# Wellposedness of the electron MHD without resistivity
for large perturbations of the uniform magnetic field

In-Jee Jeong  Department of Mathematical Sciences and Research Institute of
Mathematics, Seoul National University, Seoul, Korea 08826<EMAIL_ADDRESS>and  Sung-Jin Oh  Department of Mathematics, UC Berkeley, Berkeley, CA, USA
94720 and School of Mathematics, KIAS, Seoul, Korea 02455
<EMAIL_ADDRESS>
###### Abstract.

We prove the local wellposedness of the Cauchy problems for the electron
magnetohydrodynamics equations (E-MHD) without resistivity for possibly large
perturbations of nonzero uniform magnetic fields. While the local
wellposedness problem for (E-MHD) has been extensively studied in the presence
of resistivity (which provides dissipative effects), this seems to be the
first such result without resistivity. (E-MHD) is a fluid description of
plasma in small scales where the motion of electrons relative to ions is
significant. Mathematically, it is a quasilinear dispersive equation with
nondegenerate but nonelliptic second-order principal term. Our result
significantly improves upon the straightforward adaptation of the classical
work of Kenig–Ponce–Rolvung–Vega on the quasilinear ultrahyperbolic
Schrödinger equations, as the regularity and decay assumptions on the initial
data are greatly weakened to the level analogous to the recent work of
Marzuola–Metcalfe–Tataru in the case of elliptic principal term.

A key ingredient of our proof is a simple observation about the relationship
between the size of a symbol and the operator norm of its quantization as a
pseudodifferential operator when restricted to high frequencies. This allows
us to localize the (non-classical) pseudodifferential renormalization operator
considered by Kenig–Ponce–Rolvung–Vega, and produce instead a classical
pseudodifferential renormalization operator. We furthermore incorporate the
function space framework of Marzuola–Metcalfe–Tataru to the present case of
nonelliptic principal term.

##  1\. Introduction

In the $(3+1)$-dimensional spacetime $\mathbb{R}_{t}\times\mathbb{R}^{3}_{x}$,
the _electron magnetohydrodynamics_ system is given by

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

where ${\bf B}:\mathbb{R}_{t}\times{\mathbb{R}^{3}_{x}}\to\mathbb{R}^{3}$ is a
time-dependent vector field (magnetic field). The equation we consider is
_without resistivity_ , in the sense that it does not have the dissipative
term $\mu\Delta{\bf B}$ on the RHS of the first equation, and as a result, the
energy $\frac{1}{2}\int|{{\bf B}(t)}|^{2}\,\mathrm{d}x$ is conserved. This
equation is a fluid description of plasma in small scales, where the motion of
electrons relative to ions is significant – which is typical in low plasma
density and/or high temperature regimes, as argued in the work of Lighthill
[52] where the model was introduced. It is also the simpler small-scale limit
of Hall-MHD, which is another extended MHD model that takes the relative
motion of electrons and ions into account (see equation (Hall-MHD) below). It
is by now a well-established fact that (E-MHD) reproduces many essential
features of the Hall-MHD system and (more accurate) two-fluid plasma models
(see, e.g., [58]).

The quadratic nonlinearity $\nabla\times((\nabla\times{\bf B})\times{\bf B})$
is referred to as the Hall current term after the discovery of Hall [35], and
makes (E-MHD) a quasilinear dispersive equation. While the Hall current term
is omitted in the usual ideal MHD model, it is known to be directly
responsible for various situations including plasma confinement, collisionless
magnetic reconnection featuring $X$-configuration ([6, 67, 38, 27]), Hall-
effect thrusters ([76, 68, 62]), planetary magnetospheres including that of
the Earth’s ([26, 57]) and magnetic field dynamics in neutron stars ([33, 46,
34, 74]). In all of these situations, the effect of magnetic resistivity is
considered negligible with respect to that of the Hall current term.
Turbulence of electron-MHD system is a very active area of study in the plasma
physics literature as well ([73, 31, 15]).

On the mathematical side, the illuminating work of Jang–Masmoudi [40] (see
also [1, 59]) formally derived the Hall-MHD equations (among other equations)
from a two-species kinetic system consisting of electrons and ions, which
clarifies that the Hall term is indeed a two-species effect. We shall review
the existing mathematical literature on (E-MHD) in more detail below.

It is expected that a local wellposedness theory for (E-MHD) and related
systems would be not only useful in rigorous derivation of MHD models but also
provide a ground for a systematic study of various aforementioned physical
phenomena involving the Hall effect. In the current work, we obtain the first
local wellposedness result for (E-MHD) without resistivity, when the initial
magnetic field is a (possibly large) perturbation of a nonzero uniform
magnetic field, which we take to be ${\bf e}_{3}$ (i.e., the uniform magnetic
field of unit strength in the $x^{3}$-direction). This indeed seems to be the
usual setup for (E-MHD) in the physics literature, including the original work
of Lighthill ([52]). Furthermore, such an assumption cannot be completely
avoided: in stark contrast to the resistive case, or even the usual ideal MHD
model, the Cauchy problem for (E-MHD) and Hall-MHD is, in general, strongly
illposed for arbitrarily small and smooth (even analytic) data ([44]).
Analyzing the linearized equations near steady states with a degeneracy where
${\bf B}$ vanishes, one sees that illposedness arises from degenerate
dispersion, wherein the frequency of wave packets traveling towards degeneracy
grows indefinitely in an arbitrarily short interval of time.

Our local wellposedness result is phrased in terms of the translation-
invariant space introduced by Marzulola–Metcalfe–Tataru [54, 55, 56] in the
context of quasilinear Schrödinger equations, but adapted to (E-MHD). To
define this space, we first fix a partition $\mathcal{I}_{k}$ of $\mathbb{R}$
(for each nonnegative integer $k$) into intervals of length $2^{k}$, and a
smooth partition of unity $\\{\chi_{I}\\}_{I\in\mathcal{I}_{k}}$ such that
$\operatorname{supp}\chi_{I}\subseteq 2I$ (i.e., the interval with the same
center as $I$ but twice the length). Given $s\in\mathbb{R}$ and
$u:\mathbb{R}^{3}\to\mathbb{R}$, define

$\|{u}\|_{\ell^{1}_{\mathcal{I}}H^{s}}^{2}:=\sum_{k\geq
0}\left(2^{sk}\sum_{I\in\mathcal{I}_{k}}\|{\chi_{I}(x^{3})P_{k}u}\|_{L^{2}}\right)^{2},$

where $\\{P_{k}\\}$ are the inhomogeneous Littlewood–Paley projections (see
Section 2.1). As in [54, 55, 56], note that any translations of this norm are
equivalent. But unlike in those papers, the variable $x^{3}$ plays a
distinguished role in the definition of our norm; this is because of the
_conic directionality_ of the group velocities for the linearization of
(E-MHD) around ${\bf e}_{3}$, to be discussed in more detail below. As usual,
we extend this norm to vector-valued functions by summing up the norm of all
components.

Our main theorem may now be stated as follows.

######  Theorem 1.1 (Main theorem, simple version).

Let $s>\frac{7}{2}$ and consider a vector field ${\bf
B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfying $\nabla\cdot{\bf B}_{0}=0$.
Assume furthermore that ${\bf B}_{0}$ satisfies the following properties:

  1. (1)

Nondegeneracy. ${\bf B}_{0}(x)\neq 0$ at every point $x\in\mathbb{R}^{3}$,

  2. (2)

Asymptotic uniformity. $\|{{\bf B}_{0}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<+\infty$,

  3. (3)

Nontrapping. Every nonconstant solution $(X,\Xi)(t)$ to the Hamiltonian system
associated with $p_{{\bf B}_{0}}(x,\xi)={\bf B}_{0}(x)\cdot\xi|{\xi}|$ escapes
to $x^{3}=\pm\infty$, i.e., $X^{3}(t)\to\infty$ or $X^{3}(t)\to-\infty$ as
$t\to\infty$.

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

The solution in Theorem 1.1 belongs to the space ${\bf B}-{\bf e}_{3}\in
C([0,T);\ell^{1}_{\mathcal{I}}H^{s})$ for some $T>0$. To quantify the lifespan
$T$ guaranteed by our theorem, we need more quantitative information on ${\bf
B}_{0}$. We refer the reader to Theorem 5.1 below for a more precise
formulation. We shall see later in Corollary 5.2 that the nontrapping
assumption is satisfied for small perturbations of ${\bf e}_{3}$, which gives
the following:

######  Corollary 1.2 (Small data local wellposedness).

For a given $s>\frac{7}{2}$, there exist positive constants $\varepsilon,T$
depending only on $s$ such that for any ${\bf
B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfying $\nabla\cdot{\bf B}_{0}=0$
and $\|{{\bf B}_{0}-{\bf e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<\varepsilon$,
there exists a unique local-in-time solution to (E-MHD) on the time interval
$[0,T]$ with initial data ${\bf B}_{0}$.

The motivation for the assumptions in Theorem 1.1 is as follows. Assumption
(1) is a basic requirement that avoids, in particular, degenerate dispersion,
which may lead to illposedness [44]. To motivate Assumption (2), as well as
the distinguished role played by $x^{3}$ in the definition of
$\ell^{1}_{\mathcal{I}}H^{s}$, let us consider the linearization (E-MHD)
around ${\bf B}={\bf e}_{3}$, which is the constant-coefficient system

$\partial_{t}b+{\bf e}_{3}\cdot\nabla(\nabla\times b)=0,\quad\nabla\cdot b=0.$

Using the $L^{2}$-projections $\Pi_{\pm}(D)=$ (where $D=i^{-1}\partial_{x}$
and $m(D)$ is the Fourier multiplier with symbol $m(\xi)$) to decompose
$b=b_{+}+b_{-}$ with $b_{\pm}=\Pi_{\pm}(D)b$, this system is diagonalized and
we arrive at the equations

$\partial_{t}b_{\pm}\pm{\bf e}_{3}\cdot D|{D}|b_{\pm}=0,$

which is a dispersive equation with dispersion relation $\pm p(\xi)=\pm{\bf
e}_{3}\cdot\xi|{\xi}|$. Hence, the group velocity takes the form

$v_{\pm}(\xi):=\pm\nabla_{\xi}p(\xi)=\pm\begin{pmatrix}\frac{\xi_{1}\xi_{3}}{|{\xi}|}\\\
\frac{\xi_{2}\xi_{3}}{|{\xi}|}\\\
\frac{\xi_{1}^{2}+\xi_{2}^{3}+2\xi_{3}^{2}}{|{\xi}|}\end{pmatrix}.$

Observe that the set of all possible group velocities form a cone around ${\bf
e}_{3}$ with aperture less than $\pi$ – we call this the _conic
directionality_ of (E-MHD) (see also Lemma 4.1). In particular, the
$x^{3}$-component of the group velocity is always comparable to $|{\xi}|$,
which is why the above definition of $\ell^{1}_{\mathcal{I}}H^{s}$ – involving
localization and summation in only the $x^{3}$ variable – is the suitable
adaptation of the $l^{1}H^{s}$ space introduced in [54]. As in [54], the extra
$\ell^{1}$ summability controls the frequency evolution of wave packets in the
far-away region – see also Lemmas 4.2 and 4.7 below.

If we consider instead the linearization of (E-MHD) around an arbitrary
magnetic field $\mathring{{\bf B}}$, then the diagonalization of the principal
symbol leads to the scalar symbols $\pm p_{\mathring{{\bf
B}}}(t,x,\xi):=\pm\mathring{{\bf B}}(t,x)\cdot\xi|{\xi}|$, and the Hamiltonian
flows associated to $\pm p_{\mathring{{\bf B}}}$ are the adequate
generalization of the group velocities $v_{\pm}(\xi)$ in the constant
coefficient case. Assumption (3) ensures that, at least for a short time,
there are no trapped bicharacteristics (i.e., solution to the Hamiltonian flow
that stays in a bounded region for all times) with $\mathring{{\bf B}}={\bf
B}$, which would be an obstruction for the _local smoothing effect_ of
(E-MHD). This effect is key to our proof, and will be discussed further in
Section 1.1 below.

The system (E-MHD) may be compared with _quasilinear Schrödinger equations_ ,
which are equations of the form

$i\partial_{t}u+\sum_{j,k=1}^{d}\partial_{j}(g^{jk}(u,\overline{u},\nabla_{x}u,\nabla_{x}\overline{u})\partial_{k}u)=F(u,\overline{u},\nabla_{x}u,\nabla_{x}\overline{u})$
(1.1)

where $g$ and $F$ are smooth functions of their variables, with $g$ a real-
valued symmetric matrix. In view of the nonellipticity of the principal
symbol, (E-MHD) most closely resembles the _ultrahyperbolic_ case, i.e., when
$g$ is nondegenerate but _not_ positive definite.

Local wellposedness of the Cauchy problem for quasilinear ultrahyperbolic
Schrödinger equations was first obtained in the landmark papers of
Kenig–Ponce–Rolvung–Vega [47, 48]. However, Theorem 1.1 significantly improves
upon the straightforward adaptation of [47, 48], as the regularity and decay
assumptions on the initial data are greatly weakened to the level analogous to
the recent work of Marzuola–Metcalfe–Tataru [56] for the case when $g$ is
positive definite. Our approach to proving Theorem 1.1 builds upon these
works, but also differs from both approaches in some key aspects; see Section
1.1 below.

######  Remark 1.3 (Extension to the Hall-MHD system).

In the absence of magnetic resistivity, the (incompressible) Hall-MHD system
reads:

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

Here, ${\bf u},p$ are the fluid velocity field and the pressure, respectively,
and $\nu\geq 0$ is the kinematic viscosity. As a simple extension of Theorem
1.1, one can obtain the following result for (Hall-MHD): Under the same
assumptions as in Theorem 1.1 for ${\bf B}_{0}$, if the initial velocity field
${\bf u}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfies $\nabla\cdot{\bf
u}_{0}=0$ and ${\bf u}_{0}\in\ell^{1}_{\mathcal{I}}H^{s+1}$, then we have
local wellposedness for (Hall-MHD) with initial data $({\bf B}_{0},{\bf
u}_{0})$. We shall defer the precise statement and proof to our forthcoming
work in [43].

######  Remark 1.4 (Extension to quasilinear ultrahyperbolic Schrödinger
equations [60]).

In view of the analogies with (1.1), a natural question is whether a result
analogous to Theorem 1.1 can be also proved for (1.1) building on the ideas
put forth in this paper. This is a nontrivial problem, as the conic
directionality of $p_{\mathring{{\bf B}}}$ significantly simplifies the
analysis of (E-MHD) compared to (1.1). Nevertheless, the answer to this
question has been answered in _affirmative_ ; see the recent work of
Pineau–Taylor [60]. See also Remark 1.6 below.

######  Remark 1.5 (Possible relaxations of assumptions).

Let us discuss some ways the assumptions in Theorem 1.1 may be relaxed. Some
kind of nondegeneracy and nontrapping assumptions are always needed, but in
view of the directionality of the bicharacteristics associated with $p_{{\bf
B}_{0}}$, we may replace $\|{{\bf B}_{0}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<+\infty$ by an assumption that does
not require decay in the orthogonal direction to $x^{3}$. Moreover, we may let
${\bf B}_{0}$ tend to different nondegenerate configurations as
$x^{3}\to\pm\infty$. Some of these extensions will be explored in [43].

The remainder of the introduction is structured as follows. In Section 1.1, we
describe the main ideas of our proof of Theorem 1.1. In Section 1.2, we
discuss the previous literature on related problems. Then in Section 1.3, an
outline of the rest of the paper is provided.

###  1.1. Main ideas

We now give a discussion of the main ideas of this paper.

####  1.1.1. Basic setup and local smoothing estimate

A useful way to understand the nonlinear term $\nabla\times((\nabla\times{\bf
B})\times{\bf B})$ in (E-MHD) is to use paraproduct decomposition. To wit, we
write (E-MHD) as

$\partial_{t}{\bf B}-\nabla\times(T_{{\bf B}}\times(\nabla\times{\bf
B}))+\nabla\times(T_{(\nabla\times{\bf B})}\times{\bf B})=(\cdots),$

where $T_{{\bf A}}\times{\bf B}$ is the _(low-high) paraproduct_ defined
(using Littlewood–Paley projections) as

$T_{{\bf A}}\times{\bf B}:=\sum_{k}P_{<k-10}{\bf A}\times P_{k}{\bf B}.$

The point of this decomposition is that the terms we omitted on the RHS,
called _residual_ or _high-high interaction_ , are smoother (more precisely,
see Proposition 5.5). Moreover, $\nabla\times(T_{(\nabla\times{\bf
B})}\times{\bf B})$ on the LHS is less dangerous than the other term, since at
most one derivative falls on the high frequency factor (namely, ${\bf B}$ on
the right). As in the previous discussion of the linearized (E-MHD) around
${\bf e}_{3}$, the first two terms on the LHS can be diagonalized (to the
leading order) by introducing $b_{\pm}:=\Pi_{\pm}(D)b$ where $b={\bf B}-{\bf
e}_{3}$. We arrive at the system

${\bf L}_{{\bf B}}^{\sharp}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}:=\partial_{t}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}P_{{\bf B}}^{(2)\sharp}&0\\\ 0&-P_{{\bf
B}}^{(2)\sharp}\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}0&\mathfrak{S}^{(1)\sharp}_{{\bf B}}\\\
-\mathfrak{S}^{(1)\sharp}_{{\bf B}}&0\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}=(\cdots)$

where $P_{{\bf B}}^{(2)\sharp}$ is a diagonal anti-symmetric paradifferential
(see Section 2.3) second-order operator and $\mathfrak{S}_{{\bf
B}}^{(1)\sharp}$ is a matrix-valued symmetric paradifferential first-order
operator given by

$\displaystyle P_{{\bf B}}^{(2)\sharp}b$
$\displaystyle:=\sum_{k}\left[\frac{1}{2}{\bf
B}^{\alpha}_{<k-10}P_{k}\partial_{\alpha}|{\nabla}|b+\frac{1}{2}|{\nabla}|\partial_{\alpha}P_{k}{\bf
B}^{\alpha}_{<k-10}\right],$ $\displaystyle\mathfrak{S}_{{\bf
B}}^{(1)\sharp}b$
$\displaystyle:=\sum_{k}\left[\frac{1}{2}\left(|{\nabla}|(\partial^{\alpha}P_{<k-10}{\bf
B}_{\beta})+(\partial_{\beta}P_{<k-10}{\bf
B}^{\alpha})|{\nabla}|\right)P_{k}b^{\beta}+\frac{1}{2}[|{\nabla}|,P_{<k-10}{\bf
B}\cdot\nabla]P_{k}b^{\alpha}\right],$

and we have moved to the RHS and omitted less important terms for this
heuristic discussion. See Sections 3 and 5, as well as the proof of
Proposition 7.2, for details. Terms in $(\cdots)$ possibly loses $1$
derivative of $b$, so we need to prove an estimate that gains $1$ derivative
for this system. This is precisely the role of the _local smoothing estimate_

$\displaystyle\|{\langle{D}\rangle^{s}\vec{b}}\|_{X^{0}}$
$\displaystyle:=\|{\langle{D}\rangle^{s+\frac{1}{2}}\vec{b}}\|_{LE}+\|{\langle{D}\rangle^{s}\vec{b}}\|_{L^{\infty}L^{2}[0,T]}$
(1.2)
$\displaystyle\lesssim\|{\langle{D}\rangle^{s}\vec{b}(t=0)}\|_{L^{2}}+\|{\langle{D}\rangle^{s}\vec{g}^{(1)}}\|_{L^{1}L^{2}}+\|{\langle{D}\rangle^{s-\frac{1}{2}}\vec{g}^{(2)}}\|_{LE^{\ast}},$

where $s>0$, $\vec{b}=(b_{+},b_{-})^{\top}$, ${\bf L}_{{\bf
B}}^{\sharp}\vec{b}=\vec{g}^{(1)}+\vec{g}^{(2)}$ and $LE$ and $LE^{\ast}$ are
norms defined in (2.1)–(2.2) below. Comparing the terms involving $LE$ and
$LE^{\ast}$, one may note that this estimate indeed controls $1$ more
derivative of $\vec{b}$ compared to ${\bf L}_{{\bf B}}^{\sharp}\vec{b}$.
Heuristically, this order-$1$ local smoothing phenomenon for (E-MHD) is
expected based on the nondegeneracy, asymptotic uniformity and nontrapping
properties of ${\bf B}$ (see Theorem 1.1); see also the discussion below.

For the ensuing discussion, let us also introduce the (non-paradifferential)
linear operator

${\bf L}_{{\bf B}}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}:=\partial_{t}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}P_{{\bf B}}^{(2)}&0\\\ 0&-P_{{\bf
B}}^{(2)}\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}0&\mathfrak{S}^{(1)}_{{\bf B}}\\\
-\mathfrak{S}^{(1)}_{{\bf B}}&0\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}$

where

$\displaystyle P_{{\bf B}}^{(2)}b$ $\displaystyle:=\frac{1}{2}{\bf
B}^{\alpha}\partial_{\alpha}|{\nabla}|b+\frac{1}{2}|{\nabla}|\partial_{\alpha}{\bf
B}^{\alpha},$ $\displaystyle\mathfrak{S}_{{\bf B}}^{(1)}b$
$\displaystyle:=\frac{1}{2}\left(|{\nabla}|(\partial^{\alpha}{\bf
B}_{\beta})+(\partial_{\beta}{\bf
B}^{\alpha})|{\nabla}|\right)b^{\beta}+\frac{1}{2}[|{\nabla}|,{\bf
B}\cdot\nabla]b^{\alpha},$

which is a part of the linearization of (E-MHD) around ${\bf B}$ (see Section
3).

