii}^{i}v_{i}+\Big{(}\sum_{m\neq i}c_{im}^{i}w_{m}\Big{)}\rho_{i},$ (2.25)
$\displaystyle\partial_{s_{i}}w_{i}=$ $\displaystyle-
c_{ii}^{i}w_{i}^{2}+\Big{(}\sum_{m\neq
i}(-c_{im}^{i}+\gamma_{im}^{i})w_{m}\Big{)}w_{i}+\sum_{m\neq i,k\neq i\atop
m\neq k}\gamma_{km}^{i}w_{k}w_{m},$ (2.26)
$\displaystyle\partial_{s_{i}}v_{i}=$ $\displaystyle\Big{(}\sum_{m\neq
i}\gamma_{im}^{i}w_{m}\Big{)}v_{i}+\sum_{m\neq i,k\neq i\atop m\neq
k}\gamma_{km}^{i}w_{k}w_{m}\rho_{i}$ (2.27)

where $\partial_{s_{i}}=\lambda_{i}\partial_{x}+\partial_{t}$ and

$\displaystyle c_{im}^{i}=\nabla_{\Phi}\lambda_{i}\cdot r_{m},$ (2.28)
$\displaystyle\gamma_{im}^{i}=-(\lambda_{i}-\lambda_{m})l^{i}\cdot(\nabla_{\Phi}r_{i}\cdot
r_{m}-\nabla_{\Phi}r_{m}\cdot r_{i}),\quad m\neq i,$ (2.29)
$\displaystyle\gamma_{km}^{i}=-(\lambda_{k}-\lambda_{m})l^{i}\cdot(\nabla_{\Phi}r_{k}\cdot
r_{m}),\qquad\qquad\qquad k\neq i,\ m\neq i.$ (2.30)

######  Remark 2.2.

We call the $i^{\text{th}}$ characteristic genuinely nonlinear if
$c_{ii}^{i}\neq 0$ for all $(x,t)$ and on the contrary linearly degenerate
when $c_{ii}^{i}\equiv 0$.

######  Remark 2.3.

As will be proved in Theorem 1.1, all the coefficients $c^{i}_{im}$,
$\gamma^{i}_{im}$, $\gamma^{i}_{km}$ of the decomposed system (2.25)-(2.27)
are uniformly bounded of order $1$. The proof will be given in Section 3 and
Appendix A. Moreover, important schematic structures of 3D compressible MHD
are also exhibited via system (2.25)-(2.27).

##  3\. Proof of Theorem 1.1

In this section, we explore the structures of the decomposed system
(2.25)-(2.27) and aim to prove Theorem 1.1. These structures play a crucial
role in proving the shock formation of (2.3).

We start from analyzing $w_{1}$. Our ultimate goal is to show that $w_{1}$
blows up in a finite time. Observe that in (2.26) the equation of $w_{1}$ is
of Ricatti-type. In particular, with the following lemma we show that the
$1^{\text{st}}$ characteristic is genuinely nonlinear.

######  Lemma 3.1 (Genuine nonlinearity of $\mathcal{C}_{1}$).

For $|H_{1}|\ll 1$ and $|\Phi|\leq 2\delta\ll 1$, there holds
$c_{11}^{1}(\Phi)<0$.

###### Proof.

Plugging the formula of $r_{1}$ (2.9), $\lambda_{1}$ (2.5), the fast and the
Alfvén wave speeds $C_{f}$, $C_{a}$ (2.6) into the definition of
$c^{1}_{11}(\Phi)$ (2.28), one can verify that $c^{1}_{11}(\Phi)$ is
continuous with respect to variable $\Phi$ and parameter $H_{1}$. Since
$|H_{1}|\ll 1$ and $|\Phi|\leq 2\delta\ll 1$, it then suffices to show that
$c_{11}^{1}(0)<0$ for the case of $H_{1}=0$. With $H_{1}=0$, it holds that

$\lambda_{1}=u_{1}+\big{\\{}\frac{\mu_{0}}{\varrho}(H_{2}^{2}+H_{3}^{2})+A\gamma\varrho^{\gamma-1}\exp
S\big{\\}}^{1/2}.$ (3.1)

We hence obtain

$\begin{split}c^{1}_{11(0)}(0)=&\nabla_{\Phi}\lambda_{1}(0)\cdot r_{1}(0)\\\
=&(1,0,0,\frac{\gamma-1}{2}\sqrt{A\gamma},0,0,\frac{\sqrt{A\gamma}}{2})\cdot(-\sqrt{A\gamma},0,0,-1,0,0,0)^{T}\\\
=&-\frac{\sqrt{A\gamma}(\gamma+1)}{2}<0.\end{split}$ (3.2)

Using continuity, we also conclude that $c_{11}^{1}(\Phi)<0$. ∎

Note that $c_{11}^{1}(\Phi)<0$ is important for the Riccati-type equation
(2.26) of $w_{1}$ to develop a potential singularity.

For $w_{2},w_{6}$, the following lemma indicates the vanishing of the Riccati-
type terms.

######  Lemma 3.2.

For $c_{i2}^{i}$ and $c_{i6}^{i}$ defined in (2.28), it holds that

$c_{i2}^{i}=0,\quad c_{i6}^{i}=0\quad\text{for}\quad i=1,\cdots,7.$ (3.3)

###### Proof.

Recall the definition (2.28)

$c_{i2}^{i}=\nabla_{\Phi}\lambda_{i}\cdot r_{2}.$

Invoking the right eigenvector $r_{2}$ in Section 2, we have

$c_{i2}^{i}=\sqrt{\frac{\mu_{0}}{\varrho}}\frac{H_{3}}{H_{2}}\partial_{u_{2}}\lambda_{i}-\sqrt{\frac{\mu_{0}}{\varrho}}\partial_{u_{3}}\lambda_{i}-\frac{H_{3}}{H_{2}}\partial_{H_{2}}\lambda_{i}+\partial_{H_{3}}\lambda_{i}.$
(3.4)

Since the eigenvalues $\lambda_{i}$ are independent of $u_{2}$ and $u_{3}$, it
holds that

$c_{i2}^{i}=-\frac{H_{3}}{H_{2}}\partial_{H_{2}}\lambda_{i}+\partial_{H_{3}}\lambda_{i}.$
(3.5)

Observe that $\lambda_{i}$ in (2.5) can be written as

$\lambda_{i}=f_{i}(u_{1},\rho,H_{2}^{2}+H_{3}^{2},S).$

Substituting it back in (3.4), we then prove that the coefficients
$c_{i2}^{i}$ satisfy

$c_{i2}^{i}=-\frac{H_{3}}{H_{2}}\partial_{H_{2}^{2}+H_{3}^{2}}f_{i}\cdot
2H_{2}+\partial_{H_{2}^{2}+H_{3}^{2}}f_{i}\cdot 2H_{3}=0.$ (3.6)

In the same fashion, we deduce $c_{i6}^{i}=0$. ∎

The above lemma immediately implies that $c^{2}_{22}=0$ and $c^{6}_{66}=0$.
Moreover, by a straightforward calculation, one can easily see that
$c^{4}_{44}=0$. Hence the $2^{\text{nd}},4^{\text{th}},6^{\text{th}}$
characteristics are linearly degenerate.

######  Remark 3.1.

For $c^{i}_{ik}$, $\gamma^{i}_{im}$, $\gamma^{i}_{km}$ with other indices, we
need to prove that they are $O(1)$. This is achieved via conducting a heavy
calculation of checking all the scenarios and exploring the delicate
cancellations among them. A 15-page detailed calculation is attached in
Appendix A.

The main proposition in this section is as follows.

######  Proposition 3.1 (Estimates of coefficients).

For $|\Phi|\leq 2\delta\ll 1$, the decomposed system (2.25)-(2.27) admits
uniformly bounded non-trivial coefficients $c^{i}_{ii}$, $c^{i}_{im}$,
$\gamma^{i}_{im}$, $\gamma^{i}_{km}$, i.e.,

$\max\limits_{i,k,m}\sup\limits_{|\Phi|\leq
2\delta}\big{\\{}|c^{i}_{ii}|,|c^{i}_{im}|,|\gamma^{i}_{im}|,|\gamma^{i}_{km}|\big{\\}}\leq
C,$ (3.7)

where $C$ is a uniform constant.

###### Proof.

Let $\Gamma(\Phi)$ be the maximum of all the coefficients $\\{|c_{im}^{i}|\\}$
and $\\{|\gamma_{km}^{i}|\\}$ in (2.28)-(2.30). With the eigenvalues and
eigenvectors in Section 2, we will show

$\Gamma(\Phi)=O(1)\qquad\text{for}\qquad|\Phi|\leq 2\delta\ll 1.$ (3.8)

This is clearly true for $c^{i}_{im}$, $\gamma^{4}_{4m}$ and $\gamma^{4}_{km}$
as $\nabla_{\Phi}\lambda_{i}$, $r_{i}$ and $l_{4}$ contain only $O(1)$ terms.
But for $\gamma^{i}_{im},\gamma^{i}_{km}$ with $i\neq 4$, they might be
singular since $\frac{1}{H_{2}^{2}+H_{3}^{2}}$ appears in the corresponding
left eigenvectors. Then these potentially singular coefficients could destroy
all the arguments we construct. We must calculate all these coefficients
concretely to eliminate this danger. After a thorough 15-page calculation, we
find that all the potentially singular denominators are cancelled in the
matrix multiplication. Whence, (3.8) holds for all the coefficients (See
Appendix A).

