# Long time regularity for $3$d gravity waves
with vorticity

Daniel Ginsberg  Department of Mathematics, Brooklyn College (CUNY), 2900
Bedford Ave, Brooklyn, NY 11210<EMAIL_ADDRESS>and  Fabio
Pusateri  Department of Mathematics, University of Toronto, 40 St. George
street, Toronto, M5S 2E4, Ontario, Canada<EMAIL_ADDRESS>
###### Abstract.

We consider the Cauchy problem for the full free boundary Euler equations in
$3$d with an initial small velocity of size $O(\varepsilon_{0})$, in a moving
domain which is initially an $O(\varepsilon_{0})$ perturbation of a flat
interface. We assume that the initial vorticity is of size
$O(\varepsilon_{1})$ and prove a regularity result up to times of the order
$\varepsilon_{1}^{-1+}$, independent of $\varepsilon_{0}$.

A key part of our proof is a normal form type argument for the vorticity
equation; this needs to be performed in the full three dimensional domain and
is necessary to effectively remove the irrotational components from the
quadratic stretching terms and uniformly control the vorticity. Another
difficulty is to obtain sharp decay for the irrotational component of the
velocity and the interface; to do this we perform a dispersive analysis on the
boundary equations, which are forced by a singular contribution from the
rotational component of the velocity.

As a corollary of our result, when $\varepsilon_{1}$ goes to zero we recover
the celebrated global regularity results of Wu (Invent. Math. 2012) and
Germain, Masmoudi and Shatah (Ann. of Math. 2013) in the irrotational case.

> ###### Contents
>
>   1. 1 Introduction
>   2. 2 Strategy and main propositions
>   3. 3 Estimates for the vector potential
>   4. 4 Estimates for the vorticity
>   5. 5 Proof of Proposition 2.8
>   6. 6 Decay of the boundary variables
>   7. A Supporting material
>   8. B The boundary equations
>   9. C The elliptic system for the vector potential
>   10. D Energy estimates: proof of Proposition 2.5
>

##  1\. Introduction

We consider the classical free boundary Euler equations with gravity in three
space dimensions:

$\displaystyle(\partial_{t}+v^{k}\partial_{k})v_{i}$
$\displaystyle=-\partial_{i}p-ge_{3},$ $\displaystyle\quad\text{ in
}\mathcal{D}_{t},$ (1.1a) $\displaystyle\operatorname{div}v$
$\displaystyle=0,$ $\displaystyle\quad\text{ in }\mathcal{D}_{t},$ (1.1b)
$\displaystyle p$ $\displaystyle=0,$ $\displaystyle\quad\text{ on
}\partial\mathcal{D}_{t},$ (1.1c) $\displaystyle(1,v)$ $\displaystyle\text{ is
tangent to }\mathcal{D}=\cup_{t\geq 0}\\{t\\}\times\mathcal{D}_{t}.$ (1.1d)

We are adopting the usual convention of summing over repeated upper and lower
indexes. In what follows we set $g=1$. We assume that the boundary of the
moving domain, denoted $\partial\mathcal{D}_{t}$, is given by the graph of a
function $h$:

$\displaystyle\mathcal{D}_{t}:=\\{(x,y)\in{\mathbb{R}}^{2}\times{\mathbb{R}}\,:\,y\leq
h(t,x)\\}.$ (1.2)

This problem, and closely related models, have been studied extensively. We
will recall the local and global well-posendess theory and other results in
the literature below in Subsection 1.1.

For the moment we point out that in the irrotational case
($\omega:=\operatorname{curl}v=0$) one can construct classes of global
solutions close to a flat and still interface; see Wu [47] and Germain-
Masmoudi-Shatah [19] for the problem (1.1), and [20, 17] and the other
references given below for the case of other $3$d and $2$d models. These are
essentially the only known classes of global solutions for the initial value
problem. In this paper we are interested in the regularity question for the
Cauchy problem for general solutions with rotation, $\omega\neq 0$.

The first natural question to ask is: given an initial (divergence free)
velocity field and an initial perturbation of a flat interface of size
$\varepsilon_{0}$ (typically measured in a weighted Sobolev space), and an
initial vorticity of size $\varepsilon_{1}$, what is the maximal time of
existence and regularity of solutions? Our main result will show that if
$\varepsilon_{1}$ is the size of the initial vorticity, then the above problem
admits a solution at least until times that are (almost) of the order of
$1/\varepsilon_{1}$, uniformly in the size $\varepsilon_{0}$ of the
irrotational components of the solution. This is the natural time scale for
the evolution of the vorticity, which, in three dimensions, is a transport
equation with quadratic terms. By sending $\varepsilon_{1}$ to zero one then
also recovers the celebrated results of [19] and [47].

We first give here an informal statement, and will give a more precise one in
Theorem 2.2:

######  Theorem 1.1.

Assume that the initial height
$h(0,x):{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ and the initial (divergence-
free) velocity $v(0,x,y)$ defined on
$\mathcal{D}_{0}:=\\{(x,y)\in{\mathbb{R}}^{2}\times{\mathbb{R}}\,:\,y\leq
h(0,x)\\}$, are of size $\varepsilon_{0}$ in sufficiently regular weighted
Sobolev spaces. Assume that $\operatorname{curl}v(0,x,y)$ is of size
$\varepsilon_{1}$ in a sufficiently regular weighted Sobolev space, and that
$\operatorname{curl}v_{0}(0,x,h(x))=0$.

Then, for any fixed $\delta>0$ there exists $\bar{\varepsilon}_{0}$ and
$\bar{c}$ sufficiently small, independent of $\varepsilon_{1}$, such that, for
any $\varepsilon_{1}\leq\varepsilon_{0}\leq\bar{\varepsilon}_{0}$, the system
(1.1) has a unique classical solution $(v,h)$ with the above given initial
data111The pressure $p$ can be recovered from $v$ through an elliptic equation
in a standard way. $(v(0),h(0))$, on the interval
$[-T_{\varepsilon_{1}},T_{\varepsilon_{1}}]$ with

$T_{\varepsilon_{1}}:=\frac{\bar{c}}{\varepsilon_{1}^{1-\delta}}.$ (1.3)

###  1.1. Previous results

Studies on the free boundary Euler equations go back at least to Cauchy,
Laplace and Lagrange [14], and the analysis of (1.1), and several of its
variants, has been a very active research area in the last few decades. We
will not try to give a complete list of references here, but only mention
those results that are closer and most relevant to the present work. We direct
the interested reader to the extensive lists of references in some of the
cited works, and to the survey [30] for more background.

