# The time-like minimal surface equation in Minkowski space: low regularity
solutions

Albert Ai  Department of Mathematics, University of Wisconsin, Madison
<EMAIL_ADDRESS>,  Mihaela Ifrim  Department of Mathematics, University of
Wisconsin, Madison<EMAIL_ADDRESS>and  Daniel Tataru  Department of
Mathematics, University of California at Berkeley<EMAIL_ADDRESS>
###### Abstract.

It has long been conjectured that for nonlinear wave equations which satisfy a
nonlinear form of the null condition, the low regularity well-posedness theory
can be significantly improved compared to the sharp results of Smith-Tataru
for the generic case. The aim of this article is to prove the first result in
this direction, namely for the time-like minimal surface equation in the
Minkowski space-time. Further, our improvement is substantial, namely by $3/8$
derivatives in two space dimensions and by $1/4$ derivatives in higher
dimensions.

###### Key words and phrases:

time-like minimal surface, low regularity, normal forms, nonlinear wave
equations, paracontrolled distributions

###### 2020 Mathematics Subject Classification:

35L72, 35B65

###### Contents

  1. 1 Introduction
  2. 2 Notations, paraproducts and some commutator type bounds
  3. 3 A complete set of equations
  4. 4 Energy and Strichartz estimates
  5. 5 Control parameters and related bounds
  6. 6 Paracontrolled distributions
  7. 7 Energy estimates for the paradifferential equation
  8. 8 Energy estimates for the full equation
  9. 9 Energy and Strichartz estimates for the linearized equation
  10. 10 Short time Strichartz estimates
  11. 11 Conclusion: proof of the main result

##  1\. Introduction

The question of local well-posedness for nonlinear wave equations with rough
initial data is a fundamental question in the the study of nonlinear waves,
and which has received a lot of attention over the years. The result of Smith
and Tataru [36], proved almost 20 years ago, provides the sharp regularity
threshold for generic nonlinear wave equations in view of Lindblad’s
counterexample [28]. On the other hand, it has also been conjectured [43] that
for nonlinear wave equations which satisfy a suitable nonlinear null
condition, the result of [36] can be improved, and the well-posedness
threshold can be lowered. In this paper we provide the first result which
proves the validity of this conjecture, for a representative equation in this
class, namely the hyperbolic minimal surface equation. Further, our
improvement turns out to be substantial; precisely, we gain $3/8$ derivatives
in two space dimensions and $1/4$ derivatives in higher dimension. At this
regularity level, the Lorentzian metric $g$ in our problem is no better that
$C_{x,t}^{\frac{1}{4}+}\cap L^{2}_{t}C_{x}^{\frac{1}{2}+}$,
($C_{x,t}^{\frac{3}{8}+}\cap L^{4}_{t}C_{x}^{\frac{1}{2}+}$ in $2d$) far below
anything studied before.

Most of the ideas introduced in this paper will likely extend to other
nonlinear wave models, and open the way toward further progress in the study
of low regularity solutions.

###  1.1. The Minimal Surface Equation in Minkowski Space

Let $n\geq 2$, and ${\mathfrak{M}}^{n+2}$ be the $n+2$ dimensional Minkowski
space-time. A codimension one time-like submanifold
$\Sigma\subset{\mathfrak{M}}^{n+2}$ is called a minimal surface if it is
locally a critical point for the area functional

$\mathcal{L}=\int_{\Sigma}\,dA,$

where the area element is measured relative to the Minkowski metric. A
standard way to think of this equation is by representing $\Sigma$ as a graph
over ${\mathfrak{M}}^{n+1}$,

$\Sigma=\\{x_{n+1}=u(t,x_{1},\cdot,x_{n})\\},$

where $u$ is a real valued function

$u:D\subset{\mathfrak{M}}^{n+1}\rightarrow{\mathbb{R}},$

which satisfies the constraint

(1.1) $u_{t}^{2}<1+|\nabla_{x}u|^{2},$

expressing the condition that its graph is a time-like surface in
${\mathfrak{M}}^{n+2}$.

Then the surface area functional takes the form

(1.2) $\mathcal{L}(u)=\int\sqrt{1-u_{t}^{2}+|\nabla_{x}u|^{2}}\ dx.$

Interpreting this as a Lagrangian, the minimal surface equation can be thought
of as the associated Euler-Lagrange equation, which takes the form

(1.3) $-\frac{\partial}{\partial
t}\left(\frac{u_{t}}{\sqrt{1-u_{t}^{2}+|\nabla_{x}u|^{2}}}\right)+\sum_{i=1}^{n}\frac{\partial}{\partial
x_{i}}\left(\frac{u_{x_{i}}}{\sqrt{1-u_{t}^{2}+|\nabla_{x}u|^{2}}}\right)=0.$

Under the condition (1.1), the above equation is a quasilinear wave equation.

The left hand side of the last equation can be also interpreted as the mean
curvature of the hypersurface $\Sigma$, and as such the minimal surface
equation is alternatively described as the _zero mean curvature flow_.

In addition to the above geometric interpretation, the minimal surface
equation for time-like surfaces in the Minkowski space is also known as the
Born-Infeld model in nonlinear electromagnetism [48], as well as a model for
evolution of branes in string theory [14].

On the mathematical side, the question of global existence for small, smooth
and localized initial data was considered in work of Lindblad [29], Brendle
[7], Stefanov [38] and Wong [46]. The stability of a nonflat steady solution,
called the catenoid, was studied in [27, 9]. Some blow-up scenarios due to
failure of immersivity were investigated by Wong [47]. Minimal surfaces have
also been studied as singular limits of certain semilinear wave equations by
Jerrard [20]. The local well-posedness question fits into the similar theory
for the broader class of quasilinear wave equations, but there is also one
result which is specific to minimal surfaces, due to Ettinger [10]; this is
discussed later in the paper.

In our study of the minimal surface equation, the above way of representing it
is less useful, and instead it is better to think of it in geometric terms. In
particular the fact that the above Lagrangian (1.2) and the equation (1.3) are
formulated relative to a background Minkowski metric is absolutely non-
essential; one may instead use any flat Lorentzian metric. This is no surprise
since any two such metrics are equivalent via a linear transformation. Perhaps
less obvious is the fact that the equations may be actually written in an
identical fashion, independent of the background metric; see Remark 3.1 in
Section 3.

For full details on the structure of the equation we refer the reader to
Section 3 of the paper, but here we review the most important facts.

The main geometric object is the metric $g$ which is the trace of the
Minkowski metric in ${\mathfrak{M}}^{n+2}$ on $\Sigma$, and which, expressed
in the $(t=x_{0},x_{1},\cdots,x_{n})$ coordinates, has the form

(1.4) $g_{\alpha\beta}:=m_{\alpha\beta}+\partial_{\alpha}u\partial_{\beta}u,$

where $m_{\alpha\beta}$ denotes the Minkowski metric with signature
$(-1,1,...,1)$ in ${\mathfrak{M}}^{n+1}$. Since $\Sigma$ is time-like, this is
also a Lorentzian metric. This has determinant

(1.5)
$g:=|\det(g^{\alpha\beta})|=1+m^{\alpha\beta}\partial_{\alpha}u\,\partial_{\beta}u,$

and the dual metric is

(1.6)
$g^{\alpha\beta}:=m^{\alpha\beta}-\frac{m^{\alpha\gamma}m^{\beta\delta}\partial_{\gamma}u\,\partial_{\delta}u}{1+m^{\mu\nu}\partial_{\mu}u\,\partial_{\nu}u}.$

Here, and later in the paper, we carefully avoid raising indices with respect
to the Minkowski metric. Instead, all raised indices in this paper will be
with respect to the metric $g$.

Relative to this metric, the equation (1.3) can be expressed in the form

(1.7) $\Box_{g}u=0,$

where $\Box_{g}$ is the covariant d’Alembertian, and which in this problem
will be shown to have the simple expression

(1.8) $\Box_{g}=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}.$

An important role will also be played by the associated linearized equation,
which, as it turns out, may be easily expressed in divergence form as

(1.9)
$\partial_{\alpha}{\hat{g}}^{\alpha\beta}\partial_{\beta}v=0,\qquad{\hat{g}}^{\alpha\beta}:=g^{-\frac{1}{2}}g^{\alpha\beta}.$

Our objective in this paper will be to study the local well-posedness of the
associated Cauchy problem with initial data at $t=0$,

(1.10) $\left\\{\begin{aligned} &\Box_{g}u=0,\\\ &u(t=0)=u_{0},\\\
&u_{t}(t=0)=u_{1},\end{aligned}\right.$

where the initial data $(u_{0},u_{1})$ is taken in classical Sobolev spaces,

(1.11) $u[0]:=(u_{0},u_{1})\in\mathcal{H}^{s}:=H^{s}\times H^{s-1},$

and is subject to the constraint

(1.12) $u_{1}^{2}-|\partial_{x}u_{0}|^{2}<1.$

Here we use the following notation for the Cauchy data in (1.3) at time $t$,

$u[t]:=(u(t,\cdot),u_{t}(t,\cdot)).$

We aim to investigate the range of exponents $s$ for which local well-
posedness holds, and significantly improve the lower bound for this range.

###  1.2. Nonlinear wave equations

The hyperbolic minimal surface equation (1.3) can be seen as a special case of
more general scalar quasilinear wave equations, which have the form

(1.13) $g^{\alpha\beta}(\partial
u)\partial_{\alpha}\partial_{\beta}u=N(u,\partial u),$

where, again, $g^{\alpha\beta}$ is assumed to be Lorentzian, but without any
further structural properties, and where $u$ may be a vector valued function.
This generic equation will serve as a reference.

As a starting point, we note that the equation (1.3) (and also (1.13) if
$N=0$) admits the scaling law

$u(t,x)\to\lambda^{-1}u(\lambda t,\lambda x).$

This allows us to identify the critical Sobolev exponent as

$s_{c}=\frac{n+2}{2}.$

Heuristically, $s_{c}$ serves as a universal threshold for local well-
posedness, i.e. we have to have $s>s_{c}$. Taking a naive view, one might
think of trying to reach the scaling exponent $s_{c}$. However, this is a
quasilinear wave equation, and getting to $s_{c}$ has so far proved impossible
in any problem of this type.

As a good threshold from above, one might start with the classical well-
posedness result, due to Hughes, Kato, and Marsden [16], and which asserts
that local well-posedness holds for $s>s_{c}+1$. This applies to all equations
of the form (1.13), and can be proved solely by using energy estimates. These
have the form

(1.14) $\|u[t]\|_{\mathcal{H}^{s}}\lesssim
e^{\int_{0}^{t}\|\partial^{2}u(s)\|_{L^{\infty}}ds}\|u[0]\|_{\mathcal{H}^{s}}.$

They may also be restated in terms of quasilinear energy functionals $E^{s}$
which have the following two properties:

  1. (a)

Coercivity,

$E^{s}(u[t])\approx\|u[t]\|_{\mathcal{H}^{s}}^{2}.$

  2. (b)

Energy growth,

(1.15) $\frac{d}{dt}E^{s}(u)\lesssim\|\partial^{2}u\|_{L^{\infty}}\cdot
E^{s}(u).$

To close the energy estimates, it then suffices to use Sobolev embeddings,
which allow one to bound the above $L^{\infty}$ norm, which we will refer to
as a _control parameter_ , in terms of the $\mathcal{H}^{s}$ Sobolev norm
provided that $s>\frac{n}{2}+2$, which is one derivative above scaling.

The reason a derivative is lost in the above analysis is that one would only
need to bound $\|\partial^{2}u\|_{L^{1}L^{\infty}}$, whereas the norm that is
actually controlled is $\|\partial^{2}u\|_{L^{\infty}L^{\infty}}$; this
exactly accounts for the one derivative difference in scaling. It also
suggests that the natural way to improve the classical result is to control
the $L^{p}L^{\infty}$ norm directly. This is indeed possible in the context of
the Strichartz estimates, which in dimension three and higher give the bound

$\|\partial^{2}u\|_{L^{2}L^{\infty}}\lesssim\|u[0]\|_{\mathcal{H}^{\frac{n+3}{2}}},$

with another $\epsilon$ derivatives loss in three space dimensions. When true,
such a bound yields well-posedness for $s>\frac{n+3}{2}$, which is $1/2$
derivatives above scaling. The numerology changes slightly in two space
dimensions, where the best possible Strichartz estimate has the form

$\|\partial^{2}u\|_{L^{4}L^{\infty}}\lesssim\|u[0]\|_{\mathcal{H}^{\frac{n}{2}+\frac{7}{4}}},$

which is $3/4$ derivatives above scaling.

The difficulty in using Strichartz estimates is that, while these are well
known in the constant coefficient case [11, 22] and even for smooth variable
coefficients [21, 31], that is not as simple in the case of rough
coefficients. Indeed, as it turned out, the full Strichartz estimates are true
for $C^{2}$ metrics, see [33] ($n=2,3$), [42] (all $n$), but not, in general,
for $C^{\sigma}$ metrics when $\sigma<2$, see the counterexamples of [34, 35].
This difficulty was resolved in two stages:

  1. (i)

_Semiclassical time scales and Strichartz estimates with loss of derivatives._
The idea here, which applies even for $C^{\sigma}$ metrics with $\sigma<2$, is
that, associated to each dyadic frequency scale $2^{k}$, there is a
corresponding “ semiclassical ” time scale $T_{k}=2^{-\alpha k}$, with
$\alpha$ dependent on $\sigma$, so that full Strichartz estimates hold at
frequency $2^{k}$ on the scale $T_{k}$. Strichartz estimates with loss of
derivatives are then obtained by summing up the short time estimates with
respect to the time intervals, separately at each frequency. This idea was
independently introduced in [5] and [41], and further refined in [4] and [44].

  2. (ii)

_Wave packet coherence and parametrices._ The observation here is that in the
study of nonlinear wave equations such as (1.13), in addition to Sobolev-type
regularity for the metric, we have an additional piece of information, namely
that the metric itself can be seen as a solution to a nonlinear wave equation.
This idea was first introduced and partially exploited in [24], but was
brought to full fruition in [36], where it was shown that almost loss-less
Strichartz estimates hold for the solutions to (1.13) at exactly the correct
regularity level.

The result in [36] represents the starting point of the present work, and is
concisely stated as follows111The primary result in [36] is for the case when
$g=g(u)$, but it directly carries over to equations of the form (1.13). The
result as stated below applies equally to both cases, but if $g=g(u)$ then
$s_{c}$ is one unit lower.:

######  Theorem 1.1 (Smith-Tataru [36]).

(1.13) is locally well-posed in $\mathcal{H}^{s}$ provided that

(1.16) $s>s_{c}+\frac{3}{4},\qquad n=2,$

respectively

(1.17) $s>s_{c}+\frac{1}{2},\qquad n\geq 3.$

As part of this result, almost loss-less Strichartz estimates were obtained
both directly for the solution $u$, and more generally for the associated
linearized evolution. We will return to these estimates in Section 10 for a
more detailed statement and an in-depth discussion.

The optimality of this result, at least in dimension three, follows from work
of Lindblad [28], see also the more recent two dimensional result in [32].
However, this counterexample should only apply to “generic” models, and the
local well-posedness threshold might possibly be improved in problems with
additional structure, i.e. some form of null condition.

Moving forward, we recall that in [43], a null condition was formulated for
quasilinear equations of the form (1.13).

######  Definition 1.2 ([43]).

The nonlinear wave equation (1.13) satisfies the _nonlinear null condition_ if

(1.18) $\frac{\partial g^{\alpha\beta}(u,p)}{\partial
p_{\gamma}}\xi_{\alpha}\xi_{\beta}\xi_{\gamma}=0\qquad\text{in}\qquad
g^{\alpha\beta}(u,p)\xi_{\alpha}\xi_{\beta}=0.$

Here we use the terminology “nonlinear null condition” in order to distinguish
it from the classical null condition, which is relative to the Minkowski
metric, and was heavily used in the study of global well-posedness for
problems with small localized data, see [23] as well as the books [37, 15]. In
geometric terms, this null condition may be seen as a cancellation condition
for the self interactions of wave packets traveling along null geodesics. In
Section 3 we verify that the minimal surface equation indeed satisfies the
nonlinear null condition.

Further, it was conjectured in [43] that, for problems satisfying (1.18), the
local well-posedness threshold can be lowered below the one in [36]. This
conjecture has remained fully open until now, though one should mention two
results in [25] and [10] for the Einstein equation, respectively the minimal
surface equation, where the endpoint in Theorem 1.1 is reached but not
crossed.

The present work provides the first positive result in this direction,
specifically for the minimal surface equation. Indeed, not only are we able to
lower the local well-posedness threshold in Theorem 1.1, but in effect we
obtain a substantial improvement, namely by $3/8$ derivatives in two space
dimensions and by $1/4$ derivatives in higher dimension.

###  1.3. The main result

Our main result, stated in a succinct form, is as follows:

######  Theorem 1.3.

The Cauchy problem for the minimal surface equation (1.10) is locally well-
posed for initial data $u[0]$ in $\mathcal{H}^{s}$ which satisfies the
constraint (1.12), where

(1.19) $s>s_{c}+\frac{3}{8},\qquad n=2,$

respectively

(1.20) $s>s_{c}+\frac{1}{4},\qquad n\geq 3.$

The result is valid regardless of the $\mathcal{H}^{s}$ size of the initial
data. Here we interpret local well-posedness in a strong Hadamard sense,
including:

  * •

_existence of solutions_ in the class $u([\cdot])\in C[0,T;\mathcal{H}^{s}]$,
with $T$ depending only on the $\mathcal{H}^{s}$ size of the initial data.

  * •

_uniqueness of solutions_ , in the sense that they are the unique limits of
smooth solutions.