####  1.1.2. Previous approaches

Having set up a more concrete setting for a heuristic discussion, let us now
review the previous approaches to local wellposedness of quasilinear
dispersive equations. Indeed, in the pioneering papers of Kenig–Ponce–Vega
[49, 50, 51] and Kenig–Ponce–Rolvung–Vega [47, 48], which proved local
wellposedness of quasilinear Schrödinger equations with elliptic and
ultrahyperbolic principal terms, respectively, the main idea was to establish
local smoothing estimates for suitable class of variable coefficient linear
dispersive equations (see also [53, 14, 37] for earlier works). Translated to
our context111In particular, $b_{+}$ and $b_{-}$ in our context are analogous
to $u$ and $\bar{u}$, respectively, in [47]. The symbol for the
renormalization operator – referred to as an “integrating factor” – is denoted
by $K$., the class of equations they considered consists of non-
paradifferential operators ${\bf L}_{{\bf B}_{0}}$ around a time-independent
magnetic field ${\bf B}_{0}$ – we note that ${\bf B}_{0}$ may be taken to be
the initial magnetic field (hence our notation), since we expect ${\bf B}$ to
stay close to the initial data (in an adequate sense) on the time interval
under consideration. The proof of local smoothing estimate for ${\bf L}_{{\bf
B}_{0}}$ according to [47] (the ultrahyperbolic case, which is closer to
(E-MHD)) would proceed as follows:

Step 1: Local smoothing assuming boundedness of energy. Establish (1.2) for
${\bf L}_{{\bf B}_{0}}\vec{b}=\vec{g}^{(1)}+\vec{g}^{(2)}$ assuming

$\|{\vec{b}}\|_{L^{\infty}H^{s}[0,T]}\lesssim\|{\vec{b}(t=0)}\|_{H^{s}}+\|{\vec{g}^{(1)}}\|_{L^{1}H^{s}}+\|{\langle{D}\rangle^{s-\frac{1}{2}}\vec{g}^{(2)}}\|_{LE^{\ast}},$
(1.3)

Step 2: Boundedness of energy. Establish (1.3) for ${\bf L}_{{\bf
B}_{0}}\vec{b}=\vec{g}^{(1)}+\vec{g}^{(2)}$.

The first step involves a _positive commutator argument_ , based on the
geometric observation that every bicharacteristic $(X(t),\Xi(t))$ associated
with ${\bf B}_{0}\cdot\xi|{\xi}|$ initially in a slab of the form
$\\{\alpha<x^{3}<\beta\\}$ exits the slab at time at most
$O(\frac{\beta-\alpha}{|{\Xi(0)}|})$ (during which $|{\Xi(t)}|$ stays
comparable to $\Xi(0)$) under suitable nondegeneracy, asymptotic uniformity
(i.e., decay) and nontrapping assumptions (as in Theorem 1.1). For the second
step, we start by writing out the equations for
$\tilde{b}_{\pm}:=\langle{D}\rangle^{s}b_{\pm}$:

$\displaystyle\partial_{t}\begin{pmatrix}\tilde{b}_{+}\\\
\tilde{b}_{-}\end{pmatrix}+\left[\begin{pmatrix}P_{{\bf B}_{0}}^{(2)}&0\\\
0&-P_{{\bf B}_{0}}^{(2)}\end{pmatrix}+\begin{pmatrix}\mathfrak{C}^{s(1)}_{{\bf
B}_{0}}&0\\\ 0&-\mathfrak{C}^{s(1)}_{{\bf
B}_{0}}\end{pmatrix}+\begin{pmatrix}0&\mathfrak{S}^{(1)}_{{\bf B}_{0}}\\\
-\mathfrak{S}^{(1)}_{{\bf
B}_{0}}&0\end{pmatrix}\right]\begin{pmatrix}\tilde{b}_{+}\\\
\tilde{b}_{-}\end{pmatrix}=\langle{D}\rangle^{s}{\bf L}_{{\bf
B}}\begin{pmatrix}b_{+}\\\ b_{-}\end{pmatrix}+(\cdots),$

where $\mathfrak{C}^{s(1)}_{{\bf B}_{0}}:=[\langle{D}\rangle^{s},P_{{\bf
B}_{0}}^{(2)}]\langle{D}\rangle^{-s}$ is a diagonal _symmetric_ first-order
operator arises from commuting $\langle{D}\rangle^{s}$ with $P_{{\bf
B}_{0}}^{(2)}$. Then we control the time derivative of the $H^{s}$-energy
$\|{\tilde{b}_{+}}\|_{L^{2}}^{2}+\|{\tilde{b}_{-}}\|_{L^{2}}^{2}$ by
multiplying the above equation on the left by $(\tilde{b}_{+},\tilde{b}_{-})$
and integrating on $\mathbb{R}^{3}$. The key difficulty here is that
$\mathfrak{C}^{s(1)}_{{\bf B}_{0}}$ contributes a term involving one
derivative of $\tilde{b}_{\pm}$ without smallness (if ${\bf B}_{0}-{\bf
e}_{3}$ is large), and hence which cannot be directly handled with local
smoothing in Step 1 without circular reasoning.

The idea of [47] (a formal version of this idea goes back to [16, 25, 37]) was
to remove the problematic term $\pm\mathfrak{C}^{s(1)}_{{\bf
B}_{0}}\tilde{b}_{\pm}$ by working with a conjugation
$\operatorname{Op}(O)\tilde{b}_{\pm}$, where $\operatorname{Op}(O)$ is a
suitable pseudodifferential operator (see Section 2.3 for the notation). A
short formal computation tells us that the symbol of $O$ needs to satisfy

$\mathrm{H}_{{\bf B}_{0}}O(x,\xi)=\mathfrak{c}^{s(1)}_{{\bf
B}_{0}}(x,\xi)O(x,\xi),$ (1.4)

where $\mathrm{H}_{{\bf B}_{0}}$ is the vector field on
$\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{x}$ (geometrically,
$T^{\ast}\mathbb{R}^{3}$) associated with the Hamiltonian flow of ${\bf
B}_{0}(x)\cdot\xi|{\xi}|$ and $\mathfrak{c}^{s(1)}_{{\bf B}_{0}}(x,\xi)$ is
the principal symbol of $\mathfrak{C}^{s(1)}_{{\bf B}_{0}}$. The good news is
that such a symbol $O(x,\xi)$ is uniformly bounded under the assumptions on
${\bf B}_{0}$ in Theorem 1.1. The bad news, however, is that the derivatives
of $O(x,\xi)$ behave badly. In fact, $O(x,\xi)$ does _not_ belong to the
classical symbol class $S^{0}$, but rather only obeys

$|{\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}O(x,\xi)}|\lesssim_{\boldsymbol{\alpha},\boldsymbol{\beta}}\langle{x}\rangle^{|{\boldsymbol{\beta}}|}|{\xi}|^{-|{\boldsymbol{\beta}}|},$

which is a nonstandard symbol class considered by Craig–Kappeler–Strauss [16].
In particular, it is known that the above assumptions are _not_ sufficient to
guarantee the boundedness of the operator $\operatorname{Op}(O)$ in $L^{2}$.
Kenig–Ponce–Vega–Rolvung [47] show, nevertheless, that $O(x,\xi)$, after
suitable modifications in the construction, belongs to a refined (but still
nonstandard) class of symbols for which $L^{2}$-boundedness of
$\operatorname{Op}(O)$ and a suitable version of symbolic calculus hold. The
operator $\operatorname{Op}(O)$ is then used to carry out the proof of
boundedness of energy, thereby completing the proof of local smoothing. The
argument in [47] requires strong regularity and decay assumptions on ${\bf
B}_{0}$ (in particular, compared to Theorem 1.1).

Recently, Marzuola–Metcalfe–Tataru [54, 55, 56] revisited the local
wellposedness problem for quasilinear Schrödinger equations and greatly
improved the hypotheses on regularity and decay of initial data. Some of the
ideas introduced in [54, 55, 56] are: working with a paralinearization instead
of the usual linearization of the equation (i.e., ${\bf L}_{{\bf B}}^{\sharp}$
instead of ${\bf L}_{{\bf B}_{0}}$), introduction of a function space
framework that encodes the necessary decay in a translation-invariant manner
(see $\ell^{1}_{\mathcal{I}}H^{s}$) and a new proof of local smoothing
estimate that, in particular, does not involve the renormalization operators
$\operatorname{Op}(O)$. However, the method of [56] (the large data case)
cannot be directly applied to (E-MHD), as it uses the ellipticity of the
principal term in an important manner.

####  1.1.3. New ideas in this work

In this work, we introduce a new approach that builds upon – but is distinct
from – both [47] and [54]. A key new ingredient in our proof is a simple
observation, which is of interest on its own, that we dub the _high frequency
Calderón–Vaillancourt trick_ (see Lemma 2.10 for details):

> For $a\in S^{0}$ (classical symbol class), the $L^{2}\to L^{2}$-operator
> norm of $Op(a)$ restricted to inputs supported in high frequency (i.e.,
> $\|{Op(a)P_{>k}(D)}\|_{L^{2}\to L^{2}}$) is bounded by
> $\|{a}\|_{L^{\infty}_{x,\xi}}$, provided that the frequency threshold $k$ is
> large enough (depending on higher order bounds for $a$).

This trick is useful since, in the problem of local wellposedness, a low
frequency part of the solution is easy to deal with using the energy estimate
and the smallness of the time interval.

Our approach for proving local smoothing for ${\bf L}_{{\bf B}_{0}}$ follows
the basic roadmap of [47] sketched above, but using the high frequency
Calderón–Vaillancourt trick we get enough quantitative precision to perform a
_physical space localization_ of the renormalization operator so that (i) it
lies in the _classical_ symbol class $S^{0}$ instead of an nonstandard class
as in [47], and (ii) weaker regularity and decay assumptions for ${\bf B}_{0}$
as in Theorem 1.1 are sufficient. Concretely, our localized renormalization
operator $O_{\pm}$ for $\tilde{b}_{\pm}$ takes the form

$\displaystyle O_{\pm}(x,\xi)=\begin{cases}O(x,\xi)&|{x^{3}}|\lesssim R,\\\
\exp(\pm C_{O})&x^{3}\gtrsim R\\\ \exp(\mp C_{O})&-x^{3}\gtrsim
R,\end{cases},$

where $C_{O}>0$ is some constant (independent of $R$); $R$ is chosen
sufficiently large so that ${\bf B}_{0}$ is small in $\\{x^{3}\gtrsim R\\}$,
which we quantify by $\epsilon$; $O$ solves (1.4); and $O_{\pm}(x,\xi)$ is
chosen to be (suitably) monotonic along bicharacteristics in the transition
regions $|{x^{3}}|\simeq R$, which is crucial for the localization to be
admissible (which, in turn, is why we let $O_{\pm}$ depend on $\pm$). The
error resulting from this localization is bounded by
$C\epsilon\|{O_{\pm}\tilde{b}_{\pm}}\|_{X^{0}}$, which we must absorb using
the local smoothing estimate in Step 1.

Since $O_{\pm}(x,\xi)$ is constant in $|{x^{3}}|\gtrsim R$, we have
$O_{\pm}(x,\xi)\in S^{0}$. But the symbol bounds depend on $R$, so the
operator norm of $\operatorname{Op}(O_{\pm})$ will still depend on $R$222To
make matters worse, in reality the derivatives of $O_{\pm}$ also depend on the
maximal length $L$ of bicharacteristics inside $\\{|{x^{3}}|\lesssim R\\}$
(see Definition 4.3). But this issue is resolved similarly to the
$R$-dependence issue discussed here via the high frequency
Calderón–Vaillancourt trick.. At this point, one may worry about running into
a circular reasoning in the choice of $\epsilon$ and $R$ unless $R$ is
quantitatively related to $\epsilon$ (e.g., there is a strong decay assumption
on ${\bf B}_{0}$ analogous to [47]). Our key observation, however, is that
$\|{O_{\pm}}\|_{L^{\infty}}$ is bounded _independently_ of $\epsilon$ and $R$
(essentially since $O$ is) and hence, by the high frequency
Calderón–Vaillancourt trick (more precisely, its strengthening in Proposition
6.3), $\|{\operatorname{Op}(O_{\pm})P_{>k_{(1)}}}\|_{X^{0}\to X^{0}}$ is also
bounded _independently_ of $R$ for $k_{(1)}$ sufficiently large. As remarked
above, the contribution of the low frequencies $P_{<k_{(1)}}$ can be treated
via simpler energy estimates by choosing the time $T$ small333The frequency
threshold $k_{(1)}$ depends on $R$, but it is compensated by choosing $T$
small at the end of the argument.. This observation breaks the possible
circularity and allows us to prove boundedness of energy with only a
translation-invariant decay assumption on ${\bf B}_{0}$. Moreover, as we avoid
introducing any nonstandard symbol classes, our renormalization argument is
significantly simpler than [47]. See Section 7.5 for the detailed
argument444Although, pedantically, Section 7.5 considers the paralinearized
system, which is more difficult than the simplified problem discussed here..

In fact, we are able to carry out our argument (essentially) in the context of
_paralinearized operator_ ${\bf L}_{{\bf B}}^{\sharp}$ (see Section 7 for the
precise setting) and in the function space framework analogous to [54, 55, 56]
(see Section 2.2 and Section 6). As a result, we optimize the regularity and
decay assumptions on the initial data to the same level as [56].

######  Remark 1.6 (Extension to quasilinear ultrahyperbolic Schrödinger
equations [60]).

The conic directionality of (E-MHD) simplifies our argument, as evinced by the
form of $O_{\pm}(x,\xi)$ above (in particular, it suffices to localize the
symbol to the slab $\\{|x^{3}|\lesssim R\\}$ independent of $\xi$). In a
recent work [60] of Pineau–Taylor, the strategy of physical space localization
via high frequency Calderón–Vaillancourt trick has been extended to the
quasilinear ultrahyperbolic Schrödinger equations (1.1), where the conic
directionality no longer holds and further new ideas are required.

###  1.2. Discussions

Whistler waves versus Alfvén waves. We make a comparison between Whistler and
Alfvén waves, where the latter is the basic wave for the usual ideal MHD
system. The ideal MHD system linearized around $(0,{\bf e}_{3})$ reads

$\begin{split}\partial_{t}{\bf b}+{\bf e}_{3}\cdot\nabla{\bf
u}=0,\qquad\partial_{t}{\bf u}={\bf e}_{3}\cdot\nabla{\bf b},\end{split}$

and this can be written as

$\begin{split}\partial_{tt}{\bf b}+({\bf e}_{3}\cdot\nabla)^{2}{\bf
b}=0,\end{split}$

which is nothing but the (1+1)-dimensional wave equation in $(t,x^{3})$.
Therefore, we see that the Alfvén waves travel along ${\bf B}$ with speed
$|{{\bf B}}|$, while Whistler waves travels within a cone around ${\bf B}$
with speed proportional to $|{{\bf B}}||{\xi}|$. It is this property of
Whistler waves that is responsible for various aforementioned phenomena
observed in plasmas.

Mathematical literature on Hall and electron MHD. Regarding Hall and electron
MHD equations with magnetic resistivity, local well-posedness and temporal
decay was obtained in pioneering works [7] and [10], respectively. Most of the
subsequent mathematical literature dealt with the resistive case as well ([9,
17, 24, 12, 75, 77, 72, 71, 23, 18, 19]), and the simpler case of
$2\frac{1}{2}$-dimensional solutions have been widely studied ( [5, 61, 75,
13, 29, 21]), not only by mathematicians but also by physicists. This means
that one considers magnetic fields of the form

$\begin{split}{\bf B}=\nabla\times(\psi\,{\bf e}_{3})+\phi\,{\bf
e}_{3},\end{split}$ (1.5)

where $\psi,\phi$ are scalar valued functions of $t,x^{1},x^{2}$. In this
case, (E-MHD) reduces to

$\left\\{\begin{aligned}
\partial_{t}\psi+\nabla^{\perp}\phi\cdot\nabla\psi=0,&\\\
\partial_{t}\phi-\nabla^{\perp}\psi\cdot\nabla\Delta\psi=0.&\end{aligned}\right.$
(1.6)

(Here, $\nabla^{\perp}=(\partial_{x^{1}},\partial_{x^{2}})$ and
$\nabla^{\perp}=(-\partial_{x^{2}},\partial_{x^{1}})$. In the presence of the
velocity field, a similar ansatz can be given for it as well.) Recently, Dai
[21] added a $\Delta\phi$ term to the RHS of the $\phi$ equation in (1.6) and
obtained global wellposedness of smooth solutions near the uniform magnetic
field ${\bf B}={\bf e}_{1}$. In the same paper, it is conjectured that global
regularity persists when (1.6) has a resistivity term only in the $\psi$
equation. Moreover, we note that dyadic models for Hall- and electron-MHD
systems have been studied in [20] and [22].

In the irresistive case, Chae–Weng [11] presented smooth axisymmetric
solutions of Hall- and electron-MHD equations which blow up in finite time
(see also [28, 30, 41]).555This was done without having uniqueness
corresponding to the initial data, which is still an open problem. However,
for general smooth initial data, strong illposedness statements including
nonexistence of smooth solutions to Hall- and electron-MHD were proved in
[44], by considering degenerate magnetic fields under the
$2\frac{1}{2}$-dimensional setup. That is, for certain $C^{\infty}$-smooth and
decaying initial data to (1.6), there cannot be no associated smooth solution
in any arbitrarily short interval of time. Similar illposedness statements for
smooth data compactly supported in $\mathbb{R}^{3}$ for (E-MHD) and (Hall-MHD)
will be handled in our forthcoming work [42].

Degenerate quasilinear dispersive PDE. The nondegeneracy condition of the
principal term was assumed in the aforementioned works on quasilinear
dispersive equations. (This is the condition (1) in the statement of Theorem
1.1 in our case.) When the principal term has a point of degeneracy, analysis
of the bicharacteristic curves near such a point shows a very rapid growth of
the frequency (for instance, see [32, 45, 2, 4, 3]) in general. An argument
involving generalized energy identities and testing against wavepackets
introduced in [44] provides a robust way of establishing norm growth of the
solutions to quasilinear degenerate dispersive PDE corresponding to the
behavior of bicharacteristics. This argument was then generalized to give
strong illposedness for various families of degenerate quasilinear dispersive
PDE in [45, 8], including degenerate versions of KdV and Schrödinger
equations. For some of these equations, local wellposedness (not in Sobolev or
Hölder-type spaces, but rather in a space adapted to the degeneracy) for
certain classes of degenerate initial data has been obtained in [32, 36] based
on the Lagrangian approach. Motivations for studying degenerate dispersive
equations can be found for instance in [32, 64, 63, 66, 65].

###  1.3. Structure of the paper

In Section 2, we collect some preliminaries for the whole paper, including
notation and conventions (Section 2.1), a precise formulation of the function
space framework (Section 2.2), review of pseudodifferential and
paradifferential operators (Sections 2.3), and the high frequency
Calderón–Vaillancourt trick (Section 2.4). In Section 3, we record some core
computations of this paper, namely a reformulation of (E-MHD) as an equation
for a perturbation $b$ of a background magnetic field and also the
introduction of the variables $b_{\pm}$, with respect to which the principal
term in the linearized E-MHD is diagonalized. In Section 4, we study the
properties of the Hamiltonian flow associated with the principal symbol of the
linearized system after diagonalization. Moreover, a quantitative formulation
of the assumptions in Theorem 1.1 is given in this section (Definition 4.3).
In Section 5, we give a precise version of the main theorem (Theorem 5.1) and
reduce its proof to a local smoothing estimate for the paralinearized system
(Proposition 5.4) and nonlinear estimates for the remainder (Proposition 5.5).
Then, after proving some relevant linear and nonlinear estimates in our
function space framework in Section 6, we prove Propositions 5.4 and 5.5 in
Section 7, which is the technical heart of our paper.

### Acknowledgements

I.-J. Jeong was supported by the Samsung Science and Technology Foundation
under Project Number SSTF-BA2002-04. S.-J. Oh was partially supported by a
Sloan Research Fellowship and a National Science Foundation CAREER Grant under
NSF-DMS-1945615.

##  2\. Preliminaries

###  2.1. Notation and conventions

We use the following notation in the paper.

  * •

$\mathbb{Z}_{\geq 0}$ denotes the set of nonnegative integers.

  * •

$\langle{\cdot}\rangle=(1+(\cdot)^{2})^{\frac{1}{2}}$.
$(\cdot)_{+}=\max\\{\cdot,0\\}$.