Local well-posedness. The local well-posedness theory of the free boundary
Euler equations and several of its variants is well-understood in a variety of
different scenarios, due to the contributions of many authors. Without being
exhaustive we mention [12, 46, 47, 10, 35, 33, 11, 36, 37, 2, 9, 26, 45] and
refer the reader to [30, Section $2$] and to the book of Lannes [34, Chapter
4]. In short, for sufficiently regular Sobolev initial data, classical smooth
solutions exist on a (small) time interval $[-T,T]$ where $T$ is approximately
the minimum between the inverse of the size of the initial data (in a Sobolev
space) and the arc-chord constant of the initial interface.

We remark that among the cited works only [10, 11, 37, 35, 26, 45] treat the
full problem with rotation; for the case of constant vorticity, the paper [25]
proves an extended life span, and the recent work of Wang [42] establishes low
regularity local wellposedness. All the other works only consider the
irrotational case, customarily referred to as the ‘water waves’ problem. The
main advantage in considering the irrotational problem, as far as local
existence is concerned, is that the equations of motion can be reduced to
equations on the interface for suitable unknowns; this reduction can be done
both in Eulerian or Lagrangian coordinates.

Global irrotational solutions. In the irrotational case one can construct
global solutions to the water waves problem in the vicinity of a flat and
still interface. More precisely, for localized initial data in a weighted
Sobolev space, one can rely on dispersion and pointwise decay to prove
scattering (and modified scattering) results. We refer the reader to [48, 27,
4, 23, 28, 29, 24, 43] for the case of $1$d interfaces, and to [49, 19, 17,
44] for $2$d interfaces; see also [30, Section 3] and [15] for an overview of
these results. As a corollary of our main result, when $\varepsilon_{1}$ goes
to zero we recover the global regularity results for the irrotational problem
with gravity of Wu [49] and Germain, Masmoudi and Shatah [19]. Our work is
also related to the work by Ionescu and Lie [32] where the authors prove a
similar result for the one-fluid Euler-Maxwell system with vorticity in 3d;
one major difference in the case of [32] is that the linear decay of
irrotational solution is integrable-in-time, unlike the case of irrotational
gravity waves, which decay at the rate of $t^{-1}$ in $L^{\infty}_{x}$.

The problem with vorticity. The question of long-term regularity with
vorticity is much more delicate than in the irrotational case. This is due to
the fact that the vorticity satisfies a transport equation with a quadratic
nonlinear (stretching) term. Moreover, in the free boundary problem, the
presence of non-trivial vorticity prevents the reduction of the equations
solely to the boundary.222 We also note that the Taylor sign condition
$-\nabla_{N}p|_{\partial\mathcal{D}_{t}}>0$, which is needed for local well-
posedness, holds automatically in the irrotational case but can fail if there
is nonzero vorticity (see, for example, [41, 50]), though it holds
automatically in the small data regime we are working in here. So far, to our
knowledge, the only available results on extended lifespans are those of the
first author [21], the work [25] proving a time of existence of
$\varepsilon_{0}^{-2}$ in the case of constant vorticity in the $2d$ case, and
[40] proving an $\varepsilon_{0}^{-2}$ existence result in the case of point
vortices. Concerning the problem of finding other types of solutions with
vorticity, we mention the recent work of Ehnstrom, Walsh and Zheng [18] on
stationary solutions. Finally, we also mention Castro-Lannes [8] who proved a
well-posedness results with a new Hamiltonian formulation for shallow water
waves with vorticity, Berti-Franzoi-Maspero [5] who construct quasi-periodic
in time solutions with constant vorticity, and [6] who prove an almost global
existence result with constant vorticity on the torus.

Further references. For further references we refer the reader to the
following: the review [30] for more background on the construction of long-
time and global solutions; [7, 16] for more literature on spatially periodic
solutions; and to the review [22] for more on traveling and stationary waves
(including the case with vorticity).

Acknowledgments. D.G. is supported in part by a start-up grant from Brooklyn
College. Part of this work was completed while D.G. was partially supported by
the Simons Center for Hidden Symmetries and Fusion Energy. F.P. is supported
in part by a start-up grant from the University of Toronto, and NSERC grant
RGPIN-2018-06487. The authors would like to thank Alexandru Ionescu and
Chongchun Zeng for helpful discussions about the problem.

##  2\. Strategy and main propositions

###  2.1. General set-up and some ideas

We being by decomposing the divergence free vector field $v$ into its
rotational and irrotational parts in $\mathcal{D}_{t}$,

$v=\nabla\psi+v_{\omega},\qquad\Delta\psi=0,\quad v_{\omega}\cdot n=0,$ (2.1)

and we denote the vorticity by $\omega=\operatorname{curl}v$. The moving
boundary condition reads

$\partial_{t}h=\nabla\psi\cdot(-\nabla h,1).$

We let $\varphi(t,x):=\psi|_{\partial\mathcal{D}_{t}}=\psi(t,x,h(t,x))$ be the
trace of the velocity potential; one can reconstruct $\psi$ from $\varphi$
solving a standard elliptic problem. We also define the main boundary,
dispersive, variable

$u=h+i\Lambda^{1/2}\varphi,\qquad\Lambda:=|\nabla|.$ (2.2)

The proof of our main result will be based on several interconnected bootstrap
arguments for the quantities $\nabla\psi,h,v_{\omega},\omega$ and $u$, for the
vector potential $\beta$ associated to $v_{\omega}$ (i.e.
$-\operatorname{curl}\beta=v_{\omega}$), and/or their counterparts in the
flattened domain obtained by mapping $y\rightarrow z:=y-h(t,x)$. A high level
description of the proof is the following:

  * •

High order energy and decay. The basic starting point of our proof is weighted
energy estimates for $v$, $h$ and $\omega$. The weighted $L^{2}$-based Sobolev
norms that we use are based on the vector fields generated by the invariance
of the equation: (3d) translation and scaling333Technically these are only
approximate invariance since the domain is not translation or scaling
invariant in the vertical direction. and 2d rotations. The energy estimate
guarantees that top-order energy norms of $v$, $h$ and $\omega$ remain of size
$\varepsilon_{0}\langle t\rangle^{p_{0}}$, with $p_{0}$ a small constant, as
long as we can prove time-decay at a rate of $\langle t\rangle^{-1}$ for a
lower order weighted norm of $v$ and $h$ in $L^{\infty}$. See Proposition 2.5
for a precise statement of the energy inequality. The main efforts then go
into proving the necessary sharp decay in time. To prove this, we use two
separate arguments, one for $v_{\omega}$, and one for $u$. For these arguments
we also need high order bounds on the velocity potential on the interface,
which do not follow immediately from the $L^{2}$-orthogonality of $\nabla\psi$
and $v_{\omega}$; we give the additional arguments needed in Section 5.