  * •

_higher regularity_ , i.e. if in addition the initial data
$u[0]\in\mathcal{H}^{m}$ with $m>s$, then the solution satisfies
$u([\cdot])\in C(0,T;\mathcal{H}^{m})$, with a bound depending only on the
$\mathcal{H}^{m}$ size of the data,

$\|u([\cdot])\|_{C(0,T;\mathcal{H}^{m})}\lesssim\|u[0]\|_{\mathcal{H}^{m}}.$

  * •

_continuous dependence_ in $\mathcal{H}^{s}$, i.e. continuity of the the data
to solution map

$\mathcal{H}^{s}\ni u[0]\to u([\cdot])\in C[0,T;\mathcal{H}^{s}].$

  * •

_weak Lipschitz dependence_ , i.e. for two $\mathcal{H}^{s}$ solutions $u$ and
$v$ we have the difference bound

$\|u([\cdot])-v([\cdot])\|_{C(0,T;\mathcal{H}^{\frac{1}{2}})}\lesssim\|u[0]-v[0]\|_{\mathcal{H}^{\frac{1}{2}}}.$

In addition to the above components of the local well-posedness result, a key
intermediate role in the proof of the above theorem is played by the
Strichartz estimates, not only for the solution $u$, but also, more
importantly, for the linearized problem

(1.21) $\left\\{\begin{aligned}
&\partial_{\alpha}{\hat{g}}^{\alpha\beta}\partial_{\beta}v=0,\\\
&v(t=0)=v_{0},\\\ &v_{t}(t=0)=v_{1},\end{aligned}\right.$

as well as its paradifferential counterpart

(1.22) $\left\\{\begin{aligned}
&\partial_{\alpha}T_{{\hat{g}}^{\alpha\beta}}\partial_{\beta}v=0,\\\
&v(t=0)=v_{0},\\\ &v_{t}(t=0)=v_{1},\end{aligned}\right.$

Here the paraproducts are defined using the Weyl quantization, see Section 2.2
for more details. For later reference, we state the Strichartz estimates in a
separate theorem:

######  Theorem 1.4.

Then there exists some $\delta_{0}>0$, depending on $s$ in (1.19), (1.19) so
that the following properties hold for every solution $u$ as in Theorem 1.3:

a) The solution $u$ in Theorem 1.3 satisfies the Strichartz estimates

(1.23) $\displaystyle\|\langle D_{x}\rangle^{\frac{1}{2}+\delta_{0}}\partial
u\|_{L^{4}L^{\infty}}\lesssim 1,\qquad n=2,$ $\displaystyle\|\langle
D_{x}\rangle^{\frac{1}{2}+\delta_{0}}\partial u\|_{L^{2}L^{\infty}}\lesssim
1,\qquad n\geq 3.$

b) Both the linearized equation (1.21) and its paradifferential version (1.22)
are well-posed in $\mathcal{H}^{\frac{5}{8}}$ for $(n=2)$ respectively
$\mathcal{H}^{\frac{1}{2}}$ for $n\geq 3$, and the following Strichartz
estimates hold for each222Of course with an implicit constant which may depend
on $\delta$. $\delta>0$:

(1.24) $\displaystyle\|v\|_{L^{\infty}\mathcal{H}^{\frac{5}{8}}}+\|\langle
D_{x}\rangle^{-\frac{n}{2}-\frac{1}{4}-\delta}\partial
v\|_{L^{4}(0,1;L^{\infty})}\lesssim$ $\displaystyle\
\|v[0]\|_{\mathcal{H}^{\frac{5}{8}}}\qquad n=2,$

respectively

(1.25) $\displaystyle\|v\|_{L^{\infty}\mathcal{H}^{\frac{1}{2}}}+\|\langle
D_{x}\rangle^{-\frac{n}{2}-\frac{1}{4}-\delta}\partial
v\|_{L^{2}(0,1;L^{\infty})}\lesssim$ $\displaystyle\
\|v[0]\|_{\mathcal{H}^{\frac{1}{2}}}\qquad n\geq 3,$

We note that the Strichartz estimates in both parts (a) and (b) have
derivative losses, namely $1/8$ derivatives in the $L^{4}L^{\infty}$ bound in
two dimensions, respectively $1/4$ derivatives in higher dimensions. These
estimates only represent the tip of the iceberg. One may also consider the
inhomogeneous problem, allow source terms in dual Strichartz spaces, etc.
These and other variations which play a role in this paper are discussed in
Section 4.

To understand the new ideas in the proof of our main theorem, we recall the
two key elements of the proof of the result in [36], namely (i) the classical
energy estimates (1.15) and (ii) the nearly lossless Strichartz estimates; at
the time, the chief difficulty was to prove the Strichartz estimates.

In this paper we completely turn the tables, taking part (ii) above for
granted, and instead work to improve the energy estimates. Let us begin with a
simple observation, which is that the minimal surface equation (1.7) has a
cubic nonlinearity, which allows one to replace (1.15) with

(1.26) $\frac{d}{dt}E^{s}(u)\lesssim\|\partial
u\|_{L^{\infty}}\|\partial^{2}u\|_{L^{\infty}}\cdot E^{s}(u).$

This is what one calls a _cubic energy estimate_ , which is useful in the
study of long time solutions but does not yet help with the low regularity
well-posedness question. The key to progress lies in developing a much
stronger form of this bound, which roughly has the form333See Section 2 for
our Besov norm notations.

(1.27) $\frac{d}{dt}E^{s}(u)\lesssim\|\partial
u\|_{B^{\frac{1}{2}}_{\infty,2}}^{2}\cdot E^{s}(u),$

where the two control norms on the right are now balanced, and only require
$1/2$ derivative less than (1.26). This is what we call a _balanced energy
estimate_ , which may only hold for a very carefully chosen energy functional
$E^{s}$.

This is an idea which originates in our recent work on 2D water waves (see
[1]), where balanced energy estimates are also used in order to substantially
lower the low regularity well-posedness threshold. Going back further, this
has its roots in earlier work of the last two authors [18], [17], in the
context of trying to apply normal form methods in order to obtain long time
well-posedness results in quasilinear problems. There we have introduced what
we called the _modified energy method_ , which in a nutshell asserts that in
quasilinear problems it is far better to modify the energies in a normal form
fashion, rather than to transform the equation. It was the cubic energy
estimates of [17] which were later refined in [1] to balanced energy
estimates. Along the way, we have also borrowed and adapted another idea from
Alazard and Delort [3, 2], which is to prepare the problem with a partial
normal form transformation, and is a part of their broader concept of
paradiagonalization; that same idea is also used here.

There are several major difficulties in the way of proving energy estimates
such as (1.27):

  * •

The normal form structure is somewhat weaker in the case of the minimal
surface equation, compared to water waves. As a consequence, we have to
carefully understand which components of the equation can be improved with a
normal form analysis and which cannot, and thus have to be estimated directly.

  * •

Not only are the energy functionals $E^{s}$ not explicit, they have to be
constructed in a very delicate way, following a procedure which is reminiscent
of Tao’s renormalization idea in the context of wave-maps [40], as well as the
subsequent work [45] of the third author on the same problem.

  * •

Keeping track of symbol regularities in our energy functionals and in the
proof of the energy estimates is also a difficult task. To succeed, here we
adapt and refine a suitable notion of paracontrolled distributions, an idea
which has already been used successfully in the realm of stochastic pde’s [12,
13].

  * •

The balanced energy estimates need to be proved not only for the full
equation, but also for the associated linear paradifferential equation, as a
key intermediate step, as well as for the full linearized flow. In particular,
when linearizing, some of the favourable normal form structure (or null
structure, to use the nonlinear wave equations language) is lost, and the
proofs become considerably more complex.

Finally, the Strichartz estimates of [36] cannot be used directly here.
Instead, we are able to reformulate them in a paradifferential fashion, and to
apply them on appropriate semiclassical time scales. After interval summation,
this leads to Strichartz estimates on the unit time scale but with derivative
losses. Precisely, in our main Strichartz estimates, whose aim is to bound the
control parameters in (1.27), we end up losing essentially $1/8$ derivatives
in two space dimensions, and $1/4$ derivatives in higher dimension. These
losses eventually determine the regularity thresholds in our main result in
Theorem 1.3.

One consequence of these energy estimates is the following continuation result
for the solutions:

######  Theorem 1.5.

The $\mathcal{H}^{s}$ solution $u$ given by Theorem 1.3 can be continued for
as long as the following integral remains finite:

(1.28) $\int_{0}^{T}\|\partial
u(t)\|_{B^{\frac{1}{2}}_{\infty,2}}^{2}dt<\infty.$

###  1.4. An outline of the paper

#### Paraproducts and paradifferential calculus

The bulk of the paper is written in the language of paradifferential calculus.
The notations and some of the basic product and paracommutator bounds are
introduced in Section 2. Importantly, we use the Weyl quantization throughout;
this plays a substantial role as differences between quantizations are not
always perturbative in our analysis. Also of note, we emphasize the difference
between balanced and unbalanced bounds, so some of our $\Psi$DO product or
commutator expansions have the form

$\text{commutator}=\text{principal part}+\text{unbalanced lower
order}+\text{balanced error}.$

#### The geometric form of the minimal surface equation

While the flat d’Alembertian may naively appear to play a role in the
expansion (1.3) of the minimal surface equation, this is not at all useful,
and instead we need to adopt a geometric viewpoint. As a starting point, in
Section 2 we consider several equivalent formulations of the minimal surface
equation, leading to its geometric form in (1.7). This is based on the metric
$g$ associated to the solution $u$ by (1.4), whose dual we also compute. Two
other conformally equivalent metrics will also play a role. In the same
section we derive the linearized equation, and also introduce the associated
linear paradifferential flow.

#### Strichartz estimates

As explained earlier, Strichartz estimates play a major role in our analysis.
These are applied to several equations, namely the full evolution, the linear
paradifferential evolution and finally the linearized equation; in the present
paper, we view the bounds for the paradifferential equation as the core ones,
and the other bounds as derived bounds, though not necessarily in a directly
perturbative fashion. The Strichartz estimates admit a number of formulations:
in direct form for the homogeneous flow, in dual form for the inhomogeneous
one, or in the full form. The aim of Section 4 is to introduce all these forms
of the Strichartz estimates, as well as to describe the relations between
them, in the context of this paper. A new idea here is to allow source terms
which are time derivatives of distributions in appropriate spaces; this is
achieved by reinterpreting the wave equation as a system.

#### Control parameters in energy estimates

We begin Section 5 by defining the control parameters ${\mathcal{A}}$ and
${\mathcal{B}}$, which will play a fundamental role in our energy estimate.
Here ${\mathcal{A}}$ is a scale invariant norm, at the level of $\|\partial
u\|_{L^{\infty}}$, which will remain small uniformly in time. ${\mathcal{B}}$,
on the other hand, is time dependent and at the level of
$\||D_{x}|^{\frac{1}{2}}\partial u\|_{L^{\infty}}$, and will control the
energy growth. Typically, our _balanced cubic energy estimates_ will have the
form

$\frac{\partial E}{\partial t}\lesssim_{{\mathcal{A}}}{\mathcal{B}}^{2}E.$

To propagate energy bounds we will need to know that ${\mathcal{B}}\in
L^{2}_{t}$. Also in the same section we prove a number of core bounds for our
solutions in terms of the control parameters.

#### The multiplier method and paracontrolled distributions

Both the construction of our energies and the proof of the energy estimates
are based on a paradifferential implementation of the multiplier method, which
leads to space-time identities of the form

$\int_{0}^{T}\\!\\!\\!\\!\int_{{\mathbb{R}}^{n}}\Box_{g}u\cdot
Xu\,dxdt=\left.E_{X}(u)\right|_{0}^{T}+\int_{0}^{T}\\!\\!\\!\\!\int_{{\mathbb{R}}^{n}}R(u)\,dxdt$

in a paradifferential format, where the vector field $X$ is our multiplier and
$E_{X}$ is its associated energy, while $R(u)$ is the energy flux term which
will have to be estimated perturbatively. A fundamental difficulty is that the
multiplier $X$, which should heuristically be at the regularity level of
$\partial u$ cannot be chosen algebraically, and instead has to be constructed
in an inductive manner relative to the dyadic frequency scales. In order to
accurately quantify the regularity of $X$, in Section 6 we use and refine the
notion of paracontrolled distributions; in a nutshell, while $X$ may not be
chosen to be a function of $\partial u$, it will still have to be
paracontrolled by $\partial u$, which we denote by $X\llcurly u$.

#### Energy estimates for the paradifferential equation

The construction of the energy functionals is carried out in Section 2.2,
primarily at the level of the linear paradifferential equation, first in
$\mathcal{H}$ and then in $\mathcal{H}^{s}$. In both cases there are two
steps: first the construction of the symbol of the multiplier $X$, as a
paracontrolled distribution, and then the proof of the energy estimates. The
difference between the two cases is that $X$ is a vector field in the first
case, but a full pseudodifferential operator in the second case; because of
this, we prefer to present the two arguments separately.

#### Energy estimates for the full equation

The aim of Section 8 is to prove that balanced cubic energy estimates hold for
the full equation in all $\mathcal{H}^{s}$ spaces with $s\geq 1$. We do this
by thinking about the full equation in a paradifferential form, i.e. as a
linear paradifferential equation with a nonlinear source term, and then by
applying a normal form transformation to the unbalanced part of the source
term.

#### Well-posedness for the linearized equation

The goal of Section 9 is to establish both energy and Strichartz estimates for
$\mathcal{H}^{\frac{1}{2}}$ solutions ($\mathcal{H}^{\frac{5}{8}}$ in
dimension two) to the linearized equation. This is achieved under the
assumption that both energy and Strichartz estimates for
$\mathcal{H}^{\frac{1}{2}}$ solutions ($\mathcal{H}^{\frac{5}{8}}$ in
dimension two) for the linear paradifferential equation hold. We remark that,
while the energy estimates for the linear paradifferential equation have
already been established by this point in the paper, the corresponding
Strichartz estimates have yet to be proved.

#### Short time Strichartz estimates for the full equation

The local well-posedness result of Smith and Tataru [36] yields well-posedness
and nearly sharp Strichartz estimates on the unit time scale for initial data
which is small in the appropriate Sobolev space. Our objective in Section 10
is to recast this result as a short time result for a corresponding large data
problem. This is a somewhat standard scaling/finite speed of propagation
argument, though with an interesting twist due to the need to use homogeneous
Sobolev norms.

#### Small vs. large $\mathcal{H}^{s}$ data

In our main well-posedness prof, in order to avoid more cumbersome notations
and estimates, it is convenient to work with initial data which is small in
$\mathcal{H}^{s}$. This is not a major problem, as this is a nonlinear wave
equation which exhibits finite speed of propagation. This allows us to reduce
the large data problem to the small data problem by appropriate localizations.
This argument is carried out at the beginning of Section 11.

#### Rough solutions as limits of smooth solutions

Our sequence of modules discussed so far comes together in Section 11, where
we finally obtain our rough solutions $u$ as a limit of smooth solutions
$u^{h}$ with initial data frequency localized below frequency $2^{h}$. The
bulk of the proof is organized as a bootstrap argument, where the bootstrap
quantities are uniform energy type bounds for both $u^{h}$ and for their
increments $v^{h}=\dfrac{d}{dh}u^{h}$, which solve the corresponding
linearized equation. The main steps are as follows:

  * •

we use the short time Strichartz estimates derived from [36] for $u^{h}$ and
$v^{h}$ in order to obtain long time Strichartz estimates for $u^{h}$, which
in turn implies energy estimates for both the full equation and the
paradifferential equation, and closes one half of the bootstrap.

  * •

we combine the short time Strichartz estimates and the long time energy
estimates for the paradifferential equation in $\mathcal{H}^{\frac{1}{2}}$
($\mathcal{H}^{\frac{5}{8}}$ if $n=2$) to obtain long time Strichartz
estimates for the same paradifferential equation.

  * •

we use the energy and Strichartz estimates for the paradifferential equation
to obtain similar bounds for the linearized equation. This in turn implies
long time energy estimates for $v^{h}$, closing the second half of the
bootstrap loop.

#### The well-posedness argument

Once we have a complete collection of energy estimates and Strichartz
estimates for both the full equation and the linearized equation, we are able
to use frequency envelopes in order to prove the remaining part of the well-
posedness results, namely the strong convergence of the smooth solutions, the
continuous dependence, and the associated uniqueness property. In this we
follow the strategy outlined in the last two authors’ expository paper [19].

###  1.5. Acknowledgements

The first author was supported by the Henry Luce Foundation. The second author
was supported by a Luce Associate Professorship, by the Sloan Foundation, and
by an NSF CAREER grant DMS-1845037. The third author was supported by the NSF
grant DMS-2054975 as well as by a Simons Investigator grant from the Simons
Foundation.

This material is also based upon work supported by the National Science
Foundation under Grant No. DMS-1928930 while all three authors participated in
the program Mathematical problems in fluid dynamics hosted by the Mathematical
Sciences Research Institute in Berkeley, California, during the Spring 2021
semester.

##  2\. Notations, paraproducts and some commutator type bounds

We begin with some standard notations and conventions:

  * •

The greek indices $\alpha,\beta,\gamma,\delta$ etc. in expressions range from
$0$ to $n$, where $0$ stands for time. Roman indices $i,j$ are limited to the
range from $1$ to $n$, and are associated only to spatial coordinates.

  * •

The differentiation operators with respect to all coordinates are
$\partial_{\alpha}$, $\alpha=0,...,n$. By $\partial$ without any index we
denote the full space-time gradient. To separate only spatial derivatives we
use the notation $\partial_{x}$.

  * •

We consistently use the Einstein summation convention, where repeated indices
are summed over, unless explicitly stated otherwise.

  * •

The inequality sign $x\lesssim y$ means $x\leq Cy$ with a universal implicit
constant $C$. If instead the implicit constant $C$ depends on some parameter
$A$ then we write instead $x\lesssim_{A}y$.

###  2.1. Littlewood-Paley decompositions and Sobolev spaces

We denote the Fourier variables by $\xi_{\alpha}$ with $\alpha=0,...,n$. To
separate the spatial Fourier variables we use the notation $\xi^{\prime}$.

####  2.1.1. Littlewood-Paley decompositions

For distributions in ${\mathbb{R}}^{n}$ we will use the standard inhomogeneous
Littlewood-Paley decomposition

$u=\sum_{k=0}^{\infty}P_{k}u,$

where $P_{k}=P_{k}(D_{x})$ are multipliers with smooth symbols
$p_{k}(\xi^{\prime})$, localized in the dyadic frequency region
$\\{|\xi|\approx 2^{k}\\}$ (unless $k=0$, where we capture the entire unit
ball). We emphasize that no such decompositions are used in the paper with
respect to the time variable. We will also use the notations $P_{<k}$,
$P_{>k}$ with the standard meaning. Often we will use shorthand for the
Littlewood-Paley pieces of $u$, such as $u_{k}:=P_{k}u$ or $u_{<k}:=P_{<k}u$.

####  2.1.2. Function spaces

For our main evolution we will use inhomogeneous Sobolev space $H^{s}$, often
combined as product spaces $\mathcal{H}^{s}=H^{s}\times H^{s-1}$ for the
position/velocity components of our evolution. In the next to last section of
the paper only we will have an auxiliary use for the corresponding homogeneous
spaces $\dot{H}^{s}$, in connection with scaling analysis.