  * •

$\mathbf{1}_{E}$: characteristic function of $E$

  * •

$\chi_{>1}(\cdot),\chi_{>R}(\cdot)$: $\chi_{>1}(\cdot)$ is a smooth even
function such that $\chi_{>1}(s)=1$ for $|{s}|>1$, $\chi_{>1}(s)=0$ for
$|{s}|<\frac{1}{2}$ and nondecreasing on $(0,\infty)$.
$\chi_{>R}(\cdot)=\chi_{>1}(\tfrac{\cdot}{R})$

  * •

$\chi_{<1}(\cdot),\chi_{<R}(\cdot)$: $\chi_{<1}(\cdot)$ is a smooth even
function such that $\chi_{<1}(s)=1$ for $|{s}|<1$, $\chi_{<1}(s)=0$ for
$|{s}|>2$ and nonincreasing on $(0,\infty)$.
$\chi_{<R}(\cdot)=\chi_{<1}(\tfrac{\cdot}{R})$

#### Vector calculus and index notation

  * •

We use the vector calculus notation

$\nabla f=\begin{pmatrix}\partial_{1}f\\\ \partial_{2}f\\\
\partial_{3}f\end{pmatrix},\quad\nabla^{\top}f=\begin{pmatrix}\partial_{1}f&\partial_{2}f&\partial_{3}f\end{pmatrix},\quad\nabla\times{\bf
u}=\begin{pmatrix}\partial_{2}{\bf u}_{3}-\partial_{3}{\bf u}^{2}\\\
-\partial_{1}{\bf u}^{3}+\partial_{3}{\bf u}^{1}\\\ \partial_{1}{\bf
u}^{2}-\partial_{2}{\bf u}^{1}\end{pmatrix}.$

  * •

We also employ the index notation, where the plain greek letters
$\alpha,\beta,\gamma,\ldots$ are used for vector indices. As usual, repeated
upper and lower indices are always summed (unless otherwise specified), and we
raise and lower indices using the Kronecker delta
$\tensor{\delta}{{}^{\alpha}_{\beta}}$ (which is $1$ if $\alpha=\beta$ and $0$
otherwise). For computation, it will sometimes be useful to express a usual
vector calculus operation in the index notation. For instance, the curl is
expressed as

$(\nabla\times{\bf u})^{\alpha}=(\nabla\times)^{\alpha}_{\beta}{\bf
u}^{\beta}=\tensor{\epsilon}{{}^{\alpha\gamma}_{\beta}}\partial_{\gamma}{\bf
u}^{\beta},$

where $\tensor{\epsilon}{{}_{\alpha\beta\gamma}}$ is the Levi-Civita symbol
(totally antisymmetric with $\epsilon_{123}=1$).

  * •

We use the boldfaced greek letters
$\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\gamma},\ldots$ for multi-
indices.

#### Fourier analysis and Littlewood–Paley theory

  * •

Given a function $m:\mathbb{R}^{d}\to\mathbb{R}$, we write $m(D)$ for the
Fourier multiplier operator with multiplier $m(\xi)$; $D=i^{-1}\nabla$. For
$\sigma\in\mathbb{R}$, we write
$|{\nabla}|^{\sigma}=(-\Delta)^{\frac{\sigma}{2}}$.

  * •

In this paper, we consider the inhomogeneous Littlewood–Paley projections with
respect to the spatial frequency. Fix some
$\phi_{0}:\mathbb{R}_{+}\to\mathbb{R}$ be a non-negative, decreasing, smooth
function supported on $[0,2]$, which satisfies $\phi_{0}(\xi)=1$ for
$|\xi|\leq 1$. For $k\geq 1$, set
$\phi_{k}(\xi)=\phi_{0}(2^{-k}\xi)-\phi_{0}(2^{-k+1}\xi)$, and then we define
$P_{k}$ to be the Fourier multiplier operator with multiplier
$\phi_{k}(|\xi|)$ for each $k\geq 0$. We also write
$P_{<k}=\sum_{k^{\prime}<k}P_{k^{\prime}}$.
$P_{[k,k^{\prime})}=P_{<k^{\prime}}-P_{<k}$. Since we use the inhomogeneous
projections, it is assumed that the summation with respect to
$k,k^{\prime},\cdots$ is only for nonnegative integers.

###  2.2. Function spaces

Let $J$ be a time interval. Recall that $\mathcal{I}_{\ell}$ refers to a
partition of $\mathbb{R}$ into intervals of size $2^{\ell}$, and that
$\\{\chi_{I}\\}_{I\in\mathcal{I}_{\ell}}$ is a smooth partition of unity such
that $\operatorname{supp}\chi_{I}\subseteq 2I$. Note that for
$I,I^{\prime}\in\mathcal{I}_{\ell}$, either $I\cap I^{\prime}\neq\emptyset$ or
$\operatorname{dist}(I,I^{\prime})\gtrsim 2^{\ell}$.

The local smoothing (or local energy decay) norms are:

$\|{b}\|_{LE[J]}=\sup_{\ell\in\mathbb{Z}_{\geq
0}}\sup_{I\in\mathcal{I}_{\ell}}2^{-\frac{\ell}{2}}\|{u}\|_{L^{2}L^{2}(J\times\\{x^{3}\in
I\\})}$ (2.1)

$\|{g}\|_{LE^{\ast}[J]}=\inf_{g=\sum_{\ell\in\mathbb{Z}_{\geq
0}}g_{\ell}}\sum_{\ell\in\mathbb{Z}_{\geq
0}}\sum_{I\in\mathcal{I}_{\ell}}2^{\frac{\ell}{2}}\|{g_{\ell}}\|_{L^{2}L^{2}(J\times\\{x^{3}\in
I\\})}$ (2.2)

Then for each $k\in\mathbb{Z}_{\geq 0}$, we define the dyadic norms (i.e., a
family of norms indexed by $k\in\mathbb{Z}_{\geq 0}$) by

$\|{b}\|_{X_{k}[J]}=2^{\frac{k}{2}}\|{b}\|_{LE[J]}+\|{b}\|_{L^{\infty}L^{2}[J]}$
(2.3)
$\|{g}\|_{Y_{k}[J]}=\inf_{g=g_{1}+g_{2}}\left(2^{-\frac{k}{2}}\|{g_{1}}\|_{LE^{\ast}[J]}+\|{g_{2}}\|_{L^{1}L^{2}[J]}\right).$
(2.4)

Following [54], given any dyadic norm $\mathcal{X}_{k}$ and $r\in[1,\infty]$,
we define

$\displaystyle\|{b}\|_{\ell^{r}_{\mathcal{I}}\mathcal{X}_{k}[J]}$
$\displaystyle=\left(\sum_{I\in\mathcal{I}_{k}}\|{\chi_{I}(x^{3})b}\|_{\mathcal{X}_{k}[J]}^{r}\right)^{\frac{1}{r}},$
(2.5)

and for $r=\infty$, we replace the summation in $I\in\mathcal{I}_{k}$ by the
supremum as usual.

These dyadic spaces are slowly varying in the following sense:

######  Lemma 2.1 (Slow variance of $X_{k}$ and $Y_{k}$).

For any $k,k^{\prime}\in\mathbb{Z}_{\geq 0}$ and $1\leq r\leq\infty$, we have

$\displaystyle\|{b}\|_{X_{k}}$ $\displaystyle\lesssim
2^{\frac{1}{2}|{k-k^{\prime}}|}\|{b}\|_{X_{k^{\prime}}},$ (2.6)
$\displaystyle\|{b}\|_{\ell^{r}_{\mathcal{I}}X_{k}}$ $\displaystyle\lesssim
2^{\frac{3}{2}|{k-k^{\prime}}|}\|{b}\|_{\ell^{r}_{\mathcal{I}}X_{k^{\prime}}},$
(2.7) $\displaystyle\|{g}\|_{Y_{k}}$ $\displaystyle\lesssim
2^{\frac{1}{2}|{k-k^{\prime}}|}\|{g}\|_{Y_{k^{\prime}}},$ (2.8)
$\displaystyle\|{g}\|_{\ell^{r}_{\mathcal{I}}Y_{k}}$ $\displaystyle\lesssim
2^{\frac{3}{2}|{k-k^{\prime}}|}\|{g}\|_{\ell^{r}_{\mathcal{I}}Y_{k^{\prime}}}.$
(2.9)

Moreover, the following embedding properties hold:

######  Lemma 2.2.

For any $k\in\mathbb{Z}_{\geq 0}$, we have

$\displaystyle\|{b}\|_{\ell^{\infty}_{\mathcal{I}}X_{k}}\lesssim\|{b}\|_{X_{k}}\lesssim\|{b}\|_{\ell^{1}_{\mathcal{I}}X_{k}},$
(2.10)
$\displaystyle\|{g}\|_{\ell^{\infty}_{\mathcal{I}}Y_{k}}\lesssim\|{g}\|_{Y_{k}}\lesssim\|{g}\|_{\ell^{1}_{\mathcal{I}}Y_{k}}.$
(2.11)

We omit the proofs of Lemmas 2.1 and 2.2, which follow rather immediately from
the definitions.

For $s\in\mathbb{R}$, we define the summed-up spaces by

$\displaystyle\|{b}\|_{X^{s}[J]}^{2}$
$\displaystyle=\sum_{k}(2^{sk}\|{P_{k}b}\|_{X_{k}[J]})^{2},$
$\displaystyle\|{g}\|_{Y^{s}[J]}^{2}$
$\displaystyle=\sum_{k}(2^{sk}\|{P_{k}g}\|_{Y_{k}[J]})^{2},$
$\displaystyle\|{b}\|_{\ell^{r}_{\mathcal{I}}X^{s}[J]}^{2}$
$\displaystyle=\sum_{k}(2^{sk}\|{P_{k}b}\|_{\ell^{r}_{\mathcal{I}}X_{k}[J]})^{2},$
$\displaystyle\|{g}\|_{\ell^{r}_{\mathcal{I}}Y^{s}[J]}^{2}$
$\displaystyle=\sum_{k}(2^{sk}\|{P_{k}g}\|_{\ell^{r}_{\mathcal{I}}Y_{k}[J]})^{2},$

and so on. For $s>\frac{7}{2}$ and $r\in[1,\infty]$, observe that the
following string of embeddings hold:

$\displaystyle\ell^{r}X^{s}[J]\hookrightarrow\ell^{r}L^{\infty}H^{s}[J]\hookrightarrow\ell^{r}C_{t}C^{2,s-\frac{7}{2}}_{x}[J]\hookrightarrow
C_{t}C^{2,s-\frac{7}{2}}_{x}[J],$

where the third embedding is Morrey’s inequality. Observe also that the
following bound holds for any $s\in\mathbb{R}$ and $r\in[1,\infty]$:

$\|{b}\|_{\ell^{r}_{\mathcal{I}}X^{s}[J]}\lesssim\|{b}\|_{\ell^{r}_{\mathcal{I}}L^{\infty}H^{s}[J]}+|{J}|^{\frac{1}{2}}\|{b}\|_{\ell^{r}_{\mathcal{I}}L^{\infty}H^{s+\frac{1}{2}}[J]}.$
(2.12)

We omit the proofs of these straightforward bounds.

###  2.3. Pseudo- and paradifferential operators

In this subsection, we review the basic theory of _pseudo-_ and
_paradifferential operators_ that will be used in this paper.

####  2.3.1. Classical symbols and quantizations

Let $\mathfrak{a}(x,\xi)$ be a complex-valued smooth function on
$\mathbb{R}^{d}\times\mathbb{R}^{d}$. We say that $\mathfrak{a}$ is a (scalar-
valued) _classical symbol of order $m$_ $(m\in\mathbb{R})$, and write
$\mathfrak{a}\in S^{m}$, if

$[\mathfrak{a}]_{S^{m};N}:=\sum_{\boldsymbol{\alpha},\boldsymbol{\beta}:|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|\leq
N}\sup_{x,\xi}|{\langle{\xi}\rangle^{|{\boldsymbol{\beta}}|{-m}}\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\mathfrak{a}(x,\xi)}|<+\infty\hbox{
for every }N\geq 0.$ (2.13)

We also write $S^{-\infty}=\cap_{m\in\mathbb{R}}S^{m}$. Given a symbol
$\mathfrak{a}(x,\xi)\in S^{m}$, the corresponding _left_ (resp.  _right_)
_quantization_ is formally666The formal expressions of the form
$\iint(\cdots)e^{i\xi\cdot(x-y)}\,\mathrm{d}y\mathrm{d}\xi$ in this section
may be made precise by multiplying the integrand by $m_{\leq 0}(\epsilon
x)m_{\leq 0}(\epsilon\xi)$ and taking $\epsilon\to 0$, thanks to the
regularity bounds on the integrand originating from the classical symbol
bounds. We omit the standard details. defined as

$\displaystyle\operatorname{Op}^{(l)}(\mathfrak{a}(x,\xi))u=\mathfrak{a}(x,D)u$
$\displaystyle=\iint\mathfrak{a}(x,\xi)u(y)e^{i\xi\cdot(x-y)}\,\mathrm{d}y\frac{\mathrm{d}\xi}{(2\pi)^{d}},$
$\displaystyle\Big{(}\hbox{resp.}~{}\operatorname{Op}^{(r)}(\mathfrak{a}(x,\xi))u=\mathfrak{a}(D,x)u$
$\displaystyle=\iint\mathfrak{a}(y,\xi)u(y)e^{i\xi\cdot(x-y)}\,\mathrm{d}y\frac{\mathrm{d}\xi}{(2\pi)^{d}}\Big{)}$

for $u\in\mathcal{S}(\mathbb{R}^{d})$. Since we will mainly use left
quantization, and often simply write
$\operatorname{Op}(\mathfrak{a}(x,\xi))=\operatorname{Op}^{(l)}(\mathfrak{a}(x,\xi))$.
Note that if $a(x,\xi)$ splits into $a(x,\xi)=c(x)m(\xi)$, then $c(x)$ is
multiplied to the left (resp. right) of $m(D)$ under left (resp. right)
quantization, i.e.,

$\operatorname{Op}(\mathfrak{a}(x,\xi))=\operatorname{Op}^{(l)}(\mathfrak{a}(x,\xi))=c(x)m(D),\quad\operatorname{Op}^{(r)}(\mathfrak{a}(x,\xi)=m(D)c(x).$
(2.14)

The following basic $L^{2}$ boundedness and Gårding’s inequality are standard:

######  Proposition 2.3.

Let $\mathfrak{a}\in S^{0}$.

  1. (1)

($L^{2}$-boundedness) We have

$\|{\mathfrak{a}(x,D)}\|_{L^{2}\to L^{2}}\lesssim[\mathfrak{a}]_{S^{0};10d}.$

  2. (2)

(Gårding’s inequality) If, in addition, $\mathfrak{a}$ is real-valued and
$\mathfrak{a}\geq 0$, then

$\langle{\mathfrak{a}(x,D)u,u}\rangle\geq-C[\mathfrak{a}]_{S^{0};20d}\|{u}\|_{H^{-\frac{1}{2}}}^{2}.$

The same result holds for $\mathfrak{a}(D,x)$.

For a proof, see [70, Thm. 0.5.C] and [70, Prop. 0.7.A]. However, in this
paper (especially in Section 7), we will need a more precise understanding of
the dependence of $\|{\mathfrak{a}(x,D)}\|_{L^{2}\to L^{2}}$ on the symbol
bounds. For this purpose, we will rely on the Calderón–Vaillancourt theorem,
which is a more powerful $L^{2}$-boundedness theorem; see Section 2.4 below.

####  2.3.2. Paradifferential operators

To work in the low-regularity translation-invariant setting analogous to
Marzuola–Metcalfe–Tataru [54, 55, 56], we also employ _paradifferential
operators_. Since we also need to consider the composition and commutation of
such operators with classical pseudodifferential operators, it will be
beneficial to review some of their theory. A more systematic account of
paradifferential operators can be found in, e.g., [70].

Adopting the notation in [70], we define the symbol class
$\mathcal{B}S^{m}_{1,1}$ as

$\mathcal{B}S^{m}_{1,1}:=\\{\mathfrak{a}:[\mathfrak{a}]_{S^{m}_{1,1};N}<+\infty\hbox{
for all }N\geq
0,\,\operatorname{supp}\widehat{\mathfrak{a}}(\eta,\xi)\subseteq\\{|{\eta}|<\tfrac{1}{8}|{\xi}|\\}\\},$

where (formally)
$\widehat{\mathfrak{a}}(\eta,\xi)=\int\mathfrak{a}(x,\xi)e^{-i\eta\cdot
x}\,\mathrm{d}x$ and

$[\mathfrak{a}]_{S^{m}_{1,1};N}:=\sum_{\boldsymbol{\alpha},\boldsymbol{\beta}:|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|\leq
N}\sup_{x,\xi}|{\langle{\xi}\rangle^{-|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|-{m}}\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\mathfrak{a}(x,\xi)}|.$

An important example of a paradifferential operator is the _paraproduct_ ,
defined as follows: given $g\in L^{\infty}(\mathbb{R}^{d})$, the paraproduct
of $g$ and $u$ is

$T_{g}u=\sum_{k}P_{<k-10}gP_{k}u,$

or equivalently, $T_{g}=\operatorname{Op}(\sum_{k}P_{<k-10}g(x)P_{k}(\xi))$.
Clearly, $T_{g}\in\mathcal{B}S^{0}_{1,1}$ with

$\Big{[}\sum_{k}P_{<k-10}g(x)P_{k}(\xi)\Big{]}_{S^{0}_{1,1};N}\lesssim_{N}\|{g}\|_{L^{\infty}}.$

Another important way that paradifferential operators arise is by the
following decomposition. Given $\mathfrak{a}\in S^{m}$, consider the
decomposition $\mathfrak{a}=\mathfrak{a}^{\sharp}+\mathfrak{a}^{\flat}$, where

$\displaystyle\mathfrak{a}^{\sharp}(x,\xi)=\sum_{k}P_{<k-10}\mathfrak{a}(x,\xi)P_{k}(\xi),\quad\mathfrak{a}^{\flat}(x,\xi)=\sum_{k}P_{\geq
k-10}\mathfrak{a}(x,\xi)P_{k}(\xi).$

where $P_{<k-10}$ and $P_{\geq k-10}$ acts on the $x$-variable. It is easy to
verify that (for any $N,m_{0}\geq 0$)

$\displaystyle\mathfrak{a}^{\sharp}$
$\displaystyle\in\mathcal{B}S^{m}_{1,1}\hbox{ with
}[\mathfrak{a}^{\sharp}]_{S^{m}_{1,1};N}\leq[\mathfrak{a}]_{S^{m};N},$ (2.15)
$\displaystyle\mathfrak{a}^{\flat}$ $\displaystyle\in S^{-\infty}\hbox{ with
}[\mathfrak{a}^{\flat}]_{S^{m-m_{0}}_{1,1};N}\lesssim_{m,m_{0},N}[\mathfrak{a}]_{S^{m};N+m_{0}}.$
(2.16)

We have the following basic $L^{p}$ boundedness theorem for paradifferential
operators.

######  Proposition 2.4.

Let $\mathfrak{a}\in\mathcal{B}S^{0}_{1,1}$. Then for every $1<p<\infty$, we
have

$\|{\mathfrak{a}(x,D)}\|_{L^{p}\to
L^{p}}\lesssim[\mathfrak{a}]_{S^{0}_{1,1};10d}.$

The same result holds for $\mathfrak{a}(D,x)$.

For a proof, see [70, Prop. 3.4.F].

####  2.3.3. Symbolic calculus for pseudodifferential operators

We now recall the symbolic calculus for pseudodifferential operators. In our
argument, we need to consider composition (and commutation) of classical
pseudodifferential operators and paradifferential operators. Here, we record
precise formulas for the remainders.

We begin with composition. Let $\mathfrak{a}(x,\xi)$ and $\mathfrak{b}(x,\xi)$
be symbols that are either in $S^{m}$ or $\mathcal{B}S^{m}_{1,1}$ (with
possibly different orders $m$). Then the composition
$\mathfrak{a}(x,D)\mathfrak{b}(x,D)$ is again a left quantization of a
classical symbol, i.e.,

$\mathfrak{a}(x,D)\mathfrak{b}(x,D)=(\mathfrak{a}\circ\mathfrak{b})(x,D)$

where $(\mathfrak{a}\circ\mathfrak{b})(x,\xi)$ is formally given by

$(\mathfrak{a}\circ\mathfrak{b})(x,\xi)=e^{-i\xi\cdot
x}\mathfrak{a}(x,D)\mathfrak{b}(x,D)e^{i\xi\cdot(\cdot)}=\iint\mathfrak{a}(x,\eta)\mathfrak{b}(y,\xi)e^{i(\eta-\xi)\cdot(x-y)}\,\mathrm{d}y\frac{\mathrm{d}\eta}{(2\pi)^{d}}.$

By Taylor expansion, we have

$\begin{split}\mathfrak{a}(x,\eta)&=\mathfrak{a}(x,\xi)+\int_{0}^{1}(1-s)(\eta-\xi)_{a}(\partial_{\xi_{a}}\mathfrak{a})(x,s\eta+(1-s)\xi)\,\mathrm{d}s\end{split}$

and

$\displaystyle\mathfrak{a}(x,\eta)$
$\displaystyle=\mathfrak{a}(x,\xi)+(\eta-\xi)_{a}(\partial_{\xi_{a}}\mathfrak{a})(x,\xi)+\int_{0}^{1}(1-s)(\eta-\xi)_{a}(\eta-\xi)_{b}(\partial_{\xi_{a}}\partial_{\xi_{b}}\mathfrak{a})(x,s\eta+(1-s)\xi)\,\mathrm{d}s.$

Using
$(\eta-\xi)_{a}e^{i(\eta-\xi)\cdot(x-y)}=-i^{-1}\partial_{y^{a}}e^{i(\eta-\xi)\cdot(x-y)}$,
integrating $\partial_{y^{a}}$ by parts and noting that $\int
e^{i(\eta-\xi)\cdot(x-y)}\frac{\mathrm{d}\eta}{(2\pi)^{d}}=\delta_{0}(x-y)$,
we obtain the formal formulae

$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)$
(2.17)
$\displaystyle=-\iint\int_{0}^{1}(1-s)(\partial_{\xi_{a}}\mathfrak{a})(x,s\eta+(1-s)\xi)(\partial_{x^{a}}\mathfrak{b})(y,\xi)e^{i(\eta-\xi)\cdot(x-y)}\,\mathrm{d}s\mathrm{d}y\frac{\mathrm{d}\eta}{(2\pi)^{d}}$

and

$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)-i^{-1}\partial_{\xi_{a}}\mathfrak{a}(x,\xi)\partial_{x^{a}}\mathfrak{b}(x,\xi)$
(2.18)
$\displaystyle=-\iint\int_{0}^{1}(1-s)(\partial_{\xi_{a}}\partial_{\xi_{b}}\mathfrak{a})(x,s\eta+(1-s)\xi)(\partial_{x^{a}}\partial_{x^{b}}\mathfrak{b})(y,\xi)e^{i(\eta-\xi)\cdot(x-y)}\,\mathrm{d}s\mathrm{d}y\frac{\mathrm{d}\eta}{(2\pi)^{d}}.$

Based on these, the following result can be established.