We begin with a concrete calculations for $\gamma_{64}^{2}$. With

$\begin{split}&r_{2}=\left(\begin{array}[]{cc}0\\\ \frac{C_{a}H_{3}}{H_{1}}\\\
-\frac{C_{a}H_{2}}{H_{1}}\\\ 0\\\ -H_{3}\\\ H_{2}\\\ 0\end{array}\right),\
r_{4}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\ -\frac{\varrho}{\gamma}\\\ 0\\\
0\\\ 1\end{array}\right),\ r_{6}=\left(\begin{array}[]{cc}0\\\
\frac{C_{a}H_{3}}{H_{1}}\\\ -\frac{C_{a}H_{2}}{H_{1}}\\\ 0\\\ H_{3}\\\
-H_{2}\\\ 0\end{array}\right)\,\,\text{and}\end{split}$
$l_{2}=\Big{(}0,\frac{H_{1}H_{3}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},-\frac{H_{2}H_{1}}{2C_{a}(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0,-\frac{H_{3}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},\frac{H_{2}}{2(\text{\dashuline{$H_{2}^{2}+H_{3}^{2}$}})},0\Big{)},\\\
$

we have

$\nabla_{\Phi}r_{6}\cdot r_{4}=\Big{(}0,\frac{C_{a}H_{3}}{2\gamma
H_{1}},-\frac{C_{a}H_{2}}{2\gamma H_{1}},0,0,0,0\Big{)}^{\top},$

and hence

$\displaystyle\gamma_{64}^{2}$ $\displaystyle=$
$\displaystyle-(\lambda_{6}-\lambda_{4})l^{2}\cdot(\nabla_{\Phi}r_{6}\cdot
r_{4})$ $\displaystyle=$
$\displaystyle-(\lambda_{6}-\lambda_{4})\Big{(}\frac{H_{3}^{2}}{4\gamma(H_{2}^{2}+H_{3}^{2})}+\frac{H_{2}^{2}}{4\gamma(H_{2}^{2}+H_{3}^{2})}\Big{)}$
$\displaystyle\overset{(*)}{=}$
$\displaystyle-\frac{\lambda_{6}-\lambda_{4}}{4\gamma}=O(1).$

Note the crucial cancellation in equality $(*)$.

We use $\gamma^{2}_{26}$ to demonstrate another type of cancellation. Proceed
as above and it follows

$\nabla_{\Phi}r_{2}\cdot r_{6}-\nabla_{\Phi}r_{6}\cdot
r_{2}=\Big{(}0,-\frac{2C_{a}H_{2}}{H_{1}},-\frac{2C_{a}H_{3}}{H_{1}},0,0,0,0\Big{)}^{\top}.$

Thus

$\displaystyle\gamma_{26}^{2}$ $\displaystyle=$
$\displaystyle-(\lambda_{2}-\lambda_{6})l^{2}\cdot(\nabla_{\Phi}r_{2}\cdot
r_{6}-\nabla_{\Phi}r_{6}\cdot r_{2})$ $\displaystyle=$
$\displaystyle(\lambda_{2}-\lambda_{6})\Big{(}\frac{H_{3}H_{2}}{H_{2}^{2}+H_{3}^{2}}-\frac{H_{2}H_{3}}{H_{2}^{2}+H_{3}^{2}}\Big{)}{\overset{(**)}{=}}0.$

Note the crucial cancellation in equality $(**)$. The computation of other
coefficients, i.e., $\gamma^{i}_{im},\gamma^{i}_{km}$ with $i\neq 4$, can be
conducted similarly but with more involved expressions. Yet miraculously for
any $\gamma^{i}_{km}$, the cancellation of singular factor
$\frac{1}{H_{2}^{2}+H_{3}^{2}}$ is either in the form of $(*)$ or in the form
of $(**)$. The detailed calculations can be found in a 15-page Appendix A.
These estimates critically reveal the subtle structures of our system. ∎

Now with all the uniformly bounded coefficients at our disposal, we obtain the
structures of the system as listed in (1.5)-(1.7). This concludes the proof of
Theorem 1.1.

##  4\. Proof of shock formation (Theorem 1.2)

We prove shock formation for the 3D compressible MHD in this section. Firstly,
we deduce proper a priori estimates (Subsection 4.1). These delicate estimates
act as a powerful tool to cope with the strong nonlinear interaction terms.
Next, employing the a priori estimates, we set up and close a bootstrap
argument (Subsection 4.2). Finally, the shock formation follows from the
$L^{\infty}$-estimates obtained via the bootstrap argument (Subsection 4.3).

###  4.1. A priori estimates

In this subsection, we derive the main a priori estimates. We firstly define
the characteristic strips and a separating time $t_{0}^{(\eta)}$. Beyond
$t_{0}^{(\eta)}$ the strips under consideration will be completely separated.
The a priori estimates will also be divided into two parts: before and after
$t_{0}^{(\eta)}$. These estimates provide a manifestation of how to control
the nonlinear interactions of waves with different or almost-the-same
characteristic speeds.

Let $\mathcal{C}_{i}(z_{i})$ be the $i^{\text{th}}$ characteristic issuing
from $z_{i}$ with propagation speed $\lambda_{i}$. Define the corresponding
$i^{\text{th}}$ characteristic strip

$\mathcal{R}_{i}:=\cup_{z_{i}\in I_{0}}\mathcal{C}_{i}(z_{i}),$ (4.1)

where $I_{0}=\\{x:|x|\leq 2\eta\\}$ is the compact support of the initial data
and $\eta$ is a chosen positive parameter. We will require $\eta$ to be small
to drive the ill-posedness. See Section 5.1-5.3. Yet for the purpose of shock
formation in this section, $\eta$ is not necessary to be small.

For the planar symmetric 3D ideal MHD system (2.3), two pairs of
characteristics are of almost the same travelling speed. This is because
$H_{2}^{2}+H_{3}^{2}$ is sufficiently small, and from (2.5)(2.6) it follows

$\lambda_{2}\thickapprox\lambda_{3},\ \lambda_{5}\thickapprox\lambda_{6}.$

This is illustrated by the long-time overlapping of the corresponding
characteristic strips $\mathcal{R}_{2},\mathcal{R}_{3}$ or
$\mathcal{R}_{5},\mathcal{R}_{6}$ in the picture below. Our strategy is to
take unions of these overlapped strips and consider the propagation of waves
within the following five characteristic strips:

$\\{\mathcal{R}_{1},\mathcal{R}_{\bar{2}},\mathcal{R}_{4},\mathcal{R}_{\bar{5}},\mathcal{R}_{7}\\},$
(4.2)

where $\mathcal{R}_{\bar{2}}:=\mathcal{R}_{2}\bigcup\mathcal{R}_{3}$ and
$\mathcal{R}_{\bar{5}}:=\mathcal{R}_{5}\bigcup\mathcal{R}_{6}$. These five
strips will be completely separated after a finite separating time
$t_{0}^{(\eta)}$.

$2\eta$$-2\eta$$t=0$$t=t_{0}^{(\eta)}$$\mathcal{R}_{1}$$\mathcal{R}_{2}\cup\mathcal{R}_{3}$$\mathcal{R}_{4}$$\mathcal{R}_{5}\cup\mathcal{R}_{6}$$\mathcal{R}_{7}$

Now via estimates we first determine the separating time $t_{0}^{(\eta)}$.
Take the supremum and infimum of the eigenvalues

$\displaystyle\bar{\lambda}_{i}:=\sup_{\Phi\in
B_{2\delta}^{7}(0)}\lambda_{i}(\Phi),\quad\underline{\lambda}_{i}:=\inf_{\Phi\in
B_{2\delta}^{7}(0)}\lambda_{i}(\Phi),\quad\text{for}\quad i=1,4,7,$
$\displaystyle\bar{\lambda}_{\bar{2}}:=\sup_{\Phi\in
B_{2\delta}^{7}(0)}\\{\lambda_{2}(\Phi),\lambda_{3}(\Phi)\\},\quad\underline{\lambda}_{\bar{2}}:=\inf_{\Phi\in
B_{2\delta}^{7}(0)}\\{\lambda_{2}(\Phi),\lambda_{3}(\Phi)\\},$
$\displaystyle\bar{\lambda}_{\bar{5}}:=\sup_{\Phi\in
B_{2\delta}^{7}(0)}\\{\lambda_{5}(\Phi),\lambda_{6}(\Phi)\\},\quad\underline{\lambda}_{\bar{5}}:=\inf_{\Phi\in
B_{2\delta}^{7}(0)}\\{\lambda_{5}(\Phi),\lambda_{6}(\Phi)\\},$

and define

$\sigma:=\min_{\alpha<\beta\atop\alpha,\beta\in\\{1,\bar{2},4,\bar{5},7\\}}(\underline{\lambda}_{\alpha}-\bar{\lambda}_{\beta}).$

According to (2.5)(2.6)(2.8), when $\delta$ is sufficiently small, $\sigma$
has a uniform positive lower bound. For
$\alpha\in\\{1,\bar{2},4,\bar{5},7\\}$, $z\in I_{0}$, it follows from (2.14)
that

$z+\underline{\lambda}_{\alpha}t\leq X_{\alpha}(z,t)\leq
z+\bar{\lambda}_{\alpha}t.$

Moreover, for all $\alpha<\beta$, with
$\alpha,\beta\in\\{1,\bar{2},4,\bar{5},7\\}$, there holds

$X_{\alpha}(-2\eta,t)-X_{\beta}(2\eta,t)\geq(-2\eta+\underline{\lambda}_{\alpha}t)-(2\eta+\bar{\lambda}_{\beta}t)=-4\eta+(\underline{\lambda}_{\alpha}-\bar{\lambda}_{\beta})t\geq-4\eta+\sigma
t.$

Note that the above difference is strictly positive when

$t>t_{0}^{(\eta)}:=\frac{4\eta}{\sigma}>0.$ (4.3)

This implies that the five characteristic strips are well separated after
$t_{0}^{(\eta)}$.

