# Paralinearization of free boundary problems in fluid dynamics

Thomas Alazard

###### Abstract.

A classical topic in the mathematical theory of hydrodynamics is to study the
evolution of the free surface separating air from an incompressible perfect
fluid. The goal of this survey is to examine this problem for two important
sets of equations: the water wave equations and the Hele-Shaw equations,
including the Muskat problem. These equations are different in nature,
dispersive or parabolic, but we will see that they can be studied using
related tools. In particular, we will discuss a paradifferential approach to
these problems.

Thomas Alazard, Université Paris-Saclay, ENS Paris-Saclay, CNRS, Centre
Borelli UMR9010, avenue des Sciences, F-91190 Gif-sur-Yvette France,
<EMAIL_ADDRESS>
##  1\. Introduction

The aim of this survey article is to provide an overview of various results in
the mathematical analysis of free boundary problems in fluid dynamics.

The primary objective is to depict the evolution of a free surface, denoted as
$\Sigma(t)$, which, by simplicity, is assumed to be represented as the graph
of a function $\eta=\eta(t)$. Additionally, it is assumed that $\Sigma(t)$
undergoes transport driven by a fluid occupying the region
$\Omega(t)\subset\mathbb{R}^{N}$ located beneath $\Sigma(t)$. To be more
specific, it is assumed that the normal velocity $V_{n}$ of $\Sigma$ obeys to
the equation

(1.1) $V_{n}=u\cdot n,$

where $u\colon\Omega\to\mathbb{R}^{N}$ is the fluid velocity, and $n$
represents the normal vector to the free surface.

This is a broad formulation that encompasses various fluid dynamics problems.
Indeed, numerous equations are associated with this problem due to various
factors that influence the dynamics. These factors include the compressibility
of the fluid, its irrotational or rotational nature, whether the fluid has a
bottom or not, and whether the restoring forces are determined by gravity or
surface tension. Furthermore, a range of equations can be derived under
different asymptotic conditions. Notable examples include the Boussinesq and
Korteweg-de Vries equations, which have been extensively studied (see [92, 27,
67, 74, 91]).

This paper primarily focuses on three fundamental models: the water wave
problem, the Hele-Shaw equation, and the Muskat equation. These models differ
in their mathematical nature (dispersive versus parabolic). However, we will
show how related mathematical tools can be employed to study them.
Specifically, this paper will discuss a paradifferential approach to these
problems. In simple terms, the goal is to find a method to transform these
equations in such a way that they can be reduced to simpler equations.

###  1.1. The equations

To study these equations, we adopt a formulation which involves a set of
equations reduced to the free surface. This formulation revolves around the
displacement function $\eta$ and the trace $\psi$ of the potential flow
variable on the free surface. This particular approach has a rich historical
background, notably tracing its roots back to the works of Zakharov [109],
Craig-Sulem [52], and Lannes [73], who introduced an analysis based in the
Dirichlet-to-Neumann operator. We adopt this perspective in the following.

Consider a time-dependent domain

$\displaystyle\Omega(t)=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}\,:\,y<\eta(t,x)\\}\subset\mathbb{R}^{N}\qquad(N=d+1,\leavevmode\nobreak\
d\geq 1)$ and set
$\displaystyle\Sigma(t)=\partial\Omega(t)=\\{y=\eta(t,x)\\}.$

In this case, the normal $n$ and normal velocity $V_{n}$ are given by

$\displaystyle n$
$\displaystyle=\frac{1}{\sqrt{1+|\nabla\eta|^{2}}}\begin{pmatrix}-\nabla\eta\\\
1\end{pmatrix}$ $\displaystyle V_{n}$
$\displaystyle=n\cdot\frac{\operatorname{d}}{\operatorname{d}\\!t}\begin{pmatrix}x\\\
\eta(t,x)\end{pmatrix}=\frac{\partial_{t}\eta}{\sqrt{1+|\nabla\eta|^{2}}}.$

$n$$\Sigma(t)$$\Omega(t)$

Then, the kinematic equation $V_{n}=u\cdot n$ (see (1.1)) simplifies to

(1.2) $\partial_{t}\eta=\sqrt{1+|\nabla\eta|^{2}}\,u\cdot n.$

Assume now that $u$ is irrotational and incompressible, that is assume that

$\operatorname{curl}_{x,y}u=0\quad,\quad\operatorname{div}_{x,y}u=0.$

Then $u=\nabla_{x,y}\phi$ for some potential $\phi\colon\Omega\to\mathbb{R}$
such that $\Delta_{x,y}\phi=0$. Now, following Zakharov [108, 109], set

$\psi(t,x)=\phi(t,x,\eta(t,x))$

and introduce, following Craig-Sulem [52] and Lannes [73, 74], the Dirichlet-
to-Neumann operator by

$G(\eta)\psi=\partial_{y}\phi-\nabla\eta\cdot\nabla\phi\Big{\arrowvert}_{y=\eta}=\sqrt{1+|\nabla\eta|^{2}}\,\partial_{n}\phi\Big{\arrowvert}_{y=\eta}.$

It follows from this definition and the kinematic equation (1.2) that

(1.3) $\partial_{t}\eta=\sqrt{1+|\nabla\eta|^{2}}\,(u\cdot
n)\arrowvert_{y=\eta}=\sqrt{1+|\nabla\eta|^{2}}\,\partial_{n}\phi\arrowvert_{y=\eta}=G(\eta)\psi.$

To derive a closed set of equations, we also require an evolution equation for
$\psi$. The simplest way to obtain such an equation is to postulate that :

$\psi=-g\eta,$

where $g$ is the acceleration of gravity. Consequently, it follows from
equation (1.3) that

(1.4) $\partial_{t}\eta+gG(\eta)\eta=0.$

This equation has physical significance and is commonly referred to as the
Hele-Shaw equation (as seen in references such as [43, 21, 86]). It
corresponds to the physical scenario where the fluid velocity and pressure are
related through Darcy’s law, which states that:

$u=-\nabla_{x,y}(P+gy).$

Alternatively, one can consider a slightly more complex equation in which
$\psi$ is determined by the solution of an equation of the form:

$\psi+K(\eta)\psi=-\eta.$

This case encompasses the significant Muskat equation, which corresponds to a
Hele-Shaw problem with two phases (see [86]).

Out of the numerous conceivable relationships between $\eta$ and $\psi$, the
one that undoubtedly stands out as the most elegant is the discovery made by
Vladimir Zakharov.

######  Theorem 1.1 (Zakharov [108]).

Consider an irrotational, incompressible velocity field $u=\nabla_{x,y}\phi$
satisfying $\partial_{t}u+u\cdot\nabla_{x,y}u=-\nabla_{x,y}(P+gy)$ and
$P\arrowvert_{y=\eta}=0$. Then $\eta$ and $\psi$ are conjugated:

$\left\\{\begin{aligned} \frac{\partial\eta}{\partial
t}&=\phantom{-}\frac{\delta\mathcal{H}}{\delta\psi}\\\
\frac{\partial\psi}{\partial
t}&=-\frac{\delta\mathcal{H}}{\delta\eta}\end{aligned}\right.\quad\text{where}\quad\left\\{\begin{aligned}
&\psi(t,x)=\phi(t,x,\eta(t,x))\\\\[8.61108pt]
&\mathcal{H}=\frac{1}{2}\int_{\mathbb{R}^{d}}\psi
G(\eta)\psi\operatorname{d}\\!x+\frac{g}{2}\int_{\mathbb{R}^{d}}\eta^{2}\operatorname{d}\\!x.\end{aligned}\right.$

A popular form of the equations is the following :

(1.5) $\left\\{\begin{aligned} &\partial_{t}\eta-G(\eta)\psi=0,\\\
&\partial_{t}\psi+g\eta+\frac{1}{2}\left|\nabla\psi\right|^{2}-\frac{1}{2}\frac{\bigl{(}\nabla\eta\cdot\nabla\psi+G(\eta)\psi\bigr{)}^{2}}{1+|\nabla\eta|^{2}}=0.\end{aligned}\right.$

######  Remark 1.2.

$i)$ It is proved in [11] that this system is equivalent to the incompressible
Euler equations with free surface.

$ii)$ Brenier ([35]) related the Hele-Shaw (1.4) and water-wave equations
(1.5) by a quadratic change of time.

$iii)$ One can consider more general hamiltonians to take into account surface
tension effect or rotational flows (see [42, 64] and references therein).

A key feature of this problem is that (1.5) is not a system of PDEs (indeed,
$G(\eta)$ is a nonlocal operator). Other difficulties present themselves: the
equations are fully nonlinear (instead of semi-linear, see the thesis by Ayman
Rimah Said [90]). Eventually, the Hamiltonian does not control the dynamics.

##  2\. The Dirichlet-to-Neumann operator

One can define the Dirichlet-to-Neumann operator for rough domains. For
instance, Arendt and ter Elst ([28]) define it for bounded connected open set
$\Omega\subset\mathbb{R}^{N}$ whose boundary has a finite $(N-1)$-dimensional
Hausdorff measure (see also [102]). Here, we will consider smoother functions.
In broad terms, we aim to identify a level of regularity beyond which we can
effectively transform the equations, enabling us to conjugate them to simpler
counterparts. We begin by discussing classical results that hold for Lipschitz
free surface. Subsequently, we will consider free surfaces with slightly
higher regularity, specifically those belonging to the class $C^{1,\alpha}$ in
terms of Sobolev embedding.