######  Proposition 2.5 (Composition of pseudodifferential operators).

Let $\mathfrak{a}\in S^{m}$.

  1. (1)

Let $\mathfrak{b}\in S^{m^{\prime}}$. Then

$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)$ $\displaystyle\in
S^{m+m^{\prime}},$
$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)$
$\displaystyle\in S^{m+m^{\prime}-1},$
$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)-i^{-1}\partial_{\xi_{a}}\mathfrak{a}(x,\xi)\partial_{x^{a}}\mathfrak{b}(x,\xi)$
$\displaystyle\in S^{m+m^{\prime}-2},$

with

$\displaystyle[(\mathfrak{a}\circ\mathfrak{b})(x,\xi)]_{S^{m+m^{\prime}};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}[\mathfrak{b}]_{S^{m^{\prime}};N+10d},$
$\displaystyle[(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)]_{S^{m+m^{\prime}-1};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}[\mathfrak{b}]_{S^{m^{\prime}};N+10d},$
$\displaystyle[(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)-i^{-1}\partial_{\xi_{a}}\mathfrak{a}(x,\xi)\partial_{x^{a}}\mathfrak{b}(x,\xi)]_{S^{m+m^{\prime}-2};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}[\mathfrak{b}]_{S^{m^{\prime}};N+10d}.$

  2. (2)

Let $\mathfrak{b}\in\mathcal{B}S^{m^{\prime}}_{1,1}$ and

$\operatorname{supp}\widehat{\mathfrak{b}}(\eta,\xi)\subseteq\\{|{\eta}|<\tfrac{1}{16}|{\xi}|\\}.$
(2.19)

Then

$\displaystyle(\mathfrak{b}\circ\mathfrak{a})(x,\xi)-\mathfrak{b}(x,\xi)\mathfrak{a}(x,\xi)$
$\displaystyle\in\mathcal{B}S^{m+m^{\prime}-1}_{1,1}+S^{-\infty}.$

More precisely, the expression decomposes into
$\mathfrak{r}^{(m+m^{\prime}-1)\sharp}\in\mathcal{B}S^{m+m^{\prime}-1}_{1,1}$
and $\mathfrak{r}^{(m+m^{\prime}-1)\flat}\in S^{-\infty}$, where

$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-1)\sharp}]_{S^{m+m^{\prime}-1}_{1,1};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}[\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d},$
$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-1)\flat}]_{S^{m+m^{\prime}-1+m_{0}};N}$
$\displaystyle\lesssim_{m,m^{\prime},m_{0},N}[\mathfrak{a}]_{S^{m};N+m_{0}+10d}[\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d},$

Moreover, we have

$\displaystyle(\mathfrak{b}\circ\mathfrak{a})(x,\xi)-\mathfrak{b}(x,\xi)\mathfrak{a}(x,\xi)-i^{-1}\partial_{\xi_{a}}\mathfrak{b}(x,\xi)\partial_{x^{a}}\mathfrak{a}(x,\xi)$
$\displaystyle\in\mathcal{B}S^{m+m^{\prime}-2}_{1,1}+S^{-\infty}.$

More precisely, the expression decomposes into
$\mathfrak{r}^{(m+m^{\prime}-2)\sharp}\in\mathcal{B}S^{m+m^{\prime}-2}_{1,1}$
and $\mathfrak{r}^{(m+m^{\prime}-2)\flat}\in S^{-\infty}$, where

$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-2)\sharp}]_{S^{m+m^{\prime}-2}_{1,1};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}[\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d},$
$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-2)\flat}]_{S^{m+m^{\prime}-1+m_{0}};N}$
$\displaystyle\lesssim_{m,m^{\prime},m_{0},N}[\mathfrak{a}]_{S^{m};N+m_{0}+10d}[\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d}.$

  3. (3)

If
$\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}\in\mathcal{B}S^{m^{\prime}}_{1,1}$
for $|{\boldsymbol{\alpha}}|\leq 1$ and (2.19) holds, then

$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)$
$\displaystyle\in\mathcal{B}S^{m+m^{\prime}-1}_{1,1}+S^{-\infty}.$

More precisely, the expression decomposes into
$\mathfrak{r}^{(m+m^{\prime}-1)\sharp}\in\mathcal{B}S^{m+m^{\prime}-1}_{1,1}$
and $\mathfrak{r}^{(m+m^{\prime}-1)\flat}\in S^{-\infty}$, where

$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-1)\sharp}]_{S^{m+m^{\prime}-1}_{1,1};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}\sum_{\boldsymbol{\alpha}:|{\boldsymbol{\alpha}}|=1}[\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d},$
$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-1)\flat}]_{S^{m+m^{\prime}-1+m_{0}};N}$
$\displaystyle\lesssim_{m,m^{\prime},m_{0},N}[\mathfrak{a}]_{S^{m};N+m_{0}+10d}\sum_{\boldsymbol{\alpha}:|{\boldsymbol{\alpha}}|=1}[\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d},$

Moreover, if
$\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}\in\mathcal{B}S^{m^{\prime}}_{1,1}$
for $|{\boldsymbol{\alpha}}|\leq 2$ and (2.19) holds, then

$\displaystyle(\mathfrak{a}\circ\mathfrak{b})(x,\xi)-\mathfrak{a}(x,\xi)\mathfrak{b}(x,\xi)-i^{-1}\partial_{\xi_{a}}\mathfrak{a}(x,\xi)\partial_{x^{a}}\mathfrak{b}(x,\xi)$
$\displaystyle\in\mathcal{B}S^{m+m^{\prime}-2}_{1,1}+S^{-\infty},$

More precisely, the expression decomposes into
$\mathfrak{r}^{(m+m^{\prime}-2)\sharp}\in\mathcal{B}S^{m+m^{\prime}-2}_{1,1}$
and $\mathfrak{r}^{(m+m^{\prime}-2)\flat}\in S^{-\infty}$, where

$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-2)\sharp}]_{S^{m+m^{\prime}-2}_{1,1};N}$
$\displaystyle\lesssim_{m,m^{\prime},N}[\mathfrak{a}]_{S^{m};N+10d}\sum_{\boldsymbol{\alpha}:|{\boldsymbol{\alpha}}|=2}[\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d},$
$\displaystyle[\mathfrak{r}^{(m+m^{\prime}-2)\flat}]_{S^{m+m^{\prime}-1+m_{0}};N}$
$\displaystyle\lesssim_{m,m^{\prime},m_{0},N}[\mathfrak{a}]_{S^{m};N+m_{0}+10d}\sum_{\boldsymbol{\alpha}:|{\boldsymbol{\alpha}}|=2}[\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}]_{S^{m^{\prime}}_{1,1};N+10d}.$

Statement (1) is standard (see, e.g. [70, Prop. 0.3.C]), while Statements (2)
and (3) follow essentially from [70, Proof of Thm. 3.4.A]. These may also be
easily obtained by studying the explicit remainder (2.18) using stationary
phase.

Proposition 2.5 has immediate consequences for commutators and Poisson
brackets. Given two symbols $\mathfrak{a}$ and $\mathfrak{b}$, recall that the
commutator is
$[\mathfrak{b},\mathfrak{a}]=\mathfrak{b}\circ\mathfrak{a}-\mathfrak{a}\circ\mathfrak{b}$
and the Poisson bracket is defined by the symbol
$\\{\mathfrak{a},\mathfrak{b}\\}=(\partial_{\xi_{a}}\mathfrak{a}(x,\xi)\partial_{x^{a}}\mathfrak{b}(x,\xi)-\partial_{\xi_{a}}\mathfrak{b}(x,\xi)\partial_{x^{a}}\mathfrak{a}(x,\xi))$.
Then, as a consequence of Proposition 2.5 (1), we have the following
statement.

######  Corollary 2.6.

Let $\mathfrak{a}\in S^{m}$ and $\mathfrak{b}\in S^{m^{\prime}}$ be classical
symbols. Then, we have that

$\begin{split}[\mathfrak{a},\mathfrak{b}],\\{\mathfrak{a},\mathfrak{b}\\}\in
S^{m+m^{\prime}-1}\end{split}$

and

$\begin{split}[\mathfrak{a},\mathfrak{b}]-\frac{1}{i}\\{\mathfrak{a},\mathfrak{b}\\}\in
S^{m+m^{\prime}-2}.\end{split}$

For later use, it will be convenient to record the following specific case of
Proposition 2.5 (2)–(3), when one of the symbols takes the form of a
paraproduct.

######  Corollary 2.7.

Let $\mathfrak{a}$ be a classical symbol and $T_{g}=\sum_{k}P_{<k-10}gP_{k}$.
Then, we have

$\begin{split}\|{[\mathfrak{a},T_{g}]}\|_{L^{2}\to
L^{2}}\lesssim[\mathfrak{a}]_{S^{1};10d}\|{g}\|_{W^{1,\infty}}\end{split}$
(2.20)

and

$\begin{split}\|{[\mathfrak{a},T_{g}]-\frac{1}{i}\operatorname{Op}\\{\mathfrak{a},T_{g}\\}}\|_{L^{2}\to
L^{2}}\lesssim[\mathfrak{a}]_{S^{2};10d}\|{g}\|_{W^{2,\infty}}.\end{split}$
(2.21)

Next, we discuss the adjoint operation. Given a classical symbol
$\mathfrak{a}(x,\xi)$, it is not difficult to verify that

$\mathfrak{a}(x,D)^{\ast}=\mathfrak{a}^{\dagger}(D,x).$

This simple relation motivates the question of computing the symbol of a right
quantization $\mathfrak{a}(D,x)$ as a left-quantized pseudodifferential
operator. We have

$\mathfrak{a}(D,x)=\mathfrak{a}^{r\to\ell}(x,D),$

where $\mathfrak{a}^{r\to\ell}(x,\xi)$ is formally given by

$\displaystyle\mathfrak{a}^{r\to\ell}(x,\xi)=e^{-i\xi\cdot
x}\mathfrak{a}(D,x)e^{i\xi\cdot(\cdot)}=\iint\mathfrak{a}(y,\eta)e^{i(\eta-\xi)\cdot(x-y)}\,\mathrm{d}y\frac{\mathrm{d}\eta}{(2\pi)^{d}}.$

Using the expansion
$\mathfrak{a}(y,\eta)=\mathfrak{a}(y,\xi)+\int_{0}^{1}(\eta-\xi)^{\alpha}(\partial_{\xi_{\alpha}}\mathfrak{a})(y,s\eta+(1-s)\xi)\,\mathrm{d}s$
and proceeding as in the derivation of (2.18), we obtain the formal formula

$\displaystyle\mathfrak{a}^{r\to\ell}(x,\xi)-\mathfrak{a}(x,\xi)=\iint\int_{0}^{1}i^{-1}(\partial_{x^{a}}\partial_{\xi_{a}}\mathfrak{a})(y,s\eta+(1-s)\xi)e^{i(\eta-\xi)\cdot(x-y)}\,\mathrm{d}s\mathrm{d}y\frac{\mathrm{d}\eta}{(2\pi)^{d}}.$
(2.22)

######  Proposition 2.8 (Right to left quantization).

The following statements hold.

  1. (1)

If $\mathfrak{a}\in S^{m}$, then $\mathfrak{a}^{r\to\ell}-\mathfrak{a}\in
S^{m-1}$, with

$\displaystyle[\mathfrak{a}^{r\to\ell}-\mathfrak{a}]_{S^{m-1};N}\lesssim_{m,N}[\mathfrak{a}]_{S^{m};N+10d}.$

  2. (2)

If $\mathfrak{b},\partial_{x}\mathfrak{b}\in\mathcal{B}S^{m}_{1,1}$ with
(2.19), then
$\mathfrak{b}^{r\to\ell}-\mathfrak{b}\in\mathcal{B}S^{m-1}_{1,1}$, with

$\displaystyle[\mathfrak{b}^{r\to\ell}-\mathfrak{b}]_{S^{m-1}_{1,1};N}\lesssim_{m,N}\sum_{\boldsymbol{\alpha}:|{\boldsymbol{\alpha}}|=1}[\partial_{x}^{\boldsymbol{\alpha}}\mathfrak{b}]_{S^{m}_{1,1};N+10d}.$

Statement (1) is standard; see, e.g., [70, Prop. 0.3.B]. Statement (2) again
follows essentially from [70, Proof of Thm. 3.4.A]. It may also be easily
obtained by studying the explicit remainder (2.22) using stationary phase.

Although we will not need them in the sequel, we note that analogous formulas
where the left and right quantizations are swapped can be derived in a similar
manner.

###  2.4. $L^{2}$-boundedness, positivity and high frequency
Calderón–Vaillancourt trick

We recall the Calderón–Vaillancourt theorem, which is a key
$L^{2}$-boundedness theorem for pseudodifferential operators.

######  Theorem 2.9 (Calderón–Vaillancourt).

Let $\mathfrak{a}(x,\xi)\in C^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$
satisfy

$|{\nabla_{x}^{(n^{\prime})}\nabla_{\xi}^{(n)}\mathfrak{a}(x,\xi)}|\leq
1\quad\hbox{ for }|{n}|+|{n^{\prime}}|\leq 10d.$ (2.23)

Then $\mathfrak{a}(x,D)$ is bounded from $L^{2}(\mathbb{R}^{d})$ to
$L^{2}(\mathbb{R}^{d})$, with the norm bounded by a universal constant.

An important difference of this statement from Proposition 2.3 (1) is that
$\mathfrak{a}$ is not required to belong to the classical symbol class
$S^{m}$. We refer to [69] for a proof.

Consider now a smooth matrix-valued symbol $\mathfrak{a}$ obeying777One may of
course formulate (2.24) and Lemma 2.10 setting $A=B=1$ without losing any
generality. However, we choose the current formulation as we often work with
symbols that behave in a regular way when differentiated in $x$ or in $\xi$ as
in (2.24).

$|{\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\mathfrak{a}(x,\xi)}|\leq
c_{\boldsymbol{\alpha}\boldsymbol{\beta}}\lambda^{-|{\beta}|},\quad\operatorname{supp}\mathfrak{a}\subseteq\\{(x,\xi):\frac{1}{100}\lambda<|{\xi}|<100\lambda\\},$
(2.24)

for some constants $c_{\boldsymbol{\alpha}\boldsymbol{\beta}}$.

Of course, qualitatively, $\mathfrak{a}\in S^{0}$, so the standard $L^{2}$
boundedness theorem and sharp Gårding’s inequality apply. However, such
results do not give precise constants in relation to $\lambda$, and hence are
insufficient for our paper. Nevertheless, it turns out that the operator norm
of $\operatorname{Op}(\mathfrak{a})$ (in, say, $L^{2}$) can be directly
bounded by the size $c_{00}$ of the symbol $\mathfrak{a}$, as long as we
restrict the input to sufficiently high frequencies (which may depend on
$c_{\boldsymbol{\alpha}\boldsymbol{\beta}}$ with
$(\boldsymbol{\alpha},\boldsymbol{\beta})\neq(0,0)$). The proof is by a simple
scaling argument combined with Theorem 2.9.

######  Lemma 2.10 (High-frequency Calderón–Vaillancourt).

Let $\mathfrak{a}$ be a smooth symbol satisfying (2.24) for
$|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|\leq 100d$. If

$\lambda\geq\max_{1\leq|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|\leq
10d}\left(\frac{c_{\boldsymbol{\alpha}\boldsymbol{\beta}}}{c_{00}}\right)^{\frac{2}{|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|}},$
(2.25)

then, for some universal constant $C>0$,

$\|{\mathfrak{a}(x,D)}\|_{L^{2}\to L^{2}}\leq Cc_{00}.$ (2.26)

###### Proof.

Let us consider a rescaling $\tilde{\mathfrak{a}}$ of the symbol
$\mathfrak{a}$ of the form

$\tilde{\mathfrak{a}}(x,\xi)=c_{00}^{-1}\mathfrak{a}(\lambda_{0}^{-1}x,\lambda_{0}\xi).$

If we take $\lambda_{0}=\lambda^{\frac{1}{2}}$, then

$|{\partial_{x}^{\alpha}\partial_{\xi}^{\boldsymbol{\beta}}\tilde{\mathfrak{a}}(x,\xi)}|\leq\frac{c_{\boldsymbol{\alpha}\boldsymbol{\beta}}}{c_{00}}\left(\frac{1}{\lambda}\right)^{\frac{|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|}{2}}.$

Thus, if (2.25) holds, then $\tilde{\mathfrak{a}}$ obeys the hypothesis of
Theorem 2.9. Since $\|{\tilde{\mathfrak{a}}(x,D)}\|_{L^{2}\to
L^{2}}=c_{00}^{-1}\|{\mathfrak{a}(x,D)}\|_{L^{2}\to L^{2}}$, we see that
$\|{\mathfrak{a}(x,D)}\|_{L^{2}\to L^{2}}\leq Cc_{00}$ as desired. ∎

##  3\. Reformulation of (E-MHD)

###  3.1. Reformulation of (E-MHD) for perturbations of $\mathring{{\bf B}}$

Given a background magnetic field $\mathring{{\bf B}}$ with
$\nabla\cdot\mathring{{\bf B}}=0$, which may not necessarily be a solution to
(E-MHD), consider the decomposition of a solution ${\bf B}$ to (E-MHD) into
$\mathring{{\bf B}}$ and perturbation $b$:

${\bf B}=\mathring{{\bf B}}+b.$

In terms of $b$, (E-MHD) may be reformulated in the following way:

$\left\\{\begin{aligned} &\partial_{t}b+\nabla\times((\nabla\times
b)\times(\mathring{{\bf B}}+b))+\nabla\times((\nabla\times\mathring{{\bf
B}})\times b)=-\partial_{t}\mathring{{\bf B}}-\mathcal{P}(\mathring{{\bf
B}})\\\ &\nabla\cdot b=0,\end{aligned}\right.$ (3.1)

where $\mathcal{P}(\mathring{{\bf B}})$ is the nonlinearity of (E-MHD):

$\mathcal{P}(\mathring{{\bf B}}):=\nabla\times((\nabla\times\mathring{{\bf
B}})\times\mathring{{\bf B}})$

We call (3.1) the _perturbation equation around the background magnetic field
$\mathring{{\bf B}}$_. Note that the RHS vanishes if $\mathring{{\bf B}}$
solves (E-MHD). In this case, keeping only the linear part in $b$, we arrive
at the _linearized ( E-MHD) around the background solution $\mathring{{\bf
B}}$_:

$\left\\{\begin{aligned} &\partial_{t}b+\nabla\times((\nabla\times
b)\times\mathring{{\bf B}})+\nabla\times((\nabla\times\mathring{{\bf
B}})\times b)=0,\\\ &\nabla\cdot b=0.\end{aligned}\right.$ (3.2)

Motivated by (3.2), we introduce the following shorthand for the spatial part
of the linearized equation:

$\displaystyle\mathcal{L}_{\mathring{{\bf B}}}b=$
$\displaystyle\nabla\times((\nabla\times b)\times\mathring{{\bf
B}})+\nabla\times((\nabla\times\mathring{{\bf B}})\times b).$ (3.3)

We also introduce the decomposition $\mathcal{L}_{\mathring{{\bf
B}}}=\mathcal{P}_{\mathring{{\bf B}}}+\mathcal{Q}_{\mathring{{\bf B}}}$, where

$\mathcal{P}_{\mathring{{\bf B}}}b=\nabla\times((\nabla\times
b)\times\mathring{{\bf B}}),\quad\mathcal{Q}_{\mathring{{\bf
B}}}b=\nabla\times((\nabla\times\mathring{{\bf B}})\times b).$ (3.4)

Note that $\mathcal{P}_{\mathring{{\bf B}}}$ contains the principal term,
while $\mathcal{Q}_{\mathring{{\bf B}}}$ is lower order.