For our estimates we will use $L^{\infty}$ based control norms. In addition to
the standard $L^{\infty}$ norms, in many estimates we will use the standard
inhomogeneous $BMO$ norm, as well as its close relatives $BMO^{s}$, with norm
defined as

$\|f\|_{BMO^{s}}=\|\langle D_{x}\rangle^{s}f\|_{BMO}.$

We will also need several related $L^{\infty}$ based Besov norms
$B^{s}_{\infty,q}$, defined as

$\|u\|_{B^{s}_{\infty,q}}^{q}=\sum_{k}2^{pks}\|P_{k}u\|_{L^{\infty}}^{p}$

with the obvious changes if $p=\infty$. In particular the spaces
$B^{0}_{\infty,1}$ and $B^{\frac{1}{2}}_{\infty,2}$ will be used for our
control norms ${\mathcal{A}}$ and ${\mathcal{B}}$.

####  2.1.3. Frequency envelopes

Throughout the paper we will use the notion of _frequency envelopes_ ,
introduced by Tao (see for example [40]), which is a very useful device that
tracks the evolution of the energy of solutions between dyadic energy shells.

######  Definition 2.1.

We say that $\\{c_{k}\\}_{k\geq 0}\in\ell^{2}$ is a frequency envelope for a
function $u$ in $H^{s}$ if we have the following two properties:

a) Energy bound:

(2.1) $\|P_{k}u\|_{H^{s}}\leq c_{k},$

b) Slowly varying

(2.2) $\frac{c_{k}}{c_{j}}\lesssim 2^{c|j-k|},\quad j,k\in\mathbb{N}.$

Here $c$ is a positive constant, which is taken small enough in order to
account for energy leakage between nearby frequencies.

One can also limit from above the size of a frequency envelope, by requiring
that

$\|u\|_{H^{s}}^{2}\approx\sum c_{k}^{2}.$

Such frequency envelopes always exist, for instance one can define

$c_{k}=\sup_{j}2^{-\delta|j-k|}\|P_{j}u\|_{H^{s}}.$

The same notion can be applied to any Besov norms. In particular we will use
it jointly for the Besov norms which define our control parameters
${\mathcal{A}}$ and ${\mathcal{B}}$.

###  2.2. Paraproducts and paradifferential operators

For multilinear analysis, we will consistently use paradifferential calculus,
for which we refer the reader to [6, 30].

We begin with the simplest bilinear expressions, namely products, for which we
will use the Littlewood-Paley trichotomy

$f\cdot g=T_{f}g+\Pi(f,g)+T_{g}f,$

where the three terms capture the _low $\times$high_ frequency interactions,
the _high $\times$high_ frequency interactions and the _low $\times$high_
frequency interactions. The paraproduct $T_{f}g$ might be heuristically
thought of as the dyadic sum

$T_{f}g=\sum_{k}f_{<k-\kappa}g_{k}$

where the frequency gap $\kappa$ can be simply chosen as a universal
parameter, say $k=4$, or on occasion may be increased and used as a smallness
parameter in a large data context. However, in our context a definition such
as the above one is too imprecise, and the difference between usually
equivalent choices is nonperturbative. Also, the symmetry properties of
$T_{f}$ as an operator in $L^{2}$ are important in our energy estimates. For
this reason, we choose to work with the Weyl quantization, and we define

$\mathcal{F}(T_{f}g)(\zeta)=\int_{\xi+\eta=\zeta}\hat{f}(\eta)\chi\left(\frac{|\eta|}{\langle\xi+\frac{1}{2}\eta\rangle}\right)\hat{g}(\xi)d\xi.$

Here $\chi$ is a smooth function supported in a small ball and which equals
$1$ near the origin. With this convention, if $f$ is real then $T_{f}$ is an
$L^{2}$ self-adjoint operator.

For paraproducts we have a number of standard bounds which we list below, and
we will refer to as Coifman-Meyer estimates:

(2.3) $\|T_{f}g\|_{L^{p}}\lesssim\|f\|_{L^{\infty}}\|g\|_{L^{p}},$

(2.4) $\|T_{f}g\|_{L^{p}}\lesssim\|f\|_{L^{p}}\|g\|_{BMO},$

(2.5) $\|\Pi(f,g)\|_{L^{p}}\lesssim\|f\|_{L^{p}}\|g\|_{BMO}.$

These hold for $1<p<\infty$, but there are also endpoint results available
roughly corresponding to $p=1$ and $p=\infty$.

Paraproducts may also be thought of as belonging to the larger class of
translation invariant bilinear operators. Such operators

$f,g\to B(f,g)$

may be described by their symbols $b(\eta,\xi)$ in the Fourier space, by

$\mathcal{F}B(u,v)(\zeta)=\int_{\xi+\eta=\zeta}b(\eta,\xi)\hat{f}(\eta)\hat{g}(\xi)d\xi.$

A special class of such operators, which we denote by $L_{lh}$, will play an
important role later in the paper:

######  Definition 2.2.

a) By $L_{lh}$ we denote translation invariant bilinear forms whose symbol
$\ell_{lh}(\eta,\xi)$ is supported in $\\{|\eta|\ll|\xi|+1\\}$ and satisfies
bounds of the form

$|\partial^{i}_{\eta}\partial^{j}_{\xi}\ell_{lh}(\eta,\xi)|\lesssim\langle\xi\rangle^{-i-j}.$

We remark that in particular the bilinear form $B(f,g)=T_{f}g$ is an operator
of type $L_{lh}$, with symbol

$b(\eta,\xi)=\chi\left(\frac{|\eta|}{\langle\xi+\frac{1}{2}\eta\rangle}\right).$

Here the factor in the denominator $\xi+\eta/2$ is the average of the $g$
input frequency and the output frequency, and corresponds exactly to our use
of the Weyl calculus. The $L^{p}$ bounds and the commutator estimates for such
bilinear form mirror exactly the similar bounds for paraproducts.

###  2.3. Commutator and other paraproduct bounds

Here we collect a number of general paraproduct estimates, which are
relatively standard. See for instance Appendix B of [17] and Section 2 of [1]
for proofs of the following estimates as well as further references.

We begin with the following standard commutator estimate:

######  Lemma 2.3 ($P_{k}$ commutators).

We have

(2.6) $\|[T_{f},P_{k}]\|_{\dot{H}^{s}\to\dot{H}^{s}}\lesssim 2^{-k}\|\partial
f\|_{L^{\infty}},$

also in $L^{\infty}$.

The following commutator-type estimates are either exact reproductions of, or
closely follow, statements from Section 2 of [1]:

######  Lemma 2.4 (Para-commutators).

Assume that $\gamma_{1},\gamma_{2}<1$. Then we have

(2.7)
$\|T_{f}T_{g}-T_{g}T_{f}\|_{\dot{H}^{s}\to\dot{H}^{s+\gamma_{1}+\gamma_{2}}}\lesssim\||D|^{\gamma_{1}}f\|_{BMO}\||D|^{\gamma_{2}}g\|_{BMO}.$

######  Lemma 2.5 (Para-associativity).

For $s+\gamma_{2}\geq 0,s+\gamma_{1}+\gamma_{2}\geq 0$, and $\gamma_{1}<1$ we
have

(2.8)
$\|T_{f}\Pi(v,u)-\Pi(v,T_{f}u)\|_{\dot{H}^{s+\gamma_{1}+\gamma_{2}}}\lesssim\||D|^{\gamma_{1}}f\|_{BMO}\||D|^{\gamma_{2}}v\|_{BMO}\|u\|_{\dot{H}^{s}}.$

######  Lemma 2.6 (Para-Leibniz rule).

For the balanced Leibniz rule error

$E^{\pi}_{L}(u,v)=T_{f}\partial_{\alpha}\Pi(u,v)-\Pi(T_{f}\partial_{\alpha}u,v)-\Pi(u,T_{f}\partial_{\alpha}v)$

we have the bound

(2.9)
$\|E^{\pi}_{L}(u,v)\|_{H^{s}}\lesssim\|f\|_{BMO^{\frac{1}{2}}}\|u\|_{BMO^{-\frac{1}{2}-\sigma}}\|v\|_{H^{s+\sigma}},\qquad\sigma\in{\mathbb{R}}.$

Next, we state paraproduct estimates which also may be found in [1]:

######  Lemma 2.7 (Para-products).

Assume that $\gamma_{1},\gamma_{2}<1$, $\gamma_{1}+\gamma_{2}\geq 0$. Then

(2.10)
$\|T_{f}T_{g}-T_{fg}\|_{\dot{H}^{s}\to\dot{H}^{s+\gamma_{1}+\gamma_{2}}}\lesssim\||D|^{\gamma_{1}}f\|_{BMO}\||D|^{\gamma_{2}}g\|_{BMO}.$

######  Lemma 2.8 (Low-high para-products).

Assume that $\gamma_{1},\gamma_{2}<1$, $\gamma_{1}+\gamma_{2}\geq 0$. Then

(2.11)
$\|T_{f}T_{g}-T_{T_{f}g}\|_{\dot{H}^{s}\to\dot{H}^{s+\gamma_{1}+\gamma_{2}}}\lesssim\||D|^{\gamma_{1}}f\|_{BMO}\||D|^{\gamma_{2}}g\|_{BMO}.$

These are stated here in the more elegant homogeneous setting, but there are
obvious modifications which also apply in the inhomogeneous case. We end with
the following Moser-type result:

######  Lemma 2.9.

Let $F$ be smooth with $F(0)=0$, and $w\in H^{s}$. Set

$R(w)=F(w)-T_{F^{\prime}(w)}w.$

Then we have the estimate

(2.12) $\|R(w)\|_{H^{s+\frac{1}{2}}}\lesssim
C(\|w\|_{L^{\infty}})\|D^{\frac{1}{2}}w\|_{BMO}\|w\|_{H^{s}}.$

###  2.4. Paradifferential operators

As a generalization of paraproducts, we will also work with paradifferential
operators. Precisely, given a symbol $a(x,\xi)$ in ${\mathbb{R}}^{n}$, we
define its paradifferential Weyl quantization $T_{a}$ as the operator

$\mathcal{F}(T_{f}g)(\zeta)=\int_{\xi+\eta=\zeta}\hat{f}(\eta)\chi\left(\frac{|\eta|}{\langle\xi+\frac{1}{2}\eta\rangle}\right)\hat{a}(\eta,\xi)\hat{g}(\xi)d\xi,$

where

$\hat{a}(\eta,\xi)=\mathcal{F}_{x}a(x,\xi).$

The simplest class of symbols one can work with is $L^{\infty}S^{m}$, which
contains symbols $a$ for which

(2.13) $|\partial_{\xi}^{\alpha}a(x,\xi)|\leq
c_{\alpha}\langle\xi\rangle^{m-|\alpha|}$

for all multi-indices $\alpha$. For such symbols, the Calderon-Vaillancourt
theorem insures appropriate boundedness in Sobolev spaces,

$T_{a}:H^{s}\to H^{s-m}.$

More generally, given a translation invariant space of distributions $X$, we
can define an associated symbol class $XS^{m}$ of symbols with the property
that

(2.14) $\|\partial_{\xi}^{\alpha}a(x,\xi)\|_{X}\leq
c_{\alpha}\langle\xi\rangle^{m-|\alpha|}$

for each $\xi\in{\mathbb{R}}^{n}$. Later in the paper, we will use several
choices of symbols of this type, using function spaces which we will associate
to our problem.

##  3\. A complete set of equations

Here we aim to further describe the minimal surface equation and the
underlying geometry, and, in particular, its null structure. We also derive
the linearized equation, and introduce the paralinearization of both the main
equation and its linearization.

###  3.1. The Lorentzian geometry of the minimal surface

Starting from the expression of the metric $g$ in (1.4), the dual metric is
easily computed to be

(3.1)
$g^{\alpha\beta}:=m^{\alpha\beta}-\frac{m^{\alpha\gamma}m^{\beta\delta}\partial_{\gamma}u\partial_{\delta}u}{1+m^{\mu\nu}\partial_{\mu}u\partial_{\nu}u}.$

Also associated to the metric $g$ is its determinant

$g=\det(g_{\alpha\beta})=\det(g^{\alpha\beta})^{-1},$

and the associated volume form

$dV=\sqrt{g}\,dx.$

This can be easily computed as

$g=1+m^{\mu\nu}\partial_{\mu}u\,\partial_{\nu}u.$

In the sequel, we will always raise indices with respect to the metric $g$,
never with respect to Minkowski. In particular we will use the standard
notation

(3.2) $\partial^{\alpha}=g^{\alpha\beta}\partial_{\beta}.$

We remark that, when applied to the function $u$, this operator has nearly the
same effect as the corresponding Minkowski operator,

(3.3) $\partial^{\alpha}u=\frac{1}{g}m^{\alpha\beta}\partial_{\beta}u.$

###  3.2. The minimal surface equation

Here we rewrite the minimal surface equation in covariant form. Using the $g$
notation above and the Minkowski metric, we rewrite (1.3) as

$m^{\alpha\beta}\partial_{\alpha}(g^{-\frac{1}{2}}\partial_{\beta}u)=0,$

or equivalently

$m^{\alpha\beta}(\partial_{\alpha}\partial_{\beta}u-\frac{1}{2g}\partial_{\alpha}g\partial_{\beta}u)=0.$

Expanding the $g$ derivative, we have

(3.4)
$\partial_{\alpha}g=2m^{\mu\nu}\partial_{\mu}u\,\partial_{\alpha}\partial_{\nu}u.$

Then in the previous equation we recognize the expression for the dual metric,
and the minimal surface equation becomes

(3.5) $g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u=0.$

Using the notation (3.2), this is written in an even shorter form,

(3.6) $\partial^{\alpha}\partial_{\alpha}u=0.$

Similarly, using also (3.3), the relation (3.4) becomes

(3.7)
$\frac{1}{2g}\partial_{\alpha}g=\partial^{\nu}u\,\partial_{\alpha}\partial_{\nu}u.$

###  3.3. The covariant d’Alembertian

The covariant d’Alembertian associated to the metric $g$ has the form

$\Box_{g}=\frac{1}{\sqrt{g}}\partial_{\alpha}\sqrt{g}g^{\alpha\beta}\partial_{\beta},$

which we can rewrite as

$\displaystyle\Box_{g}$
$\displaystyle=\partial_{\alpha}g^{\alpha\beta}\partial_{\beta}+\frac{1}{2g}(\partial_{\alpha}g)g^{\alpha\beta}\partial_{\beta}$
$\displaystyle=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}+\left(\partial_{\alpha}g^{\alpha\beta}\right)\partial_{\beta}+\frac{1}{2g}(\partial_{\alpha}g)g^{\alpha\beta}\partial_{\beta}.$

Next we need to compute the two coefficients in round brackets. The second one
is given by (3.7). For the first one, for later use, we perform a slightly
more general computation where we differentiate
$g^{\alpha\beta}(\partial_{\gamma}u)$ as a function of its arguments
$p_{\gamma}:=\partial_{\gamma}u$,

(3.8) $\frac{\partial g^{\alpha\beta}}{\partial
p_{\gamma}}=-\partial^{\alpha}u\,g^{\beta\gamma}-\partial^{\beta}u\,g^{\alpha\gamma}.$

This formula follows by directly differentiating (3.1) and from (3.3),

$\frac{\partial g^{\alpha\beta}}{\partial
p_{\gamma}}=-m^{\alpha\gamma}\partial^{\beta}u\,-m^{\beta\gamma}\partial^{\alpha}u\,+2g\partial^{\alpha}\,u\partial^{\beta}u\,\partial^{\nu}u.$

We use (3.1) once again to get (3.8)

$\displaystyle\frac{\partial g^{\alpha\beta}}{\partial p_{\gamma}}$
$\displaystyle=-[g^{\alpha\gamma}+g\partial^{\alpha}u\partial^{\gamma}u]\partial^{\beta}u-[g^{\beta\gamma}+g\partial^{\gamma}u\partial^{\beta}u]\partial^{\alpha}u+2\partial^{\alpha}u\partial^{\beta}ug\partial^{\nu}u,$
$\displaystyle=-g^{\alpha\gamma}\partial^{\beta}u-g^{\beta\gamma}\partial^{\alpha}u.$

From (3.8) and chain rule, we arrive at

(3.9)
$\partial_{\gamma}g^{\alpha\beta}=-\partial^{\alpha}u\,g^{\beta\delta}\partial_{\gamma}\partial_{\delta}u-\partial^{\beta}u\,g^{\alpha\sigma}\partial_{\gamma}\partial_{\sigma}u.$

Setting $\gamma=\alpha$ and using the minimal surface equation in the (3.5)
formulation, we get

(3.10)
$\partial_{\alpha}g^{\alpha\beta}=-\partial^{\alpha}u\,g^{\beta\delta}\partial_{\alpha}\partial_{\delta}u.$

Comparing this with (3.7), we see that the last two terms in the $\Box_{g}$
expression above cancel, and we obtain the following simplified form for the
covariant d’Alembertian:

(3.11) $\Box_{g}=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}.$

In particular, we get the covariant form of the minimal surface equation for
$u$:

(3.12) $\Box_{g}u=0.$

For later use, we introduce the notation

(3.13)
$A^{\alpha}=-\partial_{\beta}g^{\alpha\beta}=\frac{1}{2g}\partial^{\alpha}g=\partial^{\beta}u\,g^{\alpha\delta}\partial_{\beta}\partial_{\delta}u.$

An interesting observation is that from here on, the Minkowski metric plays
absolutely no role:

######  Remark 3.1.

In order to introduce the minimal surface equations we have started from the
Minkowski metric $m^{\alpha\beta}$. However, the formulation (3.5) of the
equations together with the relations (3.8) provide a complete description of
the equations without any reference to the Minkowski metric, and which is in
effect valid for any other Lorentzian metric. Indeed, the equation (3.5)
together with the fact that the metric components $g^{\alpha\beta}$ are smooth
functions of $\partial u$ satisfying (3.8) are all that is used for the rest
of the paper. Thus, our results apply equally for any other Lorentzian metric
in ${\mathbb{R}}^{n+2}$.