###  2.1. The Lipschitz threshold of regularity

Assume $\eta\in W^{1,\infty}(\mathbb{R}^{d})$ and $\psi\in
H^{\frac{1}{2}}(\mathbb{R}^{d})$. Then it follows from classical arguments
from functional analysis (Poincaré type inequalities and the Lax-Milgram
theorem, see [10, 12, 13]) that there is a unique variational solution

$\phi\in
L^{2}\big{(}\operatorname{d}\\!x\operatorname{d}\\!y/(1+|y|)^{2}\big{)}\quad,\quad\nabla_{x,y}\phi\in
L^{2}(\Omega)$

to

(2.1) $\Delta_{x,y}\phi=0\quad\text{in
}\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}\,:\,y<\eta(x)\\},\qquad\phi\big{\arrowvert}_{y=\eta}=\psi.$

Given that $\phi$ is harmonic, it is possible to express the normal derivative
in terms of the tangential derivatives. By doing so, we verify that its normal
derivative on the free surface is well-defined as an element of
$H^{-\frac{1}{2}}(\mathbb{R}^{d})$. Therefore, one can define

$G(\eta)\psi=\sqrt{1+|\nabla\eta|^{2}}\,\partial_{n}\phi\big{\arrowvert}_{y=\eta}\in
H^{-\frac{1}{2}}(\mathbb{R}^{d}),$

and it follows that

$\left\|G(\eta)\right\|_{H^{\frac{1}{2}}\to H^{-\frac{1}{2}}}\leq
C\big{(}\left\|\nabla\eta\right\|_{L^{\infty}}\big{)}.$

Moreover, a famous inequality of Rellich shows that $G(\eta)$ is also well-
defined on $H^{1}(\mathbb{R}^{d})$ and that one has

(2.2) $\left\|G(\eta)\right\|_{H^{1}\to L^{2}}\leq
C\big{(}\left\|\nabla\eta\right\|_{L^{\infty}}\big{)}.$

The first inequality of the form (2.2) was established through an integration
by parts argument by Rellich [89]. Rellich’s goal was to investigate the
eigenvalues of the Laplacian within star-shaped domains. This inequality plays
a pivotal role in numerous works pertaining to elliptic partial differential
equations. Notably, such inequalities were used by Jerison and Kenig to study
the Dirichlet problem in Lipschitz domains [72] and by Verchota [104] in his
famous work on layer potentials. See the books [79, 85] for many other
applications.

Below, we present a concise (formal) proof of a refined version of this
inequality, which follows from Zakharov’s Hamiltonian formulation and
Noether’s theorem.

######  Proposition 2.1 (Noether’s theorem implies Rellich inequality).

For any $d\geq 1$,

$\int_{\mathbb{R}^{d}}\big{|}\nabla_{x,y}\phi\arrowvert_{y=\eta}\big{|}^{2}\operatorname{d}\\!x\leq
5\int_{\mathbb{R}^{d}}|\nabla\psi|^{2}\operatorname{d}\\!x.$

In particular

$\left\|G(\eta)\right\|_{H^{1}\to L^{2}}\leq
4+4\left\|\nabla\eta\right\|_{L^{\infty}}.$

###### Proof.

This generalizes an inequality proved in [2] for the case $d=1$ using the
multiplier method. Here we give a short (formal) proof which relies only on
Noether’s theorem.

It follows from the Hamiltonian formulation of the problem and its symmetries
that

$\frac{\operatorname{d}}{\operatorname{d}\\!t}\int\eta(t,x)\operatorname{d}\\!x=0\quad\text{and}\quad\frac{\operatorname{d}}{\operatorname{d}\\!t}\int\psi(t,x)\operatorname{d}\\!x=0.$

Now, remembering that

$\partial_{t}\psi+g\eta+\frac{1}{2}\left|\nabla\psi\right|^{2}-\frac{1}{2}\frac{\bigl{(}G(\eta)\psi+\nabla\eta\cdot\nabla\psi\bigr{)}^{2}}{1+|\nabla\eta|^{2}}=0$

we infer that

$\int\frac{\bigl{(}G(\eta)\psi+\nabla\eta\cdot\nabla\psi\bigr{)}^{2}}{1+|\nabla\eta|^{2}}\operatorname{d}\\!x=\int\left|\nabla\psi\right|^{2}\operatorname{d}\\!x.$

(We parenthetically mention that the previous identity can be rigorously
derived from the divergence theorem, see [2].) Now, by using the chain rule,
we verify that

$(\partial_{y}\phi)\arrowvert_{y=\eta}=\frac{G(\eta)\psi+\nabla\eta\cdot\nabla\psi}{1+|\nabla\eta|^{2}}.$

Consequently, the previous estimate implies that

(2.3)
$\int(1+|\nabla\eta|^{2})\big{(}(\partial_{y}\phi)\arrowvert_{y=\eta}\big{)}^{2}\operatorname{d}\\!x=\int\left|\nabla\psi\right|^{2}\operatorname{d}\\!x.$

On the other hand, using again the chain rule, we have

$(\nabla_{x}\phi)\arrowvert_{y=\eta}=\nabla\psi-(\partial_{y}\phi)\arrowvert_{y=\eta}\nabla\eta,$

and hence the estimate of $(\nabla_{x}\phi)\arrowvert_{y=\eta}$ follows
immediately from (2.3). ∎

We next give an elementary proof (obtained with Quoc-Hung Nguyen in 2021, see
[7]) of an optimal trace inequality.

######  Proposition 2.2 (Alazard-Nguyen).

Let $d\geq 1$, $\eta\in W^{1,\infty}(\mathbb{R}^{d})$ and $\psi\in
H^{\frac{1}{2}}(\mathbb{R}^{d})$. There exists $c>0$ such that

(2.4) $\int_{\mathbb{R}^{d}}\psi
G(\eta)\psi\operatorname{d}\\!x\geq\frac{c}{1+\left\|\nabla\eta\right\|_{\rm{BMO}}}\left\|\psi\right\|_{\dot{H}^{\frac{1}{2}}}^{2}.$

######  Remark 2.3 (A trace inequality).

Let $u\in H^{1}(\Omega)$ and set $\psi=u\arrowvert_{y=\eta}$. Denote by $\phi$
the harmonic extension of $\psi$. Then

$\displaystyle\iint_{\Omega}\left|\nabla_{x,y}u\right|^{2}\operatorname{d}\\!y\operatorname{d}\\!x\geq\iint_{\Omega}\left|\nabla_{x,y}\phi\right|^{2}\operatorname{d}\\!y\operatorname{d}\\!x=\int_{\partial\Omega}\phi\partial_{n}\phi\operatorname{d}\\!\sigma,$

where we used the divergence theorem to obtain the last equality. Now, by
definition of $G(\eta)$, we have

$\displaystyle\int_{\partial\Omega}\phi\partial_{n}\phi\operatorname{d}\\!\sigma=\int_{\mathbb{R}^{d}}\psi
G(\eta)\psi\operatorname{d}\\!x.$

Consequently, it follows from (2.4) that

$\iint_{\Omega}\left|\nabla_{x,y}u\right|^{2}\operatorname{d}\\!y\operatorname{d}\\!x\geq\frac{c}{1+\left\|\nabla\eta\right\|_{\rm{BMO}}}\left\|u\arrowvert_{y=\eta}\right\|_{\dot{H}^{\frac{1}{2}}}^{2}.$

######  Remark 2.4 (An optimal inequality).

As explained in [24], the dependence in $\left\|\nabla\eta\right\|_{\rm{BMO}}$
is _optimal_. More precisely, it is proved in the latter reference that

$\int\psi
G(\eta)\psi\operatorname{d}\\!x\geq\frac{c}{(1+\left\|\nabla\eta\right\|_{L^{\infty}})^{m}}\left\|\psi\right\|_{\dot{H}^{\frac{1}{2}}}^{2}\quad\Rightarrow\quad
m\geq 1.$

###### Proof.

For the sake of notational simplicity, we consider only the case $d=1$ (we
refer to [7] for the general case). Set $v(x,z)=\phi(x,z+\eta(x))$. Then, it
follows from the divergence theorem that

$\int_{\mathbb{R}}\psi
G(\eta)\psi\operatorname{d}\\!x=\iint_{\Omega}\left|\nabla_{x,y}\phi\right|^{2}\operatorname{d}\\!y\operatorname{d}\\!x=\iint_{\mathbb{R}^{2}_{-}}\Big{[}(\partial_{x}v-\partial_{z}v\partial_{x}\eta)^{2}+(\partial_{z}v)^{2}\Big{]}\operatorname{d}\\!z\operatorname{d}\\!x=:Q.$

On the other hand, the Plancherel theorem implies that

$\displaystyle\int_{\mathbb{R}}\psi\left|D_{x}\right|\psi\operatorname{d}\\!x=\iint_{\mathbb{R}^{2}_{-}}\partial_{z}(v\left|D_{x}\right|v)\operatorname{d}\\!z\operatorname{d}\\!x=2\iint_{\mathbb{R}^{2}_{-}}(\partial_{z}v)\left|D_{x}\right|v\operatorname{d}\\!z\operatorname{d}\\!x$
(2.5)
$\displaystyle\qquad=2\iint_{\mathbb{R}^{2}_{-}}\Big{[}(\partial_{z}v)\mathcal{H}\big{(}\partial_{x}v-(\partial_{z}v)\partial_{x}\eta\big{)}+(\partial_{z}v)\mathcal{H}\big{(}(\partial_{z}v)\partial_{x}\eta\big{)}\Big{]}\operatorname{d}\\!z\operatorname{d}\\!x,$

where we denoted by $\mathcal{H}$ the Hilbert transform. Since $\mathcal{H}$
is bounded on $L^{2}$, we have

$2\iint_{\mathbb{R}^{2}_{-}}(\partial_{z}v)\mathcal{H}\big{(}\partial_{x}v-(\partial_{z}v)\partial_{x}\eta\big{)}\operatorname{d}\\!z\operatorname{d}\\!x\leq
Q=\int_{\mathbb{R}}\psi G(\eta)\psi\operatorname{d}\\!x.$

So it remains only to estimate the second term in the left-hand side of (2.5).
To do this, observe that, since $\mathcal{H}^{*}=-\mathcal{H}$, we have

$2\iint_{\mathbb{R}^{2}_{-}}(\partial_{z}v)\mathcal{H}\big{(}(\partial_{z}v)\partial_{x}\eta\big{)}\operatorname{d}\\!z\operatorname{d}\\!x=\iint_{\mathbb{R}^{2}_{-}}(\partial_{z}v)\big{[}\mathcal{H},\partial_{x}\eta\big{]}\partial_{z}v\operatorname{d}\\!z\operatorname{d}\\!x.$