###  3.2. Diagonalization of the principal symbol

Replacing $\nabla$ by the symbol $i\xi$ in the term
$\mathcal{P}_{\mathring{{\bf B}}}b$ in (3.4), we see that the principal symbol
of the linearized (E-MHD) around $\mathring{{\bf B}}$ (i.e., (3.2)) takes the
form

$\tensor{({\bf p}_{\mathring{{\bf
B}}})}{{}^{\alpha}_{\beta}}(x,\xi)=-\tensor{\epsilon}{{}^{\gamma\alpha}_{\beta}}\mathring{{\bf
B}}^{\delta}\xi_{\delta}\xi_{\gamma}.$

Our goal in this subsection is to diagonalize ${\bf p}_{\mathring{{\bf B}}}$.
In what follows, we use the shorthand ${\bf p}={\bf p}_{\mathring{{\bf B}}}$.
Note that

$\displaystyle\tensor{{\bf p}}{{}^{\alpha}_{\gamma}}\tensor{{\bf
p}}{{}^{\gamma}_{\beta}}=$
$\displaystyle\tensor{\epsilon}{{}^{\delta\alpha}_{\gamma}}\tensor{\epsilon}{{}^{\delta^{\prime}\gamma}_{\beta}}(\mathring{{\bf
B}}^{\delta}\xi_{\delta})^{2}\xi_{\delta}\xi_{\delta^{\prime}}$
$\displaystyle=$
$\displaystyle(-\delta^{\delta\delta^{\prime}}\delta^{\alpha}_{\beta}+\delta^{\delta}_{\beta}\delta^{\alpha\delta^{\prime}})(\mathring{{\bf
B}}^{\delta}\xi_{\delta})^{2}\xi_{\delta}\xi_{\delta^{\prime}}$
$\displaystyle=$ $\displaystyle-(\mathring{{\bf
B}}^{\delta}\xi_{\delta})^{2}|{\xi}|^{2}\delta^{\alpha}_{\beta}+(\mathring{{\bf
B}}^{\delta}\xi_{\delta})^{2}\xi^{\alpha}\xi_{\beta},$

so that

$({\bf p}\mp i\mathring{{\bf B}}^{\alpha}\xi_{\alpha}|{\xi}|I)({\bf p}\pm
i\mathring{{\bf B}}^{\alpha}\xi_{\alpha}|{\xi}|I)=(\mathring{{\bf
B}}^{\alpha}\xi_{\alpha})^{2}\xi\otimes\xi.$

By simple linear algebra, we see that if we define

$\tensor{(\Pi_{0})}{{}^{\alpha}_{\beta}}=\frac{\xi^{\alpha}\xi_{\beta}}{|{\xi}|^{2}},\quad\tensor{(\Pi_{\pm})}{{}^{\alpha}_{\beta}}=\frac{1}{2}\left(\delta^{\alpha}_{\beta}\mp\frac{\tensor{\epsilon}{{}^{\gamma\alpha}_{\beta}}\xi_{\gamma}}{i|{\xi}|}-\frac{\xi^{\alpha}\xi_{\beta}}{|{\xi}|^{2}}\right),$

then

$\Pi_{s}^{2}=\Pi_{s},\quad\Pi_{s}\Pi_{s^{\prime}}=0,\quad
s,s^{\prime}\in\\{-,0,+\\},$

and $\Pi_{-}+\Pi_{0}+\Pi_{+}=I$. Moreover,

$\displaystyle{\bf p}(x,\xi)=$ $\displaystyle
ip(x,\xi)\Pi_{+}(\xi)-ip(x,\xi)\Pi_{-}(\xi),$ $\displaystyle p(x,\xi)=$
$\displaystyle p_{\mathring{{\bf B}}}(x,\xi):=\mathring{{\bf
B}}^{\alpha}\xi_{\alpha}|{\xi}|,$

which is the desired diagonalization of ${\bf p}_{\mathring{{\bf B}}}$. We
will refer to $p(x,\xi)=p_{\mathring{{\bf B}}}(x,\xi)$ as the _scalar
principal symbol_ of (3.2).

######  Remark 3.1.

The matrices $\Pi_{+}(\xi)$, $\Pi_{0}(\xi)$, $\Pi_{-}(\xi)$ are nothing but
eigenspace projections of the cross product $\xi\times$ viewed as an operator
on $\mathbb{C}^{3}(=\mathbb{R}^{3}\otimes_{\mathbb{R}}\mathbb{C})$.
Accordingly, the corresponding multipliers $\Pi_{+}(D)$, $\Pi_{0}(D)$ and
$\Pi_{-}(D)$ are $L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})$ projections that
diagonalize the curl operator $\nabla\times$. Since

$\tensor{(\Pi_{s})}{{}^{\alpha}_{\beta}}(\xi)=\overline{\tensor{(\Pi_{s})}{{}^{\alpha}_{\beta}}(-\xi)},\qquad
s\in\\{-,0,+\\},$

it follows that $\Pi_{s}(D):L^{2}(\mathbb{R}^{3};\mathbb{R}^{3})\to
L^{2}(\mathbb{R}^{3};\mathbb{R}^{3})$.

###  3.3. Linearized equation after diagonalization of the principal part

Motivated by the computation in Section 3.2, we return to the linearized
(E-MHD) equation (3.2),

$\left\\{\begin{aligned} &\partial_{t}b+\mathcal{L}_{\mathring{{\bf
B}}}b=g,\\\ &\nabla\cdot b=0,\end{aligned}\right.$

and compute the equation satisfied by the diagonalized variables
$b_{\pm}=\Pi_{\pm}(D)b$.

######  Proposition 3.2.

The system (3.2) is equivalent to the following system for the diagonalized
variables $b_{\pm}=\Pi_{\pm}(D)b$:

$\displaystyle\partial_{t}b_{\pm}\pm P_{\mathring{{\bf
B}}}^{(2)}b_{\pm}\pm\mathfrak{S}^{(1)}_{\mathring{{\bf
B}}}b_{\mp}\pm\mathfrak{A}^{(1)}_{\mathring{{\bf
B}}}b_{\pm}\pm\nabla\left(\mathfrak{U}^{(0)}_{\mathring{{\bf
B}}}(b_{\pm}+b_{\mp})\right)$ (3.5)
$\displaystyle\quad+(\nabla\times\mathring{{\bf B}})\cdot\nabla
b_{\pm}+\mathfrak{R}^{(0)}_{\mathring{{\bf B}};\pm}(b_{\pm}+b_{\mp})$
$\displaystyle=\Pi_{\pm}(D)g,$

or in the form of a system,

$\displaystyle\partial_{t}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}P_{\mathring{{\bf B}}}^{(2)}&0\\\
0&-P_{\mathring{{\bf B}}}^{(2)}\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}0&\mathfrak{S}^{(1)}_{\mathring{{\bf B}}}\\\
-\mathfrak{S}^{(1)}_{\mathring{{\bf B}}}&0\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}$ (3.6)
$\displaystyle+\begin{pmatrix}\mathfrak{A}^{(1)}_{\mathring{{\bf B}}}&0\\\
0&-\mathfrak{A}^{(1)}_{\mathring{{\bf B}}}\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+(\nabla\times\mathring{{\bf
B}})\cdot\nabla\begin{pmatrix}b_{+}\\\ b_{-}\end{pmatrix}$
$\displaystyle+\nabla\begin{pmatrix}\mathfrak{U}^{(0)}_{\mathring{{\bf
B}}}&\mathfrak{U}^{(0)}_{\mathring{{\bf B}}}\\\
-\mathfrak{U}^{(0)}_{\mathring{{\bf B}}}&-\mathfrak{U}^{(0)}_{\mathring{{\bf
B}}}\end{pmatrix}\begin{pmatrix}b_{+}\\\
b_{-}\end{pmatrix}+\begin{pmatrix}\mathfrak{R}^{(0)}_{\nabla\times\mathring{{\bf
B}};+}&\mathfrak{R}^{(0)}_{\nabla\times\mathring{{\bf B}};+}\\\
\mathfrak{R}^{(0)}_{\nabla\times\mathring{{\bf
B}};-}&\mathfrak{R}^{(0)}_{\nabla\times\mathring{{\bf
B}};-}\end{pmatrix}\begin{pmatrix}b_{+}\\\ b_{-}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\Pi_{+}(D)g\\\ \Pi_{-}(D)g\end{pmatrix},$

where $P^{(2)}_{\mathring{{\bf B}}}$ is the scalar 2nd order operator of the
form

$P^{(2)}_{\mathring{{\bf B}}}=\frac{1}{2}\left((\mathring{{\bf
B}}\cdot\nabla)|{\nabla}|+|{\nabla}|(\mathring{{\bf B}}\cdot\nabla)\right),$

$\mathfrak{S}^{(1)}_{\mathring{{\bf B}}}$ is a matrix-valued symmetric 1st
order operator of the form

$\tensor{(\mathfrak{S}^{(1)}_{\mathring{{\bf
B}}})}{{}^{\alpha}_{\beta}}=\frac{1}{2}\left(|{\nabla}|(\partial^{\alpha}\mathring{{\bf
B}}_{\beta})+(\partial_{\beta}\mathring{{\bf
B}}^{\alpha})|{\nabla}|\right)+\frac{1}{2}[|{\nabla}|,\mathring{{\bf
B}}\cdot\nabla]\delta^{\alpha}_{\beta},$

$\mathfrak{A}^{(1)}_{\mathring{{\bf B}}}$ is a matrix-valued anti-symmetric
1st order operator of the form

$\tensor{(\mathfrak{A}^{(1)}_{\mathring{{\bf
B}}})}{{}^{\alpha}_{\beta}}=\frac{1}{2}\left(|{\nabla}|(\partial^{\alpha}\mathring{{\bf
B}}_{\beta})-(\partial_{\beta}\mathring{{\bf B}}^{\alpha})|{\nabla}|\right),$

$\mathfrak{U}^{(0)}_{\mathring{{\bf B}}}$ is a covector-valued 0th order
operator of the form

$\displaystyle(\mathfrak{U}^{(0)}_{\mathring{{\bf B}}})_{\beta}=$
$\displaystyle\frac{1}{2}\frac{\partial_{\gamma}}{|{\nabla}|}(\partial^{\gamma}\mathring{{\bf
B}}_{\beta}+\partial_{\beta}\mathring{{\bf B}}^{\gamma}),$

and $\mathfrak{R}^{(0)}_{\nabla\times\mathring{{\bf B}};\pm}$ is a matrix-
valued 0th order operator of the form

$\tensor{(\mathfrak{R}^{(0)}_{\nabla\times\mathring{{\bf
B}};\pm})}{{}^{\alpha}_{\beta}}=[\tensor{\Pi_{\pm}(D)}{{}^{\alpha}_{\beta}},(\nabla\times\mathring{{\bf
B}})\cdot\nabla]-\tensor{\Pi_{\pm}(D)}{{}^{\alpha}_{\gamma}}(\partial_{\beta}(\nabla\times\mathring{{\bf
B}})^{\gamma}).$

We also make the obvious observation that each $b_{\pm}$ satisfies the
divergence-free condition

$\nabla\cdot b_{\pm}=0.$ (3.7)

###### Proof.

Note that

$\tensor{\Pi_{\pm}(D)}{{}^{\alpha}_{\beta}}=\frac{1}{2}\delta^{\alpha}_{\beta}\pm\frac{1}{2}|{\nabla}|^{-1}(\nabla\times)^{\alpha}_{\beta}+\frac{1}{2}|{\nabla}|^{-2}\nabla^{\alpha}\nabla_{\beta}.$

Recall the definition of $\mathcal{L}_{\mathring{{\bf B}}}$ in (3.3):

$\mathcal{L}_{\mathring{{\bf B}}}b=\underbrace{\nabla\times((\nabla\times
b)\times\mathring{{\bf B}})}_{\mathcal{P}_{\mathring{{\bf
B}}}b}+\underbrace{\nabla\times((\nabla\times\mathring{{\bf B}})\times
b)}_{\mathcal{Q}_{\mathring{{\bf B}}}b}$

We first compute

$\displaystyle\left(\Pi_{\pm}(D)\mathcal{P}_{\mathring{{\bf
B}}}b\right)^{\alpha}$
$\displaystyle=\left(\Pi_{\pm}(D)\left(\nabla\times((\nabla\times
b)\times\mathring{{\bf B}})\right)\right)^{\alpha}$
$\displaystyle=\left(\frac{1}{2}\delta^{\alpha}_{\beta}\pm\frac{1}{2}|{\nabla}|(\nabla\times)^{\alpha}_{\beta}\right)\left(\nabla\times((\nabla\times
b)\times\mathring{{\bf B}})\right)^{\beta}$
$\displaystyle=\frac{1}{2}\left(\mathring{{\bf
B}}\cdot\nabla(\nabla\times)^{\alpha}_{\beta}b^{\beta}-(\partial_{\gamma}\mathring{{\bf
B}}^{\alpha})(\nabla\times)^{\gamma}_{\beta}b^{\beta}\right)$
$\displaystyle\mathrel{\phantom{=}}\pm\frac{1}{2}|{\nabla}|^{-1}(-\Delta)\left((\mathring{{\bf
B}}\cdot\nabla)b^{\alpha}-\mathring{{\bf
B}}_{\beta}\partial^{\alpha}b^{\beta}\right)$
$\displaystyle\mathrel{\phantom{=}}\pm\frac{1}{2}|{\nabla}|^{-1}\partial^{\alpha}\partial_{\gamma}\left((\mathring{{\bf
B}}\cdot\nabla)b^{\gamma}-\mathring{{\bf
B}}_{\beta}\partial^{\gamma}b^{\beta}\right)$
$\displaystyle=\frac{1}{2}\left(\mathring{{\bf
B}}\cdot\nabla(\nabla\times)^{\alpha}_{\beta}b^{\beta}\pm|{\nabla}|(\mathring{{\bf
B}}\cdot\nabla)\delta^{\alpha}_{\beta}b^{\beta}\right)$
$\displaystyle\mathrel{\phantom{=}}+\frac{1}{2}\left(-(\partial_{\gamma}\mathring{{\bf
B}}^{\alpha})(\nabla\times)^{\gamma}_{\beta}b^{\beta}\pm|{\nabla}|(\partial^{\alpha}\mathring{{\bf
B}}_{\gamma})\delta^{\gamma}_{\beta}b^{\beta}\right)$
$\displaystyle\mathrel{\phantom{=}}\pm\frac{1}{2}\partial^{\alpha}|{\nabla}|^{-1}\left(\Delta(\mathring{{\bf
B}}_{\beta}b^{\beta})+(\partial_{\beta}\mathring{{\bf
B}}^{\gamma})\partial_{\gamma}b^{\beta}-\partial_{\gamma}(\mathring{{\bf
B}}_{\beta}\partial^{\gamma}b^{\beta})\right)$
$\displaystyle=\frac{1}{2}\left(\mathring{{\bf B}}\cdot\nabla|{\nabla}|(\pm
b_{\pm}\mp b_{\mp})^{\alpha}\pm|{\nabla}|(\mathring{{\bf
B}}\cdot\nabla)(b_{\pm}+b_{\mp})^{\alpha}\right)$
$\displaystyle\mathrel{\phantom{=}}+\frac{1}{2}\left(-(\partial_{\beta}\mathring{{\bf
B}}^{\alpha})|{\nabla}|(\pm b_{\pm}\mp
b_{\mp})^{\beta}\pm|{\nabla}|(\partial^{\alpha}\mathring{{\bf
B}}_{\beta})(b_{\pm}+b_{\mp})^{\beta}\right)$
$\displaystyle\mathrel{\phantom{=}}\pm\partial^{\alpha}\left(\frac{1}{2}\frac{\partial_{\gamma}}{|{\nabla}|}\left((\partial^{\gamma}\mathring{{\bf
B}}_{\beta}+\partial_{\beta}\mathring{{\bf
B}}^{\gamma})(b_{\pm}+b_{\mp})^{\beta}\right)\right)$

After rearranging terms, we obtain

$\displaystyle\left(\Pi_{\pm}(D)\mathcal{P}_{\mathring{{\bf
B}}}b\right)^{\alpha}$ $\displaystyle=\pm\frac{1}{2}(\mathring{{\bf
B}}\cdot\nabla|{\nabla}|+|{\nabla}|(\mathring{{\bf
B}}\cdot\nabla))b_{\pm}^{\alpha}$
$\displaystyle\mathrel{\phantom{=}}\pm\frac{1}{2}[|{\nabla}|,\mathring{{\bf
B}}\cdot\nabla]b_{\mp}^{\alpha}$
$\displaystyle\mathrel{\phantom{=}}\pm\frac{1}{2}\left(|{\nabla}|(\partial^{\alpha}\mathring{{\bf
B}}_{\beta})-(\partial_{\beta}\mathring{{\bf
B}}^{\alpha})|{\nabla}|\right)b_{\pm}^{\beta}$
$\displaystyle\mathrel{\phantom{=}}\pm\frac{1}{2}\left(|{\nabla}|(\partial^{\alpha}\mathring{{\bf
B}}_{\beta})+(\partial_{\beta}\mathring{{\bf
B}}^{\alpha})|{\nabla}|\right)b_{\mp}^{\beta}$
$\displaystyle\mathrel{\phantom{=}}\pm\partial^{\alpha}\left(\frac{1}{2}\frac{\partial_{\gamma}}{|{\nabla}|}\left((\partial^{\gamma}\mathring{{\bf
B}}_{\beta}+\partial_{\beta}\mathring{{\bf
B}}^{\gamma})(b_{\pm}+b_{\mp})^{\beta}\right)\right)$

On the other hand,

$\displaystyle\left(\Pi_{\pm}(D)\mathcal{Q}_{\mathring{{\bf
B}}}b\right)^{\alpha}$
$\displaystyle=\left(\Pi_{\pm}(D)\left(\nabla\times((\nabla\times\mathring{{\bf
B}})\times b)\right)\right)^{\alpha}$
$\displaystyle=\tensor{\Pi_{\pm}(D)}{{}^{\alpha}_{\beta}}\left((\nabla\times\mathring{{\bf
B}})\cdot\nabla b^{\beta}-(\partial_{\gamma}(\nabla\times\mathring{{\bf
B}})^{\alpha})b^{\gamma}\right)$ $\displaystyle=(\nabla\times\mathring{{\bf
B}})\cdot\nabla b_{\pm}^{\alpha}$
$\displaystyle\mathrel{\phantom{=}}+\tensor{[\Pi_{\pm}(D)}{{}^{\alpha}_{\beta}},(\nabla\times\mathring{{\bf
B}})\cdot\nabla](b_{\pm}+b_{\mp})^{\beta}$
$\displaystyle\mathrel{\phantom{=}}-\tensor{\Pi_{\pm}(D)}{{}^{\alpha}_{\gamma}}(\partial_{\beta}(\nabla\times\mathring{{\bf
B}})^{\gamma})(b_{\pm}+b_{\mp})^{\beta},$

which completes the proof. ∎

##  4\. Analysis of bicharacteristics for linearized equations

In this section, we assume that ${\bf B}_{0}$ is a smooth, time-independent
and divergence-free vector field on $\mathbb{R}^{3}$. In particular, it need
not be a solution to (E-MHD).

###  4.1. Basic properties of the bicharacteristic flow

The Hamiltonian flow equation associated to the scalar symbol $p_{{\bf
B}_{0}}(x,\xi)$ is

$\left\\{\begin{aligned} \dot{X}^{\alpha}=&\partial_{\xi_{\alpha}}p_{{\bf
B}_{0}}(X,\Xi)={\bf
B}_{0}^{\beta}(X)|{\Xi}|\left(\delta^{\alpha}_{\beta}+\frac{\Xi_{\beta}\Xi_{\alpha}}{|{\Xi}|^{2}}\right),\\\
\dot{\Xi}_{\alpha}=&-\partial_{x^{\alpha}}p_{{\bf
B}_{0}}(X,\Xi)=-\partial_{\alpha}{\bf
B}_{0}^{\beta}(X)\Xi_{\beta}|{\Xi}|,\end{aligned}\right.$ (4.1)

where $\alpha=1,2,3$. An integral curve $(X,\Xi)(t)$ of this flow is called a
_bicharacteristic_ ; its projection $X(t)$ to $M$ is called a _projected
characteristic curve_.

We begin by identifying the possible set of directions of $X(t)$ in relation
to ${\bf B}_{0}(X(t))$.

######  Lemma 4.1 (Cone of directions around ${\bf B}_{0}$).

Along any bicharacteristic $(X,\Xi)(t)$ associated with $\pm p_{{\bf B}_{0}}$,
we have

$|{\angle(\dot{X}(t),\pm{\bf B}_{0}(X(t)))}|\leq\tan^{-1}\frac{1}{2\sqrt{2}}.$
(4.2)

In particular, we have

$|{{\bf B}_{0}(X(t))}||{\Xi(t)}|\leq|{\dot{X}(t)}|.$ (4.3)

Moreover, if $|{{\bf B}_{0}-{\bf e}_{3}}|<\frac{1}{2}$, then

$2\max\\{|{\dot{X}^{1}(t)}|,|{\dot{X}^{2}(t)}|\\}\leq|{\Xi(t)}|\leq{12}|{\dot{X}^{3}(t)}|.$
(4.4)

###### Proof.

Without loss of generality, we fix the sign $\pm p_{{\bf B}_{0}}=+p_{{\bf
B}_{0}}$ and consider $t=0$. Let $(x,\xi)=(X,\Xi)(0)$. Since the case ${\bf
B}_{0}(x)=0$ is trivial, we may assume that ${\bf B}_{0}(x)\neq 0$. Rotate the
(local) coordinate axes so that ${\bf B}_{0}(x)$ is aligned with ${\bf
e}_{3}=\partial_{x^{3}}$; by a further rotation about the (new) $x^{3}$-axis,
we may also set $\xi_{1}=0$. Clearly, $|{\angle(\dot{X}(0),{\bf
B}_{0}(x))}|<\frac{\pi}{2}$ and

$\tan\angle(\dot{X}(0),{\bf
B}_{0}(x))=\frac{\dot{X}^{2}(0)}{\dot{X}^{3}(0)}=\frac{\xi_{2}\xi_{3}}{\xi_{2}^{2}+2\xi_{3}^{2}}.$

By symmetry, it suffices to compute the maximum of the last expression. By
elementary calculus, we have

$\max\frac{\xi_{2}\xi_{3}}{\xi_{2}^{2}+2\xi_{3}^{2}}=\max\frac{(\xi_{3}/\xi_{2})}{1+2(\xi_{3}/\xi_{2})^{2}}=\frac{1}{2\sqrt{2}},$

where the maximum occurs at $\xi_{3}/\xi_{2}=\frac{1}{\sqrt{2}}$. Next, (4.3)
follows by the computation

$|{{\bf B}_{0}(x)}|^{2}|{\xi}|^{2}\leq|{{\bf
B}_{0}(x)}|^{2}|{\xi}|^{2}\frac{(\xi_{2}^{2}+2\xi_{3}^{2})^{2}+\xi_{2}^{2}\xi_{3}^{2}}{|{\xi}|^{2}}=|{\dot{X}(0)}|^{2}.$

Finally, if $|{{\bf B}_{0}(x)-{\bf e}_{3}}|<\frac{1}{2}$, then

$|{\xi}|\leq 2(1+{\bf
B}_{0}^{3}(x))|{\xi}|\frac{|{\xi}|^{2}+\xi_{3}^{2}}{|{\xi}|^{2}}{\leq 6{\bf
B}_{0}^{3}(x)|{\xi}|\frac{|{\xi}|^{2}+\xi_{3}^{2}}{|{\xi}|^{2}}\leq
12\dot{X}^{3}(0)},$

which proves the second inequality of (4.4). The first inequality of (4.4) is
straightforward to prove. ∎

Next, we derive an evolution equation for the magnitude of the canonical
momentum $\Xi$.