Let $\theta$ be a small parameter obeying $0<\theta\ll 1$. We will prescribe
initial data of order $O(\theta)$ and define

$W_{0}^{(\eta)}:=\max_{i=1,\cdots,7}\sup_{z_{i}}|w_{i}^{(\eta)}(z_{i},0)|.$
(4.4)

We require the initial data satisfying the following conditions

$\begin{split}&W_{0}^{(\eta)}=\sup_{z_{1}}w_{1}^{(\eta)}(z_{1},0)=O(\theta),\quad\max_{i=2,3,5,6,7}\sup_{z_{i}}|w_{i}^{(\eta)}(z_{i},0)|\leq
C\eta(W_{0}^{(\eta)})^{2},\\\ &\sup_{z_{4}}|w_{4}^{(\eta)}(z_{4},0)|\leq C\eta
W_{0}^{(\eta)}.\end{split}$ (4.5)

Next, we introduce the main quantities666For notational simplicity, we
suppress the index $\eta$ in the solutions. to be estimated in our proof:

$\displaystyle S_{i}(t):=$
$\displaystyle\sup_{(z^{\prime}_{i},s^{\prime}_{i})\atop
z^{\prime}_{i}\in[-2\eta,2\eta],\ 0\leq s^{\prime}_{i}\leq
t}\rho_{i}(z^{\prime}_{i},s^{\prime}_{i}),$ $\displaystyle S(t):=$
$\displaystyle\max_{i}S_{i}(t),$ (4.6) $\displaystyle J_{i}(t):=$
$\displaystyle\sup_{(z^{\prime}_{i},s^{\prime}_{i})\atop
z^{\prime}_{i}\in[-2\eta,2\eta]\ 0\leq s^{\prime}_{i}\leq
t}|v_{i}(z^{\prime}_{i},s^{\prime}_{i})|,$ $\displaystyle J(t):=$
$\displaystyle\max_{i}J_{i}(t),$ (4.7) $\displaystyle W(t):=$
$\displaystyle\max_{i}\sup_{(x^{\prime},t^{\prime})\atop 0\leq t^{\prime}\leq
t}|w_{i}(x^{\prime},t^{\prime})|,$ $\displaystyle\bar{U}(t):=$
$\displaystyle\sup_{(x^{\prime},t^{\prime})\atop 0\leq t^{\prime}\leq
t}|\Phi(x^{\prime},t^{\prime})|,$ (4.8) $\displaystyle
V_{i}(t):=\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{i},\atop 0\leq
t^{\prime}\leq t}|w_{i}(x^{\prime},t^{\prime})|\quad\text{for}\quad i=1,4,7,$
$\displaystyle
V_{\bar{2}}(t):=\max\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{|w_{2}(x^{\prime},t^{\prime})|,|w_{3}(x^{\prime},t^{\prime})|\\},$
$\displaystyle
V_{\bar{5}}(t):=\max\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq
t}\\{|w_{5}(x^{\prime},t^{\prime})|,|w_{6}(x^{\prime},t^{\prime})|\\},$
$V(t):=\max_{i}V_{i}(t)\quad\text{for}\ i=1,\bar{2},4,\bar{5},7.$ (4.9)

Furthermore, in order to deal with the potentially harmful terms in
(1.8)-(1.11), we need to estimate the following upper bound of $w_{2},w_{3}$
(or $w_{5},w_{6}$) in the unions of the overlapped characteristic strips
$\mathcal{R}_{2}\cup\mathcal{R}_{3}$ (or
$\mathcal{R}_{5}\cup\mathcal{R}_{6}$),

$\check{W}(t):=\max\Big{\\{}\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{w_{2},w_{3}\\},\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq t}\\{w_{5},w_{6}\\}\Big{\\}}.$ (4.10)

$\bullet$ Estimates in the region $[0,t_{0}^{(\eta)}]$: In the region $t\leq
t_{0}^{(\eta)}$, the characteristic strips $\mathcal{R}_{i}$ are overlapped
and are not separated. Nevertheless, the inverse densities of characteristics
still have positive lower bounds in this region. Thus, no shock happens. In
below, for $W(t)$, $V(t)$, $S(t)$, $J(t)$ and $\bar{U}(t)$ defined in
(4.6)-(4.9), we derive a priori estimates for them before $t_{0}^{(\eta)}$.

Step 1. Estimate of $W(t)$. With $t\in[0,t_{0}^{(\eta)}]$ by (2.26) we have

$\frac{\partial}{\partial s_{i}}|w_{i}|\leq\Gamma W^{2}.$

As in [12], via comparing $w_{i}$ with solutions to the following ODE

$\left\\{\begin{array}[]{ll}\frac{d}{dt}Y=\Gamma Y^{2},\\\
Y(0)=W_{0}^{(\eta)},\end{array}\right.$

we obtain

$|w_{i}|\leq Y(t)=\frac{W_{0}^{(\eta)}}{1-\Gamma
W_{0}^{(\eta)}t}\quad\text{for}\quad t<\min\Big{\\{}\frac{1}{\Gamma
W_{0}^{(\eta)}},t_{0}^{(\eta)}\Big{\\}}.$ (4.11)

And by (4.3)(3.8), we have

$\Gamma W_{0}^{(\eta)}t_{0}^{(\eta)}=O(\eta W_{0}^{(\eta)})=O(\theta).$ (4.12)

Applying (4.12) to (4.11), for a small parameter
$\varepsilon\in(0,\frac{1}{100}]$ it holds that

$|w_{i}(x,t)|\leq(1+\varepsilon)W_{0}^{(\eta)}\quad\text{for any}\
x\in\mathbb{R}\ \text{and}\ t\in[0,t_{0}^{(\eta)}].$

According to (4.8), this implies that

$|W(t)|\leq(1+\varepsilon)W_{0}^{(\eta)}\quad\text{for}\
t\in[0,t_{0}^{(\eta)}].$ (4.13)

Step 2. Estimate of $S(t)$. From (2.25) and (3.8), $\rho_{i}$ satisfies

$\frac{\partial\rho_{i}}{\partial s_{i}}=O(\rho_{i}W).$

Integrating the above equation along $\mathcal{C}_{i}$, we deduce

$\rho_{i}(z_{i},t)=\rho_{i}(z_{i},0)\exp\big{(}O(tW(t))\big{)}.$ (4.14)

Then plugging (2.17) into (4.14), one can see that $\rho_{i}(z_{i},t)$ is
always positive. Moreover, invoking (4.3) and (4.13), it follows that

$\rho_{i}(z_{i},t)=\exp\big{(}O(\eta W_{0}^{(\eta)})\big{)},\quad\text{for}\
t\in[0,t_{0}^{(\eta)}].$ (4.15)

We can further choose sufficiently small $\theta$ such that

$1-\varepsilon\leq\exp\big{(}O(\eta W_{0}^{(\eta)})\big{)}\leq 1+\varepsilon.$
(4.16)

Thus, for $t\in[0,t_{0}^{(\eta)}]$, we have that $\rho_{i}$ obeys the
following lower and upper bounds

$1-\varepsilon\leq\rho_{i}(z_{i},t)\leq 1+\varepsilon\quad\text{and}\quad
S(t)=O(1).$ (4.17)

Step 3. Estimate of $V(t)$. First note that the initial data are compactly
supported in $I_{0}$. Formulated in characteristic coordinates
$(z_{i}^{\prime},s_{i}^{\prime})$, this means $w_{i}^{(\eta)}(z_{i},0)=0$ for
$z^{\prime}_{i}\notin I_{0}$. We then integrate (2.26) along the
characteristic $C_{i}$ and obtain

$V(t)=O(\eta[W(t)]^{2})=O(\eta[W_{0}^{(\eta)}]^{2}).$

Step 4. Estimate of $J(t)$. From (2.27) it follows

$\frac{\partial v_{i}}{\partial s_{i}}=O(S(t)[W(t)]^{2}).$

Using (4.13) and (4.17), for $t\in[0,t_{0}^{(\eta)}]$, we have

$J(t)=O(W_{0}^{(\eta)}+t[W(t)]^{2})=O(W_{0}^{(\eta)}+\eta[W_{0}^{(\eta)}]^{2})=O(W_{0}^{(\eta)}).$
(4.18)

Step 5. Estimate of $\bar{U}(t)$. The estimate for $\bar{U}$ can be derived
via the expression of wave decomposition. We integrate the formula (2.24)

$\Phi(x,t)=\int_{X_{7}(-2\eta,t)}^{x}\frac{\partial\Phi(x^{\prime},t)}{\partial
x}dx^{\prime}=\int_{X_{7}(-2\eta,t)}^{x}\sum_{k}w_{k}r_{k}(x^{\prime},t)dx^{\prime}.$
(4.19)

Since $r_{k}(\Phi)=O(1)$, (4.19) implies

$|\Phi(x,t)|=O\Big{(}\sum_{k}\int_{X_{7}(-2\eta,t)}^{X_{1}(2\eta,t)|}|w_{k}(x^{\prime},t)|dx^{\prime}\Big{)}.$
(4.20)

Employing (4.13) and the definition of $\bar{U}$ in (4.8), we get

$\bar{U}(t)=O\big{(}W(t)[-2\eta+(\bar{\lambda}_{1}-\underline{\lambda}_{7})t]\big{)}=O(\eta
W_{0}^{(\eta)})\quad\text{for any}\ t\in[0,t_{0}^{(\eta)}].$

In summary, before the separating time $t_{0}^{(\eta)}$, we derive the
following a priori estimates

$\displaystyle W(t)=O(W_{0}^{(\eta)}),\quad
V(t)=O(\eta[W_{0}^{(\eta)}]^{2}),\quad S(t)=O(1),$ (4.21) $\displaystyle
J(t)=O(W_{0}^{(\eta)}),\quad\bar{U}(t)=O(\eta W_{0}^{(\eta)}).$

$\bullet$ Estimates in the region $[t_{0}^{(\eta)},T]$: This part encodes the
key estimates of this paper. After $t_{0}^{(\eta)}$, $\mathcal{R}_{2}$ and
$\mathcal{R}_{3}$ (or $\mathcal{R}_{5}$ and $\mathcal{R}_{6}$) may overlap for
a long time due to the non-strict hyperbolicity. Therefore, we let
$\mathcal{R}_{\bar{2}}:=\mathcal{R}_{2}\bigcup\mathcal{R}_{3}$ and
$\mathcal{R}_{\bar{5}}:=\mathcal{R}_{5}\bigcup\mathcal{R}_{6}$ and investigate
the well-separated five characteristic strips
$\\{\mathcal{R}_{1},\mathcal{R}_{\bar{2}},\mathcal{R}_{4},\mathcal{R}_{\bar{5}},\mathcal{R}_{7}\\}$
in this part.