Then it suffices to recall that a classical result by Coifman-Rochberg-Weiss
[47] which states that the commutator between the Hilbert transform and a
$\rm{BMO}$ function is bounded on $L^{2}$ (see also [75]). ∎

###  2.2. Elliptic estimates

We have seen that

$\left\|G(\eta)\right\|_{H^{\frac{1}{2}}\to
H^{-\frac{1}{2}}}+\left\|G(\eta)\right\|_{H^{1}\to L^{2}}\leq
C(\left\|\nabla\eta\right\|_{L^{\infty}}).$

Many other results have been proved when $\eta$ is more regular ([84, 51, 106,
33, 73]). In particular, we have the following

######  Proposition 2.5 (from [12]).

$(i)$ For all $s>1+d/2$ and $1/2\leq\sigma\leq s$

$\left\|G(\eta)\psi\right\|_{H^{\sigma-1}}\leq
C\left(\left\|\eta\right\|_{H^{s}}\right)\left\|\psi\right\|_{H^{\sigma}}.$

$(ii)$ For all $s>1+d/2$,

$\left\|\left[G(\eta_{1})-G(\eta_{2})\right]\psi\right\|_{H^{s-\frac{3}{2}}}\leq
C\left(\left\|(\eta_{1},\eta_{2})\right\|_{H^{s+\frac{1}{2}}}\right)\left\|\psi\right\|_{H^{s}}\left\|\eta_{1}-\eta_{2}\right\|_{H^{s-\frac{1}{2}}}.$

Notice that, in terms of Sobolev embedding, case $(i)$ corresponds to a
function $\eta$ that is slightly more regular than Lipschitz. However, for
case $(ii)$, the difference $\eta_{1}-\eta_{2}$ is estimated in a norm weaker
than the Lipschitz norm.

One possible approach to prove such Sobolev estimates is to employ an
elementary change of variables, reducing the problem to a case where the
domain is a half-space. This transformation replaces the Laplace equation with
an elliptic equation with variable coefficients (see (3.6) below), depending
on the derivatives of $\eta$ (which are Hölder continuous in terms of Sobolev
embedding). A classical idea, rooted in Schauder’s method, involves freezing
the coefficient at a point, expressed as

$\operatorname{div}_{x,z}(\underbrace{A(x_{0},z_{0})}_{{\text{constant}}}\nabla_{x,z}v)=\operatorname{div}_{x,z}(\underbrace{(A(x_{0},z_{0})-A(x,z))}_{{\text{small}}}\nabla_{x,z}v).$

There are several ways to rigorously justify this. One method, explained in
the next section, involves the use of paradifferential calculus.

###  2.3. The shape derivative

Eventually, we recall the following fundamental identity which helps to
understand the dependence on $\eta$.

######  Proposition 2.6 (Lannes’ shape derivative formula).

For all $\eta,\psi,\zeta\in H^{\infty}(\mathbb{R}^{d})$, the shape derivative
of the Dirichlet-to-Neumann operator is given by

(2.6) $\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\bigl{(}G(\eta+\varepsilon\zeta)\psi-G(\eta)\psi\bigr{)}=-G(\eta)(B\zeta)-\operatorname{div}(V\zeta),$

where $V=(\nabla\phi)\arrowvert_{y=\eta}$ and
$B=(\partial_{y}\phi)\arrowvert_{y=\eta}$, with $\phi$ the harmonic extension
of $\psi$ solution to (2.1).

######  Remark 2.7.

This identity extends to functions with limited regularity; we refer to the
Lannes’ original article or his book (see [73, 74]).

##  3\. Paralinearization of the Dirichlet-to-Neumann operator

###  3.1. Pseudo- and paradifferential operators

Since the work of Kohn–Nirenberg and Hörmander it is said that that a linear
operator
$T\colon\mathcal{S}^{\prime}(\mathbb{R}^{d})\to\mathcal{S}^{\prime}(\mathbb{R}^{d})$
acting on temperate distributions is a pseudo-differential operator if we can
define it from a function $a=a(x,\xi)$ by the relation

(3.1) $T\bigl{(}e^{ix\cdot\xi}\bigr{)}=a(x,\xi)e^{ix\cdot\xi}.$

We then say that $a$ is the symbol for $T$ and we denote
$T=\operatorname{Op}(a)$. For instance, the operator associated with the
symbol $a=\sum_{\alpha}a_{\alpha}(x)(i\xi)^{\alpha}$ is simply the
differential operator $T=\sum_{\alpha}a_{\alpha}(x)\partial_{x}^{\alpha}$
(with classical notations). Another fundamental example is the case of the
operator $\left|D_{x}\right|$ defined by

$\left|D_{x}\right|e^{ix\cdot\xi}=\left|\xi\right|e^{ix\cdot\xi},$

so that $\left|D_{x}\right|=\operatorname{Op}(\left|\xi\right|)$.

The pseudo-differential calculus is a process that associates a symbol
$a=a(x,\xi)$ defined on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ with an operator
$\operatorname{Op}(a)$, in a way which allows for an understanding of the
properties of $\operatorname{Op}(a)$ (such as product, adjoint, and
boundedness on the usual spaces of functions) simply by examining the
properties of the symbols.

The application that associates an operator $\operatorname{Op}(a)$ to the
symbol $a$ is called a quantization. There are numerous known quantizations,
which are variants of (3.1). Bony’s quantization is particularly well-suited
for nonlinear problems. Its specificity lies in the quantization of symbols
with limited regularity in $x$. Bony’s theory of paradifferential operators
enables the study of the regularity of solutions to nonlinear partial
differential equations. A brief introduction is provided here, with references
to the original article by Bony [34], as well as the books by Hörmander [65],
Métivier [81], Meyer [82], and Taylor [100] for a more comprehensive
understanding of the general theory.

The idea of studying the incompressible Euler equation at the free surface
using tools derived from the analysis of singular integrals dates back to an
article by Craig-Schanz-Sulem [51]. This concept has been further developed by
Lannes [73] and Iooss-Plotnikov-Toland [71, 70]. The paradifferential analysis
of the Dirichlet-to-Neumann operator is introduced by Alazard-Métivier in [20]
and further developped by Alazard-Burq-Zuily [10, 12, 15] and Alazard-Delort
[17] (see also Remark 3.5).

###  3.2. Calderón’s analysis of the Dirichlet-to-Neumann operator

By using the Fourier transform, it is easily seen that
$G(0)=\left|D_{x}\right|$. More generally, if $\eta\in C^{\infty}$, then it is
known since Calderón ([37]) that $G(\eta)$ is an elliptic, self-adjoint
pseudo-differential operator of order $1$, whose principal symbol is

(3.2)
$\lambda(x,\xi)\mathrel{:=}\sqrt{(1+\left|\nabla\eta(x)\right|^{2})\left|\xi\right|^{2}-(\nabla\eta(x)\cdot\xi)^{2}}.$

This means that one can write $G(\eta)$ under the form

$G(\eta)\psi=(2\pi)^{-d}\int
e^{ix\cdot\xi}\lambda(x,\xi)\widehat{\psi}(\xi)\operatorname{d}\\!\xi+R(\eta)f,$

where the remainder $R(\eta)$ is of order $0$, satisfying (see Lannes [74])

$\exists K\geq 1,\forall
s\geq\frac{1}{2},\quad\left\|R(\eta)\psi\right\|_{H^{s}}\leq
C\left(\left\|\eta\right\|_{H^{s+K}}\right)\left\|\psi\right\|_{H^{s}}.$

######  Remark 3.1.

Three remarks are in order:

$i)$ Notice that $\lambda$ is well-defined for any $\eta\in
W^{1,\infty}(\mathbb{R}^{d})$.

$ii)$ If $d=1$ or $\eta=0$ then $\lambda(x,\xi)=\left|\xi\right|$ and
$\operatorname{Op}(\lambda)=\left|D_{x}\right|$.

$iii)$ The constant $K$ corresponds to a _loss of derivatives_ (see [66, 73]).

In dimension one, the fact that $\lambda=\left|\xi\right|$ implies that

$\displaystyle G(\eta)\psi=\left|D_{x}\right|\psi\leavevmode\nobreak\
+\leavevmode\nobreak\ R(\eta)\psi.$

By using paradifferential calculs, we will see that one can describe the
remainder term $R(\eta)=G(\eta)\psi-\left|D_{x}\right|\psi$.

###  3.3. Paraproduct

Paraproducts are well-known for their crucial role in the study of multilinear
Fourier multipliers (see [83, 99]). Loosely speaking, paraproducts are
restricted versions of products in which only certain types of frequency
interactions are permitted. A paraproduct can be defined very simply using the
Fourier inversion formula:

$a(x)b(x)=\frac{1}{(2\pi)^{2d}}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}e^{ix\cdot(\xi_{1}+\xi_{2})}\,\widehat{a}(\xi_{1})\,\widehat{b}(\xi_{2})\operatorname{d}\\!\xi_{1}\operatorname{d}\\!\xi_{2}.$

Let us decompose the integral in three terms

$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}=\iint_{\left|\xi_{1}+\xi_{2}\right|\sim\left|\xi_{2}\right|}+\iint_{\left|\xi_{1}+\xi_{2}\right|\sim\left|\xi_{1}\right|}+\iint_{\left|\xi_{1}\right|\sim\left|\xi_{2}\right|}$

to obtain the following decomposition of the product

$ab=T_{a}b+T_{b}a+R(a,b).$

To define these operators more precisely, let us consider a cut-off function
$\chi$ such that

$\chi(\xi_{1},\xi_{2})=1\quad\text{if}\quad\left|\xi_{1}\right|\leq\varepsilon_{1}\left|\xi_{2}\right|,\qquad\chi(\xi_{1},\xi_{2})=0\quad\text{if}\quad\left|\xi_{1}\right|\geq\varepsilon_{2}\left|\xi_{2}\right|,$

with $0<\varepsilon_{1}<\varepsilon_{2}<1$. Given two functions $a=a(x)$ and
$b=b(x)$ one defines