######  Lemma 4.2 (Formula for $\frac{\mathrm{d}}{\mathrm{d}t}|{\Xi}|$).

Along any bicharacteristic $(X,\Xi)(t)$ associated with $\pm p_{{\bf B}_{0}}$,

$\frac{\mathrm{d}}{\mathrm{d}t}|{\Xi}|=\mp{}^{({\bf
B}_{0})}\pi^{\alpha\beta}(X)\Xi_{\alpha}\Xi_{\beta},$ (4.5)

where ${}^{({\bf B}_{0})}\pi^{\alpha\beta}=\frac{1}{2}(\partial^{\alpha}{\bf
B}_{0}^{\beta}+\partial^{\beta}{\bf B}_{0}^{\alpha})$.

###### Proof.

For simplicity, we fix the sign $\pm p_{{\bf B}_{0}}=+p_{{\bf B}_{0}}$. Using
the Hamiltonian flow equation, we compute

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}|{\Xi}|^{2}=2\delta^{\alpha\beta}\Xi_{\alpha}\dot{\Xi}_{\beta}=-2\delta^{\alpha\beta}\xi_{\alpha}\partial_{\beta}{\bf
B}_{0}^{\gamma}(X)\Xi_{\gamma}|{\Xi}|=-(\nabla^{\alpha}{\bf
B}_{0}^{\gamma}+\nabla^{\gamma}{\bf
B}_{0}^{\alpha})(X)\Xi_{\alpha}\Xi_{\gamma}|{\Xi}|.$

Recalling the definition ${}^{({\bf
B}_{0})}\pi^{\alpha\beta}=\frac{1}{2}(\nabla^{\alpha}{\bf
B}_{0}^{\beta}+\nabla^{\beta}{\bf B}_{0}^{\alpha})$, (4.5) follows. ∎

###  4.2. Quantification of the assumptions

We are now ready to make the qualitative hypotheses in Theorem 1.1
quantitative – these will be the precise parameters on which the lower bound
on the lifespan $T$ depends. The following definition is central to the
precise formulation of our local wellposedness result.

######  Definition 4.3 (Quantification of asymptotic uniformity, nondegeneracy
and nontrapping).

Let $s>\frac{7}{2}$ and $R\geq 1$. Furthermore, let $\epsilon>0$ satisfy
$\epsilon<\min\\{\epsilon_{0},\alpha_{1}M\\}$ where
$\epsilon_{0},\alpha_{1}>0$ are some absolute constants that will be fixed
later. We say that ${\bf B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ belongs to
the space $\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ if the following holds:

  1. (1)

Size bound.

$\|{{\bf B}_{0}-{\bf e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<M,$ (4.6)

  2. (2)

Nondegeneracy.

$\inf_{\mathbb{R}^{3}}|{{\bf B}_{0}}|>\mu,$ (4.7)

  3. (3)

Takeuchi–Mizohota-type condition. For every nontrivial bicharacteristic
$(X,\Xi)(t)$ associated with $p_{{\bf B}_{0}}$, we have

$\int_{-\infty}^{\infty}|{\nabla{\bf B}_{0}(X(t))}||{\Xi(t)}|\,\mathrm{d}t<A,$
(4.8)

  4. (4)

$\epsilon$-asymptotic uniformity.

$\displaystyle\|{\chi_{>R}(|{x^{3}}|)({\bf B}_{0}-{\bf
e}_{3})}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<\epsilon,$ (4.9)

  5. (5)

$R$-nontrapping. For every nontrivial bicharacteristic $(X,\Xi)(t)$ associated
with $p_{{\bf B}_{0}}$, we have

$\int_{-\infty}^{\infty}\mathbf{1}_{\\{-2R<x^{3}<2R\\}}(X(t))|{\dot{X}(t)}|\,\mathrm{d}t<L.$
(4.10)

######  Remark 4.4 (On the definition of
$\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$).

Note that $M$, $\mu$ and $A$ are independent of the choice of $\epsilon$,
whereas $R$ and $L$ depend on $\epsilon$. Observe also that $\mu$, $A$, $R$
and $L$ are _not_ simply controlled in terms of the size bound $M$ in (4.6)
unless $M$ is sufficiently small. For this reason, we interpret these
parameters as quantifying various properties of ${\bf B}_{0}$ in the large
data regime. In our precise formulation of the main theorem, we will fix
$\epsilon$ depending on $M$, $\mu$ and $A$, and the time of existence $T$ will
depend only on the (resulting) parameters $s$, $M$, $\mu$, $A$, $R$, $L$ for
the initial data ${\bf B}_{0}$.

Up to a choice of a small parameter $\epsilon$, Definition 4.3 makes the
qualitative assumptions for ${\bf B}_{0}$ in Theorem 1.1 quantitative, as the
following lemma and corollary show.

######  Lemma 4.5 (Qualitative assumptions imply quantitative assumptions).

Let ${\bf B}_{0}\in{\bf e}_{3}+\ell^{1}_{\mathcal{I}}H^{s}$ satisfy (4.6) for
some $M$, as well as the hypotheses of Theorem 1.1. Then the following
statements hold.

  1. (1)

There exists $\mu>0$ such that (4.7) holds.

  2. (2)

Given any $\epsilon>0$ there exist $R$ and $L$ such that (4.9) and (4.10)
hold.

  3. (3)

There exists a constant $\epsilon_{0}=\epsilon_{0}(s)>0$ such that if (4.9)
and (4.10) hold with $\epsilon<\epsilon_{0}$, then

$\displaystyle\|{\chi_{>R}(|{x^{3}}|)({\bf B}_{0}-{\bf
e}_{3})}\|_{L^{1}_{x^{3}}L^{\infty}_{x^{1},x^{2}}\cap
L^{\infty}}+\|{\chi_{>R}(|{x^{3}}|)\nabla{\bf
B}_{0}}\|_{L^{1}_{x^{3}}L^{\infty}_{x^{1},x^{2}}}+\|{\chi_{>R}(|{x^{3}}|)\nabla^{2}{\bf
B}_{0}}\|_{L^{\infty}}<\frac{1}{100},$ (4.11)
$\displaystyle\int_{-\infty}^{\infty}|{\nabla{\bf
B}_{0}(X(t))}||{\Xi(t)}|\,\mathrm{d}t\lesssim M(1+\mu^{-1}L).$ (4.12)

######  Corollary 4.6.

Let ${\bf B}_{0}\in{\bf e}_{3}+\ell^{1}_{\mathcal{I}}H^{s}$ satisfy (4.6) for
some $M$, as well as the hypotheses of Theorem 1.1. Then the following
statements hold.

  1. (1)

There exist $\mu,A>0$ such that (4.7) and (4.8) hold.

  2. (2)

Given any $\epsilon>0$, there exist $R,L>0$ such that ${\bf
B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$.

We omit the proof of the corollary, which is obvious.

###### Proof of Lemma 4.5.

Statements (1) and (2) follow from
$\limsup_{R\to\infty}\|{\chi_{>R}(x^{3})({\bf B}_{0}-{\bf
e}_{3})}\|_{\ell^{1}H^{s}}=0$ (in view of (4.6)), as well as the nondegeneracy
and the nontrapping assumptions. For Statement (3), we choose $\epsilon_{0}$
sufficiently small compared to the constant in the Sobolev embedding
$H^{s-2}\hookrightarrow L^{\infty}$ so that (4.11) holds. To establish (4.12),
note that

$\displaystyle{\int_{-\infty}^{\infty}\mathbf{1}_{\\{-R\leq x^{3}\leq
R\\}}(X(t))}|{\nabla{\bf
B}_{0}(X(t))}||{\Xi(t)}|\,\mathrm{d}t\lesssim\mu^{-1}M{\int_{-\infty}^{\infty}\mathbf{1}_{\\{-R\leq
x^{3}\leq R\\}}(X(t))}|{\dot{X}(t)}|\,\mathrm{d}t\lesssim\mu^{-1}ML.$

where we used (4.3) in the first inequality. On the other hand, in
$\\{x^{3}>{R}\\}$, we use (4.4) (which is possible thanks to (4.11)) to
estimate

$\displaystyle{\int_{-\infty}^{\infty}\mathbf{1}_{\\{R<x^{3}\\}}(X(t))}|{\nabla{\bf
B}_{0}(X(t))}||{\Xi(t)}|\,\mathrm{d}t\lesssim\sum_{I\in\mathcal{I}:I\cap(R,\infty)\neq\emptyset}\int_{I}\|{\nabla{\bf
B}_{0}(x^{1},x^{2},x^{3})}\|_{L^{\infty}_{x^{1},x^{2}}}\,\mathrm{d}x^{3}\lesssim
M.$

The region $\\{x^{3}<{-R}\\}$ is treated similarly, which completes the proof
of (4.12). ∎

We also record some consequences of ${\bf
B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$.

######  Lemma 4.7.

Let $s>\frac{7}{2}$, $M>0$, $\mu>0$, $\epsilon>0$, $R>0$, $L>0$ and ${\bf
B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$. Provided that
$\epsilon<\epsilon_{0}$ (where $\epsilon_{0}$ is as in Lemma 4.5.(3)), we have

$\displaystyle\sup_{t,t^{\prime}\in\mathbb{R}}\frac{|{\Xi(t)}|}{|{\Xi(t^{\prime})}|}\lesssim
A,$ (4.13)
$\displaystyle\frac{1}{12}|{\Xi(t)}|\leq|{\dot{X}^{3}(t)}|\leq|{\dot{X}(t)}|\leq
2|{\Xi(t)}|\qquad\hbox{ if }X^{3}(t)>R\hbox{ or }X^{3}(t)<-R,$ (4.14)
$\displaystyle\int_{0}^{t}|{\dot{X}(t^{\prime})}|\,\mathrm{d}t^{\prime}$
$\displaystyle\lesssim\begin{cases}X^{3}(t)-X^{3}(0)&\hbox{ if
}X^{3}(0)>R\hbox{ or }X^{3}(t)<-R,\\\
L+(|{X^{3}(t)}|-R)_{+}+(|{X^{3}(0)}|-R)_{+}&\hbox{ otherwise},\end{cases}$
$\displaystyle\lesssim L+(|{X^{3}(0)}|-R)_{+}+(|{X^{3}(t)}|-R)_{+}.$ (4.15)

###### Proof.

Estimate (4.13) follows from Lemma 4.2 and (4.8). Estimate (4.14) follows from
Lemma 4.1 and (4.11). To establish (4.15), note that $X(t^{\prime})$ for
$0\leq t^{\prime}\leq t$ does not intersect $\\{-R<x^{3}<R\\}$ if $X^{3}(0)>R$
or $X^{3}(t)<-R$. Now (4.15) follows from (4.14) and (4.10). ∎

Importantly, the class $\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ is stable
under small perturbations of ${\bf B}_{0}$ in $\ell^{1}_{\mathcal{I}}H^{s}$.

######  Proposition 4.8 ($\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ is stable).

Let $s>\frac{7}{2}$, $M>0$, $\mu>0$, $A>0$, $\epsilon>0$, $R\geq 1$, $L>0$ and
$\overline{{\bf B}}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$. Then the
following holds:

  1. (1)

$\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ is an open subset of
$\mathcal{B}^{s}(M)$.

  2. (2)

Moreover, there exist $c(s,M,\mu,A),C(s,M,\mu)>0$ and if
$\epsilon<\frac{1}{2}\epsilon_{0}$ (where $\epsilon_{0}$ is as in Lemma 4.5)
and ${\bf B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfies

$\|{{\bf B}_{0}-\overline{{\bf
B}}_{0}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<c(s,M,\mu,A,\epsilon)e^{-C(s,M,\mu,A)L}$

then ${\bf
B}_{0}\in\mathcal{B}^{s}_{2\epsilon}(2M,\frac{1}{2}\mu,2{\max}\\{A,M\\},R,2L)$.

We defer the proof of this proposition until the end of Section 4.3, as we
need to develop some tools for controlling the variation of the
bicharacteristics when perturbing ${\bf B}_{0}$.

We also formulate and prove a lemma that will be needed in the proof of
Theorem 5.1 (in particular, persistence of regularity).

######  Lemma 4.9.

For $s>\frac{7}{2}$, $0<\epsilon<\epsilon_{0}$ (where $\epsilon_{0}$ is as in
Lemma 4.5.(3)) and $M,\mu,A,R,L>0$, suppose that ${\bf B}\in
C_{t}([0,T];\ell^{1}_{\mathcal{I}}H^{s})$ and ${\bf
B}(t)\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ for all $t\in[0,T]$. Then for
any $\epsilon^{\prime}>0$, there exist $R^{\prime},L^{\prime}>0$, which are
independent of $t$, such that ${\bf
B}(t)\in\mathcal{B}^{s}_{\epsilon^{\prime}}(M,\mu,A,R^{\prime},L^{\prime})$
for all $t\in[0,T]$.

###### Proof.

The case $\epsilon^{\prime}\geq\epsilon$ is trivial, so it suffices to assume
that $\epsilon^{\prime}<\epsilon$. The existence of a $t$-independent
$R^{\prime}$ such that (4.9) holds (with $\epsilon^{\prime}$ and $R^{\prime}$)
for all $t\in[0,T]$ follows from the continuity of the function
$\|{\chi_{>R}(x^{3}){\bf B}(t)}\|_{\ell^{1}_{\mathcal{I}}H^{s}}$ in $R$ and
$t$. Then, in view of Lemma 4.7, it follows that
$L^{\prime}=L+C(R^{\prime}-R)_{+}$ for some $C>0$ works. ∎

###  4.3. Variation of bicharacteristics

Here we study the linearization of (4.1) with forcing terms $F,G$:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}X^{\prime}\\\
\Xi^{\prime}\end{pmatrix}=\begin{pmatrix}\nabla_{x}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}}&\nabla_{\xi}^{\top}\nabla_{\xi}p_{{\bf B}_{0}}\\\
-\nabla_{x}^{\top}\nabla_{x}p_{{\bf
B}_{0}}&-\nabla_{\xi}^{\top}\nabla_{x}p_{{\bf
B}_{0}}\end{pmatrix}_{(x,\xi)=(X,\Xi)}\begin{pmatrix}X^{\prime}\\\
\Xi^{\prime}\end{pmatrix}+\begin{pmatrix}F\\\ G\end{pmatrix}$ (4.16)

Given a bicharacteristic $(X,\Xi)$, we need to analyze (4.16) to control the
nearby bicharacteristics. The following simple Grönwall estimate for (4.16)
will be sufficient for our purpose.

######  Lemma 4.10.

Let ${\bf B}_{0}\in\mathcal{B}_{\epsilon}^{s}(M,\mu,A,R,L)$ for
$s>\frac{7}{2}$. We have

$\displaystyle|{X^{\prime}(t)}|+\frac{1}{|{\Xi(0)}|}|{\Xi^{\prime}(t)}|$
$\displaystyle\lesssim\left(|{X^{\prime}(0)}|+\frac{1}{|{\Xi(0)}|}|{\Xi^{\prime}(0)}|+\int_{0}^{t}\left(|{F(t^{\prime})}|+\frac{1}{|{\Xi(0)}|}|{G(t^{\prime})}|\right)\,\mathrm{d}t^{\prime}\right)$
(4.17)
$\displaystyle\phantom{\lesssim}\times\exp\left(C\mu^{-1}MA\int_{0}^{t}|{\dot{X}(t^{\prime})}|\,\mathrm{d}t^{\prime}\right).$

###### Proof.

In view of Lemma 4.2 and (4.8), note that

$\sup_{t,t^{\prime}\in\mathbb{R}}\frac{|{\Xi(t)}|}{|{\Xi(t^{\prime})}|}\lesssim
A.$ (4.18)

Let $\lambda=|{\Xi(0)}|$. We rescale (4.16) in the following way:

$\displaystyle\frac{1}{\lambda}\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}X^{\prime}\\\
\frac{1}{\lambda}\Xi^{\prime}\end{pmatrix}=\begin{pmatrix}\frac{1}{\lambda}\nabla_{x}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}}&\nabla_{\xi}^{\top}\nabla_{\xi}p_{{\bf B}_{0}}\\\
-\frac{1}{\lambda^{2}}\nabla_{x}^{\top}\nabla_{x}p_{{\bf
B}_{0}}&-\frac{1}{\lambda}\nabla_{\xi}^{\top}\nabla_{x}p_{{\bf
B}_{0}}\end{pmatrix}_{(x,\xi)=(X,\Xi)}\begin{pmatrix}X^{\prime}\\\
\frac{1}{\lambda}\Xi^{\prime}\end{pmatrix}+\frac{1}{\lambda}\begin{pmatrix}F\\\
\frac{1}{\lambda}G\end{pmatrix}.$ (4.19)

In view of (4.18), the coefficient matrix of this ODE is bounded by
$C\lambda^{-1}MA|{\Xi(t)}|$. It follows by Grönwall’s inequality that

$\displaystyle|{X^{\prime}(t)}|+\frac{1}{|{\Xi(0)}|}|{\Xi^{\prime}(t)}|$
$\displaystyle\lesssim\left(|{X^{\prime}(0)}|+\frac{1}{|{\Xi(0)}|}|{\Xi^{\prime}(0)}|+\int_{0}^{t}\left(|{F(t^{\prime})}|+\frac{1}{|{\Xi(0)}|}|{G(t^{\prime})}|\right)\,\mathrm{d}t^{\prime}\right)$
$\displaystyle\phantom{\lesssim}\times\exp\left(CMA\int_{0}^{t}|{\Xi(t^{\prime})}|\,\mathrm{d}t^{\prime}\right).$

Using $|{\Xi(t^{\prime})}|\leq\mu^{-1}|{\dot{X}(t^{\prime})}|$, the desired
inequality follows. ∎

Using the preceding lemma, we obtain the following result concerning the
variation of bicharacteristics under the change of the initial conditions.

######  Proposition 4.11.

Let ${\bf B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ for
$s>\frac{7}{2}$. Given $(x,\xi)\in T^{\ast}\mathbb{R}^{3}$, denote by
$(X(t;x,\xi),\Xi(t;x,\xi))$ the bicharacteristic (i.e., the solution to (4.1))
with the initial conditions

$(X(0;x,\xi),\Xi(0;x,\xi))=(x,\xi).$

Then the following statements hold.