  * •

Estimates on $v_{\omega}$ from the vorticity. Since we work with times
$|t|\leq\varepsilon_{1}^{-1+}$, proving the needed decay for $v_{\omega}$
amounts to bounding it (almost) uniformly-in-time by $\varepsilon_{1}$. Note
that the basic energy estimates only guarantees bounds of $O(\varepsilon_{0})$
for the vorticity.

Naturally, $v_{\omega}$ can be estimated in terms of $\omega$ through a
$\mathrm{div}$-$\mathrm{curl}$ system. In practice, we relate $\omega$ and
$v_{\omega}$ by introducing the vector potential $\beta$ such that
$-\operatorname{curl}v_{\omega}=\beta$. The vector potential satisfies an
elliptic system with mixed Dirichlet and Neumann boundary conditions in the
unbounded fluid domain. When trying to obtain estimates through this elliptic
system, the limited (weighted) regularity and decay available on the geometry
need to be carefully taken into account. It turns out that, all along the
argument we need to to allow small growth for the highest norms of
$v_{\omega},\beta,\omega$, while trying to control uniformly-in-time some
lower order norms. This is essentially due to the critical nature of the
problem, relative to time-decay, and it is also the technical reason why we
need to allow for the presence of a small $\delta$ loss in our maximal time of
existence.

Flattening the domain to a half-space, and using bounds in weighted Lebesgue
spaces for the Poisson kernel we can obtain sufficiently strong bounds for
$v_{\omega}$, provided certain weighted Lebesgue norms of $\omega$ are
controlled. See Section 3.

  * •

To bound the needed weighted Lebesgue norms of $\omega$ we use the vorticity
transport equation. Here one needs to deal with the slowly decaying
contributions from the stretching terms, which are coming from the non-
integrable slow decay of the irrotational components of the solution. To
overcome this, we use a normal form type argument on the vorticity transport
equation in the full three dimensional domain. This procedure renormalizes the
vorticity equation allowing us to propagate the desired control on $\omega$.
See Section 4. These bounds on $\omega$ imply decay for $v_{\omega}$.

  * •

Finally, we need to prove decay for the irrotational components of the
solution $\nabla\psi$ and $h$; this amounts to proving decay for $u$ as in
(2.2). We start by deriving boundary equations for $u$ that extend the well-
known Zakharov-Craig-Schanz-Sulem Hamiltonian formulation [51, 13]; see (B.25)
and the simplified version in (2.38). In the general case with rotation, the
dispersive-type evolution equation for $u$ is ‘singularly’ forced by the
restriction to the boundary of the horizontal component of $v_{\omega}$.

To obtain decay for $u$ we use weighted $L^{2}$-$L^{\infty}$ estimates, and
Poincaré normal forms to remove the purely irrotational quadratic components.
To deal with the forcing and the other rotational components we use the
estimates previously established on $v_{\omega}$. Here we need to require more
(weighted) regularity for the rotational components, compared to the
regularity of the irrotational components in the $L^{\infty}_{x}$-space where
we establish time decay. Moreover, we need to pay particular attention to
small frequencies due to the singular nature of the forcing.

We will describe the above steps and the main bootstrap propositions more
precisely in Subsection 2.5 after introducing all the necessary notation and
parameters.

###  2.2. Vector fields and function spaces

In $\mathcal{D}_{t}$ we use $x=(x_{1},x_{2})$ to denote the horizontal
variables and $-\infty<y<h(t,x)$ for the vertical one. For several arguments
we will find it convenient to flatten $\partial\mathcal{D}_{t}$ with the
mapping $y\rightarrow z:=y-h(t,x)$, which transforms $\mathcal{D}_{t}$ into
the lower-half plane ${\mathbb{R}}^{2}_{x}\times\\{z<0\\}.$

We denote the standard $2$d vector fields

$\displaystyle\nabla_{x}:=(\partial_{x_{1}},\partial_{x_{2}}),\quad
S:=\frac{1}{2}t\partial_{t}+x\cdot\nabla_{x},\quad\Omega:=x\wedge\nabla_{x};$
(2.3)

we will drop the index $x$ for the gradient when there is no risk of
confusion. We denote the ‘$3$d vector fields’ in $\mathcal{D}_{t}$ as

$\displaystyle\underline{\nabla}=\nabla_{x,y}=(\partial_{x_{1}},\partial_{x_{2}},\partial_{y}),\qquad\underline{S}=S+y\partial_{y},\qquad\underline{\Omega}=\Omega.$
(2.4)

In the flattened domain ${\mathbb{R}}^{2}_{x}\times\\{z<0\\}$ we slightly
abuse notation and still denote the ‘$3$d vector fields’ by

$\displaystyle\underline{\nabla}=\nabla_{x,z}=(\partial_{x_{1}},\partial_{x_{2}},\partial_{z}),\qquad\underline{S}=S+z\partial_{z}\qquad\underline{\Omega}=\Omega.$
(2.5)

The distinction between these sets of vector fields will always be clear from
context and should generate no confusion.

Let $\Gamma$, respectively $\underline{\Gamma}$, be the collection of $2d$,
respectively $3d$ vector fields:

$\displaystyle\Gamma=(\partial_{x_{1}},\partial_{x_{2}},S,\Omega),\qquad\underline{\Gamma}=(\partial_{x_{1}},\partial_{x_{2}},\partial_{y},\underline{S},\Omega).$
(2.6)

These are respectively $4$\- and $5$-component vectors, but we will use the
same notation for multiple applications of them when this causes no confusion,
that is, we will write $\Gamma^{j}$, with the understanding that
$j\in{\mathbb{Z}}_{+}^{4}$, or $\underline{\Gamma}^{j}$ with the understanding
that $j\in{\mathbb{Z}}_{+}^{5}$.