###  3.4. The linearized equations

Our objective now is to derive the linearized minimal surface equations. We
will denote by $v$ the linearized variable. Then, by (3.8), the linearization
of the dual metric $g^{\alpha\beta}=g^{\alpha\beta}(u)$ takes the form

$\delta
g^{\alpha\beta}=-\partial^{\alpha}u\,g^{\beta\nu}\partial_{\nu}v-\partial^{\beta}u\,g^{\alpha\sigma}\partial_{\sigma}v.$

Then the linearized equation is directly computed, using the symmetry in
$\alpha$ and $\beta$, as

$g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}v-2\partial^{\alpha}u\,g^{\beta\gamma}\partial_{\gamma}v=0.$

Using the expression of $A$ in (3.13), the linearized equations take the form

(3.14)
$(g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}-2A^{\gamma}\partial_{\gamma})v=0.$

Alternatively this may also be written in a divergence form,

(3.15)
$(\partial_{\alpha}g^{\alpha\beta}\partial_{\beta}-A^{\gamma}\partial_{\gamma})v=0.$

or in covariant form,

(3.16) $\displaystyle\Box_{g}v$ $\displaystyle=2A^{\beta}\partial_{\beta}v.$

###  3.5.  Null forms and the nonlinear null condition

The primary null form which plays a role in this article is $Q_{0}$, defined
by

(3.17)
$Q_{0}(v,w):=g^{\alpha\beta}\partial_{\alpha}v\partial_{\beta}w=\partial^{\alpha}v\partial_{\alpha}w.$

Now, we verify that the nonlinear null condition (1.18) holds; for this we use
(3.8) to compute

$\frac{\partial
g^{\alpha\beta}}{\partial_{g_{\gamma}}}\xi_{\alpha}\xi_{\beta}\xi_{\gamma}=\left(-g^{\alpha\beta}\partial^{\beta}u-g^{\beta\gamma}\partial^{\alpha}u\right)\xi_{\alpha}\xi_{\beta}\xi_{\gamma},$

which vanishes on the null cone $g^{\alpha\beta}\xi_{\alpha}\xi_{\beta}=0$.

In addition we would like the contribution of $A$ to the linearized equation
to be a null form. We get

$A^{\beta}\partial_{\beta}v={\partial^{\alpha}u}\,Q_{0}(\partial_{\alpha}u,v).$

###  3.6.  Two conformally equivalent metrics

While the metric $g$ is the primary metric we use in this paper, for technical
reasons we will also introduce two additional, conformally equivalent metrics,
as follows:

(i) The metric ${\tilde{g}}$ is defined by

(3.18) ${\tilde{g}}^{\alpha\beta}:=(g^{00})^{-1}g^{\alpha\beta}.$

Then the minimal surface equation can be written as

(3.19) ${\tilde{g}}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u=0$

while the linearized equation, written in divergence form is

(3.20)
$({\tilde{g}}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}-{\tilde{A}}^{\alpha}\partial_{\alpha})v=0,$

where, still raising indices only with respect to $g$,

(3.21)
${\tilde{A}}^{\alpha}=(g^{00})^{-1}A^{\alpha}-{\tilde{g}}^{\alpha\beta}\partial_{\alpha}(\ln
g^{00})=\partial^{\beta}u{\tilde{g}}^{\alpha\delta}\partial_{\beta}\partial_{\delta}u+2\partial^{0}u{\tilde{g}}^{0\delta}{\tilde{g}}^{\alpha\beta}\partial_{\beta}\partial_{\delta}u.$

The main feature of $\tilde{g}$ is that $\tilde{g}^{00}=1$. Because of this,
it will be useful in the study of the linear paradifferential flow, in order
to prevent a nontrivial paracoefficient in front of $\partial_{0}^{2}v$ in the
equations.

(ii) The metric ${\hat{g}}$ is defined by

(3.22) ${\hat{g}}^{\alpha\beta}=g^{-\frac{1}{2}}g^{\alpha\beta}.$

Then the minimal surface equation can be written as

(3.23) ${\hat{g}}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u=0,$

which is not so useful. Instead, the advantage of using this metric is that,
using (3.13), the linearized equation can now be written in divergence form,

(3.24) $\partial_{\alpha}{\hat{g}}^{\alpha\beta}\partial_{\beta}v=0.$

This will be very useful when we study the linearized equation in
$\mathcal{H}^{\frac{1}{2}}$ (respectively $\mathcal{H}^{\frac{5}{8}}$ in two
dimensions).

###  3.7. Paralinearization and the linear paradifferential flow

A key element in our study of the minimal surface equation is the associated
linear paradifferential flow, which is derived from the linearized flow
(3.14). In inhomogeneous form, this is

(3.25)
$(\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}-T_{A^{\gamma}}\partial_{\gamma})w=f.$

Similarly we can write the paradifferential equations associated to
${\tilde{g}}$, namely

(3.26)
$(\partial_{\alpha}T_{{\tilde{g}}^{\alpha\beta}}\partial_{\beta}-T_{{\tilde{A}}^{\gamma}}\partial_{\gamma})w=f.$

as well as ${\hat{g}}$, which can be written in divergence form:

(3.27) $\partial_{\alpha}T_{{\hat{g}}^{\alpha\beta}}\partial_{\beta}w=f.$

These are all equivalent up to perturbative errors. Accordingly, we introduce
the notation

(3.28) $T_{P}=\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}$

for the paradifferential wave operator as well as its counterparts
$T_{\tilde{P}}$ and $T_{\hat{P}}$ with the metric $g$ replaced by
${\tilde{g}}$, respectively ${\hat{g}}$.

We will first use the paradifferential equation in the study of the minimal
surface problem (3.5), which we rewrite in the form

(3.29)
$(T_{g^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}-2T_{A^{\gamma}}\partial_{\gamma})u=N(u).$

A key contention of our paper is that the nonlinearity $N$ plays a
perturbative role. However, this has to be interpreted in a more subtle way,
in the sense that $N$ becomes perturbative only after a well chosen partial,
variable coefficient normal form transformation.

Secondly, we will use it in the study of the linearized minimal surface
equation, which we can write in the form

(3.30)
$\partial_{\alpha}T_{{\hat{g}}^{\alpha\beta}}\partial_{\beta}v=N_{lin}(u)v.$

Here the nonlinearity $N_{lin}$ will also play a perturbative role, in the
same fashion as above. We caution the reader that this is _not_ the
linearization of $N$.

##  4\. Energy and Strichartz estimates

Both energy and Strichartz estimates play an essential role in this paper, in
various forms and combinations. These are primarily applied first to the
linear paradifferential flow, and then to the linearized flow associated to
solutions to our main equation (1.7). Our goal here is to provide a brief
overview of these estimates.

Importantly, in this section we do not prove any energy or Strichartz
estimates. Instead, we simply provide definitions and context for what will be
proved later in the paper, and prove a good number of equivalences between
various well-posedness statements and estimates. We do this under absolutely
minimal assumptions (e.g. boundedness) on the metric $g$, in order to be able
to apply these properties easily later on. In particular there are no
commutator bounds needed or used in this section. The structure of the minimal
surface equations also plays no role here.

###  4.1. The equations

For context, here we consider a pseudo-Riemannian metric $g$ in
$I\times{\mathbb{R}}^{n}$, where $I=[0,T]$ is a time interval of unspecified
length. We will make some minimal universal assumptions on the metric $g$:

  * •

both $g$ and its inverse are uniformly bounded,

  * •

the time slices are uniformly space-like.

Associated to this metric $g$, we will consider several equations:

The linear paradifferential flow in divergence form:

(4.1) $\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}v=f,\qquad
v[0]=(v_{0},v_{1}).$

The linear paradifferential flow in non-divergence form:

(4.2) $T_{g^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}v=f,\qquad
v[0]=(v_{0},v_{1}).$

The linear flow in divergence form:

(4.3) $\partial_{\alpha}g^{\alpha\beta}\partial_{\beta}v=f,\qquad
v[0]=(v_{0},v_{1}).$

The linear flow in non-divergence form:

(4.4) $g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}v=f\qquad
v[0]=(v_{0},v_{1}).$

Several comments are in order:

  * •

As written, the above evolutions are inhomogeneous. If $f=0$ then we will
refer to them as the _homogeneous_ flows.

  * •

In the context of this paper, we are primarily interested in the metric
${\hat{g}}$, in which case the equation (4.3) represents our main linearized
flow, and (4.1) represents our main linear paradifferential flow. The metric
$g$ and the nondivergence form of the equations will be used in order to
connect our results with the result of Smith-Tataru, which will be used in our
proofs.

  * •

One may also add a gradient potential in the equations above; with the
gradient potential added there is no difference between the divergence and the
non-divergence form of the equations. We omit it in this section, as it plays
no role.

We will consider these evolutions in the inhomogeneous Sobolev spaces
$\mathcal{H}^{s}$. In order to do this uniformly, we will assume that $|I|\leq
1$; else using homogeneous spaces would be more appropriate. The exponent $s$
will be an arbitrary real number in the case of the paradifferential flows,
but will have a restricted range otherwise.

###  4.2.  Energy estimates and well-posedness for the homogeneous problem

Here we review some relatively standard definitions and facts about local
well-posedness.

######  Definition 4.1.

For any of the above flows in the homogeneous form, we say that they are
(forward) well-posed in $\mathcal{H}^{s}$ in the time interval $I=[0,T]$ if
for each initial data $u[0]\in\mathcal{H}^{s}$ there exists a unique solution
$u$ with the property that

$u([\cdot])\in C(I;\mathcal{H}^{s}).$

This corresponds to a linear estimate of the form

(4.5)
$\|v[\cdot]\|_{L^{\infty}(I;\mathcal{H}^{s})}\lesssim\|v[0]\|_{\mathcal{H}^{s}}.$

Sometimes one establishes additional bounds for the solution (e.g. Strichartz
estimates) and these are then added in to the class of solutions for which
uniqueness is established. We will comment on this where needed. If no such
assumption is used, we call this _unconditional uniqueness_.

For completeness and reference, we now state without proof a classical well-
posedness result:

######  Theorem 4.2.

Assume that $\partial g\in L^{1}(I;L^{\infty})$. Then

a) The paradiffererential flows (4.1) and (4.2) are wellposed in
$\mathcal{H}^{s}$ for all real $s$.

b) The divergence form evolution (4.3) is well-posed in $\mathcal{H}^{s}$ for
$s\in[0,1]$, and the non-divergence form evolution (4.4) is well-posed in
$\mathcal{H}^{s}$ for $s\in[1,2]$.

We remark that the metrics $g$ associated with the solutions of Smith-Tataru
satisfy the above hypothesis, but the solutions in our paper do not.

A slightly stronger form of well-posedness is to assert the existence of a
suitable (time dependent) energy functional $E^{s}$ in $\mathcal{H}^{s}$:

######  Definition 4.3.

An energy functional for either of the above problems in $\mathcal{H}^{s}$ is
a bounded quadratic form in $\mathcal{H}^{s}$ which has the following two
properties:

  1. a)

Coercivity,

(4.6) $E^{s}(v[t])\approx\|v[t]\|_{\mathcal{H}^{s}}^{2}.$

  2. b)

Bounded growth for solutions $v$ to the homogeneous equation,

(4.7) $\frac{d}{dt}E^{s}(v[t])\lesssim B(t)\|v[t]\|_{\mathcal{H}^{s}}^{2},$

where $B\in L^{1}$ depends only on $g$.

Later we will also interpret $E^{s}$ as a symmetric bilinear form in
$\mathcal{H}^{s}$. Such an interpretation is unique.

We remark that, in the context of Theorem 4.2, where $\partial g\in
L^{1}L^{\infty}$, an energy functional $E^{1}$ corresponding to $s=1$ is
classically obtained by multiplying the equation with a suitable smooth time-
like vector field and integrating by parts; we refer the reader to Section
7.2.1 where this procedure is described in greater detail. Then for $s\neq 1$
one simply defines

$E^{s}(v[0])=E^{1}(\langle D_{x}\rangle^{s-1}v[0]),$

and the corresponding control parameter $B$ may be taken as

$B(t)=\|\partial g(t)\|_{L^{\infty}}.$

###  4.3.  The wave equation as a system and the inhomogeneous problem

Switching now to the associated inhomogeneous flows, the classical set-up is
to take a source term $f\in L^{1}H^{s-1}$, and then look for solutions $v$ in
$C(I;\mathcal{H}^{s})$ as above. This is commonly done using the _Duhamel
principle_ , which is most readily applied by rewriting the wave equation as a
system. We next describe this process.

A common choice is to write the system for the pair of variables
$(v,\partial_{t}v)$. However, for us it will be more convenient to to make a
slightly different linear transformation, and use instead the pair

(4.8) $\mathbf{v}[t]:=\begin{pmatrix}v(t)\\\
g^{0\alpha}\partial_{\alpha}v(t)\end{pmatrix}:=Q\begin{pmatrix}v\\\
\partial_{t}v\end{pmatrix},\qquad Q=\begin{pmatrix}1&0\\\
g^{0j}\partial_{j}&g^{00}\end{pmatrix}$

for (4.3) and (4.4), with products replaced by paraproducts in the case of the
equation (4.1) or (4.2). For later use, we record the inverse of $Q$; this is
either

(4.9) $Q^{-1}=\begin{pmatrix}1&0\\\
-(g^{00})^{-1}g^{0j}\partial_{j}&(g^{00})^{-1}\end{pmatrix},$

or its version with products replaced by paraproducts, as needed.

The system for $\mathbf{v}$ will have the form

(4.10) $\frac{d}{dt}\mathbf{v}[t]={\mathcal{L}}\mathbf{v}[t],$

with the appropriate choice for the matrix operator ${\mathcal{L}}$. For
instance in the case of the homogeneous equation (4.3) we have

(4.11)
${\mathcal{L}}=\begin{pmatrix}-(g^{00})^{-1}g^{0j}\partial_{j}&(g^{00})^{-1}\\\
-(g^{00})^{-1}\partial_{i}g^{ij}\partial_{j}+\partial_{i}g^{i0}(g^{00})^{-1}g^{0j}\partial_{j}&-\partial_{j}(g^{00})^{-1}g^{0j}\end{pmatrix},$

which has the antisymmetry property (only for the principal part, in the non-
divergence case)

(4.12) ${\mathcal{L}}^{*}=-J{\mathcal{L}}J^{-1},\qquad J=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}.$

We will always work in settings where $Q$ is bounded and invertible in
$\mathcal{H}^{s}$. This is nearly automatic in the paradifferential case;
there we only need to make sure that the operator $T_{g^{00}}$ is invertible.
In the differential case we will have to ask that multiplication by $g$ and by
$(g^{00})^{-1}$ are bounded in $H^{s-1}$. In such settings, $\mathcal{H}^{s}$
well-posedness for our original wave equation and for the associated system
are equivalent. If a good energy functional $E^{s}$ exists for the wave
equation, then we may define an associated energy functional for the system by
setting

(4.13) $\mathbf{E}^{s}(\mathbf{v}[t])=E^{s}(Q^{-1}\mathbf{v}[t]).$

Then the properties (4.6) and (4.7) directly transfer to the homogeneous
system (4.10).

If our system is (forward) well-posed in $\mathcal{H}^{s}$, then solving it
generates a (forward) evolution operator $S(t,s)$ which is bounded in
$\mathcal{H}^{s}$ and maps the data at time $s$ to the solution at time $t$,

$S(t,s)\mathbf{v}[s]=\mathbf{v}[t].$

For the system it is easy to consider the inhomogeneous version

(4.14) $\frac{d}{dt}\mathbf{v}[t]={\mathcal{L}}\mathbf{v}[t]+\mathbf{f}[t].$

If $f\in L^{1}\mathcal{H}^{s}$ then the solution to (4.14) is given by
Duhamel’s formula,

(4.15)
$\mathbf{v}[t]=S(t,0)\mathbf{v}[0]+\int_{0}^{t}S(t,s)\mathbf{f}[s]\,ds,$

and satisfies the bound

(4.16)
$\|\mathbf{v}\|_{L^{\infty}\mathcal{H}^{s}}\lesssim\|\mathbf{v}[0]\|_{H^{s}}+\|\mathbf{f}\|_{L^{1}\mathcal{H}^{s}}.$

If we have a good energy $\mathbf{E}^{s}$ for the homogeneous system, then
Duhamel’s formula easily allows us to obtain the corresponding energy estimate
for the inhomogeneous one, namely

(4.17)
$\frac{d}{dt}\mathbf{E}^{s}(\mathbf{v}[t])\lesssim\mathbf{E}^{s}(\mathbf{v}[t],\mathbf{f}[t])+B(t)\|\mathbf{v}[t]\|_{\mathcal{H}^{s}}^{2}.$

Now we are ready to return to our original set of equations, add the source
term $f$ and reinterpret the above consequences of Duhamel’s formula there. As
in the homogeneous case, we define $\mathbf{v}[t]=Qv[t]$. Then adding the
source term $f$ in the original equation is equivalent to adding a source term
$\mathbf{f}$ in the above system. Indeed, it is readily seen that for all our
four equations, $\mathbf{f}$ is given by

(4.18) $\mathbf{f}[t]=\begin{pmatrix}0\\\ f(t)\end{pmatrix}.$

To complete the correspondence, we note that for such $\mathbf{f}$ we have

$Q^{-1}\mathbf{f}=({g^{00}})^{-1}\mathbf{f}[t].$

Then we immediately arrive at the following result:

######  Theorem 4.4.

a) Assume that either of the homogeneous paradifferential flows (4.1) or (4.2)
are well-posed in $\mathcal{H}^{s}$. Then the associated inhomogeneous flows
are well-posed in $\mathcal{H}^{s}$ for $f\in L^{1}H^{s-1}$, and the following
estimate holds

(4.19)
$\|v[\cdot]\|_{L^{\infty}(I;\mathcal{H}^{s})}\lesssim\|v[0]\|_{\mathcal{H}^{s}}+\|f\|_{L^{1}H^{s-1}}.$

In addition, if an energy functional $E^{s}$ in $\mathcal{H}^{s}$ exists, then

(4.20) $\frac{d}{dt}E^{s}(v[t])\lesssim
E^{s}(v[t],(T_{g^{00}})^{-1}\mathbf{f}[t])+B(t)\|u[t]\|_{\mathcal{H}^{s}}^{2}.$

b) The same holds for the flows (4.3) or (4.4) under the additional assumption
that multiplication by $g$ and $({g^{00}})^{-1}$ is bounded in $H^{s-1}$, with
the paraproduct replaced by the corresponding product.

For our purposes in this paper, we will also need to allow for a larger class
of source terms of the form

(4.21) $f=\partial_{t}f_{1}+f_{2}.$

To understand why this is natural, it is instructive to start from the
inhomogeneous system (4.14) and argue backward.

Above, we have used the inhomogeneous system in the case where the first
component of $\mathbf{f}$ was zero. Now we will allow for both terms to be
nonzero in $\mathbf{f}$, and derive the corresponding wave equation. For
clarity we do this in the context of the equation (4.3), for which we have
computed the corresponding operator ${\mathcal{L}}$ in (4.11); however, a
similar computation will apply in all four cases.