$\displaystyle T_{a}b$ $\displaystyle=\frac{1}{(2\pi)^{2d}}\iint
e^{ix\cdot(\xi_{1}+\xi_{2})}\chi(\xi_{1},\xi_{2})\widehat{a}(\xi_{1})\widehat{b}(\xi_{2})\operatorname{d}\\!\xi_{1}\operatorname{d}\\!\xi_{2},$
$\displaystyle T_{b}a$ $\displaystyle=\frac{1}{(2\pi)^{2d}}\iint
e^{ix\cdot(\xi_{1}+\xi_{2})}\chi(\xi_{2},\xi_{1})\widehat{a}(\xi_{1})\widehat{b}(\xi_{2})\operatorname{d}\\!\xi_{1}\operatorname{d}\\!\xi_{2},$
$\displaystyle R(a,b)$ $\displaystyle=\frac{1}{(2\pi)^{2d}}\iint
e^{ix\cdot(\xi_{1}+\xi_{2})}\bigl{(}1-\chi(\xi_{1},\xi_{2})-\chi(\xi_{2},\xi_{1})\bigr{)}\widehat{a}(\xi_{1})\widehat{b}(\xi_{2})\operatorname{d}\\!\xi_{1}\operatorname{d}\\!\xi_{2}.$

This is Bony’s decomposition of the product of two functions. One says that
$T_{a}b$ and $T_{b}a$ are paraproducts, while $R(a,b)$ is a remainder. The key
property is that a paraproduct by an $L^{\infty}$ function acts on any Sobolev
spaces $H^{s}$ with $s$ in $\mathbb{R}$. The remainder term $R(a,b)$ is
smoother than the paraproducts $T_{a}b$ and $T_{b}a$. More precisely, we have
the following results:

######  Theorem 3.2 (Bony).

For all $d\geq 1$ and all $s\in\mathbb{R}$, we have

$\displaystyle a\in L^{\infty}(\mathbb{R}^{d}),$ $\displaystyle u\in
H^{s}(\mathbb{R}^{d})$ $\displaystyle\Rightarrow$ $\displaystyle T_{a}u\in
H^{s}(\mathbb{R}^{d}).$ In addition, for all $s>0$, there holds $\displaystyle
a\in H^{s}(\mathbb{R}^{d}),$ $\displaystyle u\in H^{s}(\mathbb{R}^{d})$
$\displaystyle\Rightarrow$ $\displaystyle R(a,u)\in
H^{{2s}-\frac{d}{2}}(\mathbb{R}^{d}).$

######  Theorem 3.3 (Coifman–Meyer).

If $a\in W^{1,\infty}(\mathbb{R}^{d})$ and $u\in L^{2}(\mathbb{R}^{d})$ then

$au-T_{a}u\in H^{1}(\mathbb{R}^{d}).$

###  3.4. Paralinearization formula

We are now ready to state a paralinearization formula for the Dirichlet-to-
Neumann operator; more precisely a paradifferential refinement of the
linearization formula given by Proposition 2.6. For the sake of simplicity, we
consider only the case $d=1$ which allows to state a result involving only
paraproducts. The case $d\geq 2$ requires in addition to introduce
paradifferential operators.

######  Theorem 3.4 (from [20]).

Let $d=1$ and $3<\gamma<s$. Then

(3.3) $G(\eta)\psi=\left|D_{x}\right|(\psi-
T_{B}\eta)-\partial_{x}\bigl{(}T_{V}\eta\bigr{)}+F$

where $V=(\partial_{x}\phi)\arrowvert_{y=\eta}$,
$B=(\partial_{y}\phi)\arrowvert_{y=\eta}$ ($\phi$ is given by (2.1)) and $F$
is a smooth quadratic remainder:

(3.4) $\left\|F\right\|_{H^{s+\gamma-4}}\leq
C\left(\left\|\eta\right\|_{C^{\gamma}}\right)\left\\{\left\|\psi\right\|_{C^{\gamma}}\left\|\eta\right\|_{H^{s}}+\left\|\eta\right\|_{C^{\gamma}}\big{\|}\psi\big{\|}_{H^{s}}\right\\}.$

######  Remark 3.5.

Many extensions of this results have been proved by many authors. To name a
few, we refer to Alazard-Burq-Zuily [10, 12], Alazard-Delort [17], de Poyferré
[53], de Poyferré-Nguyen [56, 55], Berti-Delort [29], Ai [3, 4], Wang [105],
Zhu [112], Fan Zheng [110], Chengyang Shao [94]; additional references are
given in Section 4.

The statement involves only paraproducts, but we will explain that the proof
involves, in an essential way, two other kinds of operators: paracomposition
operators and paradifferential operators. Before that, we begin by analyzing a
simple case where we consider only the quadratic terms in the Taylor expansion
of the Dirichlet-to-Neumann operator.

###  3.5. A toy quadratic model.

We here study the quadratic terms in the Taylor expansion of the Dirichlet-
Neumann operator $G(\eta)$ with respect to the free surface elevation $\eta$.
Craig, Schanz and Sulem (see [51] and [98, Chapter $11$]) have shown that one
can expand the Dirichlet-to-Neumann operator as a sum of pseudo-differential
operators and gave precise estimates for the remainders. This Taylor expansion
can be obtained by using Lannes’ shape derivative formula. In particular, we
have the following result (proved in Chapter $2$ in [17]).

######  Proposition 3.6.

Set

$G_{\leq
2}(\eta)f=\left|D_{x}\right|f{-\left|D_{x}\right|(\eta\left|D_{x}\right|f)-\partial_{x}(\eta\partial_{x}f)}.$

Let $(s,\gamma,\mu)\in\mathbb{R}^{3}$ be such that

$s-1/2>\gamma\geq 14,\quad s\geq\mu\geq
5,\quad\gamma\not\in\frac{1}{2}\mathbb{N},$

and consider $(\eta,\psi)\in H^{s+\frac{1}{2}}(\mathbb{R})\times
H^{s}(\mathbb{R})$. There exists a non decreasing function
$C\colon\mathbb{R}\rightarrow\mathbb{R}$ such that,

(3.5) $\left\|G(\eta)\psi-G_{\leq 2}(\eta)\psi\right\|_{H^{\mu-1}}\\\ \leq
C(\left\|\eta\right\|_{C^{\gamma}})\left\|\eta\right\|_{C^{\gamma}}\Bigl{\\{}\left\|\psi\right\|_{C^{\gamma}}\left\|\eta\right\|_{H^{s}}+\left\|\eta\right\|_{C^{\gamma}}\big{\|}\psi\big{\|}_{H^{\mu+\frac{1}{2}}}\Bigr{\\}}.$

We will now paralinearize the quadratic term and prove Theorem 3.4 for this
model case. Let us write
$\left|D_{x}\right|(\eta\left|D_{x}\right|f)+\partial_{x}(\eta\partial_{x}f)$
as a bilinear Fourier multiplier:

$\displaystyle\left|D_{x}\right|(\eta\left|D_{x}\right|f)+\partial_{x}(\eta\partial_{x}f)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\iint
e^{ix(\xi_{1}+\xi_{2})}m(\xi_{1},\xi_{2})\widehat{\eta}(\xi_{1})\widehat{f}(\xi_{2})\operatorname{d}\\!\xi_{1}\operatorname{d}\\!\xi_{2}$
$\displaystyle m(\xi_{1},\xi_{2})$
$\displaystyle=\left|\xi_{1}+\xi_{2}\right|\left|\xi_{2}\right|-(\xi_{1}+\xi_{2})\xi_{2}.$

Now the key observation is that

$\left|\xi_{1}\right|<\left|\xi_{2}\right|\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ (\xi_{1}+\xi_{2})\xi_{2}>0\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ m(\xi_{1},\xi_{2})=0.$

Consequently

$\left|D_{x}\right|(T_{\eta}\left|D_{x}\right|f)+\partial_{x}(T_{\eta}\partial_{x}f)=0$

and hence

$\displaystyle G_{\leq 2}(\eta)f$
$\displaystyle=\left|D_{x}\right|f-\left|D_{x}\right|T_{|D_{x}|f}\eta-\partial_{x}(T_{\partial_{x}f}\eta)-\left|D_{x}\right|R(|D_{x}|f,\eta)-\partial_{x}R(\partial_{x}f,\eta)$
$\displaystyle=\left|D_{x}\right|(f-T_{|D_{x}|f}\eta)-\partial_{x}(T_{\partial_{x}f}\eta)+F_{2}\quad\text{where}$
$\displaystyle F_{2}$
$\displaystyle=-\left|D_{x}\right|R(|D_{x}|f,\eta)-\partial_{x}R(\partial_{x}f,\eta).$

Now, directly from Theorem 3.2, we see that $F_{2}$ satisfies an estimate
analogous to (3.4).

###  3.6. The good unknown of Alinhac

In this section, we aim to explain the significance of the quantity

$\omega=\psi-T_{B}\eta$

which plays a crucial role in the paralinearization formula (3.3) for the
Dirichlet-to-Neumann operator. This unknown was initially introduced by
Alazard-Métivier in [20] where it was associated with the concept of the ”good
unknown” of Alinhac. The latter plays a pivotal role in the analysis of
surface waves, particularly in the work of Alinhac on shock waves and
rarefaction waves for systems of conservation laws [26, 25] (we refer to Majda
[77, 76] and Métivier [80]). It is noteworthy that, in the context of water-
wave problems, a variation (at the linearized equation level, rather than the
paradifferential level) was already employed by Lannes [73] and Trakhinin
[103].

Here, we aim to clarify that the appearance of $\omega$ is inherent when
introducing the paracomposition operator of Alinhac [25], which is associated
with the change of variables that flattens the boundary $\\{y=\eta(x)\\}$ of
the domain. This approach optimally captures the limited smoothness of the
coordinate transformation.