  1. (1)

We have

$\displaystyle|{\partial_{x}X(t;x,\xi)}|+|{\xi}||{\partial_{\xi}X(t;x,\xi)}|+|{\xi}|^{-1}|{\partial_{x}\Xi(t;x,\xi)}|+|{\partial_{\xi}\Xi(t;x,\xi)}|$
(4.20)
$\displaystyle\lesssim\exp\left(C\mu^{-1}MA(L+(|{X^{3}(t)}|-R)_{+}+(|{X^{3}(0)}|-R)_{+})\right).$

  2. (2)

Assume, in addition, that ${\bf B}_{0}=P_{<k_{0}}{\bf B}_{0}$. Then we have

$\displaystyle|{\xi}|^{|{\boldsymbol{\beta}}|}|{\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}X(t;x,\xi)}|+|{\xi}|^{|{\boldsymbol{\beta}}|-1}|{\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\Xi(t;x,\xi)}|$
(4.21)
$\displaystyle\lesssim_{\boldsymbol{\alpha},\boldsymbol{\beta}}2^{(|{\boldsymbol{\alpha}}|-1)_{+}k_{0}}A^{(|{\boldsymbol{\beta}}|-1)_{+}}\exp\left(C_{\boldsymbol{\alpha},\boldsymbol{\beta}}\mu^{-1}MA(L+(|{X^{3}(t)}|-R)_{+}+(|{X^{3}(0)}|-R)_{+})\right).$

###### Proof.

Observe that $(\partial_{x^{j}}X(t;x,\xi),\partial_{x^{j}}\Xi(t;x,\xi))$
solves (4.16) with

$(X^{\prime}(0),\Xi^{\prime}(0),F,G)=({\bf e}_{j},0,0,0).$

Similarly, $(\partial_{\xi_{j}}X(t;x,\xi),\partial_{\xi_{j}}\Xi(t;x,\xi))$
solves (4.16) with

$(X^{\prime}(0),\Xi^{\prime}(0),F,G)=(0,{\bf e}_{j},0,0).$

Hence (4.20) follows from Lemma 4.10 and (4.15). Note that (4.20) is simply
(4.21) with $|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|=1$. We summarize
the proof of (4.21) in the case
$|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|=2$; the general case follows
from induction. Note that
$(\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}X,\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\Xi)$
solves (4.16) with

$(\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}X(0),\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\Xi(0),F,G)=(0,0,F_{\boldsymbol{\alpha},\boldsymbol{\beta}},G_{\boldsymbol{\alpha},\boldsymbol{\beta}}),$

for some
$F_{\boldsymbol{\alpha},\boldsymbol{\beta}},G_{\boldsymbol{\alpha},\boldsymbol{\beta}}$,
where the initial conditions vanish since
$\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}x=\partial_{x}^{\boldsymbol{\alpha}}\partial_{\xi}^{\boldsymbol{\beta}}\xi=0$
if $|{\boldsymbol{\alpha}}|+|{\boldsymbol{\beta}}|>1$. Moreover, as in the
proof of Lemma 4.10, we have, for
$|{\boldsymbol{\alpha}^{\prime}}|+|{\boldsymbol{\beta}^{\prime}}|\geq 2$,

$\displaystyle|{\xi}|^{|{\boldsymbol{\beta}^{\prime}}|-2}|{\partial_{x}^{\boldsymbol{\alpha}^{\prime}}\partial_{\xi}^{\boldsymbol{\beta}^{\prime}}p_{({\bf
B}_{0})_{<k_{0}}}(X,\Xi)}|\lesssim_{\boldsymbol{\alpha}^{\prime},\boldsymbol{\beta}^{\prime}}2^{(|{\boldsymbol{\alpha}^{\prime}}|-2)_{+}k_{0}}A^{(|{\boldsymbol{\beta}^{\prime}}|-2)_{+}}A^{2}M.$
(4.22)

When $|\boldsymbol{\alpha}|=2$ and $|\boldsymbol{\beta}|=0$,
$|F_{\boldsymbol{\alpha},\boldsymbol{\beta}}|$ and
$|\xi|^{-1}|G_{\boldsymbol{\alpha},\boldsymbol{\beta}}|$ are bounded by

$\begin{split}|F_{\boldsymbol{\alpha},\boldsymbol{\beta}}|&\lesssim|\nabla_{x}^{2}\nabla_{\xi}p(X,\Xi)||\nabla_{x}X|^{2}+|\xi||\nabla_{x}\nabla_{\xi}^{2}p(X,\Xi)||\nabla_{x}X|(|\xi|^{-1}|\nabla_{x}\Xi|)+|\xi|^{2}|\nabla_{\xi}^{3}p(X,\Xi)|(|\xi|^{-1}|\nabla_{x}\Xi|)^{2}\end{split}$

and

$\begin{split}|\xi|^{-1}|G_{\boldsymbol{\alpha},\boldsymbol{\beta}}|&\lesssim|\xi|^{-1}|\nabla_{x}^{3}p(X,\Xi)||\nabla_{x}X|^{2}+|\nabla_{x}^{2}\nabla_{\xi}p(X,\Xi)||\nabla_{x}X|(|\xi|^{-1}|\nabla_{x}\Xi|)+|\xi||\nabla_{\xi}^{2}\nabla_{x}p(X,\Xi)|(|\xi|^{-1}|\nabla_{x}\Xi|)^{2}\end{split}$

where $p=p_{({\bf B}_{0})_{<k_{0}}}$. Then, we simply apply (4.20) to bound
$|\nabla_{x}X|,|\xi|^{-1}|\nabla_{x}\Xi|$ and (4.22) to bound the derivatives
of $p$. This gives

$\begin{split}\int_{0}^{t}\left(|F_{\boldsymbol{\alpha},\boldsymbol{\beta}}|+|\xi|^{-1}|G_{\boldsymbol{\alpha},\boldsymbol{\beta}}|\right)\,\mathrm{d}t^{\prime}\lesssim_{\boldsymbol{\alpha},\boldsymbol{\beta}}2^{k_{0}}\mu^{-1}MA(L+(|{X^{3}(t)}|-R)_{+}+(|{X^{3}(0)}|-R)_{+}),\end{split}$

which gives (4.21) after applying Lemma 4.10 in this case. The other cases of
$|\boldsymbol{\alpha}|=|\boldsymbol{\beta}|=1$ and
$|\boldsymbol{\alpha}|=0,|\boldsymbol{\beta}|=2$ can be handled similarly. ∎

We are also ready to give a proof of Proposition 4.8.

###### Proof of Proposition 4.8.

We need to establish the five properties listed in Definition 4.3 for ${\bf
B}_{0}$, under the assumption $\|{{\bf B}_{0}-\overline{{\bf
B}}_{0}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<\delta$ with some choice of
$\delta=c(s,M,\mu,A,\epsilon)e^{-C(s,M,\nu,A)L}>0$.

To begin with, (4.6) is trivial for $\delta<M$ and (4.7) is immediate for
$\delta<\mu/2$ from the embedding $\ell^{1}_{\mathcal{I}}H^{s}\subset
L^{\infty}$. Similarly, (4.9) follows from taking $\delta\leq c(R)\epsilon$.
From now on, we shall assume that $\delta$ is sufficiently small with respect
to $(M,\mu,R,\epsilon)$, so that in particular (4.6), (4.7) and (4.9) holds.

Proof of (4.10). We now proceed to prove (4.10), by taking $\delta$ even
smaller. We fix some $(x,\xi)$ with $\xi\neq 0$ and $-2R<x^{3}<2R$. The latter
can be assumed (by a translation of $t$ variable) without loss of generality
since if either $x^{3}\geq 2R$ or $x^{3}\leq-2R$, $\dot{X}^{3}\gtrsim 1$ as
long as $X^{3}(t)\geq 2R$ or $X^{3}(t)\leq-2R$ holds, respectively. Here
$X(t),\Xi(t)$ is the bicharacteristic curve corresponding to $\overline{{\bf
B}_{0}}$, and we shall take the time interval $\\{t_{0}<t<t_{1}\\}$ where
$X^{3}(t_{0})=-2R,X^{3}(t_{1})=2R$. Note that $t_{0}<0<t_{1}$.

Let us introduce ${\bf B}_{0}^{(\sigma)}:=\sigma{\bf
B}_{0}+(1-\sigma)\overline{{\bf B}}_{0}$ and denote
$(X^{(\sigma)},\Xi^{(\sigma)})(t;x,\xi)$ the bicharacteristics with initial
conditions $(x,\xi)$ associated with ${\bf B}_{0}^{(\sigma)}$. For simplicity,
let us write $(\bar{X},\bar{\Xi})=(X^{(0)},\Xi^{(0)})(t;x,\xi)$. Then
$(X^{(\sigma)\prime},\Xi^{(\sigma)\prime}):=\partial_{\sigma}(X^{(\sigma)},\Xi^{(\sigma)})$
obeys the equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}X^{(\sigma)\prime}\\\
\Xi^{(\sigma)\prime}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\nabla_{x}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}^{(\sigma)}}&\nabla_{\xi}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}^{(\sigma)}}\\\ -\nabla_{x}^{\top}\nabla_{x}p_{{\bf
B}_{0}^{(\sigma)}}&-\nabla_{\xi}^{\top}\nabla_{x}p_{{\bf
B}_{0}^{(\sigma)}}\end{pmatrix}_{(x,\xi)=(X^{(\sigma)},\Xi^{(\sigma)})}\begin{pmatrix}X^{(\sigma)\prime}\\\
\Xi^{(\sigma)\prime}\end{pmatrix}+\begin{pmatrix}\nabla_{\xi}p_{{\bf
B}_{0}-\overline{{\bf B}}_{0}}\\\ -\nabla_{x}p_{{\bf B}_{0}-\overline{{\bf
B}}_{0}}\end{pmatrix}_{(x,\xi)=(X^{(\sigma)},\Xi^{(\sigma)})}$

with zero initial conditions. We are going to restrict the variable $t$ to
$[t_{0},t_{1}]$ and estimate $(X^{(\sigma)\prime},\Xi^{(\sigma)\prime})$,
under the following bootstrap assumptions:

$\begin{split}\sup_{\sigma\in[0,1]}\sup_{t:|\bar{X}^{3}(t)|<2R}|X^{\sigma}(t)-\bar{X}(t)|\leq\min\left\\{\frac{L}{100},{\frac{1}{4}}\right\\},\qquad\sup_{\sigma\in[0,1]}\sup_{t:|\bar{X}^{3}(t)|<2R}\frac{|\Xi^{\sigma}(t)|}{|\bar{\Xi}(t)|}\leq
10.\end{split}$ (4.23)

From our assumption $R\geq 1$ in Definition 4.3, (4.23) implies in particular
that $|X^{\sigma}(t)-\bar{X}(t)|\leq R/4$. We now proceed as in the proof of
Lemma 4.10. To begin with, we estimate the coefficient matrix, after rescaling
the equation as in (4.19). Writing for simplicity $p=p_{{\bf
B}_{0}^{(\sigma)}}(t,X^{(\sigma)},\Xi^{(\sigma)})$, we obtain under (4.23)
that

$\begin{split}|\nabla_{\xi}^{2}p|+|\xi|^{-2}|\nabla^{2}_{x}p|+|\xi|^{-1}|\nabla_{x}\nabla_{\xi}p|\leq
100C_{0}AM|\xi|^{-1}|\bar{\Xi}|.\end{split}$

Then, to estimate the forcing terms, we write for simplicity $p_{{\bf
B}_{0}-\overline{{\bf B}}_{0}}=p_{{\bf B}_{0}-\overline{{\bf
B}}_{0}}(t,X^{(\sigma)},\Xi^{(\sigma)})$ and obtain that

$\begin{split}|\nabla_{\xi}p_{{\bf B}_{0}-\overline{{\bf B}}_{0}}|\leq
C_{1}\delta|\Xi^{(\sigma)}|\leq
10C_{1}\delta|\bar{\Xi}|,\qquad\int_{0}^{t}|\nabla_{\xi}p_{{\bf
B}_{0}-\overline{{\bf B}}_{0}}|\,\mathrm{d}t^{\prime}\leq
20C_{1}\delta\mu^{-1}\int_{0}^{t}|\dot{\bar{X}}|\,\mathrm{d}t^{\prime}.\end{split}$

Similarly, using $|\Xi^{(\sigma)}|\leq 10|\bar{\Xi}|$ and
$|\xi|^{-1}|\bar{\Xi}|\leq A$, we obtain

$\begin{split}\int_{0}^{t}|\nabla_{x}p_{{\bf B}_{0}-\overline{{\bf
B}}_{0}}|\,\mathrm{d}t^{\prime}\leq 100C_{2}\delta
A\mu^{-1}\int_{0}^{t}|\dot{\bar{X}}|\,\mathrm{d}t^{\prime}.\end{split}$

With these bounds, following the proof of Lemma 4.10 gives

$\begin{split}|X^{(\sigma)\prime}|+|\xi|^{-1}|\Xi^{(\sigma)\prime}|\leq
C(20C_{1}+100C_{2})A\delta\mu^{-1}\int_{0}^{t}|\dot{\bar{X}}|\,\mathrm{d}t^{\prime}\,\exp\left(100CC_{0}AM\mu^{-1}\int_{0}^{t}|\dot{\bar{X}}|\,\mathrm{d}t^{\prime}\right)=:\delta^{*}.\end{split}$
(4.24)

By taking $\delta$ small in the form

$\begin{split}\delta\leq
c_{0}\mu(AL(A+L^{-1}))^{-1}\exp\left(-\frac{1}{c_{0}}AM\mu^{-1}L\right),\end{split}$

we may obtain in particular that
$|X^{(\sigma)\prime}|+|\xi|^{-1}|\Xi^{(\sigma)\prime}|\leq\min\\{\frac{1}{100CA},\frac{L}{200}\\}$,
where $C>0$ is the implicit constant from the estimate (4.13). This gives for
all $0<\sigma\leq 1$ that

$\begin{split}|\Xi^{(\sigma)}-\bar{\Xi}|\leq|\xi|\int_{0}^{\sigma}|\xi|^{-1}|\Xi^{(\sigma^{\prime})\prime}|\,\mathrm{d}\sigma^{\prime}\leq\frac{|\xi|}{100CA}\leq\frac{|\bar{\Xi}|}{2}\qquad\mbox{and}\qquad\frac{|\bar{\Xi}|}{2}\leq|\Xi^{(\sigma)}|\leq\frac{3|\bar{\Xi}|}{2}.\end{split}$
(4.25)

This in particular justifies the bootstrap assumption (4.23) for
$\Xi^{(\sigma)}$. Similarly, we have

$\begin{split}|X^{(\sigma)}-\bar{X}|\leq\int_{0}^{\sigma}|X^{(\sigma^{\prime})\prime}|\,\mathrm{d}\sigma^{\prime}\leq\min\left\\{\frac{L}{200},\frac{1}{100CA}\right\\}\end{split}$
(4.26)

and this justifies (4.23) for $X^{(\sigma)}-\bar{X}$, since $A\geq\frac{1}{C}$
follows from (4.13). The estimates (4.25), (4.26) are valid as long as
$t\in[t_{0},t_{1}]$.

Now we split the integral (4.10) for $\sigma=1$ as

$\left[\int_{-\infty}^{t_{0}}+\int_{t_{0}}^{t_{1}}+\int_{t_{1}}^{\infty}\right]\mathbf{1}_{\\{-2R<x^{3}<2R\\}}(X^{(1),3}(t))|{\dot{X}^{(1)}(t)}|\,\mathrm{d}t.$
(4.27)

The last integral in $\\{t\geq t_{1}\\}$ (which may be empty) can be easily
estimated as follows. We know at $t=t_{1}$ that $X^{(1),3}\geq
2R-\frac{R}{2}>R$ and therefore we have that for $t\geq t_{1}$,
$\dot{X}^{(1),3}\geq 20|\dot{X}^{(1)}|$. But then

$\begin{split}\int_{t_{1}}^{\infty}\mathbf{1}_{\\{-2R<x^{3}<2R\\}}(X^{(1)}(t))|{\dot{X}^{(1),3}(t)}|\,\mathrm{d}t\leq
20\int_{2R-\frac{L}{200}<X^{3}(t)<2R}\dot{X}^{3}(t)\,\mathrm{d}t<\frac{L}{4}.\end{split}$

The integral in $t\in(-\infty,t_{0})$ can be estimated in the same manner.
Finally, to estimate the second integral, we write

$\begin{split}\int_{t_{0}}^{t_{1}}\mathbf{1}_{\\{-2R<x^{3}<2R\\}}(X^{(1),3}(t))|{\dot{X}^{(1)}(t)}|\,\mathrm{d}t\leq\int_{0}^{1}\left[\int_{t_{0}}^{t_{1}}|{\dot{X}^{(\sigma)\prime}(t)}|\,\mathrm{d}t\right]\mathrm{d}\sigma.\end{split}$
(4.28)

To estimate the RHS of (4.28), we return to the equation for
$\dot{X}^{(\sigma)\prime}$:

$\begin{split}\dot{X}^{(\sigma)\prime}=\nabla_{x}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}^{(\sigma)}}(X^{(\sigma)},\Xi^{(\sigma)})X^{(\sigma)\prime}+\nabla_{\xi}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}^{(\sigma)}}(X^{(\sigma)},\Xi^{(\sigma)})\Xi^{(\sigma)\prime}+\nabla_{\xi}p_{{\bf
B}_{0}-\overline{{\bf B}}_{0}}(X^{(\sigma)},\Xi^{(\sigma)}).\end{split}$

We note that the last term is bounded by

$\begin{split}|\nabla_{\xi}p_{{\bf B}_{0}-\overline{{\bf
B}}_{0}}(X^{(\sigma)},\Xi^{(\sigma)})|\leq C\delta|\Xi^{(\sigma)}|\leq
C\mu^{-1}\delta|\dot{X}^{(\sigma)}|.\end{split}$

For the other terms, using (4.24) and (4.25) we obtain that

$\begin{split}|\nabla_{x}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}^{(\sigma)}}(X^{(\sigma)},\Xi^{(\sigma)})X^{(\sigma)\prime}|\leq
CM\mu^{-1}\delta^{*}|\dot{X}^{(\sigma)}|\end{split}$
$\begin{split}|\nabla_{\xi}^{\top}\nabla_{\xi}p_{{\bf
B}_{0}^{(\sigma)}}(X^{(\sigma)},\Xi^{(\sigma)})\Xi^{(\sigma)\prime}|\leq
CM|\xi|\delta^{*}\leq CMA\mu^{-1}\delta^{*}|\dot{X}^{(\sigma)}|.\end{split}$

By taking $\delta$ smaller depending on $\mu,M,A$ if necessary (this makes
$\delta^{*}$ smaller as well), we have

$\begin{split}|\dot{X}^{(\sigma)\prime}|\leq\frac{1}{10}|\dot{X}^{(\sigma)}|.\end{split}$

This allows us to write

$\begin{split}I(\sigma):=\int_{t_{0}}^{t_{1}}|{\dot{X}^{(\sigma)}(t)}|\,\mathrm{d}t\leq\int_{t_{0}}^{t_{1}}|{\dot{X}^{(0)}(t)}|\,\mathrm{d}t+\frac{1}{10}\int_{t_{0}}^{t_{1}}\int_{0}^{\sigma}|\dot{X}^{(\sigma^{\prime})}(t)|\,\mathrm{d}\sigma^{\prime}\mathrm{d}t<L+\frac{1}{10}\int_{0}^{\sigma}I(\sigma^{\prime})\,\mathrm{d}\sigma^{\prime}.\end{split}$

Note that $I(0)<L$. We may bootstrap the hypothesis $I(\sigma)<\frac{5}{4}L$;
indeed, if we assume that it holds for some $\sigma\in[0,\sigma_{0})$, then

$\begin{split}I(\sigma_{0})<L+\frac{1}{10}\sigma_{0}\frac{5}{4}L\leq\frac{9}{8}L.\end{split}$

Therefore, we have that $I(1)<\frac{5}{4}L$, which allows us to conclude that

$\begin{split}\left[\int_{-\infty}^{t_{0}}+\int_{t_{0}}^{t_{1}}+\int_{t_{1}}^{\infty}\right]\mathbf{1}_{\\{-2R<x^{3}<2R\\}}(X^{(1),3}(t))|{\dot{X}^{(1)}(t)}|\,\mathrm{d}t<2L.\end{split}$

Proof of (4.8). We use the notation from above; fix some $(x,\xi)$ and denote
$(X^{(\sigma)}(t),\Xi^{(\sigma)}(t))$ be the bicharacteristic curve
corresponding to ${\bf B}_{0}^{(\sigma)}$. The goal is to prove

$\begin{split}\int_{-\infty}^{\infty}|\nabla{\bf
B}_{0}^{(1)}(X^{(1)}(t))||\Xi^{(1)}(t)|\,\mathrm{d}t<2\max\\{A,M\\}.\end{split}$
(4.29)

In the region $\\{t_{1}<t\\}$ (the region $\\{t<t_{0}\\}$ can be handled
similarly), we proceed as in the proof of Lemma 4.5; using (4.4), we can bound

$\displaystyle{\int_{-\infty}^{\infty}\mathbf{1}_{\\{R<x^{3}\\}}(X^{(1)}(t))}|{\nabla{\bf
B}^{(1)}_{0}(X^{(1)}(t))}||{\Xi^{(1)}(t)}|\,\mathrm{d}t\lesssim\sum_{I\in\mathcal{I}:I\cap(R,\infty)\neq\emptyset}\int_{I}\|{\nabla{\bf
B}^{(1)}_{0}(x^{1},x^{2},x^{3})}\|_{L^{\infty}_{x^{1},x^{2}}}\,\mathrm{d}x^{3}\lesssim\epsilon.$
(4.30)

Therefore, by taking a small absolute constant $\alpha_{1}>0$ (recall
$\epsilon\leq\alpha_{1}M$ from Definition 4.3), we can guarantee that

$\begin{split}\int_{(-\infty,t_{0}]\cup[t_{1},\infty)}|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t\leq
M.\end{split}$ (4.31)

To handle the integral in the region $\\{t_{0}\leq t\leq t_{1}\\}$, we write

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}\sigma}\int_{t_{0}}^{t_{1}}|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t=I+II+III,\end{split}$

where

$\begin{split}I=\int_{t_{0}}^{t_{1}}|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))|\frac{\Xi^{(\sigma)}(t)\cdot\Xi^{(\sigma)\prime}(t)}{|\Xi^{(\sigma)}(t)|}\,\mathrm{d}t,\end{split}$
$\begin{split}II=\int_{t_{0}}^{t_{1}}\frac{\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t)):\nabla({\bf B}_{0}-\overline{{\bf
B}}_{0})(X^{(\sigma)}(t))}{|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))|}|\Xi^{(\sigma)}(t)|\,\mathrm{d}t,\end{split}$
$\begin{split}III=\int_{t_{0}}^{t_{1}}\frac{1}{2}\frac{X^{(\sigma)\prime}(t)\cdot\nabla|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))|^{2}}{|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))|}|\Xi^{(\sigma)}(t)|\,\mathrm{d}t.\end{split}$

We may bound each term similarly as in the proof of (4.12) from Lemma 4.5,
using (4.24). We begin with

$\begin{split}|II|\lesssim\int_{t_{0}}^{t_{1}}|\nabla({\bf
B}_{0}-\overline{{\bf
B}}_{0})(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t\lesssim\frac{\delta}{\mu}\int_{t_{0}}^{t_{1}}|{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t\lesssim\frac{\delta}{\mu}L.\end{split}$

Next, from (4.24) we obtain $|\Xi^{(\sigma)\prime}(t)|\lesssim
A\delta^{*}|\Xi^{(\sigma)}(t)|$, which gives

$\begin{split}|I|\lesssim\int_{t_{0}}^{t_{1}}|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)\prime}(t)|\,\mathrm{d}t\lesssim
A\delta^{*}\frac{M}{\mu}L.\end{split}$

Lastly, using (4.24) we bound

$\begin{split}|III|\lesssim\int_{t_{0}}^{t_{1}}|X^{(\sigma)\prime}(t)||\nabla^{2}{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t\lesssim\frac{M}{\mu}\delta^{*}\int_{t_{0}}^{t_{1}}|{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t\lesssim\delta^{*}\frac{M}{\mu}L.\end{split}$

Therefore, by taking $\delta$ smaller (which makes $\delta^{*}$ smaller as
well),

$\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}\sigma}\int_{t_{0}}^{t_{1}}|\nabla{\bf
B}_{0}^{(\sigma)}(X^{(\sigma)}(t))||\Xi^{(\sigma)}(t)|\,\mathrm{d}t\right|\leq\frac{A}{2},\end{split}$

which together with (4.31) gives (4.29). This finishes the proof. ∎

##  5\. Precise formulation and proof of the main theorem

###  5.1. Precise formulation of the local wellposedness result

We are now ready to give a precise formulation of the main local wellposedness
result.