We begin by defining

(4.22) $v(t)=\mathbf{v}_{1}(t)$

as our candidate for the wave equation solution. Then the first equation of
the system reads

(4.23)
$\partial_{t}v=-(g^{00})^{-1}g^{0j}\partial_{j}v+(g^{00})^{-1}\mathbf{v}_{2}+\mathbf{f}_{1},$

or equivalently

$g^{0\alpha}\partial_{\alpha}v=\mathbf{v}_{2}+g^{00}\mathbf{f}_{1}.$

Differentiating this with respect to time we obtain

$\displaystyle\partial_{\beta}g^{\beta\alpha}\partial_{\alpha}v=$
$\displaystyle\
\partial_{j}g^{j\alpha}\partial_{\alpha}v+\partial_{t}\mathbf{v}_{2}+\partial_{t}g^{00}\mathbf{f}_{1}$
$\displaystyle=$ $\displaystyle\
\partial_{j}g^{jk}\partial_{k}v+\partial_{j}g^{j0}\partial_{t}v+\partial_{t}\mathbf{v}_{2}+\partial_{t}g^{00}\mathbf{f}_{1}.$

Finally we substitute $\partial_{t}v$ from (4.23) and
$\partial_{t}\mathbf{v}_{2}$ from the second equation of the system. We
already know the right hand side should vanish if $\mathbf{f}=0$, so it
suffices to track the $\mathbf{f}$ terms. Then we easily obtain the desired
equation for $v$:

(4.24)
$\partial_{\beta}g^{\beta\alpha}\partial_{\alpha}v=\partial_{\alpha}g^{\alpha
0}\mathbf{f}_{1}+\mathbf{f}_{2}.$

Comparing this with (4.21), we obtain the correspondence between the source
terms for the wave equation and the system:

(4.25) $f_{1}=g^{00}\mathbf{f}_{1},\qquad
f_{2}=\partial_{k}g^{k0}\mathbf{f}_{1}+\mathbf{f}_{2}.$

We also record here the correspondence between the solutions, in the form

(4.26)
$\mathbf{v}_{1}=v,\qquad\mathbf{v}_{2}=g^{0\alpha}\partial_{\alpha}v-g^{00}\mathbf{f}_{1},$

noting that this is no longer homogeneous, as in (4.8).

The last step in our analysis is to reinterpret the bounds (4.16) and (4.17)
in terms of $v$ and $f$. To do this we make the assumption that multiplication
by $g$ and $(g^{00})^{-1}$ is bounded in both $H^{s}$ and $H^{s-1}$. Then from
(4.16) we get

$\displaystyle\|v\|_{L^{\infty}\mathcal{H}^{s}}\lesssim$ $\displaystyle\
\|\mathbf{v}\|_{L^{\infty}\mathcal{H}^{s}}+\|\mathbf{f}_{1}\|_{L^{\infty}\mathcal{H}^{s}}$
$\displaystyle\lesssim$ $\displaystyle\
\|\mathbf{v}[0]\|_{\mathcal{H}^{s}}+\|\mathbf{f}\|_{L^{1}\mathcal{H}^{s}}+\|\mathbf{f}_{1}\|_{L^{\infty}\mathcal{H}^{s}}$
$\displaystyle\lesssim$ $\displaystyle\
\|v[0]\|_{\mathcal{H}^{s}}+\|f_{1}\|_{L^{1}H^{s-1}}+\|f_{2}\|_{L^{1}H^{s}\cap
L^{\infty}\mathcal{H}^{s}}.$

Similarly, from (4.17) and (4.13) we obtain the energy bound

(4.27)
$\frac{d}{dt}E^{s}(Q^{-1}\mathbf{v}[t])\lesssim\mathbf{E}^{s}(Q^{-1}\mathbf{v}[t],Q^{-1}\mathbf{f}[t])+B(t)\|v[t]\|_{\mathcal{H}^{s}}^{2}.$

Here we use (4.9) and (4.26) to compute

$Q^{-1}\mathbf{v}[t]=v[t]-\begin{pmatrix}0\\\
\mathbf{f}_{1}\end{pmatrix}=v[t]-\begin{pmatrix}0\\\
(g^{00})^{-1}f_{1}\end{pmatrix}:=v[t]-\tilde{v}[t],$

respectively, using also (4.25),

(4.28) $\displaystyle\tilde{f}[t]:=Q^{-1}\mathbf{f}[t]=$ $\displaystyle\
Q^{-1}\begin{pmatrix}(g^{00})^{-1}f_{1}\\\
f_{2}-\partial_{k}g^{k0}(g^{00})^{-1}f_{1}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\ \begin{pmatrix}(g^{00})^{-1}f_{1}\\\
(g^{00})^{-1}(f_{2}-\partial_{k}g^{k0}(g^{00})^{-1}f_{1}-g^{0k}\partial_{k}(g^{00})^{-1}f_{1})\end{pmatrix}.$

Thus we obtain the following natural extension of Theorem 4.4 above:

######  Theorem 4.5.

a) Assume that the homogeneous evolution (4.4) or (4.3) is well-posed in
$\mathcal{H}^{s}$, and that multiplication by $g$ and $(g^{00})^{-1}$ is
bounded in $H^{s}$ and in $H^{s-1}$. Consider the evolution (4.4) with a
source term $f$ of the form

$f=\partial_{t}f_{1}+f_{2},\qquad f_{1}\in L^{1}H^{s}\cap CH^{s-1},\qquad
f_{2}\in L^{1}H^{s-1}.$

Then a unique solution $u\in C(I,\mathcal{H}^{s})$ exists. If in addition the
homogeneous problem admits an energy functional $E^{s}$ as in Definition 4.3
then we have the energy estimate

(4.29) $\frac{d}{dt}E^{s}(v[t]-\tilde{v}[t])\lesssim
E^{s}(v[t]-\tilde{v}[t],\tilde{f}[t])+B(t)\|v[t]-\tilde{v}[t]\|_{\mathcal{H}^{s}}^{2}$

with $\tilde{v}$ and $\tilde{f}$ defined above and $B$ as in (4.7).

b) The same result applies for the paradifferential equations (4.1),
respectively (4.2), where all instances of $g$ above are replaced by the
corresponding paraproducts $T_{g}$.

We remark that in the situations where we apply this result, the mapping
properties for $g$ and $(g^{00})^{-1}$ will be fairly straightforward to
verify. In the paradifferential case, for instance, the continuity of $g$ will
suffice.

###  4.4. A duality argument

Duality plays an important role in many estimates for evolution equations. We
will also use duality considerations in this paper for several arguments. We
restrict the discussion below to the problems written in divergence form, as
this is what we will use later in the paper. However, similar versions may be
formulated in the nondivergence case.

At heart, this is based on the following identity, which in the context of the
operator $\partial_{\alpha}g^{\alpha\beta}\partial_{\beta}$ is written as
follows:

(4.30)
$\int_{0}^{T}\\!\\!\\!\\!\int_{{\mathbb{R}}^{n}}\partial_{\alpha}g^{\alpha\beta}\partial_{\beta}v\cdot
w-v\cdot\partial_{\alpha}g^{\alpha\beta}\partial_{\beta}w\,dxdt=\left.\int
g^{0j}\partial_{j}v\cdot w-v\cdot g^{0j}\partial_{j}w\,dx\right|_{0}^{T}.$

This holds for any test functions $v$ and $w$. The integral on the right can
be viewed as a duality relation between $u[t]$ and $v[t]$,

$B(u[t],v[t])=\int g^{0j}\partial_{j}u\cdot v-u\cdot g^{0j}\partial_{j}v\,dx.$

Precisely, assuming that $g:H^{s-1}\to H^{s-1}$ and that $g^{00}$ is
invertible, this expression has the following two properties

  1. (1)

Boundedness,

$B:\mathcal{H}^{s}\times\mathcal{H}^{1-s}\to{\mathbb{R}}.$

  2. (2)

Coercivity,

$\sup_{\|v\|_{\mathcal{H}^{1-s}}\leq 1}B(u,v)\approx\|u\|_{\mathcal{H}^{s}}.$

A standard consequence of this relation is the following property:

######  Proposition 4.6.

The evolutions (4.3), respectively (4.1) are forward well-posed in
$\mathcal{H}^{s}$ iff they are backward well-posed in $\mathcal{H}^{1-s}$.

We remark that in the context of this paper forward and backward well-
posedness are almost identical, so for us this property says that well-
posedness in $\mathcal{H}^{s}$ and $\mathcal{H}^{1-s}$ are equivalent.

The above proposition may be equivalently reformulated as the corresponding
result for the system (4.10). It will be more convenient to view it in this
context. To do this, we reinterpret the above duality, in terms of the
associated system (4.14). In view of the symmetry property (4.12), we have the
relation

(4.31)
$\int_{0}^{T}\\!\\!\\!\\!\int_{{\mathbb{R}}^{n}}(\partial_{t}-{\mathcal{L}})\mathbf{v}\cdot
J\mathbf{w}-J\mathbf{v}\cdot(\partial_{t}-{\mathcal{L}})\mathbf{w}\,dxdt=\left.\int\mathbf{v}\cdot
J\mathbf{w}\,dx\ \right|_{0}^{T},$

where the corresponding duality relation is

(4.32) ${\mathbf{B}}({\mathbf{u}},\mathbf{v})=\int\mathbf{v}\cdot
J\mathbf{w}\,dx,$

which provides the duality between $\mathcal{H}^{s}$ and $\mathcal{H}^{1-s}$.
Incidentally, a consequence of (4.31) is the duality relation

$S(t,s)=S(s,t)^{\ast},$

where the duality between $\mathcal{H}^{s}$ and $\mathcal{H}^{1-s}$ is the one
given by the bilinear form ${\mathbf{B}}$ above. This can be used to construct
the backward evolution in $\mathcal{H}^{1-s}$ given the forward evolution in
$\mathcal{H}^{s}$, and vice-versa. The full equivalence argument is standard,
and is omitted.

###  4.5. Strichartz estimates

Here we discuss several versions of Strichartz estimates, as well as the
connection between them.

####  4.5.1. Estimates for homogeneous equations

In the context of this paper, these have the form

(4.33) $\|v\|_{S^{r}}+\|\partial_{t}v\|_{S^{r-1}}\lesssim\|v[0]\|_{H^{r}},$

where for the Strichartz space $S$ we will consider two different choices:

  1. i)

Almost lossless estimates, akin to those established in Smith-Tataru [36]. The
corresponding Strichartz norms, denoted by $S=S_{ST}$ are defined as

(4.34) $\displaystyle\|v\|_{S_{ST}^{r}}=\|v\|_{L^{\infty}H^{r}}+\|\langle
D_{x}\rangle^{r-\frac{3}{4}-\delta}v\|_{L^{4}L^{\infty}},\qquad n=2,$
$\displaystyle\|v\|_{S_{ST}^{r}}=\|v\|_{L^{\infty}H^{r}}+\|\langle
D_{x}\rangle^{r-\frac{n-1}{2}-\delta}v\|_{L^{2}L^{\infty}},\qquad n\geq 3.$

Here the loss of derivatives is measured by $\delta>0$, which is an
arbitrarily small parameter.

  2. ii)

Estimates with derivative losses, precisely the type that will be established
in this paper. The corresponding Strichartz norms, denoted by $S=S_{AIT}$ are
defined as

(4.35) $\displaystyle\|v\|_{S_{AIT}^{r}}=\|v\|_{L^{\infty}H^{r}}+\|\langle
D_{x}\rangle^{r-\frac{3}{4}-\frac{1}{8}-\delta}v\|_{L^{4}L^{\infty}},\qquad
n=2,$ $\displaystyle\|v\|_{S_{AIT}^{r}}=\|v\|_{L^{\infty}H^{r}}+\|\langle
D_{x}\rangle^{r-\frac{n-1}{2}-\frac{1}{4}-\delta}v\|_{L^{2}L^{\infty}},\qquad
n\geq 3.$

Here $\delta>0$ is again an arbitrarily small parameter, but we allow for an
additional loss of derivatives in the endpoint (Pecher) estimate, namely $1/8$
derivatives in two space dimensions and $1/4$ in higher dimensions.

These estimates can be applied to any of the four equations discussed in this
section. There are also appropriate counterparts for the corresponding system
(4.10), which have the form

(4.36)
$\|\mathbf{v}_{1}\|_{S^{r}}+\|\mathbf{v}_{2}\|_{S^{r-1}}\lesssim\|\mathbf{v}[0]\|_{\mathcal{H}^{r}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

Under very mild assumptions on $g$, these are equivalent to the ones for the
corresponding wave equation:

######  Proposition 4.7.

The Strichartz estimates (4.33) for the homogeneous wave equation are
equivalent to the Strichartz estimates (4.36) for the associated system.

We also remark on a very mild extension of the estimate (4.33) to the
inhomogeneous case. Precisely, if (4.33) holds then we also have the
inhomogeneous bound

(4.37)
$\|v\|_{S^{r}}+\|\partial_{t}v\|_{S^{r-1}}\lesssim\|v[0]\|_{H^{r}}+\|f\|_{L^{1}H^{r-1}}.$

This follows in a straightforward manner by the Duhamel formula, see the
discussion in Section 4.3.

We conclude the discussion of the Strichartz estimates for the homogeneous
equation with a simple but important case, which will be useful for us in the
sequel, and applies in particular to the solutions in [36].

######  Proposition 4.8.

Assume that $\partial g\in L^{1}L^{\infty}$ and that the Strichartz estimates
for the homogeneous equation (4.4) hold in $\mathcal{H}^{1}$. Then the
Strichartz estimates for the homogeneous equation hold in $\mathcal{H}^{r}$
for all $r\in{\mathbb{R}}$ for both paradifferential flows (4.1) and (4.2).

We remark that the implicit constant in these Strichartz estimates depends on
the implicit constant in the Strichartz estimate in the hypothesis and on the
bound for $\|\partial g\|_{L^{1}L^{\infty}}$. Later when we apply this result
we will have uniform control over both, so we obtain uniform control over the
$\mathcal{H}^{r}$ Strichartz norm.

###### Proof.

It will be easier to work with the inhomogeneous bound (4.37), as it is more
stable with respect to perturbations. We divide the proof into several steps,
all of which are relatively standard.

_Step 1:_ We start with the case $r=1$ with the additional assumption
$g^{00}=-1$. Then the second equation in (4.2) can be seen as a perturbation
of (4.4) with an $L^{1}L^{2}$ source term. Hence the bound (4.37) for (4.4)
implies the same bound for (4.2).

_Step 2:_ Next, assuming still that $g^{00}=-1$, we extend the bound (4.37)
for (4.2) to all integers $r$ by conjugating by $\langle
D_{x}\rangle^{\sigma}$ with $\sigma=r-1$, where we can estimate perturbatively
the commutator

(4.38) $\|[T_{g^{\alpha\beta}},\langle
D_{x}\rangle^{\sigma}]\partial_{\alpha}\partial_{\beta}\langle
D_{x}\rangle^{-\sigma}v\|_{L^{1}L^{2}}\lesssim\|\partial
g\|_{L^{1}L^{\infty}}\|\partial v\|_{L^{\infty}L^{2}}.$

_Step 3:_ We multiply by $(T_{g^{00}})^{-1}$ to reduce the problem with
nonconstant $g^{00}$ to the case when $g^{00}=-1$. At the conclusion of this
step, we have the bound (4.37) for (4.2) for all $r$.

_Step 4:_ We commute the paracoefficients $T_{g^{\alpha\beta}}$ inside
$\partial_{\alpha}$ perturbatively, in order to obtain the bound (4.37) for
(4.1) for all $r$. ∎

####  4.5.2. Dual Strichartz estimates

Here one considers the corresponding inhomogeneous problems, with source terms
in dual Strichartz spaces. The estimates have the form

(4.39)
$\|v\|_{L^{\infty}\mathcal{H}^{r}}\lesssim\|v[0]\|_{\mathcal{H}^{r}}+\|f\|_{(S^{1-r})^{\prime}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

Classically, these are obtained by duality from the homogeneous estimates, as
follows:

######  Proposition 4.9.

If the homogeneous estimates (4.33) hold in $\mathcal{H}^{r}$ for the forward
(backward) evolution then the dual estimates (4.39) hold in
$\mathcal{H}^{1-r}$ for the backward (forward) evolution.

However, one can do better than this by going instead through the system form
of the equations (4.14). The dual estimates for (4.14) have the form

(4.40)
$\|\mathbf{v}\|_{L^{\infty}\mathcal{H}^{r}}\lesssim\|\mathbf{v}[0]\|_{\mathcal{H}^{r}}+\|\mathbf{f}_{1}\|_{(S^{-r})^{\prime}}+\|\mathbf{f}_{2}\|_{(S^{-r+1})^{\prime}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

These are directly obtained from the homogeneous estimates for the system
(4.10) via the duality (4.31):

######  Proposition 4.10.

If the homogeneous estimates hold in $\mathcal{H}^{r}$ for the forward
(backward) evolution (4.10) then the dual estimates hold in
$\mathcal{H}^{1-r}$ for the backward (forward) evolution (4.14).

One can now further return to the original inhomogeneous equation with a
source term as in (4.21), and use the correspondence (4.25) and (4.26), in
order to transfer the dual bounds back. These dual estimates, which represent
a generalization of (4.39), have the form

(4.41)
$\|v\|_{L^{\infty}\mathcal{H}^{r}}\lesssim\|v[0]\|_{\mathcal{H}^{r}}+\|f_{1}\|_{L^{\infty}H^{r-1}\cap(S^{-r})^{\prime}}+\|f_{2}\|_{(S^{1-r})^{\prime}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

We obtain the following strengthening of Proposition 4.9:

######  Proposition 4.11.

If the homogeneous estimates (4.33) hold in $\mathcal{H}^{r}$ for the forward
(backward) evolution then the dual estimates (4.41) hold in
$\mathcal{H}^{1-r}$ for the backward (forward) evolution.