To study the elliptic equation $\Delta_{x,y}\phi=0$ in
$\Omega=\\{(x,y)\in\mathbb{R}^{d+1}\,;\,y<\eta(x)\\}$, a classical strategy is
to reduce the problem to the half-space $\mathbb{R}^{d}\times(-\infty,0)$. To
do so, the simplest change of coordinates is

$\kappa\colon(x,z)\mapsto(x,z+\eta(x)).$

Then $\phi(x,y)$ solves the Laplace equation $\Delta_{x,y}\phi=0$ if and only
if $v=\phi\circ\kappa=\phi(x,z+\eta(x))$ is a solution of $Pv=0$ in $z<0$,
where the operator $P$ is defined by

(3.6)
$P=(1+|\nabla\eta|^{2})\partial_{z}^{2}+\Delta^{2}-2\nabla\eta\cdot\nabla\partial_{z}-(\Delta\eta)\partial_{z}.$

The boundary condition on $\phi\arrowvert_{y=\eta(x)}$ becomes
$v(x,0)=\psi(x)$ and $G(\eta)$ is given by

$G(\eta)\psi=\bigl{[}(1+|\nabla\eta|^{2})\partial_{z}v-\nabla\eta\cdot\nabla
v\bigr{]}\big{\arrowvert}_{z=0}.$

We can now explain the main difficulty to work with a diffeomorphism with
limited regularity. Denote $D=-i\partial$ and introduce

$p(x,\xi,\zeta)=(1+|\nabla\eta(x)|^{2})\zeta^{2}+|\xi|^{2}-2\nabla\eta(x)\cdot\xi\zeta+i(\Delta\eta(x))\zeta.$

Observe that $P=-p(x,D_{x},D_{z})$ and write $T_{p}$ as a short notation for

$T_{1+|\nabla(x)|^{2}}D_{z}^{2}+|D_{x}|^{2}-2T_{\nabla\eta}\cdot
D_{x}D_{z}+T_{\Delta\eta}\partial_{z}.$

Since $p(x,D_{x},D_{z})v=0$, by using Theorem 3.2, we get hat $T_{p}v=f_{1}$
where $f_{1}$ is continuous in $z$ with values in $H^{s-2}$ if $\eta$ is in
$H^{s}$ and if in addition the first and second order derivatives in $x,z$ of
$v$ are bounded.

Now, the main observation is that one can associate to the diffeomorphism
$\kappa$ a paracomposition operator, denoted by $\kappa^{*}$, such that
$T_{p}(\kappa^{*}\phi)=f_{2}$ where $f_{2}$ is a smoother source term. More
precisely, where $f_{2}$ is continuous in $z$ with values in
$H^{s+\gamma-K(d)}$, if $\eta$ is in $H^{s}$ and if the derivatives in $x,z$
of order less than $\gamma$ of $v$ are bounded (the main difference between
$f_{1}$ and $f_{2}$ is that one cannot obtain a better regularity for $f_{1}$
by assuming that $v$ is smoother).

We do not define $\kappa^{*}$ here. Instead let us describe the two main
properties of paracomposition operators (see the article [25]). First, up to a
smooth remainder, there holds

$\kappa^{*}\phi=\phi\circ\kappa-T_{\phi^{\prime}\circ\kappa}\kappa$

where we denote by $\phi^{\prime}$ the differential of $\phi$. On the other
hand, one can use a symbolic calculus formula to compute the commutator of
$\kappa^{*}$ with a paradifferential operator. The latter formula implies that

$\kappa^{*}\Delta-T_{p}\kappa^{*}$

is a smoothing operator (which means here an operator bounded from $H^{\mu}$
to $H^{\mu+m}$ for any real number $\mu$, where $m$ is a fixed positive number
depending on the regularity of $\kappa$). Since $\Delta_{x,y}\phi=0$, this
means that $T_{p}\bigl{(}\phi\circ\kappa-
T_{\phi^{\prime}\circ\kappa}\kappa\bigr{)}$ is a smooth remainder term as
claimed above.

Now observe that

$\omega=\bigl{(}\phi\circ\kappa-
T_{\phi^{\prime}\circ\kappa}\kappa\bigr{)}\big{\arrowvert}_{z=0}.$

This explains why the good unknown of Alinhac $\omega$ enters into the
analysis of free boundary problems.

###  3.7. Elliptic factorization

Inspired by the theory of Alinhac’s paracomposition operators, let us
introduce

$u=\phi\circ\kappa-
T_{\phi^{\prime}\circ\kappa}\kappa=v-T_{\partial_{z}v}\eta,\quad\text{where}\quad
v(x,z)=\phi(x,z+\eta(x)).$

As explained in §3.6, $u$ satisfies a paradifferential elliptic equation with
a smooth remainder term:

$T_{1+\left|\nabla\eta\right|^{2}}\partial_{z}^{2}u+\Delta
u-2T_{\nabla\eta}\cdot\nabla\partial_{z}u-T_{\Delta\eta}\partial_{z}u\sim 0$

where we say that $f\sim 0$ provided that

$f\in C_{z}^{0}\big{(}[-1,0];H_{x}^{{s+\gamma}-K(d)}(\mathbb{R}^{d})\big{)}$

for some constant $K$ depending only on the dimension $d$ (we stated Theorem
3.4 only for $d=1$, but we explain here the strategy of the proof in arbitrary
dimension $d$).

To prove Theorem 3.4, the next step is to compute the normal derivatives of
$u$ at the boundary in terms of tangential derivatives. To do so, we adapt the
strategy introduced by Calderón [37] to this paradifferential setting. This
requires to introduce paradifferential operators.

Given a symbol $a=a(x,\xi)$, the paradifferential operator $T_{a}$ of Bony is
defined by symbol smoothing:

(3.7) $T_{a}=\operatorname{Op}(\sigma)\quad\text{ with
}\widehat{\sigma}(\theta,\eta)=\chi(\theta,\eta)\widehat{a}(\theta,\eta)$

where $\widehat{a}(\theta,\eta)=\int e^{-ix\cdot\theta}a(x,\eta)\,dx$ is the
Fourier transform of $a$ with respect to the first variable and where $\chi$
is the cut-off function already introduced to define paraproducts (see §3.3).

Then one may use symbolic calculus for paradifferential operators to factor
out the second order elliptic operator as the product of two operators of
order one. Specifically, we want to get the existence of two symbols $a,A$
such that

$(\partial_{z}-T_{a})(\partial_{z}-T_{A})u\sim 0.$

To do so, we seek $a$ and $A$ under the form

$a=\sum a_{1-j},\qquad A=\sum A_{1-j}$

(where the indices refer to the order of the symbols) such that

$\displaystyle\sum\frac{1}{i^{\alpha}\alpha!}\
\partial_{\xi}^{\alpha}a_{m}\,\partial_{x}^{\alpha}A_{\ell}=-\frac{\left|\xi\right|^{2}}{1+|\nabla\eta|^{2}},$
$\displaystyle\sum
a_{m}+A_{m}=\frac{1}{1+|\nabla\eta|^{2}}\left(2i\nabla\eta\cdot\xi+\Delta\eta\right).$

This system can be solved by induction (see [20] for the details).

Then we introduce $w\mathrel{:=}(\partial_{z}-T_{A})u$ solution to

$\partial_{z}w-T_{a}w\sim 0.$

Since this is a parabolic evolution equation in $z$ (seen with initial data in
$z<0$), we conclude that

$(\partial_{z}u-T_{A}u)\arrowvert_{z=0}=w(0)\sim 0.$

This allows to express $\partial_{z}u$ on the boundary $\\{z=0\\}$ in terms of
$T_{A}u$, and hence in terms of tangential derivatives, which is the key
ingredient to prove Theorem 3.4.

##  4\. Applications

Numerous applications of Theorem 3.4 have been explored in the analysis of the
Cauchy problem for water-waves. However, due to the vast scope of this topic,
which was extensively covered in lectures by Daniel Tataru and Sijue Wu at the
Abel Symposium, we refrain from an in-depth discussion here. For comprehensive
references on this subject, we direct the reader to their texts and Chapter 1
in [7].

Meanwhile, we highlight only articles which are directly related to the
preceding paradifferential analysis of the Dirichlet-to-Neumann operator. A
series of recent papers delve into the study of the Cauchy problem with rough
initial data, starting with [10, 12] and extending through [14, 56, 55, 54, 3,
4, 93]. Recently, this paradifferential approach experiences significant
enhancement through the utilization of balanced energy estimates introduced by
Ai, Ifrim, and Tataru ([5, 6]).

The paradifferential approach has also found application in the examination of
normal form transformations, following the seminal work of Shatah [95] and Wu
[107] (see [16, 17, 57, 58, 105, 5, 6, 68, 68, 30, 31, 60, 59]). Additionally,
longstanding questions posed by small divisor problems in the theory of
standing water waves (see [87, 71, 69]) can be analyzed through this
perspective (see [20, 8, 32]). Furthermore, applications extend to the proof
of dispersive properties of water-waves ([45, 10, 3, 4]), encompassing results
related to control theory [9, 111].

In this section, we present two illustrative examples: one concerning the
proof of Sobolev energy estimates for the Hele-Shaw equation, and the second
regarding the conjugacy of the capillary water-wave equations to a constant
coefficient equation.

###  4.1. The Cauchy problem for the Hele-Shaw equation

######  Theorem 4.1 (from [44, 78, 21, 86]).

The Cauchy problem for the Hele-Shaw equation $\partial_{t}\eta+G(\eta)\eta=0$
is well-posed (localy in time) on the sub-critical Sobolev spaces
$H^{s}(\mathbb{T}^{d})$ for $s>1+d/2$.

Let us explain the strategy of the proof. The first step is to quasilinearize
the Hele-Shaw equation. To do so, one uses Lannes’ shape derivative foruma
(see Proposition 2.6) to prove the following

######  Lemma 4.2.

Let $\phi=\phi(t,x,y)$ denote the harmonic extension of $\eta$ in
$\\{y<\eta(t,x)\\}$ and set

$a=1-(\partial_{y}\phi)\arrowvert_{y=\eta}\quad,\quad
V=-(\nabla_{x}\phi)\arrowvert_{y=\eta}.$

Then

(4.1) $\displaystyle\partial_{t}V+V\cdot\nabla
V+aG(\eta)V+\frac{\gamma}{a}V=0\quad\text{where}$
$\displaystyle\gamma=\frac{1}{1+|\nabla\eta|^{2}}\Big{(}G(\eta)(a^{2}+V^{2})-2aG(\eta)a-2V\cdot
G(\eta)V\Big{)}.$

Then, one proves that the previous equation for $V$ is parabolic. This amounts
to prove that the coefficient $a$ is positive.