Step 1. Estimate of $S(t)$. Based on the estimates we obtain in the region
$[0,t_{0}^{(\eta)}]$, we move forward to derive estimates in the region
$t\in[t_{0}^{(\eta)},T]$ with $T$ a potential shock formation time. Let us
start with estimating $S(t)$, i.e., the supremum of inverse densities. For
$\alpha=1,4,7$, if $(x,t)\in\mathcal{R}_{\alpha}$, by (2.25), we have

$\frac{\partial\rho_{\alpha}}{\partial s_{\alpha}}=O(J_{\alpha}+VS_{\alpha}).$
(4.22)

Thus, integrating (4.22) along the characteristic $\mathcal{C}_{\alpha}$, one
can obtain that

$S_{\alpha}(t)=O(1+tJ+tVS)\quad\text{with}\ \alpha=1,4,7.$ (4.23)

Employing (1.5), we note that $\rho_{3}w_{2}$ is absent in the equation of
$\rho_{3}$ in (2.25), neither is $\rho_{5}w_{6}$ in the equation of
$\rho_{5}$. Therefore, the estimate (4.23) also holds for $\alpha=3,5$. Thus,

$S_{\alpha}(t)=O(1+tJ+tVS)\quad\text{for}\quad\alpha=1,3,4,5,7.$ (4.24)

We then deal with $\rho_{2}$. In equation (2.25)

$\frac{\partial\rho_{2}}{\partial s_{2}}=O(J_{2}+VS_{2}+\rho_{2}w_{3}),$

a potentially harmful nonlinear term $\rho_{2}w_{3}$ pops up. When
$(x,t)\in\mathcal{R}_{2}$, it is possible that the point $(x,t)$ also belongs
to $\mathcal{R}_{3}$. So we can not use $V$ to bound $w_{3}$ directly as in
(4.22). Our strategy is to estimate $w_{3}$ in
$\mathcal{R}_{2}\cup\mathcal{R}_{3}$ via $\check{W}$. With this approach, we
obtain

$S_{2}(t)=O(1+tJ+tVS+t\check{W}S).$ (4.25)

Likewise, $S_{6}(t)$ shares the same bound as in (4.25). Together with (4.24),
consequently, $S(t)$ obeys

$S(t)=O(1+tJ+tVS+t\check{W}S).$ (4.26)

Step 2. Estimate of $J(t)$. For $(x,t)\in\mathcal{R}_{\alpha}$ with
$\alpha=1,4,7$, it follows from (2.27) that

$\frac{\partial v_{\alpha}}{\partial
s_{\alpha}}=O(VJ_{\alpha}+V^{2}S_{\alpha}).$

Via integration along the characteristics, we derive the following estimates:

$J_{\alpha}(t)=O(W_{0}^{(\eta)}+tVJ_{\alpha}+tV^{2}S_{\alpha})\quad\text{for}\
\alpha=1,4,7.$ (4.27)

For $\\{v_{\alpha}(x,t)\\}_{\alpha=2,3,5,6}$ and
$(x,t)\in\mathcal{R}_{\alpha}$, it holds that

$\displaystyle\frac{\partial v_{2}}{\partial
s_{2}}=O\Big{(}VJ_{2}+S_{2}V^{2}+J_{2}w_{3}+VS_{2}w_{3}\Big{)},$ (4.28)
$\displaystyle\frac{\partial v_{3}}{\partial
s_{3}}=O\Big{(}VJ_{3}+S_{3}V^{2}+J_{3}w_{2}+VS_{3}w_{2}\Big{)},$ (4.29)
$\displaystyle\frac{\partial v_{5}}{\partial
s_{5}}=O\Big{(}VJ_{5}+S_{5}V^{2}+J_{5}w_{6}+VS_{5}w_{6}\Big{)},$ (4.30)
$\displaystyle\frac{\partial v_{6}}{\partial
s_{6}}=O\Big{(}VJ_{6}+S_{6}V^{2}+J_{6}w_{5}+VS_{6}w_{5}\Big{)}.$ (4.31)

In the same manner as to prove (4.25), we have

$\\{J_{\alpha}\\}_{\alpha=2,3,5,6}=O(W_{0}^{(\eta)}+tVJ+tV^{2}S+tJ\check{W}+tVS\check{W}).$
(4.32)

Combining (4.27) and (4.32), we conclude that

$J=O(W_{0}^{(\eta)}+tVJ+tV^{2}S+tJ\check{W}+tVS\check{W}).$ (4.33)

Step 3. Estimate of $V(t)$. Among all a priori estimates, this step is the
most delicate. We shall utilize the bi-characteristic coordinates to cope with
various situations. We firstly estimate $w_{i}(x,t)$ for $i=1,4,7$ and
$(x,t)\notin\mathcal{R}_{i}$. From (2.26), it holds for $w_{i}$ that

$\frac{\partial w_{i}}{\partial{s_{i}}}=O(V^{2})+O\Big{(}\sum_{k\neq
i}w_{k}\Big{)}V+O\Big{(}\sum_{m\neq i,k\neq i,m\neq
k\atop(m,k)\neq(2,3),(m,k)\neq(5,6)}w_{m}w_{k}\Big{)}+{\underbrace{O\Big{(}(\lambda_{2}-\lambda_{3})\dbox{w_{2}w_{3}}+(\lambda_{5}-\lambda_{6})\dbox{w_{5}w_{6}}\Big{)}}_{\text{containing
weak interaction}}}.$ (4.34)

Note that $\mathcal{C}_{i}$ starts from $z_{i}\notin I_{0}$ and terminates at
$(x,t)\notin\mathcal{R}_{i}$. When $t^{\prime}\geq t_{0}^{(\eta)}$, every
point $\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}$ on $\mathcal{C}_{i}$
falls either in
$\big{(}\mathbb{R}\times[t_{0}^{(\eta)},t]\big{)}\setminus\bigcup_{k=1,\cdots,7}\mathcal{R}_{k}$
or in $\mathcal{R}_{k}$ for some $k\neq i$.

$t=0$$t=t_{0}^{(\eta)}$$2\eta$$-2\eta$$\mathcal{R}_{i}$$\mathcal{R}_{k}$$(x,t)$$z_{i}$

We denote
$I_{k}^{i}=\\{t^{\prime}\in[t_{0}^{(\eta)},t]:(x,t^{\prime})\in\mathcal{C}_{i}\bigcap\mathcal{R}_{k}\\}$
for $k\neq i$. Integrating (4.34) along $\mathcal{C}_{i}$, with
$w_{i}^{(\eta)}(z_{i},0)=0$, we deduce

$\begin{split}w_{i}(x,t)=&O\Big{(}tV^{2}+t\check{W}^{2}+V\sum_{k\neq
i}\int_{0}^{t}w_{k}\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}dt^{\prime}\Big{)}\\\
=&O\Big{(}tV^{2}+t\check{W}^{2}+V\sum_{k\neq
i}\int_{0}^{t_{0}^{(\eta)}}w_{k}\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}dt^{\prime}\Big{)}\\\
&+O\Big{(}V\sum_{k\neq
i}\int_{t_{0}^{(\eta)}}^{t}w_{k}\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}dt^{\prime}\Big{)}\\\
=&O\Big{(}tV^{2}+t\check{W}^{2}+\eta[W_{0}^{(\eta)}]^{2}+V\sum_{k\neq
i}\underbrace{\int_{I_{k}^{i}}w_{k}\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}dt^{\prime}}_{M}\Big{)}.\end{split}$
(4.35)

For the last equality we use the fact $V(t)\leq W(t)=O(W_{0}^{(\eta)})$ for
$t\leq t_{0}^{(\eta)}$.

$t=0$$t=t_{0}^{(\eta)}$$2\eta$$-2\eta$$\mathcal{R}_{i}$$\mathcal{R}_{k}$$\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}$$z_{i}$$y_{k}$

Now we need to bound the integral $M$ in (4.35). When
$\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}\in I_{k}^{i}$ for some
$k\neq i$ (See the above figure), we employ the bi-characteristic coordinates
to estimate the integral $M$

$\begin{split}&\int_{I_{k}^{i}}w_{k}\big{(}X_{i}(z_{i},t^{\prime}),t^{\prime}\big{)}dt^{\prime}\\\
=&O\Big{(}\int_{y_{k}\in[-2\eta,2\eta]}\bigg{|}\frac{\rho_{k}\big{(}y_{k},t^{\prime}(y_{i},y_{k})\big{)}}{\lambda_{i}-\lambda_{k}}w_{k}\big{(}y_{k},t^{\prime}(y_{i},y_{k})\big{)}\bigg{|}dy_{k}\Big{)}\\\
=&O(\eta J_{k}).\end{split}$ (4.36)

Here we use the fact that $\lambda_{i}-\lambda_{k}\neq 0$ for $i=1,4,7$. Then,
together with (4.35) and (4.36), for $i=1,4,7$, we obtain

$w_{i}(x,t)=O(\eta[W_{0}^{(\eta)}]^{2}+tV^{2}+t\check{W}^{2}+\eta
VJ)\quad\text{for any}\ (x,t)\notin\mathcal{R}_{i}.$ (4.37)

Next, we estimate other $w_{i}$, i.e., $\\{V_{i}\\}_{i=2,3,5,6}$. Here we
provide the details of the estimate for $V_{3}$. $\\{V_{j}\\}_{j=2,5,6}$ can
be handled in a similar way. For $w_{3}(x,t)$, with
$(x,t)\notin\mathcal{R}_{2}\cup\mathcal{R}_{3}$, we have

$\frac{\partial w_{3}}{\partial{s_{3}}}=O\bigg{(}V^{2}+\Big{(}\sum_{k\neq
2,k\neq 3}w_{k}\Big{)}V+\sum_{m,k\neq 3,m\neq
k,\atop(m,k)\neq(5,6)}w_{m}w_{k}+\dbox{(\lambda_{5}-\lambda_{6})w_{5}w_{6}}\bigg{)}.$
(4.38)

Integrating $\partial_{s_{3}}w_{3}$ along $\mathcal{C}_{3}$, it is possible
that some points $(x^{\prime},t^{\prime})\in\mathcal{C}_{3}$ lie in the region
$\mathcal{R}_{5}\cup\mathcal{R}_{6}$ for $t_{0}^{(\eta)}\leq t^{\prime}<t$.
Taking into account of this scenario, we derive the following estimate