####  4.5.3.  Full (retarded) Strichartz estimates

Here we combine the homogeneous and dual Strichartz estimates in a single
bound for the inhomogeneous problem. The classical form is

(4.42)
$\|v\|_{S^{r}}+\|\partial_{t}v\|_{S^{r-1}}\lesssim\|v[0]\|_{\mathcal{H}^{r}}+\|f\|_{(S^{1-r})^{\prime}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

However, here we need to take the extra step where we allow source terms of
the form $f=\partial_{t}f_{1}+f_{2}$, and then the estimates have the form

(4.43)
$\|v\|_{S^{r}}+\|\partial_{t}v\|_{S^{r-1}}\lesssim\|v[0]\|_{\mathcal{H}^{r}}+\|f_{1}\|_{S^{r-1}\cap(S^{-r})^{\prime}}+\|f_{2}\|_{(S^{1-r})^{\prime}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

As we will see, this is closely related to the corresponding bound for the
associated inhomogeneous system (4.14):

(4.44)
$\|\mathbf{v}_{1}\|_{S^{r}}+\|\mathbf{v}_{2}\|_{S^{r-1}}\lesssim\|\mathbf{v}[0]\|_{\mathcal{H}^{r}}+\|\mathbf{f}_{1}\|_{(S^{-r})^{\prime}}+\|\mathbf{f}_{2}\|_{(S^{-r+1})^{\prime}},\qquad
S\in\\{S_{ST},S_{AIT}\\}.$

Our main result here is as follows:

######  Theorem 4.12.

Consider either the equation (4.3) or (4.1). If the homogeneous problem is
well-posed forward in $\mathcal{H}^{r}$ and backward in $\mathcal{H}^{1-r}$
and satisfies the homogeneous Strichartz estimates (4.42) in both cases, then
the solutions to the associated forward inhomogeneous problem with source term
$f=\partial_{t}f_{1}+f_{2}$ satisfy the bounds (4.43).

###### Proof.

The proof consists of a number of steps:

Step 1: If the homogeneous problem is well-posed forward in $\mathcal{H}^{r}$
and satisfies the homogeneous Strichartz estimates (4.36), then so does the
corresponding system, see Proposition 4.7.

Step 2: If the homogeneous problem is well-posed backward in
$\mathcal{H}^{1-r}$ and satisfies the homogeneous Strichartz estimates, then
so does the corresponding system. By duality, the inhomogeneous system is
well-posed forward in $H^{r}$ and satisfies the dual Strichartz bounds (4.40).

Step 3: We represent the forward $\mathcal{H}^{r}$ solution by the Duhamel
formula

$\mathbf{v}[t]=S(t,0)\mathbf{v}[0]+\int_{0}^{t}S(t,s)\mathbf{f}(s)\,ds.$

The first term represents the solution to the homogeneous equation, and is
estimated by (4.36). For the second term we have two bounds at our disposal:
the dual bound where we fix $t$ and estimate the output in $\mathcal{H}^{s}$
in terms of the input in the dual Strichartz space, and the homogeneous bound
where we fix $s$, set $\mathbf{f}(s)\in\mathcal{H}^{r}$ and estimate the
output as a function of $t$ in the Strichartz space. Concatenating the two, we
get the restricted bound

(4.45)
$\|\mathbf{v}_{1}\|_{S^{r}(J)}+\|\mathbf{v}_{2}\|_{S^{r-1}(J)}\lesssim\|\mathbf{f}_{1}\|_{(S^{-r})^{\prime}(I)}+\|\mathbf{f}_{2}\|_{(S^{-r+1})^{\prime}(I)},\qquad
S\in\\{S_{ST},S_{AIT}\\},$

where the source $\mathbf{f}$ is supported in an interval $I$ and the output
$\mathbf{v}$ is measured in an interval $J$ so that $I$ precedes $J$. In two
dimensions we can now apply the Christ-Kiselev lemma [8] (or the
$U^{p}$-$V^{p}$ spaces, see [26]) to get the full estimate. In three and
higher dimensions we have a slight problem which is that neither method
applies for bounds from $L^{2}_{t}$ to $L^{2}_{t}$. However in our case this
is not an issue, because our estimates allow for at least a loss of $\delta$
derivatives. Then we can afford to interpolate the two endpoints and use the
Christ-Kiselev lemma for bounds from $L^{2-}_{t}$ to $L^{2+}_{t}$ and then
return to the endpoint setting by Bernstein’s inequality in space and Holder’s
inequality in time, all at the expense of an arbitrarily small increase in the
size $\delta$ of the loss.

Step 4. We transfer the estimate (4.44) back to the original system via the
correspondence (4.25), (4.26), in order to obtain (4.43). ∎

We conclude with a corollary of Theorem 4.12, which will be used later in the
paper and follows by combining this result with Proposition 4.8:

######  Corollary 4.13.

Assume that $\partial g\in L^{1}L^{\infty}$ and that the Strichartz estimates
for the homogeneous equation (4.4) hold in $\mathcal{H}^{1}$. Then the full
Strichartz estimates (4.43) hold in $\mathcal{H}^{r}$ for all
$r\in{\mathbb{R}}$ for both paradifferential flows (4.1) and (4.2).

##  5\. Control parameters and related bounds

###  5.1.  Control parameters

Here we introduce our main control parameters associated to a solution $u$ to
the minimal surface equation, which serve to bound the growth of energy for
both solutions to the minimal surface flow and for its linearization. We will
use two such primary quantities, ${\mathcal{A}}$ and ${\mathcal{B}}$, which
are defined as $L^{\infty}$ based Besov norms of the solution $u$, as follows:

(5.1) ${\mathcal{A}}=\sup_{t\in[0,T]}\sum_{k}\|P_{k}\partial
u\|_{L^{\infty}},$

respectively

(5.2) ${\mathcal{B}}(t)=\left(\sum_{k}2^{k}\|P_{k}\partial
u\|_{L^{\infty}}^{2}\right)^{\frac{1}{2}}.$

In connection with ${\mathcal{A}}$, we will also need the slightly stronger
variant ${\mathcal{A}^{\sharp}}\gtrsim{\mathcal{A}}$,

(5.3)
${\mathcal{A}^{\sharp}}=\sup_{t\in[0,T]}\sum_{k}2^{\frac{k}{2}}\|P_{k}\partial
u\|_{L^{2n}},\qquad 2<p<\infty.$

Here the choice of the exponent $2n$ is in no way essential, though it does
provide some minor simplifications in one or two places.

In a nutshell, the energy functionals we construct later in the paper will be
shown to satisfy _cubic balanced_ bounds of the form

(5.4) $\frac{dE}{dt}\lesssim_{{\mathcal{A}^{\sharp}}}{\mathcal{B}}^{2}E,$

which guarantee that energy bounds can be propagated for as long as
${\mathcal{A}^{\sharp}}$ remains finite and ${\mathcal{B}}$ remains in
$L^{2}_{t}$. One should compare these bounds with the classical energy
estimates, which have the form

(5.5) $\frac{dE}{dt}\lesssim_{{\mathcal{A}}}\|\partial^{2}u\|_{L^{\infty}}E,$

and which require an extra half derivative in the control parameter.

We continue with a few comments concerning our choice of control parameters:

  * •

Here ${\mathcal{A}}$ and ${\mathcal{A}^{\sharp}}$ are critical norms for $u$,
which may be described using the Besov notation as capturing the norm

${\mathcal{A}}=\|\partial
u\|_{L^{\infty}_{t}B^{0}_{\infty,1}},\qquad{\mathcal{A}^{\sharp}}=\|\partial
u\|_{L^{\infty}_{t}B^{\frac{1}{2}}_{2n,1}}.$

In a first approximation, the reader should think of ${\mathcal{A}}$ as simply
capturing the $L^{\infty}$ norm of $\partial u$; the slightly stronger Besov
norm above is needed for minor technical reasons, and allows us to work with
scale invariant bounds. Often we will simply rely on the simpler
$L^{\infty}$-bound, since

(5.6) $\|\partial u\|_{L^{\infty}}\lesssim{\mathcal{A}}.$

  * •

The control norm ${\mathcal{B}}$, taken at fixed time, is $1/2$ derivative
above scaling, and may also be described using the Besov notation as

${\mathcal{B}}(t)=\|\partial u(t)\|_{B^{\frac{1}{2}}_{\infty,2}}.$

Again, in a first approximation one should simply think of it as $\|\partial
u\|_{BMO^{\frac{1}{2}}}$, which in effect suffices for most of the analysis.
Indeed, we have

(5.7) $\|\partial u\|_{BMO^{\frac{1}{2}}}\lesssim{\mathcal{B}}.$

  * •

Given the choice of these control parameters, it is not difficult to see that
our energy estimates of the form (5.4) are invariant with respect to scaling.
This by itself does not mean much; even the classical energy estimates, of the
form (5.5), are scale invariant, but much less useful for low regularity well-
posedness. What is important here is that our energy estimates are _cubic_ and
_balanced_.

  * •

The fact that our control norms are based on uniform, rather than
$L^{2}$-bounds, particularly at the level of ${\mathcal{B}}$, is also
critical. This is what allows us to use Strichartz estimates to further
improve the low regularity well-posedness threshold in our results.

For bookkeeping reasons we will use a joint frequency envelope
$\\{c_{k}\\}_{k}$ for the dyadic components of both ${\mathcal{A}^{\sharp}}$
and ${\mathcal{B}}$, so that

(i) $\\{c_{k}\\}_{k}$ is normalized in $\ell^{2}$ and slowly varying,

$\sum c_{k}^{2}=1;$

(ii) We have control of dyadic Littlewood-Paley pieces as follows for
$\partial u$:

(5.8) $2^{-\frac{k}{2}}\|P_{k}\partial
u\|_{L^{2n}}\lesssim{\mathcal{A}^{\sharp}}c_{k}^{2},\qquad
2^{\frac{k}{2}}\|P_{k}\partial u\|_{L^{\infty}}\lesssim{\mathcal{B}}c_{k}.$

A-priori, these frequency envelopes depend on time. However, at the conclusion
of the paper, we will see that for our rough solutions they can be taken to be
independent of time, essentially equal to appropriate $L^{2}$-type frequency
envelopes for the initial data.

###  5.2. Related bounds

We will frequently need to use bounds which are similar to (5.8) in nonlinear
expressions, so it is convenient to have a notation for the corresponding
space:

######  Definition 5.1.

The space $\mathfrak{C}_{0}$ is the Banach space of all distributions $v$
which satisfy the bounds

(5.9) $\|v\|_{L^{\infty}}\leq C,\qquad 2^{\frac{k}{2}}\|P_{k}v\|_{L^{2n}}\leq
C{\mathcal{A}^{\sharp}}c_{k}^{2},\qquad
2^{\frac{k}{2}}\|P_{k}v\|_{L^{\infty}}\leq C{\mathcal{B}}c_{k}$

with the norm given by the best constant $C$ in the above inequalities.

For this space we have the following algebra and Moser-type result:

######  Lemma 5.2.

a) The space $\mathfrak{C}_{0}$ is closed with respect to multiplication and
para-multiplication. In particular $\mathfrak{C}_{0}$ is an algebra.

b) Let $F$ be a smooth function, and $v\in\mathfrak{C}_{0}$. Then
$F(v)\in\mathfrak{C}_{0}$. In particular if $\|v\|_{\mathfrak{C}_{0}}\lesssim
1$ then $F(v)$ satisfies

(5.10)
$\|F(v)\|_{\mathfrak{C}_{0}}\lesssim_{{\mathcal{A}}}\|v\|_{\mathfrak{C}_{0}}.$

In particular the above result applies to the metrics $g$, ${\tilde{g}}$ and
${\hat{g}}$, all of which are smooth functions of $\partial u$, and thus
belong to $\mathfrak{C}_{0}$.

###### Proof.

a) We first estimate the $\mathfrak{C}_{0}$ norm for the paraproduct $T_{f}g$
for $f,g\in\mathfrak{C}_{0}$. This is straightforward, using the $L^{\infty}$
bound for $f$, for both the second and the third norms in (5.9). It remains to
obtain a pointwise bound, for which we change the summation order in the
Littlewood-Paley expansion to obtain

$\|T_{f}g\|_{L^{\infty}}\lesssim\sum_{k}\|P_{k}f\|_{L^{\infty}}\|P_{>k}g\|_{L^{\infty}}\lesssim\|f\|_{\mathfrak{C}_{0}}\|g\|_{L^{\infty}}.$

It now remains to estimate $\Pi(f,g)$ in $\mathfrak{C}_{0}$. The uniform bound
is almost identical to the one above. For the ${\mathcal{A}^{\sharp}}$ norm we
use Bernstein’s inequality

$2^{\frac{k}{2}}\|P_{k}\Pi(f,g)\|_{L^{2n}}\lesssim\sum_{j\geq
k}2^{k}\|f_{j}g_{j}\|_{L^{n}}\lesssim\sum_{j\geq
k}2^{k}\|f_{j}\|_{L^{p}}\|g_{j}\|_{L^{p}}\lesssim\sum_{j\geq
k}2^{k-j}c_{j}^{2}{\mathcal{A}^{\sharp}}^{2}\|f\|_{\mathfrak{C}_{0}}\|g\|_{\mathfrak{C}_{0}},$

and now the $j$ summation is straightforward.

For the ${\mathcal{B}}$ norm, on the other hand, we estimate

$\|P_{k}\Pi(f,g)\|_{L^{\infty}}\lesssim\sum_{j\geq
k}\|f_{j}g_{j}\|_{L^{\infty}}\lesssim\sum_{j\geq
k}\|f_{j}\|_{L^{\infty}}\|g_{j}\|_{L^{\infty}}\lesssim\sum_{j\geq
k}2^{\frac{j}{2}}c_{j}{\mathcal{A}^{\sharp}}{\mathcal{B}}\|f\|_{\mathfrak{C}_{0}}\|g\|_{\mathfrak{C}_{0}},$

and again the $j$ summation is straightforward.

b) To prove the Moser inequality we use a continuous Littlewood-Paley
decomposition, which leads to the expansion

$F(v)=F(v_{0})+\int_{0}^{\infty}F^{\prime}(v_{<j})v_{j}\,dj.$

To estimate $P_{k}F(v)$ we consider several cases:

i) $j=k+O(1)$. Then $c_{j}\approx c_{k}$, $F^{\prime}(v_{<j})$ is directly
bounded in $L^{\infty}$ and our bounds are straightforward.

ii) $j<k-4$. Then we can insert an additional localization,

$P_{k}(F^{\prime}(v_{<j})v_{j})=P_{k}(\tilde{P}_{k}F^{\prime}(v_{<j})v_{j}),$

where we gain from the frequency difference

(5.11) $\|P_{k}F^{\prime}(v_{<j})\|_{L^{\infty}}\lesssim 2^{-N(k-j)},$

which more than compensates for the difference (ratio) between $c_{j}$ and
$c_{k}$.

iii) $j>k+4$. In this case we reexpand $F^{\prime}(v_{<j})$ and write

$F^{\prime}(v_{<j})v_{j}=F^{\prime}(v_{0})v_{j}+\int_{0}^{\infty}F^{\prime\prime}(v_{<l})v_{l}v_{j}\,dl.$

We further separate into two cases:

(iii.1) $l=j+O(1)$. Then we simply bound $F^{\prime\prime}(v_{<l})$ in
$L^{\infty}$, and estimate first for the ${\mathcal{A}^{\sharp}}$ bound using
Bernstein’s inequality

$2^{\frac{k}{2}}\|P_{k}(F^{\prime\prime}(v_{<l})v_{l}v_{j})\|_{L^{2n}}\lesssim
2^{k}\|v_{l}v_{j}\|_{L^{n}}\lesssim
2^{k}\|v_{l}\|_{L^{2n}}\|v_{j}\|_{L^{2n}}\lesssim
2^{k-j}c_{j}^{2}{\mathcal{A}^{\sharp}}^{2},$

where the $j$ and $l$ integrations are trivial. Next we estimate for the
${\mathcal{B}}$-bound

$\|P_{k}(F^{\prime\prime}(v_{<l})v_{l}v_{j})\|_{L^{\infty}}\lesssim\|v_{l}\|_{L^{\infty}}\|v_{j}\|_{L^{\infty}}\lesssim
2^{-\frac{j}{2}}{\mathcal{A}}{\mathcal{B}}c_{j},$

again with easy $j$ and $l$ integrations.

(iii.2) $l<j-4$. Then we can insert another frequency localization,

$P_{k}(F^{\prime\prime}(v_{<l})v_{l}v_{j})=P_{k}(\tilde{P}_{j}F^{\prime\prime}(v_{<l})v_{l}v_{j}),$

and repeat the computation in (b.ii) but using (5.11) to account for the
difference between $l$ and $j$.

∎

In order to avoid tampering with causality, the Littlewood-Paley projections
we use in this paper are purely spatial. This is more of a choice between
different evils than a necessity; see for instance the alternate choice made
in [36]. A substantial but worthwhile price to pay is that on occasion we will
need to separately estimate double time derivatives, in a somewhat imperfect
but sufficient fashion.

A good starting point in this direction is to think of bounds for second
derivatives of our solution $u$. If at least one of the derivatives is
spatial, then this is straightforward:

(5.12) $\|P_{<k}\partial_{x}\partial u\|_{L^{\infty}}\lesssim
2^{\frac{k}{2}}{\mathcal{B}}c_{k}.$

However, matters become more complex if instead we look at the second time
derivative of $u$. The natural idea is to use the main equation (3.5) to
estimate $\partial_{t}^{2}u$, by writing of spatial derivatives,

$\partial_{t}^{2}u=-\sum_{(\alpha,\beta)\neq(0,0)}{\tilde{g}}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u.$

If one takes this view, the main difficulties we face are with the high-high
interactions in this expression. But these high-high interactions have the
redeeming feature that they are balanced, so they will often play a
perturbative role. This leads us to define a corrected expression as follows:

######  Definition 5.3.

We denote

(5.13)
$\hat{\partial}_{t}^{2}u=\partial_{t}^{2}u+\sum_{(\alpha,\beta)\neq(0,0)}\Pi({\tilde{g}}^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u).$

More generally, by $\widehat{\partial_{\alpha}\partial_{\beta}}u$ we denote
${\partial_{\alpha}\partial_{\beta}}u$ if $(\alpha,\beta)\neq(0,0)$.

With this notation, we have

######  Lemma 5.4.

Assume that $u$ solves the equation (3.5). Then for its second time derivative
we have the decomposition

(5.14) $\partial_{t}^{2}u=\hat{\partial}_{t}^{2}u+\pi_{2}(u),$

where the two components satisfy the uniform bounds

(5.15) $\|P_{<k}\hat{\partial}_{t}^{2}u\|_{L^{\infty}}\lesssim
2^{\frac{k}{2}}{\mathcal{B}}c_{k},\qquad\|P_{<k}\hat{\partial}_{t}^{2}u\|_{L^{\infty}}\lesssim
2^{k}{\mathcal{A}}c_{k}^{2},$

respectively

(5.16)
$\|\pi_{2}(u)\|_{L^{\infty}}\lesssim_{{\mathcal{A}}}{\mathcal{B}}^{2},\qquad\|\pi_{2}(u)\|_{L^{n}}\lesssim{\mathcal{A}^{\sharp}}^{2}.$

One should compare this with the easier direct bound (5.12) for spatial
derivatives; the good part $\hat{\partial}_{t}^{2}u$ satisfies a similar
bound, but the error $\pi_{2}(u)$ does not. Later, when such expressions are
involved, we will systematically peel off perturbatively the error, and always
avoid differentiating it further.