Let $W^{s,p}=W^{s,p}(D;{\mathbb{C}}^{m})$, with $H^{s}=W^{s,2}$ be the
standard Sobolev spaces with $D$ a (sufficiently) smooth domain in
${\mathbb{R}}^{3}$, or the plane ${\mathbb{R}}^{2}$. We define the following
basic spaces:

$\displaystyle X^{r,p}_{k}(\Omega)$
$\displaystyle:=\big{\\{}f\,:\,\sum_{|j|\leq
k}{\|\underline{\Gamma}^{j}f\|}_{W^{r,p}(\Omega)}<\infty\big{\\}},\qquad
X^{r}_{k}:=X^{r,2}_{k}$ (2.7) $\displaystyle Z^{r,p}_{k}({\mathbb{R}}^{2})$
$\displaystyle:=\big{\\{}f\,:\,\sum_{|j|\leq
k}{\|\Gamma^{j}f\|}_{W^{r,p}({\mathbb{R}}^{2})}<\infty\big{\\}},\qquad
Z^{r}_{k}:=Z^{r,2}_{k}.$ (2.8)

We denote by ${\|\cdot\|}_{X^{r,p}_{k}(\Omega)}$ and
${\|\cdot\|}_{Z^{r,p}_{k}({\mathbb{R}}^{2})}$ the respective norms. We will
often omit the domain when it is clear from context.

The above spaces play the following roles: $X$ is the space where we measure
the velocity field in the whole fluid domain, while $Z$ is the space where we
measure the boundary quantities $h$ and $\varphi$ as functions of
$x\in{\mathbb{R}}^{2}$ (and time $t$).

Besides these basic spaces, in due course we will also introduce other
weighted spaces based on mixed $L^{q}_{z}L^{p}_{x}$ Lebesgue spaces in the
flat domain; see for example those appearing in Proposition 2.14.

###  2.3. Initial data and main theorem

####  2.3.1. Parameters: smallness and regularity

Let

$\displaystyle\varepsilon_{1}\ll\varepsilon_{0},\qquad
0<3p_{0}<\delta<1/100,\qquad
T_{\varepsilon_{1}}:=\bar{c}\varepsilon_{1}^{-1+\delta},$ (2.9)

for some sufficiently small absolute constant $\bar{c}>0$ (to be determined in
the course of the proof) and consider three (even) integers $N_{0},N_{1},N$
such that

$\displaystyle N_{0}\gg N_{1}\geq\frac{N_{0}}{2}+10,\qquad N:=N_{1}+12.$
(2.10)

These numbers are associated to various regularities and bounds for the main
unknowns in the problem:

  * •

$N_{0}$ corresponds to the maximum number of derivatives and vector fields
that we control on the velocity field and on the height in $L^{2}$.

  * •

$N_{1}$ corresponds to the maximum number of derivatives and vector fields for
which we prove the sharp decay rate of $(1+|t|)^{-1}$ in $L^{\infty}$ for the
irrotational part of the velocity field and the height $h$.

  * •

$N$ corresponds to the maximum number of derivatives and vector fields of the
rotational components of the solution that we control (almost) uniformly by
$\varepsilon_{1}$ on a time-scale of order (almost) $\varepsilon_{1}^{-1}$.

####  2.3.2. Initial assumptions and main theorem

We assume that the initial velocity and height satisfy

$\displaystyle\sum_{r+k\leq
N_{0}}{\|v_{0}\|}_{X^{r}_{k}(\mathcal{D}_{0})}+\sum_{r+k\leq
N_{0}}{\|h_{0}\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\leq\varepsilon_{0}.$ (2.11)

For the vorticity, we assume that it satisfies the $L^{p}$-type bounds of high
order

$\displaystyle\sum_{|r|+|k|\leq
N_{0}-20}{\|\nabla_{x,y}^{r}\underline{\Gamma}^{k}\omega_{0}\|}_{\mathcal{W}(\mathcal{D}_{0})}$
$\displaystyle\leq\varepsilon_{0},\qquad\mathcal{W}:=L^{2}\cap L^{6/5}$ (2.12)

and $L^{p}$-type bounds of smaller size $\varepsilon_{1}$ for lower order
norms:

$\displaystyle\sum_{|r|+|k|\leq
N}{\|\nabla_{x,y}^{r}\underline{\Gamma}^{k}\omega_{0}\|}_{\mathcal{W}(\mathcal{D}_{0})}$
$\displaystyle\leq\varepsilon_{1}.$ (2.13)

######  Remark 2.1.

If we define $W_{0}(x,z)=\omega_{0}(x,z+h(0,x))$, the transformed initial
vorticity in the flat domain, then (2.12)-(2.13) imply the analogous bounds

$\displaystyle\sum_{|r|+|k|\leq
N_{0}-20}{\|\nabla^{r}_{x,z}\underline{\Gamma}^{k}W_{0}\|}_{\mathcal{W}({\mathbb{R}}^{2}_{x}\times\\{z<0\\})}\leq
C\varepsilon_{0},\qquad\sum_{|r|+|k|\leq
N}{\|\nabla^{r}_{x,z}\underline{\Gamma}^{k}W_{0}\|}_{\mathcal{W}({\mathbb{R}}^{2}_{x}\times\\{z<0\\})}\leq
C\varepsilon_{1},$ (2.14)

for some absolute constant $C>0$.