######  Lemma 4.3.

$a>0$.

This is the analogue of a famous result by Wu [106] for the water wave
equations. The proof is readily obtained by noticing that the function
$y-\phi$ is harmonic and vanishes on the boundary. Therefore, the Hopf-Zaremba
principle gives $\partial_{n}(y-\phi)>0$, equivalent to $a>0$.

To analyze (4.1), we also have to compare the two first order operators:
$G(\eta)$ and $A=V\cdot\nabla$. The key point is that, even if $A$ is of order
$1$, its real part is of lower order. Indeed

${(A+A^{*})f=-(\operatorname{div}V)f}\quad\text{is of order $0$},$

where the fact that it is of order $0$ means for instance that it is bounded
from $L^{2}$ to $L^{2}$. However, this holds only if $V$ is
$W^{1,\infty}_{x}$, which corresponds to the case $s>d/2+2$. To handle the
general case $s>d/2+1$, we observe that, for such $s$, $A+A^{*}$ is not of
order $0$ in general, but it is of order strictly smaller than $1$. Namely, it
is of order $1-\varepsilon$ provided that $V$ is
$L^{\infty}_{t}(C^{0,\varepsilon})$. This is enough to see $A$ as a
subprincipal term: indeed, for a parabolic evolution equation, by using
classical interpolation arguments, one can see any operator of order strictly
smaller than the order of the dissipative term as subprincipal.

The last step of the proof is to paralinearize $G(\eta)$. Here, we cannot rely
on Theorem 3.4 since we merely assume that $s>d/2+1$. Instead, we need a
microlocal description of the Dirichlet-to-Neumann operator in the whole range
of $C^{s}$ domains, $s>1$. The latter analysis was done in [12], where the
following result is proved.

######  Lemma 4.4.

Let $d\geq 1$ and $s>1+\frac{d}{2}$. For any $\sigma$ and $\varepsilon$ such
that

$\frac{1}{2}\leq\sigma\leq s-\frac{1}{2},\quad
0<\varepsilon\leq\frac{1}{2},\qquad\varepsilon<s-1-\frac{d}{2},$

the remainder $R(\eta)f\mathrel{:=}G(\eta)f-T_{\lambda}f$ satisfies

$\left\|R(\eta)f\right\|_{H^{\sigma-1+\varepsilon}}\leq
C\bigl{(}\|\eta\|_{H^{s}}\bigr{)}\left\|f\right\|_{H^{\sigma}}.$

Recall that $\lambda$ is defined by (3.2) and $T_{\lambda}$ by (3.7). In
particular, if $d=1$ then $T_{\lambda}=\left|D_{x}\right|$ (up to a smoothing
operator).

Then by combining the previous arguments, one can give a fairly direct proof
of Sobolev energy estimates (using classical interpolation arguments).

One advantage of this approach is that it gives a blow-up criterion (indeed,
such blow-up criteria are systematically discussed by Taylor in his books for
various nonlinear equations ([100, 101]) who deduces them from applications of
the paradifferential calculus).

######  Lemma 4.5.

Let $\alpha\in(0,1)$. Consider a regular solution $h$ as given by Theorem 4.1,
and define by $T^{*}$ its lifespan. Then the following alternative holds:
either $T^{*}=+\infty$ or

(4.2) $\limsup_{t\to
T^{*}}\left\|\eta(t)\right\|_{C^{1,\alpha}(\mathbb{R}^{d})}=+\infty.$

Furthermore, by using completely different techniques (namely using the
Boundary Harnack Inequality), in [18] it is proved that one can control some
Hölder norm of the slope in terms of its $L^{\infty}$-norm and the distance
from the initial surface. Since the latter are easily bounded (by maximum
principles and comparison arguments), we conclude that the Cauchy problem is
well-posed globally in time.

######  Theorem 4.6 (from [18]).

Consider an integer $d\geq 1$ and a real number $s>d/2+1$. For any initial
data $\eta_{0}$ in $H^{s}(\mathbb{R}^{d})$, the Cauchy problem

$\partial_{t}\eta+G(\eta)\eta=0,\quad\eta\arrowvert_{t=0}=\eta_{0},$

has a unique global solution

$\eta\in C^{0}([0,+\infty);H^{s}(\mathbb{R}^{d}))\cap
C^{\infty}((0,+\infty)\times\mathbb{R}^{d}).$

###  4.2. Reduction to constant coefficients

Oversimplifying, one can think of the water-wave equations with surface
tension as

${Pu=\frac{\partial u}{\partial
t}+V(u)\partial_{x}u+i\left|D_{x}\right|^{\frac{3}{4}}\big{(}c(u)\left|D_{x}\right|^{\frac{3}{4}}u\big{)}=0}$

where $x\in\mathbb{T}$ and

$V(u)=\operatorname{Re}\left(\langle D_{x}\rangle^{-N}u\right)$

with $N$ as large as we want (for smooth enough initial data). This is a toy
model that mimics many features of the water-wave equations once written in
terms of $\eta$ and the good unknown $\omega$ (see [10]). In this section, the
goal is to explain the role of two other transformations (introduced in [8]
and further used in [9]) which can be used to further simplify the equations.

Firstly, one may use a change of variables to flatten the coefficients.
Namely, we introduce in [8] a change of variables preserving the $L^{2}$-norm
in $x$, of the form

(4.3)
$h(t,x)\mapsto(1+\partial_{x}\kappa(t,x))^{\frac{1}{2}}\,h(t,x+\kappa(t,x))$

which allows to replace $P$ by

${Q=\partial_{t}+W\partial_{x}+i\left|D_{x}\right|^{\frac{3}{2}}+R},\quad\text{$R$
is of order zero}$

where one can further assume that $\int_{\mathbb{T}}W(t,x)\,dx=0$. Notice that
this is nontrivial since the equation is nonlocal and since it implicitly
includes a cancellation of the term of order $1/2$ (this is the effect of
considering a change of variables that preserves the $L^{2}$-norm, namely the
prefactor $(1+\partial_{x}\kappa(t,x))^{\frac{1}{2}}$ in (4.3)).

To study
${\partial_{t}+W\partial_{x}+i\left|D_{x}\right|^{\frac{3}{2}}+R^{\prime}}$,
we then seek an operator $A$ such that

$\big{[}A,i\left|D_{x}\right|^{\frac{3}{2}}\big{]}+W\partial_{x}A\quad\text{is
a zero order operator}.$

In [8], we find an operator of the form

$A=\operatorname{Op}\big{(}q(t,x,\xi)e^{i\beta(t,x)|\xi|^{\frac{1}{2}}}\big{)}$

with

$\beta=\beta_{0}(t)+\frac{2}{3}\partial_{x}^{-1}W.$

Then

${\big{(}\partial_{t}+W\partial_{x}+i\left|D_{x}\right|^{\frac{3}{2}}\big{)}A=A\big{(}\partial_{t}+i\left|D_{x}\right|^{\frac{3}{2}}+R^{\prime\prime}\big{)}}$

with $R^{\prime\prime}$ of order $0$.

It is worth noting that $A$ belongs to $\operatorname{Op}S^{0}_{\rho,\rho}$
with $\rho=1/2$, known as the exotic class (refer to Coifman and Meyer [46]).
The fact that $A$ belongs to this symbol class without symbolic calculus
reflects the fact that the problem is quasi-linear ([90]). In contrast, for
the Benjamin-Ono equation, a similar conjugation is observed with
$A\in\operatorname{Op}S^{0}_{1,0}$.

##  5\. The Muskat problem

The Muskat equation serves as a two-fluid model in porous media, analogous to
the Hele-Shaw equation. In this context, we examine a time-dependent free
surface $\Sigma(t)$ that delineates two fluid domains, denoted as
$\Omega_{1}(t)$ and $\Omega_{2}(t)$. For simplicity, we restrict our
consideration to two-dimensional fluids, so that

$\displaystyle\Omega_{1}(t)$
$\displaystyle=\left\\{(x,y)\in\mathbb{R}\times\mathbb{R}\,;\,y>f(t,x)\right\\},$
$\displaystyle\Omega_{2}(t)$
$\displaystyle=\left\\{(x,y)\in\mathbb{R}\times\mathbb{R}\,;\,y<f(t,x)\right\\},$
$\displaystyle\Sigma(t)$
$\displaystyle=\partial\Omega_{1}(t)=\partial\Omega_{2}(t)=\\{y=f(t,x)\\}.$

Let us introduce the density $\rho_{i}$, the velocity $v_{i}$, and the
pressure $P_{i}$ in the domain $\Omega_{i}$ for $i=1,2$. It is assumed that
the velocities $v_{1}$ and $v_{2}$ adhere to Darcy’s law. The equations
governing the motion are then given by:

$\displaystyle v_{i}$ $\displaystyle=-\nabla(P_{i}+\rho_{i}gy)\qquad$ in
$\displaystyle\Omega_{i},$ $\displaystyle\operatorname{div}v_{i}$
$\displaystyle=0$ in $\displaystyle\Omega_{i},$ $\displaystyle P_{1}$
$\displaystyle=P_{2}$ on $\displaystyle\Sigma,$ $\displaystyle v_{1}\cdot n$
$\displaystyle=v_{2}\cdot n$ on $\displaystyle\Sigma,$
$\displaystyle\partial_{t}f$
$\displaystyle=\sqrt{1+(\partial_{x}f)^{2}}\,v_{2}\cdot n$

where $g$ is the gravity and $n$ is the outward unit normal to $\Omega_{2}$ on
$\Sigma$,

$n=\frac{1}{\sqrt{1+(\partial_{x}f)^{2}}}\begin{pmatrix}-\partial_{x}f\\\
1\end{pmatrix}.$

###  5.1. The Córdoba-Gancedo formulation

It has been long recognized that the Muskat problem can be effectively reduced
to a parabolic evolution equation for the unknown function $f$ (see [36, 61,
88, 97]). The formulation (5.1) used in the subsequent discussion stems from
the work of Córdoba and Gancedo [48]. In their study of this problem employing
contour integrals, they achieved a beautiful reformulation of the Muskat
equation in terms of finite differences.