######  Theorem 5.1 (Main theorem).

Given $s>\frac{7}{2}$, $M>0$, $\mu>0$ and $A>0$, there exist positive
constants $\epsilon=\epsilon(s,M,\mu,A)$, $c(s,M,\mu,A)$ and $C(s,M,\mu,A)$
such that the following holds. Consider a vector field ${\bf
B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfying $\nabla\cdot{\bf B}_{0}=0$.
Assume furthermore that

${\bf B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$

for some $R,L>0$. Then the Cauchy problem for (E-MHD) with ${\bf B}(t=0)={\bf
B}_{0}$ is locally wellposed on $[0,T]$, where

$T=c(s,M,\mu,A)\exp(-C(s,M,\mu,A)L),$

in the sense that the following holds:

  1. (1)

Existence and uniqueness. There exists a unique solution ${\bf B}$ with ${\bf
B}-{\bf e}_{3}\in\ell^{1}_{\mathcal{I}}X^{s}[0,T]$.

  2. (2)

Continuous dependence. The solution map ${\bf B}_{0}\mapsto{\bf B}$ is
continuous as a map
$\ell^{1}_{\mathcal{I}}H^{s}\to\ell^{1}_{\mathcal{I}}X^{s}[0,T]$;

  3. (3)

Weak Lipschitz dependence. The solution map ${\bf B}_{0}\mapsto{\bf B}$ is
Lipschitz continuous as a map
$\ell^{1}_{\mathcal{I}}L^{2}\to\ell^{1}_{\mathcal{I}}X^{0}[0,T]$;

  4. (4)

Persistence of regularity. If ${\bf B}_{0}-{\bf
e}_{3}\in\ell^{1}_{\mathcal{I}}H^{s+m}$ for $m\in\mathbb{Z}_{\geq 0}$, then
${\bf B}-{\bf e}_{3}\in\ell^{1}_{\mathcal{I}}X^{s+m}[0,T]$.

  5. (5)

Frequency envelope bound. Let $c_{k}$ be a _$\delta_{1}$ -admissible frequency
envelope_ (i.e., $\\{c_{k}\\}_{k\in\mathbb{Z}_{\geq 0}}$ is a sequence of
positive numbers obeying $\max\left\\{c_{k}/c_{j},c_{j}/c_{k}\right\\}\leq
2^{\delta_{1}|{k-j}|}$) with $0<\delta_{1}<s$. If $\|{P_{k}({\bf B}_{0}-{\bf
e}_{3})}\|_{\ell^{1}_{\mathcal{I}}H^{s}}\leq c_{k}$ for $k\in\mathbb{Z}_{\geq
0}$, then $\|{P_{k}({\bf B}-{\bf
e}_{3})}\|_{\ell^{1}_{\mathcal{I}}X^{s}}\lesssim c_{k}$ for all
$k\in\mathbb{Z}_{\geq 0}$.

Here, $\delta_{1}$ is a universal small positive constant to be determined in
the proof below.

As a direct consequence of the above, we have local wellposedness for small
initial data.

######  Corollary 5.2.

Given $s>\frac{7}{2}$, there exist positive constants
$\varepsilon^{*}=\varepsilon^{*}(s)$ and $T=T(s)$ such that the following
holds. For ${\bf B}_{0}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ satisfying
$\nabla\cdot{\bf B}_{0}=0$ and $\|{{\bf B}_{0}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<\varepsilon^{*}$, the Cauchy problem
for (E-MHD) with ${\bf B}(t=0)={\bf B}_{0}$ is locally wellposed on $[0,T]$,
in the same sense of Theorem 5.1.

To obtain Corollary 5.2 from Theorem 5.1, it suffices to note that for ${\bf
B}_{0}$ satisfying $\|{{\bf B}_{0}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}H^{s}}<\varepsilon^{*}$, we have ${\bf
B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L)$ with
$M=1,\mu=1/2,A=1,R=1,L=10$ as long as $\varepsilon^{*}$ is taken to be
sufficiently small. Then $T>0$ is provided by Theorem 5.1.

###  5.2. Paralinearization of (E-MHD) and the resulting error

A basic ingredient of the proof is a wellposedness theorem for the
_paralinearization of ( 3.1)_, which refers to the following system:

$\displaystyle\partial_{t}b-\nabla\times(T_{{\bf B}}\times(\nabla\times
b))+\nabla\times(T_{\nabla\times{\bf B}}\times b)$ $\displaystyle=g,$ (5.1)
$\displaystyle\nabla\cdot b=\nabla\cdot g$ $\displaystyle=0,$ (5.2)

where

$\displaystyle\nabla\times(T_{{\bf B}}\times(\nabla\times b))$
$\displaystyle=\sum_{k}\nabla\times(P_{<k-10}{\bf B}\times(\nabla\times
P_{k}b)),$ $\displaystyle\nabla\times(T_{\nabla\times{\bf B}}\times b)$
$\displaystyle=\sum_{k}\nabla\times((\nabla\times P_{<k-10}{\bf B})\times
P_{k}b).$

We also introduce the _paralinearization error_ ,

$G(b)=\nabla\times(b\times(\nabla\times
b))-\nabla\times(T_{b}\times(\nabla\times b))+\nabla\times(T_{\nabla\times
b}\times b).$ (5.3)

As its name suggests, $G(b)$ is precisely the error term that arises when
writing (E-MHD) (or (3.1)) in the form (5.1)–(5.2) (we omit the
straightforward proof):

######  Lemma 5.3.

If ${\bf B}$ solves (E-MHD) on $(0,T)$, then $b:={\bf B}-{\bf e}_{3}$ solves
(5.1)–(5.2) with ${\bf B}={\bf e}_{3}+b$ and $g=G(b)$ on $(0,T)$, and vice
versa.

The following two results will be proved in Section 7:

######  Proposition 5.4 (Wellposedness of the paralinearized system).

Let $s_{1}>\frac{7}{2}$ and $0<T\leq 1$. Assume that, for some
$M_{1},M_{1}^{\prime},\mu,A,\epsilon_{1},R,L>0$, we have

$\displaystyle{\bf
B}(0)\in\mathcal{B}^{s_{1}}_{\epsilon_{1}}(M_{1},\mu,A,R,L),$ (5.4)
$\displaystyle\|{{\bf B}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}\leq M_{1},$ (5.5)
$\displaystyle{\|{\partial_{t}{\bf
B}}\|_{\ell^{1}_{\mathcal{I}}L^{\infty}H^{s_{1}-2}[0,T]}\leq M_{1}^{\prime}.}$
(5.6)

Let $b$ and $g$ satisfy (5.1)–(5.2) on $(0,T)$. Then, for any $\sigma\geq 0$,
the following holds: if

$\epsilon_{1}\leq\epsilon_{(1)}(\sigma,s_{1},M_{1},M_{1}^{\prime},\mu,A),\quad
T\leq
c_{(1)}(\sigma,s_{1},M_{1},M_{1}^{\prime},\mu,A)\exp(-C_{(1)}(\sigma,s_{1},M_{1},M_{1}^{\prime},\mu,A)L)$

then we have

$\|{b}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}[0,T]}\leq
C_{(2)}(\sigma,s_{1},M_{1},M_{1}^{\prime},\mu,A)\exp(C_{(2)}(\sigma,s_{1},M_{1},M_{1}^{\prime},\mu,A)L)\left(\|{b_{0}}\|_{\ell^{1}_{\mathcal{I}}H^{\sigma}}+\|{g}\|_{\ell^{1}_{\mathcal{I}}Y^{\sigma}[0,T]}\right).$
(5.7)

######  Proposition 5.5 (Bounds for the paralinearization error).

Let $s_{1}>\frac{7}{2}$ and
$b,\overline{b}\in\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]$ with

$\|{b}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}<M_{1},\qquad\|{\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}<M_{1}.$

Then for any $\sigma\geq 0$, we have

$\displaystyle\|{G(b)}\|_{\ell^{1}_{\mathcal{I}}Y^{\sigma}[0,T]}$
$\displaystyle\lesssim_{\sigma,s_{1}}TM_{1}\|{b}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}[0,T]},$
(5.8)
$\displaystyle\|{G(b)-G(\overline{b})}\|_{\ell^{1}_{\mathcal{I}}Y^{\sigma}[0,T]}$
$\displaystyle\lesssim_{\sigma,s_{1}}TM_{1}\|{b-\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}[0,T]}.$
(5.9)

Moreover, for $0\leq\sigma<s_{1}-1$, we have

$\displaystyle\|{\nabla\times(T_{b-\overline{b}}\times(\nabla\times
b))-\nabla\times(T_{\nabla\times(b-\overline{b})}\times
b)}\|_{\ell^{1}_{\mathcal{I}}Y^{\sigma}}\lesssim_{s_{1}-1-\sigma}T^{\frac{s_{1}-1-\sigma}{s_{1}-1}}M_{1}\|{b-\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}[0,T]}.$
(5.10)

Finally, we also have the fixed time bounds

$\|{\nabla\times(T_{b}\times(\nabla\times
b))(t)}\|_{\ell^{1}_{\mathcal{I}}H^{s_{1}-2}}+\|{\nabla\times(T_{\nabla\times
b}\times
b)(t)}\|_{\ell^{1}_{\mathcal{I}}H^{s_{1}-2}}+\|{G(b(t))}\|_{\ell^{1}_{\mathcal{I}}H^{s_{1}-2}}\lesssim_{s_{1}}M_{1}^{2}.$
(5.11)

In the next subsection, we will utilize these propositions to prove Theorem
5.1.

###  5.3. Proof of local wellposedness

We now show how Theorem 5.1 follows from Propositions 5.4 and 5.5. We follow
the streamlined approach of [54, 55, 56] (see also [39]), which does not rely
on Nash–Moser or viscosity method.

###### Proof of Theorem 5.1.

Throughout the proof, we will assume that

${\bf B}_{0}\in\mathcal{B}^{s}_{\epsilon}(M,\mu,A,R,L),\quad T\leq
c(s,M,\mu,A)\exp(-C(s,M,\mu,A)L),$ (5.12)

where $c(s,M,\mu,A)$ and $C(s,M,\mu,A)$, as well as $\epsilon(s,M,\mu,A)$, are
to be fixed at the end of Step 2 below.

Step 0: Setup for the proof of existence. The proof of existence is based on
an iteration argument. We begin with $b^{(-1)}=0$. With ${\bf
B}^{(n-1)}:=b^{(n-1)}+{\bf e}_{3}$, we define $b^{(n)}$ by solving

$\displaystyle\partial_{t}b^{(n)}-\nabla\times(T_{{\bf
B}^{(n-1)}}\times(\nabla\times b^{(n)}))+\nabla\times(T_{\nabla\times{\bf
B}^{(n-1)}}\times b^{(n)})$ $\displaystyle=G(b^{(n-1)})$ (5.13)
$\displaystyle\nabla\cdot b^{(n)}$ $\displaystyle=0,$

with the initial conditions $b^{(n)}(0)={\bf B}_{0}-{\bf e}_{3}$.

Note that the difference between two consecutive iterates solves the following
equation:

$\displaystyle\partial_{t}(b^{(n+1)}-b^{(n)})-\nabla\times(T_{{\bf
B}^{(n)}}\times(\nabla\times(b^{(n+1)}-b^{(n)})))+\nabla\times(T_{\nabla\times{\bf
B}^{(n)}}\times(b^{(n+1)}-b^{(n)}))$ (5.14)
$\displaystyle=G(b^{(n-1)})-G(b^{(n)})+\nabla\times(T_{b^{(n)}-b^{(n-1)}}\times(\nabla\times
b^{(n)}))-\nabla\times(T_{\nabla\times(b^{(n)}-b^{(n-1)})}\times b^{(n)}).$

Step 1: Iteration. Fix a choice of $s_{1}=s_{1}(s)$ such that
$\frac{7}{2}<s_{1}<s$ (e.g., $s_{1}=\frac{1}{2}(\frac{7}{2}+s)$), and
construct a sequence of iterates as above. We choose $M_{1}=C_{1}M$,
$\epsilon_{1}=C_{1}\epsilon$, and $M_{1}^{\prime}=C_{1}^{2}M(1+M)$ with
$C_{1}=C_{1}(s)$ a suitably large constant (to be fixed at the end of this
step). In particular, by taking $C_{1}$ to be sufficiently large, we may
ensure that (5.4) holds. We also take

$c(s,M,\mu,A)\leq c_{(1)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A),\quad
C(s,M,\mu,A){\geq}C_{(1)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A),$ (5.15)

where $c_{(1)}$ and $C_{(1)}$ are from Proposition 5.4.

We will propagate the following bounds by induction on $n$:

  1. (1)

$\|{{\bf B}^{(n)}-{\bf e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}\leq
M_{1}$.

  2. (2)

$\|{{\bf B}^{(n)}-{\bf e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s}[0,T]}\leq
2C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)e^{C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)L}M$.

  3. (3)

$\|{\partial_{t}{\bf
B}^{(n)}}\|_{\ell^{1}_{\mathcal{I}}L^{\infty}H^{s_{1}-2}[0,T]}\leq
M_{1}^{\prime}$.

Indeed, these bounds trivially hold for ${\bf B}^{(-1)}={\bf B}_{0}$. Now,
assuming that (1), (2) and (3) hold for ${\bf B}^{(n)}$, we will show that
${\bf B}^{(n+1)}$ obeys the same bounds as well. In view of (5.12) and (5.15),
we may apply Propositions 5.4 and 5.5 to (5.13) for $b^{(n+1)}={\bf
B}^{(n+1)}-{\bf e}_{3}$. We obtain

$\|{{\bf B}^{(n+1)}-{\bf e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s}[0,T]}\leq
C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)e^{C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)L}(M+C(s,s_{1})TM_{1}M),$

where $C(s,s_{1})$ is a constant arising from applying Proposition 5.5. Taking
$c$ to be even smaller (so that $T$ is small), we may ensure that (2) holds.
To establish (1) and (3), we begin by noting that

$\|{\partial_{t}{\bf
B}^{(n+1)}}\|_{\ell^{1}_{\mathcal{I}}L^{\infty}H^{s_{1}-2}[0,T]}\lesssim(1+M_{1})\|{{\bf
B}^{(n+1)}-{\bf e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}+M_{1}^{2}$

by (5.11) (in Proposition 5.5), the induction hypothesis and the simple
algebraic computation

$\displaystyle\nabla\times(T_{{\bf B}^{(n)}}\times(\nabla\times
b^{(n+1)}))-\nabla\times(T_{\nabla\times{\bf B}^{(n)}}\times b^{(n+1)})$
$\displaystyle=\nabla\times({\bf e}_{3}\times(\nabla\times
b^{(n+1)}))+\nabla\times(T_{b^{(n)}}\times(\nabla\times
b^{(n+1)}))-\nabla\times(T_{\nabla\times b^{(n)}}\times b^{(n+1)}).$

Splitting

$\displaystyle{\bf B}^{(n+1)}-{\bf e}_{3}$ $\displaystyle=P_{<k}({\bf
B}^{(n+1)}-{\bf e}_{3})+P_{\geq k}({\bf B}^{(n+1)}-{\bf e}_{3})$
$\displaystyle=P_{<k}({\bf B}_{0}-{\bf
e}_{3})+\int_{0}^{t}P_{<k}\partial_{t}{\bf
B}^{(n+1)}(t^{\prime})\,\mathrm{d}t^{\prime}+P_{\geq k}({\bf B}^{(n+1)}-{\bf
e}_{3}),$

then applying the preceding bounds and (2.12), we obtain

$\displaystyle\|{{\bf B}^{(n+1)}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}$
$\displaystyle\leq\left\|{P_{<k}({\bf B}_{0}-{\bf
e}_{0})+\int_{0}^{t}P_{<k}\partial_{t}{\bf
B}^{(n+1)}(t^{\prime})\,\mathrm{d}t^{\prime}}\right\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}$
$\displaystyle\mathrel{\phantom{=}}+\|{P_{\geq k}({\bf B}^{(n+1)}-{\bf
e}_{3})}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}$
$\displaystyle\lesssim(1+T^{\frac{1}{2}}2^{\frac{k}{2}})\left\|{P_{<k}({\bf
B}_{0}-{\bf e}_{0})}\right\|_{\ell^{1}_{\mathcal{I}}H^{s_{1}}}$
$\displaystyle\mathrel{\phantom{=}}+(1+T^{\frac{1}{2}}2^{\frac{k}{2}})T2^{2k}\left((1+M_{1})\|{{\bf
B}^{(n+1)}-{\bf
e}_{3}}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}[0,T]}+M_{1}^{2}\right)$
$\displaystyle\mathrel{\phantom{=}}+2^{-(s-s_{1})k}2C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)e^{C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)L}M.$

Note that $s-s_{1}>0$ by our choice. Taking $k$ sufficiently large (depending
on $s,M,\mu,A$, to handle the term on the last line) and $c(s,M,\mu,A)$
smaller (so that $T$ is small depending on $k$), and $C_{1}(s)$ sufficiently
large (so that $M_{1}$ and $M_{1}^{\prime}$ are large compared to $M$ and
$M(1+M)$, respectively), we may ensure that (1) and (3) hold.

Next, we establish the convergence of ${\bf B}^{(n)}=b^{(n)}+{\bf e}_{3}$.
Indeed, applying Propositions 5.4 and 5.5 to the difference equation (5.14)
with $\sigma=0$, we obtain

$\|{b^{(n+1)}-b^{(n)}}\|_{\ell^{1}_{\mathcal{I}}X^{0}[0,T]}<C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)e^{C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)L}C(s,s_{1})TM_{1}\|{b^{(n)}-b^{(n-1)}}\|_{\ell^{1}_{\mathcal{I}}X^{0}[0,T]}.$

Taking $c_{(1)}$ sufficiently small (so that $T$ is small), we may ensure that
${\bf B}^{(n)}=b^{(n)}+{\bf e}_{3}$ is a Cauchy sequence in
$\ell^{1}_{\mathcal{I}}X^{0}[0,T]$; let us denote the limit by ${\bf B}$.
Observing that (1) and (2) hold for the limit $P_{<k}({\bf B}-{\bf e}_{3})$
for every $k\in\mathbb{Z}_{\geq 0}$, it follows that ${\bf B}-{\bf
e}_{3}\in\ell^{1}_{\mathcal{I}}X^{s}[0,T]$ with the bounds (1) and (2).
Moreover, $b={\bf B}-{\bf e}_{3}$ solves

$\displaystyle\partial_{t}b-\nabla\times(T_{{\bf B}}\times(\nabla\times
b))+\nabla\times(T_{\nabla\times{\bf B}}\times b)$ $\displaystyle=G(b)$ (5.16)
$\displaystyle\nabla\cdot b$ $\displaystyle=0,$

with the initial condition $b(0)={\bf B}_{0}-{\bf e}_{3}$, as desired.

We end this step by noting that, taking $c(s,M,\mu,A)$ sufficiently small (so
that $T$ is small), we have

${\bf
B}(t)\in\mathcal{M}^{s_{1}}_{2\epsilon_{1}}(2M_{1},\tfrac{1}{2}\mu,2\max\\{A,M_{1}\\},R,2L)\quad\hbox{
for all }t\in[0,T].$ (5.17)

To see this, fix $t\in[0,T]$. From the splitting

$\displaystyle{\bf B}^{(n+1)}-{\bf B}_{0}$
$\displaystyle=\int_{0}^{t}P_{<k}\partial_{t}{\bf
B}^{(n+1)}(t^{\prime})\,\mathrm{d}t^{\prime}+P_{\geq k}({\bf B}^{(n+1)}-{\bf
e}_{3})-P_{\geq k}({\bf B}_{0}-{\bf e}_{3}),$

we may apply (2) and (3) for ${\bf B}^{(n+1)}$, as well as (5.12), to bound

$\displaystyle\|{{\bf B}^{(n+1)}(t)-{\bf
B}_{0}}\|_{\ell^{1}_{\mathcal{I}}H^{s_{1}}[0,T]}\lesssim
T2^{2k}M_{1}^{\prime}+2^{-(s-s_{1})k}\left(2C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)e^{C_{(2)}(s,s_{1},M_{1},M_{1}^{\prime},\mu,A)L}M+M\right).$

Taking $k$ sufficiently large (depending on $s,M,\mu,A$) and $c(s,M,\mu,A)$
small enough (depending on $k$, so that $T$ is small), we may apply
Proposition 4.8 to conclude (5.17).

Step 2: Uniqueness and weak Lipschitz dependence. Let ${\bf B}$ be the
solution constructed in Step 1, and let $\overline{{\bf
B}}\in\ell^{1}_{\mathcal{I}}X^{s}[0,T]$ be a (possibly different) solution to
the initial value problem. We will show that, in fact, ${\bf B}=\overline{{\bf
B}}$.

Introducing the shorthands $b={\bf B}-{\bf e}_{3}$ and
$\overline{b}=\overline{{\bf B}}-{\bf e}_{3}$, note that the difference