$\begin{split}w_{3}(x,t)=&O\Big{(}tV^{2}+tV\check{W}+t\check{W}^{2}+V\sum_{k=1,4,7}\int_{0}^{t}w_{k}\big{(}X_{2}(z_{2},t^{\prime}),t^{\prime}\big{)}dt^{\prime}\Big{)}\\\
=&O\Big{(}tV^{2}+\eta[W_{0}^{(\eta)}]^{2}+V\sum_{k=1,4,7}\int_{I_{k}^{3}}w_{k}\big{(}X_{2}(z_{2},t^{\prime}),t^{\prime}\big{)}dt^{\prime}\Big{)}.\end{split}$
(4.39)

Similar to (4.35) and (4.36), it then follows

$V_{3}=O\left(\eta[W_{0}^{(\eta)}]^{2}+tV^{2}+tV\check{W}+t\check{W}^{2}+\eta
VJ\right).$ (4.40)

Thus, we conclude from (4.37) and (4.40)

$V=O\left(\eta[W_{0}^{(\eta)}]^{2}+tV^{2}+tV\check{W}+t\check{W}^{2}+\eta
VJ\right).$ (4.41)

Step 4. Estimate of $\check{W}(t)$. Now we proceed to estimate

$\check{W}(t)=\max\Big{\\{}\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{2}\cup\mathcal{R}_{3},\atop
0\leq t^{\prime}\leq
t}\\{w_{2},w_{3}\\},\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{5}\cup\mathcal{R}_{6},\atop
0\leq t^{\prime}\leq t}\\{w_{5},w_{6}\\}\Big{\\}},$

i.e., the supremum of $w_{2},w_{3}$ (or $w_{5},w_{6}$) taken in the unions of
the corresponding overlapped characteristic strips
$\mathcal{R}_{2}\cup\mathcal{R}_{3}$ (or
$\mathcal{R}_{5}\cup\mathcal{R}_{6})$. Note that the strong interaction of
$w_{2},w_{3}$ (or $w_{5},w_{6}$) will appear in the following estimates. And
their characteristic speeds are almost-the-same. We focus on estimating
$w_{3}(x,t)$ with $(x,t)\in\mathcal{R}_{2}\cup\mathcal{R}_{3}$. Recall the
equation of $w_{3}$,

$\frac{\partial
w_{3}}{\partial{s_{3}}}=O\Big{[}w_{3}^{2}+\boxed{w_{2}w_{3}}+w_{2}\Big{(}\sum_{k\neq
2,k\neq 3}w_{k}\Big{)}+w_{3}\Big{(}\sum_{k\neq 2,k\neq
3}w_{k}\Big{)}+\sum_{m,k\neq 2,3,m\neq k}w_{m}w_{k}\Big{]}.$ (4.42)

The marked term $\boxed{w_{2}w_{3}}$ corresponds to the strong nonlinear
interaction. Integrating (4.42), it holds that

$w_{3}(x,t)=O\left(\eta[W_{0}^{(\eta)}]^{2}+t\check{W}^{2}+tV\check{W}+tV^{2}\right),\quad\text{for}\quad(x,t)\in\mathcal{R}_{2}\cup\mathcal{R}_{3}.$
(4.43)

The estimates for $w_{2},w_{5},w_{6}$ can be obtained in a same manner as for
$w_{3}$. And hence we prove

$\check{W}(t)=O(\eta[W_{0}^{(\eta)}]^{2}+t\check{W}^{2}+tV\check{W}+tV^{2}).$
(4.44)

Step 5. Estimate of $\bar{U}(t)$. Finally, the estimate of $\Phi$ can be
deduced via the aforementioned estimates. If $(x,t)$ does not belong to any
characteristic strip, one can go back to (4.20) and obtain

$\bar{U}(t)=O\big{(}(-2\eta+(\bar{\lambda}_{1}-\underline{\lambda}_{7})t)V\big{)}.$
(4.45)

If $(x,t)\in\mathcal{R}_{k}$ for some $k$, we have

$\begin{split}|\Phi(x,t)|=&\Big{|}\int_{X_{k}(-2\eta,t)}^{x}\frac{\partial\Phi(x^{\prime},t)}{\partial
x}dx^{\prime}\Big{|}=\Big{|}\int_{X_{k}(-2\eta,t)}^{x}\sum_{k}w_{k}r_{k}(x^{\prime},t)dx^{\prime}\Big{|}\\\
\leq&\int_{X_{k}(-2\eta,t)}^{X_{k}(2\eta,t)}|w_{k}(x^{\prime},t)|dx^{\prime}=O\Big{(}\int_{-2\eta}^{2\eta}|w_{k}(z_{k},t)|\rho_{k}dz_{k}\Big{)}=O(\eta
J).\end{split}$ (4.46)

Combining (4.45) and (4.46), we therefore obtain

$\bar{U}(t)=O(\eta J+\eta V+tV),\quad\text{for any}\ t\in[t_{0}^{(\eta)},T].$
(4.47)

In conclusion, for all $t\in[0,T]$, we derive the following a priori estimates

$\displaystyle S=$ $\displaystyle O(1+tVS+tJ+tS\check{W}),$ (4.48)
$\displaystyle J=$ $\displaystyle
O(W_{0}^{(\eta)}+tVJ+tV^{2}S+t\check{W}J+tVS\check{W}),$ (4.49) $\displaystyle
V=$ $\displaystyle O(\eta[W_{0}^{(\eta)}]^{2}+tV^{2}+\eta
VJ+tV\check{W}+t\check{W}^{2}),$ (4.50) $\displaystyle\check{W}=$
$\displaystyle O(\eta[W_{0}^{(\eta)}]^{2}+t\check{W}^{2}+tV\check{W}+tV^{2}),$
(4.51) $\displaystyle\bar{U}=$ $\displaystyle O(\eta J+\eta V+tV).$ (4.52)

These a priori estimates will be used in a bootstrap argument in the next
subsection. After closing the bootstrap argument, we will obtain the upper
bounds for all the quantities defined in (4.6)-(4.10).

###  4.2. $L^{\infty}$-estimates and bootstrap argument

In this subsection, we design a bootstrap argument to bound $S$, $J$, $V$,
$\check{W}$ and $\bar{U}$. More precisely, employing the desired a priori
estimates (4.48)-(4.52), we obtain the following proposition:

######  Proposition 4.1 ($L^{\infty}$-estimates).

For $t\in[0,T_{\eta}^{*})$ with $T_{\eta}^{*}$ being the (hypothetical) shock
formation time satisfying

$T_{\eta}^{*}\leq\frac{C}{W_{0}^{(\eta)}},$ (4.53)

we have

$\displaystyle S(t)=$ $\displaystyle O(1),$ (4.54) $\displaystyle J(t)=$
$\displaystyle O(W_{0}^{(\eta)}),$ (4.55) $\displaystyle V(t)=$ $\displaystyle
O\left(\eta[W_{0}^{(\eta)}]^{2}\right),$ (4.56) $\displaystyle\check{W}(t)=$
$\displaystyle O\left(\eta[W_{0}^{(\eta)}]^{2}\right),$ (4.57)
$\displaystyle\bar{U}(t)=$ $\displaystyle O(W_{0}^{(\eta)}).$ (4.58)

######  Remark 4.1.

A posteriori, we will prove that the shock formation time $T_{\eta}^{*}$
verifies (4.53) in the next Subsection 4.3. In (4.78), we will also specify
the value of the constant $C$ in (4.53).

###### Proof.

When $t=0$, by (2.17) and definitions in (4.6)-(4.10), it holds

$S(0)=1,\quad J(0)=W_{0}^{(\eta)},\quad V(0)=0.$

To obtain (4.54)-(4.56), with $\theta$ small, we impose bootstrap assumptions:

$\displaystyle tV\leq$ $\displaystyle\theta^{\frac{1}{2}},$ (4.59)
$\displaystyle t\check{W}\leq$ $\displaystyle\theta^{\frac{1}{2}},$ (4.60)
$\displaystyle J\leq$ $\displaystyle\theta^{\frac{1}{2}}.$ (4.61)

Applying (4.59) and (4.60) to (4.48), we have

$\displaystyle S=O(1+tJ+\theta^{\frac{1}{2}}S)\quad\text{when $\theta$ is
small, this implies}\quad S=O(1+tJ).$ (4.62)

To control $J(t)$, we appeal to (4.49). Using (4.59)(4.60)(4.62), we obtain

$\displaystyle\text{via plugging \eqref{y33}\eqref{wca} into
\eqref{364}},\quad J$
$\displaystyle=O(W_{0}^{(\eta)}+\theta^{\frac{1}{2}}J+\theta^{\frac{1}{2}}VS).$
$\displaystyle\text{Absorb $\theta^{\frac{1}{2}}J$, we have}\quad J$
$\displaystyle=O(W_{0}^{(\eta)}+\theta^{\frac{1}{2}}VS)$ by (4.62),
$\displaystyle=O(W_{0}^{(\eta)}+\theta^{\frac{1}{2}}V+\theta J).$
$\displaystyle\text{Absorb $\theta J$, it holds}\quad J$
$\displaystyle=O(W_{0}^{(\eta)}+\theta^{\frac{1}{2}}V).$ (4.63)

We proceed to estimate $V(t)$. Applying (4.59)-(4.61) to (4.50), we get

$\displaystyle
V=O\Big{(}\eta[W_{0}^{(\eta)}]^{2}+\theta^{\frac{1}{2}}V+\eta\theta^{\frac{1}{2}}V+\theta^{\frac{1}{2}}\check{W}\Big{)}.$

For $0<\theta\ll 1$, the second and third terms on the right can be absorbed
by the left, hence it infers

$V=O\Big{(}\eta[W_{0}^{(\eta)}]^{2}+\theta^{\frac{1}{2}}\check{W}\Big{)}.$
(4.64)

We then deal with $\check{W}$, invoking (4.59)(4.60)(4.64) into (4.51), we
deduce

$\displaystyle\text{via plugging \eqref{y33}\eqref{wca} into
\eqref{wc},}\quad\check{W}$
$\displaystyle=O(\eta(W_{0}^{(\eta)})^{2}+\theta^{\frac{1}{2}}\check{W}+\theta^{\frac{1}{2}}V).$
$\displaystyle\text{Absorb $\theta^{\frac{1}{2}}\check{W}$, we
have}\quad\check{W}$
$\displaystyle=O(\eta(W_{0}^{(\eta)})^{2}+\theta^{\frac{1}{2}}V)$ by (4.64),
$\displaystyle=O(\eta[W_{0}^{(\eta)}]^{2}+\theta^{\frac{1}{2}}\eta(W_{0}^{(\eta)})^{2}+\theta\check{W}).$
$\displaystyle\text{Absorb $\theta\check{W}$, it holds}\quad\check{W}$
$\displaystyle=O(\eta(W_{0}^{(\eta)})^{2}).$ (4.65)

This finishes the proof of (4.57). We then substitute (4.65) into (4.64),
hence (4.56) can be verified. Furthermore, with (4.53), we have

$tV=O(\eta W_{0}^{(\eta)})=O(\theta)<\theta^{\frac{1}{2}},$ (4.66)
$t\check{W}=O(\eta W_{0}^{(\eta)})=O(\theta)<\theta^{\frac{1}{2}}.$ (4.67)

Note that (4.66) and (4.67) improve the bootstrap assumptions (4.59) and
(4.60).