To be more specific, they derived the following concise formulation of the
Muskat equation:

(5.1)
$\displaystyle\partial_{t}f=\frac{\rho}{2\pi}\int_{\mathbb{R}}\frac{\partial_{x}\Delta_{\alpha}f}{1+\left(\Delta_{\alpha}f\right)^{2}}\operatorname{d}\\!\alpha,$

where $\rho=\rho_{2}-\rho_{1}$ is the difference of the densities of the two
fluids and $\Delta_{\alpha}f$ is the slope

(5.2)
$\displaystyle\Delta_{\alpha}f(t,x)=\frac{f(t,x)-f(t,x-\alpha)}{\alpha}\cdot$

We assume $\rho_{2}>\rho_{1}$ (which means that the heavier fluid is below the
lighter one), allowing us to set $\rho=2$ without loss of generality. The
well-defined nature of the right-hand side is not immediately apparent, and we
will address this question in the subsequent part of this section.

With the current formulation, it is readily verified that the problem is
invariant by the following scaling :

$f(t,x)\mapsto\frac{1}{\lambda}f\left(\lambda t,\lambda x\right).$

Consequently, the two natural critical spaces are the following homogeneous
spaces: $\dot{H}^{\frac{3}{2}}(\mathbb{R})$ and
$\dot{W}^{1,\infty}(\mathbb{R})$.

Utilizing this formulation, Cameron [38] established the existence of a
modulus of continuity for the derivative, resulting in a global existence
theorem with the sole requirement that the product of the maximal and minimal
slopes remains bounded by 1. More recently, Abedin and Schwab also derived a
modulus of continuity in [1] through Krylov-Safonov estimates. Additionally,
Córdoba and Lazar, employing a novel formulation of the Muskat equation
involving oscillatory integrals [50], demonstrated the global well-posedness
of the Muskat equation in time. This holds under conditions that the initial
data is suitably smooth and that the $\dot{H}^{3/2}(\mathbb{R})$-norm is
sufficiently small (extended to the 3D case in [62]). We will explain in this
section a sharp result, proved with Quoc-Hung Nguyen ([23, 22]) which
established that the Cauchy problem is well-posed on the endpoint Sobolev
space $H^{3/2}(\mathbb{R})$, which is optimal concerning the equation’s
scaling. Indeed, blow-up results for certain sufficiently large data were
demonstrated by Castro, Córdoba, Fefferman, Gancedo, and López-Fernández [39,
40, 41]–a contrast to the findings presented in Theorem 3.6 for the Hele-Shaw
equation, which is the one-phase version of the Muskat problem.

###  5.2. The nonlinearity

Besides its esthetic aspect, the formulation (5.1) allows to apply tools at
interface of nonlinear partial differential equations and harmonic analysis.
For instance, one may think of the methods centering around the study of the
Hilbert transform $\mathcal{H}$ and Riesz potentials, or Besov and Triebel-
Lizorkin spaces. Recall that the Hilbert transform $\mathcal{H}$ is defined by

$\widehat{\mathcal{H}u}(\xi)=-i\frac{\xi}{\left|\xi\right|}\hat{u}(\xi).$

It follows that the fractional Laplacian
$\left|D_{x}\right|=(-\partial_{xx})^{\frac{1}{2}}$ satisfies

$\left|D_{x}\right|=\partial_{x}\mathcal{H}.$

Alternatively, $\mathcal{H}$ can be defined by a singular integral:

(5.3)
$\mathcal{H}f(x)=\frac{1}{\pi}\mathrm{pv}\int_{\mathbb{R}}\frac{f(y)}{x-y}\operatorname{d}\\!y,$

that is

$\displaystyle\mathcal{H}f(x)$
$\displaystyle=\frac{1}{\pi}\mathrm{pv}\int_{\mathbb{R}}\frac{f(y)}{x-y}\operatorname{d}\\!y=\frac{1}{\pi}\lim_{\varepsilon\to
0}\int_{\varepsilon<\left|y\right|<\frac{1}{\varepsilon}}\frac{f(x-y)}{y}\operatorname{d}\\!y$
$\displaystyle=-\frac{1}{\pi}\operatorname{pv}\int_{\mathbb{R}}\frac{f(x)-f(x-\alpha)}{\alpha}\operatorname{d}\\!\alpha=-\frac{1}{\pi}\operatorname{pv}\int_{\mathbb{R}}\Delta_{\alpha}f\operatorname{d}\\!\alpha.$

It follows that the fractional Laplacian
$\left|D_{x}\right|=\partial_{x}\mathcal{H}$ satisfies

$\left|D_{x}\right|f=-\frac{1}{\pi}\operatorname{pv}\int_{\mathbb{R}}\partial_{x}\Delta_{\alpha}f\operatorname{d}\\!\alpha.$

Consequently, by writing

$\frac{\partial_{x}\Delta_{\alpha}f}{1+\left(\Delta_{\alpha}f\right)^{2}}=\partial_{x}\Delta_{\alpha}f-(\partial_{x}\Delta_{\alpha}f)\frac{(\Delta_{\alpha}f)^{2}}{1+\left(\Delta_{\alpha}f\right)^{2}},$

we see that the Muskat equation can be written under the form

$\partial_{t}f+\left|D_{x}\right|f=\mathcal{T}(f)f\quad\text{where}\quad\mathcal{T}(f)f=-\frac{1}{\pi}\,\int_{\mathbb{R}}(\partial_{x}\Delta_{\alpha}f)\frac{(\Delta_{\alpha}f)^{2}}{1+\left(\Delta_{\alpha}f\right)^{2}}\operatorname{d}\\!\alpha.$

The next proposition states that the map $f\mapsto\mathcal{T}(f)f$ is locally
Lipschitz from $H^{\frac{3}{2}}(\mathbb{R})$ to $L^{2}(\mathbb{R})$.

######  Proposition 5.1.

Consider the operator

$\mathcal{T}(f)g=-\frac{1}{\pi}\int_{\mathbb{R}}\Delta_{\alpha}g_{x}\
\frac{(\Delta_{\alpha}f)^{2}}{1+(\Delta_{\alpha}f)^{2}}\operatorname{d}\\!\alpha,$

where $g_{x}\mathrel{:=}\partial_{x}g$.

  1. (1)

For all $f$ in $H^{1}(\mathbb{R})$ and all $g$ in
$H^{\frac{3}{2}}(\mathbb{R})$, the function

$\alpha\mapsto\Delta_{\alpha}g_{x}\
\frac{(\Delta_{\alpha}f)^{2}}{1+(\Delta_{\alpha}f)^{2}}$

belongs to $L^{1}_{\alpha}(\mathbb{R};L^{2}_{x}(\mathbb{R}))$. Consequently,
$\mathcal{T}(f)g$ belongs to $L^{2}(\mathbb{R})$. Moreover, there is a
constant $C$ such that

(5.4) $\left\|\mathcal{T}(f)g\right\|_{L^{2}}\leq
C\left\|f\right\|_{\dot{H}^{1}}\left\|g\right\|_{\dot{H}^{\frac{3}{2}}}.$

  2. (2)

There exists a constant $C>0$ such that, for all functions $f_{1},f_{2}$ in
$H^{1}(\mathbb{R})$ and for all $g$ in $H^{\frac{3}{2}}(\mathbb{R})$,

$\left\|(\mathcal{T}(f_{1})-\mathcal{T}(f_{2}))g\right\|_{L^{2}}\leq
C\left\|f_{1}-f_{2}\right\|_{\dot{H}^{1}}\left\|g\right\|_{\dot{H}^{\frac{3}{2}}}.$

  3. (3)

The map $f\mapsto\mathcal{T}(f)f$ is locally Lipschitz from
$H^{\frac{3}{2}}(\mathbb{R})$ to $L^{2}(\mathbb{R})$.

The previous proposition is an example of properties that are very simple to
prove using the definition of Sobolev spaces in terms of finite differences.
In particular, it relies on the fact that, for $s\in(0,1)$, the homogeneous
Sobolev norm $\left\|\cdot\right\|_{\dot{H}^{s}}$ is equivalent to the
Gagliardo semi-norm

(5.5) $\left\|u\right\|_{\dot{H}^{s}}^{2}\mathrel{:=}\frac{1}{4\pi
c(s)}\underset{\mathbb{R}\times\mathbb{R}}{\iint}\frac{\left|u(x)-u(y)\right|^{2}}{\left|x-y\right|^{2s}}\frac{\operatorname{d}\\!x\operatorname{d}\\!y}{\left|x-y\right|}\quad\text{with}\quad
c(s)=\int_{\mathbb{R}}\frac{1-\cos(h)}{\left|h\right|^{1+2s}}\operatorname{d}\\!h.$

Using the previous point of view, one can give an elementary proof of a
paralinearization formula for the nonlinearity $\mathcal{T}(f)f$.

The goal here is decompose the nonlinearity into several pieces having
different roles. Since it is a problem in fluid dynamics, one expects to
extract from the nonlinearity at least two terms: a convective term of the
form $V\partial_{x}f$ and an elliptic component of the form
$\gamma\left|D_{x}\right|f$, that is an equation of the form

${\partial_{t}f+V\partial_{x}f+\gamma\left|D_{x}\right|f=R(f)}$

for some coefficients $V$ and $\gamma$ depending on $\partial_{x}f$.

To reach this goal, a standard strategy is to use a paradifferential analysis.
For the Muskat equation, this idea was implemented independently in [19] and
[86]. These two approaches are very different: the approach in [86] applies
for many physical equations, while the approach in [19] is adapted to study
critical problem, as we discuss below.

To do this, following Shnirelman ([96]), we consider a simpler version of
paraproducts.