###### Proof.

The main ingredient here is the Littlewood-Paley decomposition. For expository
simplicity we prove (5.15) at fixed frequency $k$. Using the notation in
(5.13) we can rewrite equation (3.19) as

(5.17)
$\hat{\partial}_{t}^{2}u+\sum_{(\alpha,\beta)\neq(0,0)}T_{{\tilde{g}}^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u+T_{\partial_{\alpha}\partial_{\beta}u}{\tilde{g}}^{\alpha\beta}=0.$

To finish the proof we consider the expression above localized at frequency
$k$, and evaluated in the $L^{\infty}$-norm

(5.18) $\displaystyle\|P_{k}\hat{\partial}_{t}^{2}u\|_{L^{\infty}}$
$\displaystyle\leq\|P_{<k}({\tilde{g}}^{\alpha\beta})P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}+\|P_{<k}(\partial_{\alpha}\partial_{\beta}u)P_{k}({\tilde{g}}^{\alpha\beta})\|_{L^{\infty}}.$

We bound each of the terms separately. For the second we use that
${\tilde{g}}$ is bounded in $L^{\infty}$, and get

$\|P_{<k}(\partial_{\alpha}\partial_{\beta}u)P_{k}({\tilde{g}}^{\alpha\beta})\|_{L^{\infty}}\leq
2^{\frac{k}{2}}{\mathcal{B}}c_{k}.$

For the first term we rely on the same procedure, and hence finish the proof
of the first bound in (5.15). The second bound in (5.15) has as a starting
point the decomposition in (5.18), only that this time we want to bound the
RHS terms using the control norm ${\mathcal{A}}$. Here we use Lemma 5.2, part
b), and the algebra property of $L^{\infty}$ so that

$\|P_{<k}(\partial_{\alpha}\partial_{\beta}u)P_{k}({\tilde{g}}^{\alpha\beta})\|_{L^{\infty}}\leq\|P_{<k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}\|P_{k}({\tilde{g}}^{\alpha\beta})\|_{L^{\infty}}\leq
2^{k}{\mathcal{A}}c_{k}^{2},$

where for both factors we used (5.10) in order to arrive at the result.

The last bound to prove is (5.16), where because of the balanced frequencies
we can easily even out the derivatives balance and estimate each of the
factors using the ${\mathcal{B}}$ norm. Explicitly,
${\tilde{g}}^{\alpha\beta}$ is in $\mathfrak{C}_{0}$ by Lemma 5.2 and hence,
we get that for $(\alpha,\beta)\neq(0,0)$:

$\displaystyle\|\Pi({\tilde{g}}^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}$
$\displaystyle\leq\sum_{k}\|P_{k}({\tilde{g}}^{\alpha\beta})P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}$
$\displaystyle\leq\sum_{k}2^{-\frac{k}{2}}\|2^{\frac{k}{2}}P_{k}({\tilde{g}}^{\alpha\beta})\|_{L^{\infty}}2^{\frac{k}{2}}\|2^{-\frac{k}{2}}P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}$
$\displaystyle\lesssim_{{\mathcal{A}}}{\mathcal{B}}^{2}.$

which is the first bound in (5.16). The second bound in (5.16) is similar, but
replacing the $L^{\infty}$ norms with $L^{2n}$ norms. ∎

The above lemma motivates narrowing the space $\mathfrak{C}_{0}$, in order to
also include information about $\partial_{t}v$. For later use, we also define
two additional closely related spaces.

######  Definition 5.5.

a) The space $\mathfrak{C}$ is the space of distributions $v$ which satisfy
(5.9) and, in addition, $\partial_{t}v$ admits a decomposition
$\partial_{t}v=w_{1}+w_{2}$ so that

(5.19) $\|P_{k}w_{1}\|_{L^{\infty}}\leq
C2^{\frac{k}{2}}{\mathcal{B}}c_{k},\qquad\|w_{2}\|_{L^{\infty}}\leq
C{\mathcal{B}}^{2},$

endowed with the norm defined as the best possible constant $C$ in (5.9) and
in the above inequality relative to all such possible decompositions.

b) The space $\mathfrak{DC}$ consists of all functions $f$ which admit a
decomposition $f=f_{1}+f_{2}$ so that

(5.20) $\|P_{k}f_{1}\|_{L^{\infty}}\leq
C2^{\frac{k}{2}}{\mathcal{B}}c_{k},\qquad\|f_{2}\|_{L^{\infty}}\leq
C{\mathcal{B}}^{2},$

endowed with the norm defined as the best possible constant $C$ in the above
inequality relative to all such possible decompositions.

c) The space $\partial_{x}\mathfrak{DC}$ consists of functions $f$ which admit
a decomposition $f=f_{1}+f_{2}$ so that

(5.21) $\|P_{k}f_{1}\|_{L^{\infty}}\leq
C2^{\frac{3k}{2}}{\mathcal{B}}c_{k},\qquad\|P_{k}f_{2}\|_{L^{\infty}}\leq
C2^{k}{\mathcal{B}}^{2},$

endowed also with the corresponding norm.

We remark that, by definition, we have the simple inclusions

(5.22)
$\mathfrak{C}\subset\mathfrak{C}_{0},\qquad\partial:\mathfrak{C}\to\mathfrak{DC},\qquad\partial_{x}:\mathfrak{DC}\to\partial_{x}\mathfrak{DC}.$

Based on what we have so far, we begin by identifying some elements of these
spaces:

######  Lemma 5.6.

We have

(5.23) $\|\partial u\|_{\mathfrak{C}}\lesssim
1,\qquad\|\partial^{2}u\|_{\mathfrak{DC}}\lesssim 1.$

###### Proof.

The bounds in (5.23) are trivial unless both derivatives are time derivatives,
in which case it follows directly from the previous Lemma 5.4. ∎

The Moser estimates of Lemma 5.2 may be extended to this setting to include
all smooth functions of $\partial u$:

######  Lemma 5.7.

a) We have the bilinear multiplicative relations

(5.24) $\mathfrak{C}_{0}\cdot\mathfrak{DC}\to\mathfrak{DC},\qquad
T_{\mathfrak{C}_{0}}\cdot\mathfrak{DC}\to\mathfrak{DC},\qquad
T_{\mathfrak{DC}}\mathfrak{C}_{0}\to{\mathcal{A}}\mathfrak{DC},$

as well as

(5.25)
$T_{\mathfrak{DC}}\mathfrak{C}_{0}\to{\mathcal{B}}^{2}L^{\infty},\qquad\Pi(\mathfrak{DC},\mathfrak{C}_{0})\to{\mathcal{B}}^{2}L^{\infty}.$

b) The space $\mathfrak{C}$ is closed under multiplication and para-
multiplication; in particular it is an algebra.

c) Let $F$ be a smooth function, and $v\in\mathfrak{C}$. Then
$F(v)\in\mathfrak{C}$. In particular if $\|v\|_{\mathfrak{C}}\lesssim 1$ then
$F(v)$ satisfies

(5.26) $\|F(v)\|_{\mathfrak{C}}\lesssim_{{\mathcal{A}}}\|v\|_{\mathfrak{C}}.$

d) In addition we also have the paralinearization error bound

(5.27) $\|\partial R(v)\|_{L^{\infty}}\lesssim{\mathcal{B}}^{2},\qquad
R(v)=F(v)-T_{F^{\prime}(v)}v.$

Here part (a) is the main part, after which parts (b) and (c) become immediate
improvements of Lemma 5.2. But the new interesting bound is the one in part
(d), where, notably, we also bound the time derivative of $R(v)$.

###### Proof.

a) Let $z\in\mathfrak{C}_{0}$ and $w\in\mathfrak{DC}$ with the decomposition
$w=w_{1}+w_{2}$ as in (5.20). We skip the first bound in (5.24), as it is a
consequence of the rest of the estimates in (5.24) and (5.25), and first
consider the paraproduct $T_{z}w$. We will bound the contributions of $w_{1}$
and $w_{2}$ in the same norms as $w_{1}$, respectively $w_{2}$. Precisely, we
have

$\|P_{k}T_{z}w_{1}\|_{L^{\infty}}\lesssim\|z\|_{L^{\infty}}\|\tilde{P}_{k}w_{1}\|_{L^{\infty}}\lesssim\|z\|_{\mathfrak{C}_{0}}2^{\frac{k}{2}}{\mathcal{B}}c_{k}\|w\|_{\mathfrak{DC}},$

respectively

$\|T_{z}w_{2}\|_{L^{\infty}}\lesssim\sum_{k}\|z_{k}\|_{L^{\infty}}\|P_{>k}w_{2}\|_{L^{\infty}}\lesssim\|z\|_{\mathfrak{C}_{0}}\|w_{1}\|_{L^{\infty}}.$

Next we consider $T_{w}z$, where we have two choices. The first choice is to
use only the ${\mathcal{A}}$ component of the $\mathfrak{C}_{0}$ norm of $z$,
and prove the last bound in (5.24). Precisely, we have

$\|P_{k}T_{w_{1}}z\|_{L^{\infty}}\lesssim\|w_{1,<k}\|_{L^{\infty}}\|\tilde{P}_{k}z\|_{L^{\infty}}\lesssim
2^{\frac{k}{2}}{\mathcal{B}}c_{k}\|w\|_{\mathfrak{DC}}\cdot{\mathcal{A}}\|z\|_{\mathfrak{C}_{0}},$

respectively

$\|T_{w_{2}}z\|_{L^{\infty}}\lesssim\sum_{k}\|w_{2,<k}\|_{L^{\infty}}\|P_{k}z\|_{L^{\infty}}\lesssim\|w_{2}\|_{L^{\infty}}\cdot{\mathcal{A}}\|z\|_{\mathfrak{C}_{0}}.$

Alternatively, we can use the ${\mathcal{B}}$ component of the
$\mathfrak{C}_{0}$ norm of $z$ in the bound for the $w_{1}$ component,

$\|T_{w_{1}}z\|_{L^{\infty}}\lesssim\sum_{k}\|w_{1,<k}\|_{L^{\infty}}\|z_{k}\|_{L^{\infty}}\lesssim\sum_{k}2^{\frac{k}{2}}{\mathcal{B}}c_{k}\|w\|_{\mathfrak{DC}}\cdot
2^{-\frac{k}{2}}{\mathcal{B}}c_{k}\|z\|_{\mathfrak{C}_{0}}\lesssim{\mathcal{B}}^{2}\|w\|_{\mathfrak{DC}}\|z\|_{\mathfrak{C}_{0}},$

which leads to the first bound in (5.25).

It remains to consider the second bound in (5.25), where we have

$\|\Pi(w_{1},z)\|_{L^{\infty}}\lesssim\sum_{k}\|w_{1,k}\|_{L^{\infty}}\|z_{k}\|_{L^{\infty}}\lesssim\sum_{k}2^{\frac{k}{2}}{\mathcal{B}}c_{k}\|w\|_{\mathfrak{DC}}\cdot
2^{-\frac{k}{2}}{\mathcal{B}}c_{k}\|z\|_{\mathfrak{C}_{0}}\lesssim{\mathcal{B}}^{2}\|w\|_{\mathfrak{DC}}\|z\|_{\mathfrak{C}_{0}},$

respectively

$\|\Pi(w_{2},z)\|_{L^{\infty}}\lesssim\sum_{k}\|w_{2,k}\|_{L^{\infty}}\|z_{k}\|_{L^{\infty}}\lesssim\|w_{2}\|_{L^{\infty}}\cdot{\mathcal{A}}\|z\|_{\mathfrak{C}_{0}}\lesssim
A{\mathcal{B}}^{2}\|w\|_{\mathfrak{DC}}\cdot{\mathcal{A}}\|z\|_{\mathfrak{C}_{0}}.$

b) Compared with part (a) of Lemma (5.2), it remains to estimate the time
derivative of products and paraproducts. Using Leibniz’s rule, this reduces
directly to the multiplicative bounds in (a).

c) Compared with part (b) of Lemma (5.2) it remains to estimate

$\partial_{0}F(v)=F^{\prime}(v)\partial_{0}v$

in $\mathfrak{DC}$. By Lemma (5.2) we have $F^{\prime}(v)\in\mathfrak{C}_{0}$,
while $\partial_{0}v\in\mathfrak{DC}$. Then we can bound the product in
$\mathfrak{DC}$ by part (a) of this Lemma.

d) We have

$\partial R=(F^{\prime}(v)-T_{F^{\prime}(v)})\partial v-T_{\partial
F^{\prime}(v)}v=\Pi(F^{\prime}(v),\partial v)+T_{\partial
v}F^{\prime}(v)-T_{\partial F^{\prime}(v)}v.$

Now it suffices to use (5.26) for both $v$ and $F^{\prime}(v)$. ∎

Applying the above lemma shows that $F(\partial u)\in\mathfrak{C}$, and in
particular all components of the metrics $g$, ${\tilde{g}}$ and ${\hat{g}}$
are in $\mathfrak{C}$. We also have $F(\partial
u)\partial^{2}u\in\mathfrak{DC}$, which in particular shows that the gradient
potentials $A$ and ${\tilde{A}}$ belong to $\mathfrak{DC}$.

We will use part (d) when $w=\partial u$ and $F=g$, in which case (2.12) reads

(5.28) $\|\partial R(\partial
u)\|_{L^{\infty}}\lesssim_{\mathcal{A}}{\mathcal{B}}^{2},\qquad
R=g^{\alpha\beta}+T_{\partial^{\alpha}ug^{\beta\gamma}}\partial_{\gamma}u+T_{\partial^{\beta}ug^{\alpha\gamma}}\partial_{\gamma}u.$

We remark that a similar $H^{s}$ type bound for the same $R$ is provided by
(2.12), namely

(5.29) $\|R(\partial
u)\|_{H^{s-\frac{1}{2}}}\lesssim_{\mathcal{A}}{\mathcal{B}}\|\partial
u\|_{H^{s-1}}.$

The next lemma provides us with the primary example of elements of the space
$\partial_{x}\mathfrak{DC}$:

######  Lemma 5.8.

We have

(5.30)
$\|\partial_{\alpha}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{\partial_{x}\mathfrak{DC}}\lesssim
1.$

###### Proof.

The bound in (5.30) is trivial if at least two derivatives are spatial, and
follows from (5.23) unless all indices are zero. It remains to consider the
case $\alpha=\beta=\gamma=0$. Here we rely on the earlier decomposition (5.17)
to which we further apply a $\partial_{t}$:

(5.31)
$\partial_{t}\hat{\partial}_{t}^{2}u=-\sum_{(\alpha,\beta)\neq(0,0)}\left(T_{\partial_{t}\tilde{g}^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u+T_{\tilde{g}^{\alpha\beta}}\partial_{t}\partial_{\alpha}\partial_{\beta}u+T_{\partial_{t}\partial_{\alpha\beta}u}{\tilde{g}}^{\alpha\beta}+T_{\partial_{\alpha}\partial_{\beta}u}\partial_{t}{\tilde{g}}^{\alpha\beta}\right).$

We now investigate each term separately. We begin with the first term, which
needs to be bounded in the $\partial_{x}\mathfrak{DC}$ norm given in (5.21).
We have

$\|P_{k}T_{\partial_{t}\tilde{g}^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u\|_{L^{\infty}}\leq\|P_{<k}(\partial_{t}\tilde{g}^{\alpha\beta})\|_{L^{\infty}}\|P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}.$

The term that contains the time derivative falling onto the metric will be
bounded using the Moser estimate Lemma 5.7. Explicitly, we have
${\tilde{g}}^{\alpha\beta}\in\mathfrak{C}$, and due to Lemma 5.7, part $c)$,
we get $\partial_{t}{\tilde{g}}^{\alpha\beta}\in\mathfrak{DC}$ which allows us
to decompose it as in (5.20),
$\partial_{t}{\tilde{g}}^{\alpha\beta}={\tilde{g}}_{1}^{\alpha\beta}+{\tilde{g}}_{2}^{\alpha\beta}$
where

$\|P_{<k}\partial_{t}{\tilde{g}}^{\alpha\beta}\|_{L^{\infty}}\leq\|{\tilde{g}}_{1}^{\alpha\beta}\|_{L^{\infty}}+\|P_{<k}{\tilde{g}}_{2}^{\alpha\beta}\|_{L^{\infty}}\leq
C{\mathcal{B}}^{2}+C2^{\frac{k}{2}}{\mathcal{B}}c_{k}.$

We now turn to the last bound which we can estimate in two ways

$\|P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}\leq
2^{\frac{k}{2}}\|2^{-\frac{k}{2}}P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}\leq
2^{\frac{k}{2}}{\mathcal{B}}c_{k},$

or

$\|P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}\leq
2^{k}\|2^{-k}P_{k}(\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}\leq
2^{k}{\mathcal{A}}c_{k}^{2}.$

Putting together the bounds we have, leads to

$\|P_{k}T_{\partial_{t}{\tilde{g}}^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u\|_{L^{\infty}}\leq
C2^{k}{\mathcal{A}}{\mathcal{B}}^{2}c_{k}^{2}+C2^{k}{\mathcal{B}}^{2}c_{k}^{2}.$

We now bound the second term in (5.31)

$\|T_{{\tilde{g}}^{\alpha\beta}}\partial_{t}\partial_{\alpha}\partial_{\beta}u\|_{L^{\infty}}\leq\|{\tilde{g}}^{\alpha\beta}\|_{L^{\infty}}\|\partial_{t}\partial_{\alpha}\partial_{\beta}u\|_{L^{\infty}}.$

For the last term in the above estimate we know that $(\alpha,\beta)\neq(0,0)$
hence there are two cases to consider: (i) we have either $\alpha=0$ or
$\beta=0$, but not both zero, which overalls means we need to bound
$\partial_{t}^{2}\partial_{x}u$, or (ii) we have both $\alpha,\beta\neq 0$, in
which case we need a pointwise bound for $\partial_{t}\partial^{2}_{x}u$.
However, both cases can be handled in the same way if we observe that
$\partial_{x}(\partial_{x}\partial_{t}u)$ and
$\partial_{x}(\partial^{2}_{t}u)$ are elements in $\partial_{x}\mathfrak{DC}$;
this is a direct consequence of $\partial^{2}u\in\mathfrak{DC}$ as shown in
(5.23), followed by the inclusion in (5.22).