Next, for $J(t)$ and $S(t)$, it follows from (4.63)(4.56) that the estimates
(4.55) and (4.54) are valid:

$J=O(W_{0}^{(\eta)}+\theta^{\frac{1}{2}}\eta[W_{0}^{(\eta)}]^{2})=O(W_{0}^{(\eta)})=O(\theta)<\theta^{\frac{1}{2}},$
(4.68) $S=O(1+tW_{0}^{(\eta)})=O(1).$ (4.69)

And (4.68) improves the bootstrap assumption (4.61).

Finally, by (4.54)-(4.57) we deduce the estimate for $\bar{U}$

$\bar{U}(t)=O(\eta
W_{0}^{(\eta)}+\eta^{2}W_{0}^{(\eta)}+\eta[W_{0}^{(\eta)}]^{2})=O(\eta
W_{0}^{(\eta)})=O(\theta),\ t\in[0,T_{\eta}^{*}].$ (4.70)

Furthermore, since $\bar{U}$ is the supremum of $\Phi$ as defined in (4.8),
with sufficiently small $\theta$, we then conclude that the solution $\Phi$
always stays in $B_{\delta}^{7}(0)$.

∎

###  4.3. Shock formation

We will finish the proof of Theorem 1.2 in this subsection. In particular, we
will show that in $\mathcal{R}_{1}$ the inverse density $\rho_{1}\to 0$ and
$w_{1}\to\infty$ as $t\to T_{\eta}^{*}$ while $\Phi$ is uniformly bounded.
Hence a shock happens at $T_{\eta}^{*}$.

###### Proof of Theorem 1.2.

Our first step is to control $v_{1}$. We integrate (2.27)

$\frac{\partial v_{1}}{\partial s_{1}}=O\Big{(}\sum_{m\neq
1}w_{m}\Big{)}v_{1}+\underbrace{O\Big{(}\sum_{m,k\neq 1,m\neq
k}w_{m}w_{k}\Big{)}}_{\text{containing weak interaction}}\rho_{1}$

along $\mathcal{C}_{1}$ and obtain

$v_{1}(z_{1},t)=w_{1}^{(\eta)}(z_{1},0)+O(tVJ+tV^{2}S)=w_{1}^{(\eta)}(z_{1},0)+O(\eta[W_{0}^{(\eta)}]^{2}).$
(4.71)

In particular, taking $z_{0}\in I_{0}$ such that
$w_{1}^{(\eta)}(z_{0},0)=W_{0}^{(\eta)}$, then for sufficiently small
$\theta$, the above formula yields

$(1-\varepsilon)W_{0}^{(\eta)}\leq
v_{1}(z_{0},t)\leq(1+\varepsilon)W_{0}^{(\eta)}.$ (4.72)

We then estimate $\rho_{1}$. Using (2.25)

$\frac{\partial\rho_{1}}{\partial
s_{1}}=c_{11}^{1}(\Phi)v_{1}+O\Big{(}\sum_{k\neq 1}w_{k}\Big{)}\rho_{1},$

and the fact $c_{11}^{1}(\Phi)<0$, we obtain

$-c_{11}^{1}(\Phi)v_{1}-\Big{|}O\Big{(}\sum_{k\neq
1}w_{k}\Big{)}\Big{|}\rho_{1}\leq\frac{\partial\rho_{1}}{\partial s_{1}}\leq-
c_{11}^{1}(\Phi)v_{1}+\Big{|}O\Big{(}\sum_{k\neq
1}w_{k}\Big{)}\Big{|}\rho_{1}.$ (4.73)

We will then employ the Grönwall’s inequality to derive upper and lower bounds
of $\rho_{1}$. Firstly, by (4.70), we have $|\Phi|=O(\eta
W_{0}^{(\eta)})=O(\theta)\leq\delta$. Setting $\theta$ to be sufficiently
small, we further have

$(1-\varepsilon)|c_{11}^{1}(0)|\leq-
c_{11}^{1}(\Phi)\leq(1+\varepsilon)|c_{11}^{1}(0)|.$ (4.74)

For the coefficient $O\Big{(}\sum_{k\neq 1}w_{k}\Big{)}$ in (4.73), using bi-
characteristic coordinates (as in Subsection 4.2), we deduce

$\begin{split}\int_{0}^{t}\sum_{k\neq
1}|w_{k}(X_{1}(z_{1},t^{\prime}),t^{\prime})|dt^{\prime}=\sum_{k\neq
1}\int_{-2\eta}^{2\eta}\frac{|v_{k}|}{\lambda_{1}-\lambda_{k}}dy_{k}=O(\eta
W_{0}^{(\eta)}+\eta J)=O(\eta W_{0}^{(\eta)})=O(\theta).\end{split}$

Here we use the fact that the factor $\frac{1}{\lambda_{1}-\lambda_{k}}$
arising from the bi-characteristic transformation is of $O(1)$. For
$0<\theta\ll 1$, the above estimate implies

$1-\varepsilon\leq\exp{\Big{(}\int_{0}^{t}\pm|O\big{(}\sum_{k\neq
1}w_{k}(X_{1}(z_{1},t^{\prime}),t^{\prime})\big{)}|dt^{\prime}\Big{)}}\leq
1+\varepsilon.$ (4.75)

Applying the Grönwall’s inequality to (4.73) and combining with (4.74)-(4.75),
we get

$\begin{split}&(1-\varepsilon)(1-(1+\varepsilon)^{2}|c_{11}^{1}(0)|\int_{0}^{t}v_{1}(z_{1},t^{\prime})dt^{\prime})\\\
\leq&\rho_{1}(z_{1},t)\\\
\leq&(1+\varepsilon)(1-(1-\varepsilon)^{2}|c_{11}^{1}(0)|\int_{0}^{t}v_{1}(z_{1},t^{\prime})dt^{\prime}).\end{split}$
(4.76)

By the first inequality of (4.72), we then arrive at

$\rho_{1}(z_{0},t)\geq(1-\varepsilon)\Big{(}1-(1+\varepsilon)^{3}|c_{11}^{1}(0)|tW_{0}^{(\eta)}\Big{)},$
(4.77)

which shows that $\rho_{1}(z_{0},t)>0$ when
$t<\frac{1}{(1+\varepsilon)^{3}|c_{11}^{1}(0)|W_{0}^{(\eta)}}$. Meanwhile, the
second inequality of (4.72) implies

$\rho_{1}(z_{0},t)\leq(1+\varepsilon)\Big{(}1-(1-\varepsilon)^{3}|c_{11}^{1}(0)|tW_{0}^{(\eta)}\Big{)}.$

Together with (4.77), we conclude that there exists a finite positive time
$T_{\eta}^{*}$ (shock formation time) such that

$\lim_{t\rightarrow
T_{\eta}^{*}}\rho_{1}(z_{0},t)=0\quad\text{and}\quad\lim_{t\rightarrow
T_{\eta}^{*}}w_{1}(z_{0},t)=\lim_{t\rightarrow
T_{\eta}^{*}}\frac{v_{1}}{\rho_{1}}(z_{0},t)=\infty.$

And $T_{\eta}^{*}$ obeys

$\frac{1}{(1+\varepsilon)^{3}|c_{11}^{1}(0)|W_{0}^{(\eta)}}\leq
T_{\eta}^{*}\leq\frac{1}{(1-\varepsilon)^{3}|c_{11}^{1}(0)|W_{0}^{(\eta)}}.$
(4.78)

As we mentioned in Remark 4.1, this verifies the condition (4.53). ∎

Recall that the estimates of $V(t)$ and $\check{W}(t)$ in Section 4.2 show
that the unknowns $\\{w_{i}(z,t)\\}_{i=2,3,5,6}$ are bounded in the whole
space before the shock formation time $T_{\eta}^{*}$, and
$\\{w_{i}(z,t)\\}_{i=4,7}$ are bounded when
$(z,t)\notin\\{\mathcal{R}_{i}\\}_{i=4,7}$. In fact, now we prove that
$\\{w_{i}(z,t)\\}_{i=4,7}$ are also regular in
$\\{\mathcal{R}_{i}\\}_{i=4,7}$. To show this, we introduce

$\check{W}_{i}^{\prime}(t):=\sup_{(z,t^{\prime})\in\mathcal{R}_{i}\atop 0\leq
t^{\prime}\leq t}\\{w_{i}(z,t^{\prime})\\}\quad\text{for}\quad i=4,7$ (4.79)

and prove the following:

######  Proposition 4.2.

For $t\in[0,T_{\eta}^{*})$ with the shock formation time $T_{\eta}^{*}$
satisfying (4.78), we have $\check{W}_{i}^{\prime}(t)=O(W_{0}^{(\eta)})$.