We can now state a more precise version of our main result:

######  Theorem 2.2.

Assume (2.11)-(2.13) and fix $\delta\in(0,1/100)$ and $3p_{0}<\delta$. Assume
that $\omega_{0}$ vanishes on the boundary of444Since the boundary is a
material curve, this condition is preserved in time. $\mathcal{D}_{0}$. Then,
there exists $\overline{\varepsilon_{0}}$ and $\bar{c}>0$ such that, for any
$\varepsilon_{1}\leq\varepsilon_{0}\leq\overline{\varepsilon_{0}}$, there
exists a unique solution of (1.1) with initial conditions $v(t=0)=v_{0}$ and
$h(t=0)=h_{0}$ satisfying (2.11)-(2.13), that remains regular for $|t|\leq
T_{\varepsilon_{1}}=\bar{c}\varepsilon_{1}^{-1+\delta}$ and satisfies
following: the $L^{2}$ bounds

$\displaystyle\sum_{r+k\leq
N_{0}}{\|v(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}+\sum_{r+k\leq
N_{0}-1}{\|\omega(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}\lesssim\varepsilon_{0}\langle
t\rangle^{p_{0}},$ (2.15)

and

$\displaystyle\sum_{r+k\leq
N_{0}}{\|h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\lesssim\varepsilon_{0}\langle
t\rangle^{p_{0}},$ (2.16)

and the decay bounds

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\|\nabla\nabla\psi(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}\lesssim\varepsilon_{0}\langle
t\rangle^{-1},$ (2.17) $\displaystyle\sum_{r+k\leq N_{1}-5}{\|\nabla
v_{\omega}(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta},$ (2.18)

and

$\displaystyle\sum_{r+k\leq
N_{1}}{\|h(t)\big{\|}}_{Z^{r,\infty}_{k}({\mathbb{R}}^{2})}\lesssim\varepsilon_{0}\langle
t\rangle^{-1}.$ (2.19)

###  2.4. Main a priori assumptions

In this subsection we list all the main a priori assumptions that we are going
to make. For convenience, some of these assumptions are stated in the domain
$\mathcal{D}_{t}$, while others are stated in the flattened domain
${\mathbb{R}}^{2}_{x}\times\\{z<0\\}$ and some are in terms of the boundary
variables. Then, in Subsection 2.5 we are going to explain how all these a
priori assumptions are bootstrapped on an interval $[0,T]$ with $T\leq
T_{\varepsilon_{1}}$, and also provide some of the main elliptic-type bounds
that are needed for the arguments.

\- A priori assumption in $\mathcal{D}_{t}$. We make the following a priori
assumptions on the high-order energy ($L^{2}$-based) norms of the velocity,
vorticity, and height:

$\displaystyle\sup_{[0,T]}\,\langle t\rangle^{-p_{0}}\Big{(}\sum_{r+k\leq
N_{0}}{\|v(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}+\sum_{r+k\leq
N_{0}-1}{\|\omega(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}+\sum_{r+k\leq
N_{0}}{\|h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\Big{)}\leq
2c_{E}\varepsilon_{0}$ (2.20)

where $c_{E}$ is an absolute constant to be chosen large enough.

######  Remark 2.3.

Note how we let the highest order energy norms grow like $\langle
t\rangle^{p_{0}}$, where $p_{0}$ is the parameter in (2.9); this parameter can
be chosen of the form $C\varepsilon_{0}$ for a suitable absolute constant
$C>0$. We will however prove uniform bounds (almost) of $O(\varepsilon_{1})$
on a lower number $N$ of derivatives and vector fields of the vorticity
components, essentially propagating the bound (2.13).

We also assume a priori decay bounds on the velocity in the interior:

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\|v(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}\leq
2c_{v}\big{(}\varepsilon_{0}\langle t\rangle^{-1}+\varepsilon_{1}\langle
t\rangle^{\delta}\big{)},\qquad t\in[0,T].$ (2.21)

Note that we make decay assumptions (and prove decay bounds) on $v$ and not
just on $\nabla v$, which would be sufficient for the sole purpose of closing
standard energy estimates in Sobolev spaces without vector fields (see
Proposition 2.5); these stronger bounds are also needed in other parts of the
proof.

\- A priori assumptions on the boundary variables. We assume sharp pointwise
decay bounds for the ‘boundary variables’ $(h,\varphi)$:

$\displaystyle\sup_{[0,T]}\,\langle t\rangle\,\sum_{r+k\leq
N_{1}}{\big{\|}u(t)\big{\|}}_{Z^{r,\infty}_{k}({\mathbb{R}}^{2})}\leq
2c_{B}\varepsilon_{0},\qquad u:=h+i|\nabla|^{1/2}\varphi,$ (2.22)

where $c_{B}$ is an absolute constant to be chosen large enough.

\- A priori assumptions on the vorticity in the flat domain. Some of the main
parts of our argument are performed in the flattened domain
${\mathbb{R}}^{2}_{x}\times\\{z<0\\}$. We denote the vorticity in the
flattened coordinates as

$\displaystyle W(t,x,z):=\omega(t,x,z+h(t,x)),\qquad
W_{0}(x,z):=\omega_{0}(x,z+h_{0}(x)),$ (2.23)

and we will bootstrap three main a priori bounds on it. For this purpose we
introduce the weighted Lebesgue spaces $\mathcal{X}^{n}$ defined by the norm
(see (2.12))

$\displaystyle\begin{split}{\|f\|}_{\mathcal{X}^{n}}:=\sum_{|r|+|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\,f\big{\|}}_{\mathcal{W}({\mathbb{R}}^{2}_{x}\times\\{z<0\\})},\qquad\mathcal{W}=L^{2}\cap
L^{6/5}.\end{split}$ (2.24)

The first two main a priori bounds on $W$ are

$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\leq
2c_{L}\varepsilon_{1},$ $\displaystyle\qquad t\in[0,T],\quad j=0,1,$ (2.25)
$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\leq
2c_{H}\varepsilon_{0}^{j}\varepsilon_{1}\langle t\rangle^{\delta},$
$\displaystyle\qquad t\in[0,T],\quad j=0,1,$ (2.26)

where $c_{L}<c_{H}$ are some absolute constants to be chosen large enough (we
use the same one for $j=0$ or $1$). In (2.26), the growth rate $\delta$ is the
parameter in (2.9). We also assume a high-order (weak) bound

$\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{0}-20}}\leq
2c_{W}\varepsilon_{0}\langle t\rangle^{2p_{0}},\quad t\in[0,T].$ (2.27)

######  Remark 2.4.

Note how we are propagating bounds for $W$ (hence for the vorticity $\omega$)
of the order $\varepsilon_{1}$ with a small growth factor at a level of vector
fields larger than $N_{1}$; we choose $N=N_{1}+12$ for concreteness. Along
with this, we also bootstrap a lower norm with the sharp bound of
$\varepsilon_{1}$. The need to proceed with this two tier bootstrap is again
attributable to the growth of the highest order weighted energies.