######  Proposition 5.2 ([19]).

Let $0<\varepsilon<1/2$. Given a bounded function $a=a(x)$, denote by
$\tilde{T}_{a}\colon{H}^{1+\varepsilon}(\mathbb{R})\rightarrow{H}^{1+\varepsilon}(\mathbb{R})$
the paraproduct-type operator defined by

${\tilde{T}_{a}g=\Lambda^{-(1+\varepsilon)}(a\Lambda^{1+\varepsilon}g)}\quad,\quad\Lambda=(I-\partial_{xx})^{1/2}.$

Then

${\mathcal{T}(f)f=\tilde{T}_{\frac{f_{x}^{2}}{1+f_{x}^{2}}}\left|D_{x}\right|f+\tilde{T}_{V(f)}\partial_{x}f+R_{\varepsilon}(f)}$

where $R_{\varepsilon}$ is of order $<1$:

$\left\|R_{\varepsilon}(f)\right\|_{H^{1+\varepsilon}}\leq\mathcal{F}\big{(}\left\|f\right\|_{{H}^{\frac{3}{2}+\varepsilon}}\big{)}\left\|f\right\|_{{H}^{\frac{3}{2}+\varepsilon}}\left\|f\right\|_{H^{2+{\frac{\varepsilon}{2}}}}.$

To prove this proposition, the approach in [19] exploits the formulation (5.1)
of the Muskat equation to give such a paradifferential decomposition in a
direct manner. The desired decomposition of the nonlinearity is achieved by
splitting the coefficient into its odd and even components:

$\displaystyle\mathcal{O}\left(\alpha,\cdot\right)$
$\displaystyle=\frac{1}{2}\bigg{(}\frac{1}{1+\left(\Delta_{\alpha}f\right)^{2}}-\frac{1}{1+\left(\Delta_{-\alpha}f\right)^{2}}\bigg{)},$
$\displaystyle\mathcal{E}\left(\alpha,\cdot\right)$
$\displaystyle=\frac{1}{2}\bigg{(}\frac{1}{1+\left(\Delta_{\alpha}f\right)^{2}}+\frac{1}{1+\left(\Delta_{-\alpha}f\right)^{2}}\bigg{)}.$

Since $\Delta_{\alpha}f(x)\to f_{x}(x)$ when $\alpha\to 0$, we further
decompose $\mathcal{E}\left(\alpha,\cdot\right)$ as follows

${\mathcal{E}\left(\alpha,\cdot\right)=\frac{1}{1+(\partial_{x}f)^{2}}+\left(\mathcal{E}\left(\alpha,\cdot\right)-\frac{1}{1+(\partial_{x}f)^{2}}\right)}.$

By considering that
${\displaystyle{\left|D_{x}\right|f=-\frac{1}{\pi}\int_{\mathbb{R}}\partial_{x}\Delta_{\alpha}f\operatorname{d}\\!\alpha}}$
we obtain the following decomposition of the nonlinearity:

(5.6)
${\partial_{t}f+\frac{1}{1+(\partial_{x}f)^{2}}\left|D_{x}\right|f+V(f)\partial_{x}f+R=0}$

with

$\displaystyle
V=-\frac{1}{\pi}\int_{\mathbb{R}}\frac{\mathcal{O}\left(\alpha,\cdot\right)}{\alpha}\operatorname{d}\\!\alpha,\qquad
R=-\frac{1}{\pi}\int_{\mathbb{R}}\frac{f_{x}(\cdot-\alpha)}{\alpha}\left(\frac{1}{1+(\Delta_{\alpha}f)^{2}}-\frac{1}{1+f_{x}^{2}}\right)\operatorname{d}\\!\alpha.$

###  5.3. Estimate of the critical norm

Proposition 5.2 enables the establishment of well-posedness for the Cauchy
problem on sub-critical Sobolev spaces. However, to address the Cauchy problem
on critical spaces, additional ideas are requisite.

The first estimation of the critical Sobolev $H^{3/2}$-norm was obtained in
the work of Córdoba and Lazar ([50]), who applied methods from harmonic
analysis to establish the well-posedness of the Cauchy problem for certain
initial data, where the slope can be arbitrarily large. Specifically, they
proved the following result.

######  Theorem 5.3 (Córdoba and Lazar).

The Cauchy problem for the Muskat equation is well-posed globally in time for
all initial data $f_{0}\in H^{\frac{5}{2}}(\mathbb{R})$ such that

$(1+\left\|\partial_{x}f_{0}\right\|_{L^{\infty}}^{4})\left\|f_{0}\right\|_{\dot{H}^{\frac{3}{2}}}\ll
1.$

Their main technical result is a key a priori estimate in the critical Sobolev
space, which reads:

${\frac{\operatorname{d}}{\operatorname{d}\\!t}\left\|f\right\|_{\dot{H}^{\frac{3}{2}}}^{2}+{\int_{\mathbb{R}}\frac{(\partial_{xx}f)^{2}}{1+(\partial_{x}f)^{2}}\operatorname{d}\\!x}}\lesssim\left(\left\|f\right\|_{\dot{H}^{\frac{3}{2}}}+\left\|f\right\|_{\dot{H}^{\frac{3}{2}}}^{2}\right)\left\|f\right\|_{\dot{H}^{2}}^{2}.$

While the statement of this estimate is straightforward, its proof is very
intricate due to the non-Banach algebra structure of
$H^{\frac{1}{2}}(\mathbb{R})$. Córdoba and Lazar addressed this challenge in
their proof, which relies on a clever reformulation of the equation in terms
of oscillatory integrals and the systematic application of Besov spaces
techniques. Further insights into this approach can be found in the works of
Gancedo-Lazar [62] and Granero-Bellinchón and Scrobogna [63].

The question then arises of how to derive a global existence result from their
inequality. The underlying idea is quite straightforward: the objective is to
absorb the right-hand side with the left-hand side. This necessitates
controlling the denominator $1+(\partial_{x}f)^{2}$ from below. To achieve
this, an integration by parts argument is employed, tracing back to the work
of Córdoba and Gancedo [49] (see also Cameron [38] and Gancedo-Lazar [62],
among others). This yields

$\frac{\operatorname{d}}{\operatorname{d}\\!t}\lVert\partial_{x}f\rVert_{L^{\infty}}\lesssim\lVert
f\rVert_{H^{2}}^{2}.$

By combining these two inequalities, it follows that there exists $c_{0}>0$
small enough, so that

$\frac{\operatorname{d}}{\operatorname{d}\\!t}\lVert
f\rVert_{\dot{H}^{\frac{3}{2}}}^{2}+c_{0}\int_{\mathbb{R}}(\partial_{xx}f)^{2}\,dx\leq
0,$

thereby estimating the $L^{\infty}_{t}(H^{3/2}_{x})$-norm.

We proceed to discuss two improvements: global existence for small data,
without the need for initial data to be smooth, and secondly, local well-
posedness for large data in the critical space. These improvements are based
on two distinct techniques.

###  5.4. A null-type property

In [23, 22], the first idea is to prove a (weak) null-type property. By this
we mean the proof of an inequality similar to Córdoba-Lazar but with the key
difference that the right-hand side involves the weak coercive quantity given
by the left-hand side.

######  Theorem 5.4 ([23]).

The Cauchy problem is well-posed, globally in time, for all initial data
$f_{0}\in H^{\frac{3}{2}}(\mathbb{R})$ such that
${\left\|f_{0}\right\|_{\dot{H}^{\frac{3}{2}}}\ll 1.}$

The proof relies on a null-type property, enabling to prove that

$\frac{\operatorname{d}}{\operatorname{d}\\!t}\left\|f\right\|_{\dot{H}^{\frac{3}{2}}}^{2}+{{{\int_{\mathbb{R}}\frac{(\partial_{xx}f)^{2}}{1+(\partial_{x}f)^{2}}\operatorname{d}\\!x}}}\lesssim\left(1+\|f\|_{\dot{H}^{\frac{3}{2}}}^{7}\right)\|f\|_{\dot{H}^{\frac{3}{2}}}{{\int_{\mathbb{R}}\frac{(\partial_{xx}f)^{2}}{1+(\partial_{x}f)^{2}}\operatorname{d}\\!x}.}$

This estimate implies at once a global well-posedness result for small data.
Indeed, this part is perturbative in character, and one can absorb the right-
hand side by the left-hand side, provided that $\|f\|_{\dot{H}^{\frac{3}{2}}}$
is small enough: if $\|f\|_{\dot{H}^{\frac{3}{2}}}$ is small enough then

$\frac{\operatorname{d}}{\operatorname{d}\\!t}\left\|f\right\|_{\dot{H}^{\frac{3}{2}}}^{2}+c_{0}\int_{\mathbb{R}}\frac{(\partial_{xx}f)^{2}}{1+(\partial_{x}f)^{2}}\operatorname{d}\\!x\leq
0.$

###  5.5. Paralinearization and enhanced regularity

The main result in [23] is

######  Theorem 5.5 ([23]).

The Cauchy problem is locally well-posed for any initial data in the critical
space $H^{\frac{3}{2}}(\mathbb{R})$.

To prove this result, we do not estimate the $H^{\frac{3}{2}}(\mathbb{R})$
norm, but a quantity which depends on the initial data. The idea is to exploit
an enhanced decay of the Fourier transform. Specifically, we use the fact
that, for all $f_{0}\in H^{\frac{3}{2}}(\mathbb{R})$ there is a weight
$\kappa_{0}$ such that

$\displaystyle{\left\|f_{0}\right\|_{\mathcal{H}^{\frac{3}{2},\kappa_{0}}}^{2}\mathrel{:=}\int_{\mathbb{R}}(1+\left|\xi\right|)^{3}\kappa_{0}(|\xi|)^{2}\big{|}\hat{f_{0}}(\xi)\big{|}^{2}\operatorname{d}\\!\xi<+\infty}\quad\text{and}$
$\displaystyle{\lim_{|\xi|\to+\infty}\kappa_{0}(|\xi|)=+\infty}.$

We then estimate the
$L^{\infty}_{t}(\mathcal{H}^{\frac{3}{2},\kappa_{0}})$-norm of $f$ using the
previous decomposition (5.6) of the nonlinearity, by

  * •

commuting $\left|D_{x}\right|\kappa_{0}(D_{x})$ with the equation (5.6)

  * •

taking the $L^{2}_{x}$ scalar product with
$\left|D_{x}\right|^{2}\kappa_{0}(D_{x})f$ and integrating in time.

The proof is very technical and we refer to the introduction of the original
article [23] for more explanations.

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