###### Proof.

We study $w_{4}$ and $w_{7}$ separately. For $w_{4}$, since $c^{4}_{44}=0$, it
verifies

$\frac{\partial w_{4}}{\partial{s_{4}}}=O\Big{(}\sum_{k\neq
4}w_{k}\Big{)}w_{4}+\underbrace{O\Big{(}\sum_{m\neq 4,k\neq 4\atop m\neq
k}w_{m}w_{k}\Big{)}}_{\text{containing weak interaction}}.$ (4.80)

Let $(z,t)\in\mathcal{R}_{4}$ and $t\in[t_{0}^{(\eta)},T_{\eta}^{*}]$,
integrating (4.80) along $\mathcal{C}_{4}$, we obtain

$w_{4}(z,t)\leq O(W_{0}^{(\eta)}+tV\check{W}_{4}^{\prime}+tV^{2}).$ (4.81)

In view of the facts $tV=O(\theta)$ and
$V=O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}$, we hence get

$\check{W}_{4}^{\prime}\leq O(W_{0}^{(\eta)}).$ (4.82)

For $w_{7}$, even though its evolution equation is of Riccati-type, its
initial datum is designed to be much more smaller than that of $w_{1}$. This
guarantees that $w_{7}(z,t)$ is regular when $t\leq T_{\eta}^{*}$. Invoking
the estimate of $V$, it holds that

$\begin{split}\frac{\partial
w_{7}}{\partial{s_{7}}}=&-c^{7}_{77}w_{7}^{2}+O\Big{(}\sum_{k\neq
7}w_{k}\Big{)}w_{7}+\underbrace{O\Big{(}\sum_{m\neq 7,k\neq 7\atop m\neq
k}w_{m}w_{k}\Big{)}}_{\text{containing weak interaction}}\\\
=&-c^{7}_{77}w_{7}^{2}+O(V)w_{7}+O(V^{2})\\\
=&-c^{7}_{77}w_{7}^{2}+O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}w_{7}+O\big{(}\eta^{2}[W_{0}^{(\eta)}]^{4}\big{)}.\end{split}$
(4.83)

Equivalently,

$\frac{d}{ds}\Big{[}\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}s\Big{)}w_{7}\Big{]}=-\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}s\Big{)}c^{7}_{77}w_{7}^{2}+\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}s\Big{)}O\big{(}\eta^{2}[W_{0}^{(\eta)}]^{4}\big{)}.$
(4.84)

Since

$\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}s\Big{)}\leq\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}T_{\eta}^{*}\Big{)}\leq\exp\Big{(}O\big{(}\eta
W_{0}^{(\eta)}\big{)}\Big{)}=O(e^{\theta}),$ (4.85)

by choosing $\theta$ sufficiently small, we have

$1-\varepsilon\leq\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}s\Big{)}\leq
1+\varepsilon.$ (4.86)

Thus, back to (4.84), it holds

$\begin{split}\frac{d}{ds}\Big{[}\exp\Big{(}O\big{(}\eta[W_{0}^{(\eta)}]^{2}\big{)}s\Big{)}w_{7}\Big{]}\leq
Cw_{7}^{2}+O\big{(}\eta^{2}[W_{0}^{(\eta)}]^{4}\big{)}.\end{split}$ (4.87)

Integrating (4.87) along $\mathcal{C}_{7}$, we obtain

$\begin{split}\check{W}_{7}^{\prime}&\leq
O\big{(}w_{7}(z,0)+t\check{W}_{7}^{\prime
2}+t\eta^{2}[W_{0}^{(\eta)}]^{4}\big{)}\\\ &\leq
O\big{(}[W_{0}^{(\eta)}]^{2}+t\check{W}_{7}^{\prime
2}+\eta^{2}[W_{0}^{(\eta)}]^{3}\big{)}\\\ &\leq
O\big{(}[W_{0}^{(\eta)}]^{2}+t\check{W}_{7}^{\prime 2}\big{)}.\end{split}$
(4.88)

Now we introduce an additional bootstrap assumption

$t\check{W}_{7}^{\prime}\leq\theta^{\frac{1}{2}}.$ (4.89)

Then by (4.88), it holds

$\check{W}_{7}^{\prime}\leq O\big{(}[W_{0}^{(\eta)}]^{2}\big{)}$ (4.90)

and the bootstrap assumption (4.89) can be improved by

$t\check{W}_{7}^{\prime}\leq
O(W_{0}^{(\eta)})=O(\theta)<\theta^{\frac{1}{2}}.$ (4.91)

∎

Moreover, we show that the shock formation only occurs in $\mathcal{R}_{1}$ by
deriving positive lower bounds for $\rho_{2},\cdots,\rho_{7}$. This concludes
the proof of Theorem 1.2.

######  Proposition 4.3.

For $t\leq T_{\eta}^{*}$, we have $\\{\rho_{i}\\}_{i=2,\cdots,7}$ are all
bounded away from zero in the whole $(x,t)$-plane, i.e., $\rho_{i}>C>0$ for a
uniform constant $C$.

###### Proof.

To derive a uniform lower bound for $\rho_{i}$, we start from estimates inside
$\mathcal{R}_{i}$. Similarly to (4.76), via estimating (2.25), we have

$\begin{split}\rho_{i}(z_{i},t)\geq(1-\varepsilon)\Big{(}1-(1+\varepsilon)^{2}|c_{ii}^{i}(0)|\int_{0}^{t}|v_{i}(z_{i},t^{\prime})|dt^{\prime}\Big{)}.\end{split}$

Employing the estimates of $v_{i}$, one can then bound $\rho_{i}$ from below
in $\mathcal{R}_{i}$.

We then prove that $\\{\rho_{i}\\}_{i=1,\cdots,7}$ obey a positive lower bound
outside $\mathcal{R}_{i}$. For the equation (2.25) of $\rho_{i}$, applying the
Grönwall’s inequality and estimates obtained in Section 4, one can obtain that
for any $z_{i}\notin[\eta,2\eta],\ t\in[t_{0}^{(\eta)},T_{\eta}^{*})$,

$\rho_{i}(z_{i},t)\geq\exp\Big{(}-\int_{0}^{t}\Big{|}O\Big{(}\sum_{k}w_{k}(z_{i},s)\Big{)}\Big{|}ds\Big{)}\geq
1-\varepsilon.$ (4.92)

Interested readers are also referred to Proposition 8.1 in [3] for a more
detailed discussion. ∎

##  5\. Proof of $H^{2}$ ill-posedness (Theorem 1.3)

In this section, we use the above shock formation to prove $H^{2}$ ill-
posedness for 3D ideal MHD equations (1.1). We first construct a family of
initial data $w_{i}^{(\eta)}(z,0)$, which will lead to the ill-posedness. For
each $\eta$, we will prove that there is a corresponding shock formation at
finite time $T_{\eta}^{*}$. The meaning of the ill-posedness is twofold:
Firstly, the lifespan $T_{\eta}^{*}$ will vanish when $\eta\to 0$. Moreover,
the $L^{2}$ norm of $w_{1}$ will blow up at $T_{\eta}^{*}$.

###  5.1. Initial data designed for proving ill-posedness

Our first step is to construct the initial data. Counterexamples of local
well-posedness will evolve from them.

We design a family of initial data $w_{1}^{(\eta)}(z,0)$ satisfying

$w_{1}^{(\eta)}(z,0)=\theta\int_{\mathbb{R}}\zeta_{\frac{\eta}{10}}(y)|\ln(z-y)|^{\alpha}\chi(z-y)dy\quad\text{with}\
0<\alpha<\frac{1}{2}.$ (5.1)

Here, $\eta>0$ is a small parameter and $\chi$ is the characteristic function

$\chi(z)=\left\\{\begin{array}[]{ll}1,\quad
z\in[\frac{6}{5}\eta,\frac{9}{5}\eta],\\\ 0,\quad
z\notin(\frac{6}{5}\eta,\frac{9}{5}\eta),\end{array}\right.$

and $\zeta_{\frac{\eta}{10}}(z)$ is a test function in
$C_{0}^{\infty}(\mathbb{R})$ verifying

$\text{supp}\
\zeta_{\frac{\eta}{10}}(z)=\\{z:|z|\leq\frac{\eta}{10}\\},\quad\text{and}\quad\int_{\mathbb{R}}\zeta_{\frac{\eta}{10}}(z)dz=1.$

For such initial datum, we present here the estimate of its $H^{1}$ norm in
the following lemma.

######  Lemma 5.1.

Let $0<\alpha<\frac{1}{2}$.We construct a function $\hat{w}_{1}$ as below.

$\hat{w}_{1}(x,x_{2},x_{3})=w_{1}^{(\eta)}(x,0):=\theta\int_{\mathbb{R}}\zeta_{\frac{\eta}{10}}(y)|\ln(x-y)|^{\alpha}\chi(x-y)dy\
{\text{in}\ B_{\frac{\eta}{2}}^{3}},$ (5.2)

where $B_{\frac{\eta}{2}}^{3}$ denotes the 3D unit ball of radius
$\frac{\eta}{2}$ centered at the origin. Then, we have that $\hat{w}_{1}\in
H^{1}(\mathbb{R}^{3})$ satisfying

$\|\hat{w}_{1}\|_{H^{1}(\mathbb{R}^{3})}\lesssim\theta\sqrt{\eta}(1+|\ln\frac{6\eta}{5}|^{\alpha}+|\ln\frac{9\eta}{5}|^{\alpha}\big{)}.$
(5.3)

###### Proof.

See Lemma A.1 in [3] for a detailed similar calculation. ∎

For $k=2,\cdots,7$, choose

$w_{k}^{(\eta)}(z,0)=\theta^{2}\int_{\mathbb{R}}\zeta_{\frac{\eta}{10}}(y)\chi(z-y)dy.$
(5.4)

One can verify that these initial data satisfy all the requirements in Section
4.

Recall that

$\Phi_{x}(x,t)=\sum_{k=1}^{7}w_{k}(x,t)r_{k}(\Phi).$ (5.5)

Now we go back to the original system (2.3) by reversing the process of wave
decomposition. The initial data of the original variables $\Phi$ are denoted
as $\Phi(x,0)=\hat{\Phi}_{0}(x,x_{2},x_{3})$. According to (5.5), the initial
data satisfy

$\partial_{x}\hat{\Phi}_{0}(x,x_{2},x_{3})=\sum_{k=1}^{7}\hat{w}_{k}(x,x_{2},x_{3})r_{k}\big{(}\Phi(x,0)\big{)}.$
(5.6)

Since $r_{k}$ constructed in (2.9) is Lipschitz continuous in $\Phi$, by a
standard ODE argument, given the aforementioned $\hat{w}_{k}(x,x_{2},x_{3})$,
one can solve the ODE system (5.6) for $\hat{\Phi}_{0}$. Moreover, because
$r_{k}(\Phi)\in L^{\infty}(B_{\delta}^{7}(0))$ and $\hat{w}_{k}\in
H^{1}(\mathbb{R}^{3})$, we have $\hat{\Phi}_{0}\in H^{2}(\mathbb{R}^{3})$.
These constitute the $H^{2}$ initial data for the 3D ideal compressible MHD
equations (2.3).