We now explain our overall strategy for recovering all these assumptions and
obtaining Theorem 1.1.

###  2.5. Strategy of the proof and main propositions

The proof of our Theorem 1.1 proceeds in a several steps based on some key
propositions. Note that the order in which the various intermediate results
are presented here is not the same as that of the sections in which the proofs
are given, but follows what we believe to be a more reader-friendly
description.

### Step 1: Energy estimates and other high-order norms

We begin with an energy estimate that controls the increment of the top-order
weighted norms.

######  Proposition 2.5 (Top order energy inequality).

Assume that (2.20) holds and recall the definition of the spaces (2.7) and
(2.8). Then there exist energy functionals $\mathcal{E}_{r,k}$ such that:

  * •

We have

$\displaystyle\begin{split}&\mathcal{E}_{r,k}(t)\approx{\|v(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}^{2}+{\|\omega(t)\|}_{X^{r-1}_{k}(\mathcal{D}_{t})}^{2}+{\|h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}^{2},\end{split}$
(2.28)

  * •

If we define

$\displaystyle\mathcal{E}_{0}(t):=\sum_{r+k\leq
N_{0}}\mathcal{E}_{r,k}(t)\approx\sum_{r+k\leq
N_{0}}{\|v(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}^{2}+{\|\omega(t)\|}_{X^{r-1}_{k}(\mathcal{D}_{t})}^{2}+{\|h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}^{2},$
(2.29)

then, for all $t\in[0,T]$,

$\displaystyle\frac{d}{dt}\mathcal{E}_{0}(t)\lesssim
Z_{0}(t)\cdot\mathcal{E}_{0}(t)$ (2.30)

where

$\displaystyle Z_{0}(t):=\sum_{r+k\leq
N_{0}/2+4}{\|v(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}+{\|h(t)\|}_{Z^{r+2,\infty}_{k}({\mathbb{R}}^{2})}.$
(2.31)

Note that the initial assumptions (2.11)-(2.13) imply

$\displaystyle\mathcal{E}_{0}(0)\lesssim\varepsilon_{0}.$ (2.32)

$L^{2}$-based energy estimates are a fairly standard result for this problem,
see for example [10, 19, 36, 37, 47] for energy estimates in standard Sobolev
spaces without vector fields. Estimates with vector fields are also
essentially standard although, to the best of our knowledge, the estimates in
Proposition 2.5 do not appear in the literature exactly as stated. In the
irrotational setting [48, 49] prove estimates with vector fields for gravity
waves, [20] proves estimates for the problem with surface tension and no
gravity, and [17] proves estimates for the gravity-capillary problem using
only the rotation vectorfield (since the problem is not scaling invariant);
energy estimates including the scaling vector field are also proved in some
lower dimensional cases [27, 29]. In section D we give a brief sketch of the
main ingredients needed in order to carry out the proof of the energy estimate
with vector fields in our setting.

As a consequence of the main energy inequality we obtain the following
standard result:

######  Proposition 2.6 (Decay implies Energy bootstrap).

Assume (2.11), and that, for $T\leq T_{\varepsilon_{1}}$, the a priori decay
assumptions (2.22)-(2.21) hold. Then, there exists $c_{E}$ large enough such
that

$\displaystyle\sup_{[0,T]}\,\langle t\rangle^{-p_{0}}\Big{(}\sum_{r+k\leq
N_{0}}{\|v(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}+\sum_{r+k\leq
N_{0}-1}{\|\omega(t)\|}_{X^{r}_{k}(\mathcal{D}_{t})}+\sum_{r+k\leq
N_{0}}{\|h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\Big{)}\leq
c_{E}\varepsilon_{0}.$ (2.33)

###### Proof of Proposition 2.6.

The a priori assumptions (2.21)-(2.22) directly imply that

$\displaystyle Z_{0}(t)\lesssim\varepsilon_{0}\langle
t\rangle^{-1}+\varepsilon_{1}\langle t\rangle^{\delta}.$

This and (2.30), together with (2.29) and (2.32), give

$\displaystyle\begin{split}\mathcal{E}_{0}(t)&\leq
C\mathcal{E}_{0}(0)\exp\Big{(}C\int_{0}^{t}\big{(}\varepsilon_{0}\langle
s\rangle^{-1}+\varepsilon_{1}\langle s\rangle^{\delta}\big{)}\,ds\Big{)}\leq
C\varepsilon_{0}^{2}\langle t\rangle^{C\varepsilon_{0}}\end{split}$

having used the definition of $T_{\varepsilon_{1}}$ from (2.9) to bound
uniformly the time integral of $\varepsilon_{1}\langle s\rangle^{\delta}$.
This implies (2.33) provided the absolute constant $c_{E}$ is chosen large
enough. ∎

Proposition 2.6 closes the bootstrap for the norm in (2.20). The main efforts
in our proof are then dedicated to bootstraping the a priori decay bounds
(2.22) and (2.21). Before moving on to explain how to obtain these, we give
the bootstrap for the control of the high-order norm of $W$, see (2.27), and
how this is used to bound $|\nabla|^{1/2}\varphi$ in the next two
propositions.

######  Proposition 2.7.

Let $W$ be defined as in (2.23) and let $\mathcal{X}^{n}$ be the space defined
in (2.24). Under the assumptions (2.25) and (2.27) on $W$, the decay
assumptions (2.21) and (2.22) on $v$ and $h$, and the a priori energy bound
(2.20), we have, for all $t\in[0,T]$,

$\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{0}-20}}\leq
c_{W}\varepsilon_{0}\langle t\rangle^{2p_{0}}.$ (2.34)

The proof of Proposition 2.7 is given in Section 5 (see Proposition 5.2).
Using Proposition 2.7 we can obtain bounds on the vector potential in the flat
domain

$\displaystyle V_{\omega}(t,x,z)=v_{\omega}(t,x,z+h(t,x));$ (2.35)

this can be done in appropriate spaces via elliptic estimates for $\alpha$
such that $\operatorname{curl}\alpha\approx V_{\omega}$; see (3.3) and (3.7)
for the exact definition. Then, using $\nabla\psi=v-v_{\omega}$ and basic
trace estimates we can obtain the following:

######  Proposition 2.8 (Bounds on the velocity potential).

Under the a priori assumptions (2.27) and (2.25) on $W$, and the a priori
assumptions (2.20) and (2.22) on $h$ and $v$, it holds

$\displaystyle\sup_{[0,T]}\,\langle t\rangle^{-3p_{0}}\sum_{r+k\leq
N_{0}-20}{\big{\|}\underline{\Gamma}^{k}\nabla_{x,z}^{r}\nabla_{x,z}\Psi(t)\big{\|}}_{L^{2}({\mathbb{R}}^{2}_{x}\times\\{z<0\\})}\leq
c_{P}^{\prime}\varepsilon_{0},$ (2.36)

for some suitably large absolute constant $c_{P}^{\prime}>c_{E}+c_{W}$. In
particular,

$\displaystyle\sup_{[0,T]}\,\langle t\rangle^{-3p_{0}}\sum_{r+k\leq
N_{0}-20}{\big{\|}|\nabla|^{1/2}\varphi(t)\big{\|}}_{Z^{r}_{k}({\mathbb{R}}^{2})}\leq
c_{P}\varepsilon_{0}.$ (2.37)

for some suitably large absolute constant $c_{P}>c_{E}+c_{W}$.

######  Remark 2.9.

The choice of the growth rate $\langle t\rangle^{3p_{0}}$ in (2.36) is
dictated by the nature of the argument that we use; this necessitates changing
variables from the moving domain to the flat one and, therefore, taking into
account the growth of the high-order weighted norm of $h$. This growth could
be avoided by bootstrapping additional low norms of the irrotational
components.

Also note that the bound in Proposition 2.8 for the simple $L^{2}$ norm of
$\nabla\Psi$ (or $|\nabla|^{1/2}\varphi$) is a direct consequence of the Hodge
decomposition. However, the bound with vector fields requires some non-trivial
arguments, including Proposition 2.7 and elliptic type estimate similar to
those in Section 3 (see also Proposition 2.14 below).

We give details for the proof of Proposition 2.8 in Section 5.

### Step 2: Decay estimates on the boundary and in the interior

Our next main step is the proof of decay for the boundary dispersive variable
$u=h+i|\nabla|^{1/2}\varphi$. First, we derive an equation for $u$ by adapting
the classical Zakharov-Craig-Schanz-Sulem formulation; see also [21] and [8].
More precisely, in Lemma B.3 we obtain that

$\displaystyle\partial_{t}u+i|\nabla|^{1/2}u$
$\displaystyle=B_{0}-i|\nabla|^{-3/2}\nabla\cdot\partial_{t}P_{\omega}+\cdots$
(2.38)

where $B_{0}$ denotes quadratic terms in $h$ and $\varphi$ and $P_{\omega}$ is
the restriction to the interface of the horizontal components of $v_{\omega}$:

$\displaystyle
P_{\omega}(t,x):=\big{(}(v_{\omega})_{1},(v_{\omega})_{2}\big{)}(t,x,h(t,x))=\big{(}(V_{\omega})_{1},(V_{\omega})_{2}\big{)}(t,x,0).$
(2.39)

The “$\cdots$” in (2.38) denote other quadratic terms that involve at least
one $P_{\omega}$, plus other cubic terms, and we disregard them here for the
sake of the discussion. Note that (2.38) is forced in a singular way by
$\partial_{t}P_{\omega}$; this creates some technical difficulties. See
Section B for full details of the derivation of (2.38).

One can see that in order to prove decay for $u$ through the Duhamel’s formula
associated to (B.25) we need, among other things, strong enough control on
$P_{\omega}$, at a level of (weighted) regularity which is higher than that of
the space in which $u$ decays (see (2.22)). We will obtain these estimates on
$P_{\omega}$ in the next step.

Based on (2.38) and suitable assumptions on $P_{\omega}$, we can recover decay
for $u$:

######  Proposition 2.10 (Sharp decay of the irrotational component).

Assume a priori that (2.22) holds, together with

$\displaystyle\sup_{[0,T]}\,\langle t\rangle^{-3p_{0}}\sum_{r+k\leq
N_{0}-20}{\|u(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\lesssim\varepsilon_{0}.$
(2.40)

Moreover, assume that for some $N\geq N_{1}+11$ we have, for all $t\leq
T_{\varepsilon_{1}}$,

$\displaystyle\sum_{r+k\leq
N}{\big{\|}\partial_{t}^{j}P_{\omega}(t)\big{\|}}_{Z^{r}_{k}({\mathbb{R}}^{2})}\leq
c^{\prime}_{H}\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta},\quad j=0,1.$ (2.41)

Then, for all $t\leq T$,

$\displaystyle\sum_{r+k\leq
N_{1}}{\|u(t)\|}_{Z^{r,\infty}_{k}({\mathbb{R}}^{2})}\leq
c_{B}\varepsilon_{0}\langle t\rangle^{-1}.$ (2.42)

for some large enough absolute constant $c_{B}>c^{\prime\prime}_{H}$. Here
$c^{\prime\prime}_{H}>c_{H}^{\prime}$ is the constant in (2.48).

######  Remark 2.11.

In our proof of Proposition 2.10 in Section 6, we will actually show a
slightly stronger bound than (2.42), with an $\ell^{1}$ sum over frequencies:

$\displaystyle\sum_{\ell\in{\mathbb{Z}}}\sum_{r+k\leq
N_{1}}{\|P_{\ell}u(t)\|}_{Z^{r,\infty}_{k}({\mathbb{R}}^{2})}\leq
c_{B}\varepsilon_{0}\langle t\rangle^{-1};$ (2.43)

see Remark 6.1. This technical improvement helps to show decay for
$\nabla\psi$; see Lemma 2.12.

Note that the assumption (2.40) is directly guaranteed by (2.8). Proposition
2.10 then recovers the a priori decay assumption (2.22) closing the bootstrap
provided $c_{B}$ is large enough. The proof of Proposition 2.10 is given in
Section 6 and uses:

  * -

The boundary evolution equations in the presence of vorticity (see Lemma B.1);

  * -

A Klainerman-Sobolev type estimate for the flow of $e^{it|\nabla|^{1/2}}$ (see
Lemma A.1);

  * -

Normal form arguments to deal with the quadratic irrotational terms that have
slow decay;

  * -

The estimate (2.41) to bound the nonlinear and forcing terms involving the
rotational component $P_{\omega}$; particular care needs to be put into
handling small frequencies here, due to the singular nature of the operator
acting on $P_{\omega}$ in (2.38).