Finally, the third and fourth terms in (5.31) can be treated in the same way
the first term in (5.31) was shown to be bounded. ∎

We continue with another, slightly more subtle balanced bound:

######  Lemma 5.9.

For $g,h\in\mathfrak{C}$ define

$r=(T_{g}T_{h}-T_{gh})\partial^{2}u.$

Then we have the balanced bound

(5.32) $\|r\|_{L^{\infty}}\lesssim{\mathcal{B}}^{2}.$

###### Proof.

For $\partial^{2}u$ we use the $\mathfrak{DC}$ decomposition as in (5.20),

$\partial^{2}u=f_{1}+f_{2}.$

We begin with the contribution $r_{1}$ of $f_{1}$, which we expand as

$r_{1}=\sum_{k}(T_{g}T_{h}-T_{gh})f_{1,k}.$

This vanishes unless the frequencies $k_{1}$, $k_{2}$ of $g$ and $h$ are
either

(i) $k_{1},k_{2}\leq k$ and $\max\\{k_{1},k_{2}\\}=k+O(1)$, or

(ii) $k_{1}=k_{2}>k+O(1)$.

Then we use the ${\mathcal{A}}$ component of the $\mathfrak{C}_{0}$ norm for
the lower frequency and and the ${\mathcal{B}}$ component for the lower
frequency to estimate

$\displaystyle\|r_{1}\|_{L^{\infty}}\lesssim$ $\displaystyle\
\sum_{k}\|f_{1,k}\|_{L^{\infty}}\left(\sum_{k_{1}<k+O(1)}\|g_{k_{1}}\|_{L^{\infty}}\sum_{k_{2}=k+O(1)}\|h_{k_{2}}\|_{L^{\infty}}\right.$
$\displaystyle\,+\left.\sum_{k_{1}=k+O(1)}\|g_{k_{1}}\|_{L^{\infty}}\sum_{k_{2}<k+O(1)}\|h_{k_{2}}\|_{L^{\infty}}+\sum_{k_{1}=k_{2}\geq
k+O(1)}\|g_{k_{1}}\|_{L^{\infty}}\|h_{k_{2}}\|_{L^{\infty}}\right)$
$\displaystyle\lesssim$ $\displaystyle\
2^{\frac{k}{2}}{\mathcal{B}}c_{k}({\mathcal{A}}\cdot
2^{-\frac{k}{2}}{\mathcal{B}}c_{k}+2^{-\frac{k}{2}}{\mathcal{B}}c_{k}\cdot{\mathcal{A}}+{\mathcal{A}}\cdot
2^{-k}{\mathcal{B}}c_{k})$ $\displaystyle\lesssim$ $\displaystyle\
{\mathcal{A}}{\mathcal{B}}^{2},$

as needed. ∎

As already discussed in the introduction, the paradifferential wave operator

(5.33) $T_{P}=\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta},$

as well as its counterparts $T_{\tilde{P}}$ and $T_{\hat{P}}$ with the metric
$g$ replaced by ${\tilde{g}}$, respectively ${\hat{g}}$, play an important
role in our context.

Throughout the paper, we will interpret various objects related to $u$ as
approximate solutions for the $T_{P}$ equation. We provide several results of
this type, where we use our control parameters ${\mathcal{A}},{\mathcal{B}}$
in order to estimate the source term in the paradifferential equation for both
$u$ and for its derivatives.

######  Lemma 5.10.

We have

(5.34) $\|T_{P}u\|_{L^{\infty}}\lesssim{\mathcal{B}}^{2},$

as well as the similar bounds for $T_{\tilde{P}}$ and $T_{\hat{P}}$.

###### Proof.

We first prove the bound (5.34), and for this we begin with the
paradifferential equation associated to the minimal surface equation (3.5)

$T_{g^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u+T_{\partial_{\alpha}\partial_{\beta}u}g^{\alpha\beta}+\Pi(g^{\alpha,\beta},\partial_{\alpha}\partial_{\beta}u)=0,$

and further isolate the part we are interested in estimating

$\partial_{\beta}T_{g^{\alpha\beta}}\partial_{\alpha}u-T_{\partial_{\beta}g^{\alpha\beta}}\partial_{\alpha}u+T_{\partial_{\alpha}\partial_{\beta}u}g^{\alpha\beta}+\Pi(g^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u)=0.$

The bound we want relies on getting bounds for the following terms

$\|T_{P}u\|_{L^{\infty}}=\|\partial_{\beta}T_{g^{\alpha\beta}}\partial_{\alpha}u\|_{L^{\infty}}\leq\|T_{\partial_{\beta}g^{\alpha\beta}}\partial_{\alpha}u\|_{L^{\infty}}+\|T_{\partial_{\alpha}\partial_{\beta}u}g^{\alpha\beta}\|_{L^{\infty}}+\|\Pi(g^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}.}$

However, the bounds for all of these terms rely on the use of the fact that
$g^{\alpha\beta}\in\mathfrak{C}$, $\partial
g^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u\in\mathfrak{DC}$
(consequence of Lemma 5.7), as well as on the bound given by Lemma 5.4.
Precisely the estimate (5.25) implies that

$\|T_{\partial_{\beta}g^{\alpha\beta}}\partial_{\alpha}u\|_{L^{\infty}}+\|T_{\partial_{\alpha}\partial_{\beta}u}g^{\alpha\beta}\|_{L^{\infty}}+\|\Pi(g^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u)\|_{L^{\infty}}\lesssim{\mathcal{B}}^{2}.$

Similar bounds will be obtained for $T_{\hat{P}}$ and $T_{\tilde{P}}$ by the
use of the same results mentioned in the proof of bound (5.34).

∎

We next consider similar bounds for derivatives of $u$. Here we will
differentiate between space and time derivatives. We begin with spatial
derivatives:

######  Lemma 5.11.

We have

(5.35) $\|P_{<k}T_{P}\partial_{x}u\|_{L^{\infty}}\lesssim
2^{k}{\mathcal{B}}^{2},$

as well as the similar bounds for $T_{\tilde{P}}$ and $T_{\hat{P}}$.

###### Proof.

For this proof we rely on the previous Lemma 5.10 and on Lemma 5.4 . This
becomes obvious after we commute the $\partial_{x}$ across the $T_{P}$
operator

$P_{k}T_{P}\partial_{x}u=\partial_{x}P_{k}\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}u-P_{k}\partial_{\alpha}T_{\partial_{x}g^{\alpha\beta}}\partial_{\beta}u$

The first term on the RHS of the identity above is bounded using (5.34) as
follows

$\|\partial_{x}P_{k}\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}u\|_{L^{\infty}}\lesssim
2^{k}\|\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}u\|_{L^{\infty}}\lesssim
2^{k}{\mathcal{B}}^{2}.$

Here we took advantage of the $\partial_{x}$ accompanied by the frequency
projector $P_{k}$. A similar advantage will not present itself for the last
term, where we need to distribute the $\alpha$ derivative

$P_{k}\partial_{\alpha}T_{\partial_{x}g^{\alpha\beta}}\partial_{\beta}u=P_{k}T_{\partial_{\alpha}\partial_{x}g^{\alpha\beta}}\partial_{\beta}u+P_{k}T_{\partial_{x}g^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u:=e_{1}+e_{2}.$

We bound $e_{1}$ using Lemma 5.7, by placing $\partial_{x}\partial
g^{\alpha\beta}\in\partial_{x}\mathfrak{DC}$ which means it will admit a
decomposition as follows

$\partial_{x}\partial g^{\alpha\beta}=f_{1}+f_{2},$

where

$\|P_{k}f_{1}\|_{L^{\infty}}\lesssim
2^{\frac{3k}{2}}{\mathcal{B}}c_{k},\quad\|P_{k}f_{2}\|_{L^{\infty}}\lesssim
2^{k}{\mathcal{B}}^{2}.$

Thus, we get

$\|e_{1}\|_{L^{\infty}}\lesssim\left(\|P_{k}f_{1}\|_{L^{\infty}}+\|P_{k}f_{2}\|_{L^{\infty}}\right)\|\partial_{\beta}u\|_{L^{\infty}}\lesssim\left(2^{k}{\mathcal{B}}^{2}+2^{\frac{3k}{2}}{\mathcal{B}}c_{k}\right)\|\partial_{\beta}u\|_{L^{\infty}},$

which leads to the desired bound once we estimate the last term accordingly.
The bounds can be one of the following

$\|\partial_{\beta}u\|_{L^{\infty}}\lesssim
2^{-\frac{k}{2}}\|2^{\frac{k}{2}}\partial_{\beta}u\|_{L^{\infty}}\lesssim
2^{-\frac{k}{2}}{\mathcal{B}}\qquad\mbox{ or
}\qquad\|\partial_{\beta}u\|_{L^{\infty}}\lesssim{\mathcal{A}}.$

For the first term in the bracket we use the control norm ${\mathcal{A}}$, and
for the second term we use the ${\mathcal{B}}$ norm bound.

For $e_{2}$ we use the decomposition in Lemma 5.4 for
$\partial_{\alpha}\partial_{\beta}u$ and for $g$ we use the fact that
$g\in\mathfrak{C}_{0}$ where we can use either the ${\mathcal{A}}$ bound or
the ${\mathcal{B}}$ bound. The computations are similar to the case of
$e_{1}$.

The bounds for $T_{\tilde{P}}$ and $T_{\hat{P}}$ follow from the exact
argument as the one used above in the $T_{P}$ case.

∎

######  Lemma 5.12.

a) We have

(5.36) $T_{P}\partial u\in\partial({\mathcal{B}}^{2}L^{\infty}),$

i.e. there exists a representation

(5.37) $T_{P}\partial
u=\partial_{\alpha}f^{\alpha},\qquad|f|\lesssim{\mathcal{B}}^{2}.$

b) We also have

(5.38)
$\|P_{k}\partial_{\alpha}T_{g^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{L^{\infty}}+\|P_{k}\partial_{\gamma}T_{g^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\alpha}}u\|_{L^{\infty}}\lesssim
2^{k}{\mathcal{B}}^{2}.$

Similar results hold with $g$ replaced by ${\tilde{g}}$ or ${\hat{g}}$.

###### Proof.

a) We write

$\displaystyle\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}\partial u=$
$\displaystyle\
\partial\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}u-\partial_{\alpha}T_{\partial
g^{\alpha\beta}}\partial_{\beta}u.$

Here by (5.25) we have

$\|T_{\partial g}\partial u\|_{L^{\infty}}\lesssim{\mathcal{B}}^{2},$

so we get

$\displaystyle\partial_{\alpha}T_{g^{\alpha\beta}}\partial_{\beta}\partial u=$
$\displaystyle\ \partial
T_{g^{\alpha\beta}}\partial_{\alpha}\partial_{\beta}u+\partial({\mathcal{B}}^{2}L^{\infty})=\partial\Pi(g^{\alpha\beta},\partial_{\alpha}\partial_{\beta}u)+\partial
T_{\partial_{\alpha}\partial_{\beta}u}g^{\alpha\beta}+\partial({\mathcal{B}}^{2}L^{\infty}),$

where we can use again the previous lemma.

b) The first step here is to reduce to the case of the metric ${\tilde{g}}$.
Each of the other two metrics may be written in the form $h{\tilde{g}}$, with
$h=h(\partial u)$. Then we can write

(5.39)
$\partial_{\alpha}T_{h{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\gamma}}u=T_{h}\partial_{\alpha}T_{{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\gamma}}u-T_{\partial_{\alpha}h}T_{{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\gamma}}u+\partial_{\alpha}(T_{h{\tilde{g}}^{\alpha\beta}}-T_{h}T_{{\tilde{g}}^{\alpha\beta}})\widehat{\partial_{\beta}\partial_{\gamma}}u.$

The first term corresponds to our reduction, and the remaining terms need to
be estimated perturbatively. This is straightforward unless $\alpha=0$, so we
focus now on this case.

For the middle term in (5.39) we can use the last part of (5.26) in Lemma 5.7
for $\partial_{0}h$. Using a $\mathfrak{DC}$ decomposition for it,
$\partial_{0}h=h_{1}+h_{2}$, we can match the two terms with the two pointwise
bounds for $\widehat{\partial_{\beta}\partial_{\gamma}}u$, namely

$\|P_{k}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{L^{\infty}}\lesssim
2^{k}{\mathcal{A}}c_{k}^{2},\qquad\|P_{k}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{L^{\infty}}\lesssim
2^{\frac{k}{2}}{\mathcal{B}}c_{k}.$

This yields

$\displaystyle\|P_{k}T_{\partial_{\alpha}h}T_{{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{L^{\infty}}\lesssim$
$\displaystyle\
\|h_{1}\|_{L^{\infty}}\cdot\|\tilde{P}_{k}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{L^{\infty}}+\|h_{2}\|_{L^{\infty}}\cdot\|\tilde{P}_{k}\widehat{\partial_{\beta}\partial_{\gamma}}u\|_{L^{\infty}}$
$\displaystyle\lesssim$ $\displaystyle\ {\mathcal{B}}^{2}\cdot
2^{k}{\mathcal{A}}c_{k}^{2}+2^{\frac{k}{2}}{\mathcal{B}}c_{k}\cdot
2^{\frac{k}{2}}{\mathcal{B}}c_{k},$

where the frequency envelope provides the summation with respect to $k$.

For the last expression in (5.39) we distribute $\partial_{0}$. If it falls on
the main term, then we can combine the bound (5.30) with the para-composition
bound in Lemma 2.4 exactly as in the proof of Lemma 5.9. If it falls on $h$
(or similarly on ${\tilde{g}}$) then we use the same decomposition as above
for $\partial_{0}h$. For the $h_{1}$ contribution we have a direct bound
without using any cancellations, while for $h_{2}$ we use again Lemma 2.4.

We continue with the second444Note that these two steps are interchangeable.
step, which is to switch $\partial_{\alpha}$ and $\partial_{\gamma}$. For
fixed $\alpha$ and $\gamma$, we write

(5.40)
$\partial_{\alpha}T_{{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\gamma}}u=\partial_{\gamma}T_{{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\alpha}}u+f,$

where $f$ satisfies

(5.41) $\|P_{<k}f\|_{L^{\infty}}\lesssim 2^{k}{\mathcal{B}}^{2}.$

This is trivial if $\alpha=\gamma=0$. If both are nonzero, or if one of them
is zero but $\beta\neq 0$, then there is no hat correction and this is a
straightforward commutator bound. It remains to discuss the case when
$\beta=0$ and exactly one of $\alpha$ and $\gamma$ are zero, say $\gamma=0$.
Then we need to consider the difference

$\displaystyle f=$ $\displaystyle\ \partial_{\alpha}T_{{\tilde{g}}^{\alpha
0}}\widehat{\partial_{0}\partial_{0}}u-\partial_{0}T_{{\tilde{g}}^{\alpha
0}}\partial_{0}\partial_{\alpha}u$ $\displaystyle=$ $\displaystyle\
\partial_{\alpha}T_{{\tilde{g}}^{\alpha 0}}\Pi(u,\partial_{x}\partial
u)+T_{\partial_{\alpha}{\tilde{g}}^{\alpha
0}}\partial_{0}\partial_{0}u-T_{\partial_{0}{\tilde{g}}^{\alpha
0}}\partial_{0}\partial_{\alpha}u,$

which can be estimated as in (5.41) using the fact that $\alpha\neq 0$ and the
bound (5.15) for the second time derivative of $u$, respectively the similar
bound (5.26) (third estimate) for $\partial_{0}{\tilde{g}}$.

It remains to examine the expression

$g_{\gamma}=\partial_{\gamma}T_{{\tilde{g}}^{\alpha\beta}}\widehat{\partial_{\beta}\partial_{\alpha}}u,$

where, unlike above, we take advantage of the summation with respect to
$\alpha$ and $\beta$. Then, using the $u$ equation, we have

$g_{\gamma}=\partial_{\gamma}T_{\partial_{\alpha}\partial_{\beta}u}{\tilde{g}}^{\alpha\beta}.$

The term where both $\alpha$ and $\beta$ are zero vanishes, so this can be
estimated directly as in (5.41) if $\gamma\neq 0$, and using either (5.15) or
(5.26) (third estimate) for $\partial_{0}{\tilde{g}}$, otherwise.

∎

##  6\. Paracontrolled distributions

To motivate this section, we start from the classical energy estimates for the
wave equation, which are obtained using the multiplier method. Precisely, one
multiplies the equation $\Box_{g}u=f$ by $Xu$ and simply integrates by parts.
Here $X$ is any regular time-like vector field. In the next section, we prove
energy estimates for the paradifferential equation (3.25), by emulating this
strategy at the paradifferential level. The challenge is then to uncover a
suitable vector field $X$. Unlike the classical case, here not every time-like
vector field $X$ will suffice. Instead $X$ must be carefully chosen, and in
particular it will inherently have a limited regularity.

Since the metric $g$ is a function of $\nabla u$, scaling considerations
indicate that the vector field $X$ should be at the same regularity level.
Naively, one might hope to have an explicit expression $X=X(\nabla u)$ for our
vector field. Unfortunately, seeking such an $X$ eventually leads to an
overdetermined system. At the other extreme, one might enlarge the class of
$X$ to all distributions which satisfy the same $H^{s}$ and Besov norms as
$\partial u$, which is essentially the class of functions which satisfy
(5.26). While this class will turn out to contain the correct choice for $X$,
it is nevertheless too large to allow for a clean implementation of the
multiplier method.

Instead, there is a more subtle alternative, namely to have the vector $X$ to
be _paracontrolled_ by $\partial u$. This terminology was originally
introduced by Gubinelli, Imkeller and Perkowski [12] in connection to Bony’s
calculus, in order to study stochastic pde problems, see also [13]. However,
similar constructions have been carried earlier in the renormalization
arguments e.g. for wave maps, in work of Tao [40], Tataru [45] and Sterbenz-
Tataru [39]; the last reference used the name _renormalizable_ for the
corresponding class of distributions.

In the standard usage, this is more of a principle than an exact notion, which
needs to be properly adapted to one’s purposes. For our own objective here, we
provide a very precise definition of this notion, which is exactly tailored to
the problem at hand.

###  6.1.  Definitions and key properties

######  Definition 6.1.

We say that a function $z$ is paracontrolled by $\partial u$ in a time
interval $I$ if it admits a representation555Such a representation might not
be unique in general, though later in the paper we often identify specific
choices for the paracoefficients. of the form

(6.1) $z=T_{a}\partial u+r,$

where the vector field $a$ and the error $r$ have the following properties:

(i) bounded para-coefficients $a$:

(6.2) $\|a\|_{\mathfrak{C}}\leq C.$