###  5.2. Estimate for $\partial_{z_{1}}\rho_{1}$

In this subsection, we derive a uniform bound for
$|\partial_{z_{1}}\rho_{1}|$. As we will see in Subsection 5.3, this upper
bound plays a crucial role in proving the desired ill-posedness result.

For a fixed $\eta>0$, we employ characteristic and bi-characteristic
coordinates introduced in (2.14)-(2.16) for $i\neq j$:

$(x,t)=\big{(}X_{i}(z_{i},s_{i}),s_{i}\big{)}=\big{(}X_{i}(y_{i},t^{\prime}(y_{i},y_{j})),t^{\prime}(y_{i},y_{j})\big{)}=\big{(}X_{j}(y_{j},t^{\prime}(y_{i},y_{j})),t^{\prime}(y_{i},y_{j})\big{)}.$

As discussed in Section 2, the transformations between these coordinates
satisfy

$\partial_{y_{i}}=\frac{\rho_{i}}{\lambda_{j}-\lambda_{i}}\partial_{s_{j}}=\partial_{z_{i}}+\frac{\rho_{i}}{\lambda_{j}-\lambda_{i}}\partial_{s_{i}}.$
(5.7)

Along $\mathcal{C}_{1}$ we fix $y_{1}$ and set $y_{7}$ as a parameter. From
(5.7), we have

$\partial_{y_{1}}=\partial_{z_{1}}+\frac{\rho_{1}}{\lambda_{7}-\lambda_{1}}\partial_{s_{1}}=\partial_{z_{1}}-\frac{\rho_{1}}{2C_{f}}\partial_{s_{1}},$
(5.8)

where $C_{f}$ is the fast wave speed defined in (2.6). We now state the main
result of this subsection.

######  Proposition 5.1.

Fix $\eta$, for all $z_{i}\in\mathbb{R}$ and any $t<T^{*}$, there holds
$|\partial_{z_{1}}\rho_{1}|\leq C_{\eta}$ for a positive constant $C_{\eta}$.

###### Proof.

Using (5.8), we get

$\partial_{z_{1}}\rho_{1}=\partial_{y_{1}}\rho_{1}+\frac{\rho_{1}}{2C_{f}}\partial_{s_{1}}\rho_{1}.$
(5.9)

To bound $\partial_{z_{1}}\rho_{1}$, we start with controlling
$\partial_{y_{1}}\rho_{1}$. Let

$\tau_{1}^{(7)}:=\partial_{y_{1}}\rho_{1},\quad\pi_{1}^{(7)}:=\partial_{y_{1}}v_{1}.$

Since

$\partial_{y_{7}}=\frac{\rho_{7}}{\lambda_{1}-\lambda_{7}}\partial_{s_{1}}=\frac{\rho_{7}}{2C_{f}}\partial_{s_{1}},$
(5.10)

we have that $\tau_{1}^{(7)}$ obeys

$\begin{split}\partial_{y_{7}}\tau_{1}^{(7)}=&\partial_{y_{1}}\partial_{y_{7}}\rho_{1}=\partial_{y_{1}}\Big{(}\frac{\rho_{7}}{2C_{f}}\partial_{s_{1}}\rho_{1}\Big{)}=\partial_{y_{1}}\Big{(}\frac{\rho_{1}\rho_{7}}{2C_{f}}\sum_{m}c_{1m}^{1}w_{m}\Big{)}\\\
=&\frac{\rho_{7}}{2C_{f}}\Big{(}c_{11}^{1}\partial_{y_{1}}v_{1}+\sum_{m\neq
1}c_{1m}^{1}w_{m}\partial_{y_{1}}\rho_{1}\Big{)}+\frac{\rho_{1}}{2C_{f}}\Big{(}\sum_{m\neq
7}c_{1m}^{1}w_{m}\partial_{y_{1}}\rho_{7}+c_{17}^{1}\partial_{y_{1}}v_{7}\Big{)}\\\
&+\frac{\rho_{1}\rho_{7}}{2C_{f}}\Big{(}\sum_{m\neq 1,m\neq
7}c_{1m}^{1}\frac{1}{\rho_{m}}\partial_{y_{1}}v_{m}-\sum_{m\neq 1,m\neq
7}c_{1m}^{1}\frac{w_{m}}{\rho_{m}}\partial_{y_{1}}\rho_{m}\Big{)}\\\
&-\frac{\partial_{y_{1}}C_{f}}{2C_{f}^{2}}\rho_{1}\rho_{7}\sum_{m}c_{1m}^{1}w_{m}+\frac{\rho_{1}\rho_{7}}{2C_{f}}\sum_{m}\partial_{y_{1}}c_{1m}^{1}w_{m}.\end{split}$
(5.11)

We still need to estimate
$\partial_{y_{1}}C_{f},\partial_{y_{1}}c_{1m}^{1},\partial_{y_{1}}\rho_{m}$
and $\partial_{y_{1}}v_{m}$ in (5.11). Note that for $C_{f}$, its derivative
satisfies

$\begin{split}\partial_{y_{1}}C_{f}=&\nabla_{\Phi}(\lambda_{1}-u_{1})\cdot\partial_{y_{1}}\Phi=\nabla_{\Phi}\lambda_{1}\cdot[\partial_{s_{7}}X_{7}\partial_{y_{1}}t^{\prime}\partial_{x}\Phi+\partial_{y_{1}}t^{\prime}\partial_{t}\Phi]\\\
=&\nabla_{\Phi}(\lambda_{1}-u_{1})\cdot\Big{[}\lambda_{7}\frac{\rho_{1}}{\lambda_{7}-\lambda_{1}}\sum_{k}w_{k}r_{k}+\frac{\rho_{1}}{\lambda_{7}-\lambda_{1}}(-A(\Phi)\sum_{k}w_{k}r_{k})\Big{]}\\\
=&O\Big{(}v_{1}+\rho_{1}\sum_{k\neq 1}w_{k}\Big{)}.\end{split}$ (5.12)

Similarly, for $c^{1}_{1m}$, we also have

$\begin{split}\partial_{y_{1}}c_{1m}^{1}=&\nabla_{\Phi}c_{1m}^{1}\cdot\partial_{y_{1}}\Phi=\nabla_{\Phi}c_{1m}^{1}\cdot[\partial_{s_{7}}X_{7}\partial_{y_{1}}t^{\prime}\partial_{x}\Phi+\partial_{y_{1}}t^{\prime}\partial_{t}\Phi]\\\
=&\nabla_{\Phi}c_{1m}^{1}\cdot\Big{[}\lambda_{7}\frac{\rho_{1}}{\lambda_{7}-\lambda_{1}}\sum_{k}w_{k}r_{k}+\frac{\rho_{1}}{\lambda_{7}-\lambda_{1}}(-A(\Phi)\sum_{k}w_{k}r_{k})\Big{]}\\\
=&O\Big{(}v_{1}+\rho_{1}\sum_{k\neq 1}w_{k}\Big{)}.\end{split}$ (5.13)

By (5.7), it holds

$\partial_{y_{1}}\rho_{m}=\frac{\rho_{1}}{\lambda_{m}-\lambda_{1}}\partial_{s_{m}}\rho_{m}=O\Big{(}\rho_{m}v_{1}+\rho_{1}\rho_{m}\sum_{k\neq
1}w_{k}\Big{)}\quad\text{when}\quad m\neq 1,$ (5.14)

and

$\partial_{y_{1}}v_{m}=\frac{\rho_{1}}{\lambda_{m}-\lambda_{1}}\partial_{s_{m}}v_{m}=O\Big{(}\rho_{m}v_{1}\sum_{k\neq
1}w_{k}+\rho_{1}\rho_{m}\sum_{j\neq 1,k\neq 1\atop j\neq
k}w_{j}w_{k}\Big{)}\quad\text{when}\quad m\neq 1.$ (5.15)

Together with the estimates obtained in Proposition 4.1, (5.12)-(5.15) imply
that (5.11) could be rewritten as

$\begin{split}\partial_{y_{7}}\tau_{1}^{(7)}=B_{11}^{\eta}\tau_{1}^{(7)}+B_{12}^{\eta}\pi_{1}^{(7)}+B_{13}^{\eta},\end{split}$
(5.16)

where $B_{11}^{\eta},B_{12}^{\eta},B_{13}^{\eta}$ are uniformly bounded
constants depending on $\eta$.

In the same fashion, we obtain the evolution equation for $\pi_{1}^{(7)}$:

$\begin{split}\partial_{y_{7}}\pi_{1}^{(7)}=&\frac{\rho_{7}}{2C_{f}}\Big{(}\sum_{p\neq
1,q\neq 1\atop p\neq q}\gamma_{pq}^{1}w_{p}w_{q}\tau_{1}^{(7)}+\sum_{p\neq
1}\gamma_{1p}^{1}w_{p}\pi_{1}^{(7)}\Big{)}\\\
&-\frac{\partial_{y_{1}}C_{f}}{4C_{f}^{2}}\Big{(}\sum_{p\neq
1}\gamma_{1p}^{1}w_{p}v_{1}\rho_{7}+\sum_{p\neq 1,q\neq 1\atop p\neq
q}\gamma_{pq}^{1}w_{p}w_{q}\rho_{7}\rho_{1}\Big{)}\\\
&+\frac{\rho_{7}\rho_{1}}{2C_{f}}\Big{(}\sum_{p\neq
1}\partial_{y_{1}}\gamma_{1p}^{1}w_{p}w_{m}+\sum_{p\neq 1,q\neq 1\atop p\neq
q}\partial_{y_{1}}\gamma_{pq}^{1}w_{p}w_{q}\Big{)}\\\
&+\frac{\rho_{1}}{2C_{f}}\Big{(}\sum_{p\neq
1}\gamma_{1p}^{1}w_{p}w_{1}+\sum_{p\neq 1,q\neq 1\atop p\neq
q}\gamma_{pq}^{1}w_{p}w_{q}\Big{)}\partial_{y_{1}}\rho_{7}\\\