The assumption (2.41) will follow from a fixed point argument which
essentially constructs and bounds $V_{\omega}$. See in particular the
conclusion of Lemmas 3.6 and 3.7.

Before moving on to the next main step in the proof, we add here the estimates
that recover the bootstrap assumption (2.21) on the decay of the velocity in
the interior. These are obtained in an elliptic way at fixed time $t$ from
other bounds that are bootstrapped. For convenience we split these estimate
into two lemmas:

######  Lemma 2.12 (Decay of the irrotational component).

Assume that (2.43) holds, together with the a priori bounds (2.20) and (2.22)
on $h$. Then, for all $t\leq T$ we have

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\|\nabla\psi(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}\leq
c_{i}\varepsilon_{0}\langle t\rangle^{-1}.$ (2.44)

for some $c_{i}>c_{B}$.

######  Lemma 2.13 (Decay of the rotational component).

Assume that the bounds (2.47) on $V_{\omega}$ hold for $t\in[0,T]$, with
$T\leq T_{\varepsilon_{1}}$. Then, for all $t\leq T$, we have

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\|v_{\omega}(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}\leq
c_{r}\varepsilon_{1}\langle t\rangle^{\delta},$ (2.45)

for some $c_{r}>c_{H}^{\prime}$.

Since $v=\nabla\psi+v_{\omega}$, Lemmas 2.12 and 2.13 recover the bootstrap
assumption (2.21) provided $c_{v}>c_{i}+c_{r}$ is chosen large enough. The
proofs of these lemmas are given in Subsection 5.4.

### Step 3: Estimates for the rotational part of the velocity

In Section 3 we prove estimates for $v_{\omega}$ which in particular imply the
assumption (2.41) used in Proposition 2.10. We first consider the vector
potential $\beta$ such that $-\operatorname{curl}\beta=v_{\omega}$. $\beta$
satisfies an elliptic system, $\Delta\beta=\omega$, with mixed Dirichlet and
Neumann boundary conditions; see Lemma 3.1. We then work in the flat half-
space by considering $\alpha(t,x,z):=\beta(t,x,z+h(t,x))$, and $V_{\omega}$ as
in (2.35). Then, $\alpha$ satisfies an elliptic system with $h$-dependent
coefficients, which is forced by the vorticity $W$. Assuming suitable bounds
on the forcing $W$, a fixed point argument, which relies on (weighted)
estimates for the Poisson kernel, gives estimates for $\alpha$ in weighted
$L^{p}_{z}L^{q}_{x}$ spaces. From the bounds obtained on $\alpha$ we can then
directly deduce estimates for $V_{\omega}$ in similar spaces. The main steps
are contained in Proposition 3.12, which constructs and bounds $\alpha$, and
its direct consequence Lemma 3.7, which gives bounds for $V_{\omega}$. We
summarize these results in the following statement:

######  Proposition 2.14 (Bounds for $V_{\omega}$).

Assume that (2.20)-(2.22) hold, and that $W$ is given so that (2.25) and
(2.26) hold, that is, for all $t\in[0,T]$, and for $j=0,1$

$\displaystyle\begin{split}&{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\leq
c_{L}\varepsilon_{1},\qquad{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\leq
c_{H}\varepsilon_{0}^{j}\varepsilon_{1}\langle t\rangle^{\delta},\end{split}$
(2.46)

where $\mathcal{X}^{n}=\mathcal{X}^{n}({\mathbb{R}}^{3}_{-})$ is the space
defined in (2.24).

Then, for all $t\in[0,T]$, and for $j=0,1$, we have the bounds

$\displaystyle\begin{split}{\|\partial_{t}^{j}V_{\omega}(t)\|}_{Y^{N_{1}-10-j}}&\leq
c_{H}^{\prime}\varepsilon_{1},\\\
{\|\partial_{t}^{j}V_{\omega}(t)\|}_{Y^{N_{1}+12-j}}&\leq
c_{H}^{\prime}\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta},\end{split}$ (2.47)

for some large enough absolute constant $c_{H}^{\prime}>c_{H}$, where
$Y^{n}=Y^{n}({\mathbb{R}}^{3}_{-})$ is the space defined by the norm

$\displaystyle{\|g\|}_{Y^{n}}=\sum_{|r|+|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}g\big{\|}}_{Y^{0}},$
$\displaystyle{\|g\|}_{Y^{0}}:={\big{\|}|\nabla|^{1/2}g\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}+{\|g\|}_{L^{\infty}_{z}L^{2}_{x}}+{\big{\|}\nabla_{x,z}g\big{\|}}_{L^{2}_{z}L^{2}_{x}}.$

In particular, we also have, for all $t\in[0,T_{\varepsilon_{1}}]$, and some
$c_{H}^{\prime\prime}>c_{H}^{\prime}$,

$\displaystyle\sum_{r+k\leq
N_{1}+12-j}{\big{\|}\partial_{t}^{j}V_{\omega}(t,\cdot,0)\big{\|}}_{Z^{r}_{k}({\mathbb{R}}^{2})}\leq
c_{H}^{\prime\prime}\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta},\qquad j=0,1.$ (2.48)

Note that the bound (2.48) for the restriction of $V_{\omega}$ to $z=0$
follows from (2.47) and the definitions of the spaces $Y^{n}$ and $Z^{r}_{k}$,
and implies the validity of the assumption (2.41).

To conclude the proof of our result, we then only need to prove that the
bounds in (2.46) hold true, that is, we need to close the bootstrap for the a
priori assumptions (2.25) and (2.26). This is done in the last main step.

### Step 4: Estimates for the vorticity

Our last main step is to bootstrap the estimates (2.25)-(2.26). The main point
is to obtain estimates of size essentially $\varepsilon_{1}$, comparable to
the size of the initial vorticity. While this is a natural bound to expect it
is not straightforward to obtain, as we will explain below. Notice that the
basic energy estimate (2.33) only gives a bound on $\omega$ of the order
$O(\varepsilon_{0})$. This is the main result:

######  Proposition 2.15.

Assume that the initial conditions (2.13) holds, and the a priori assumptions
(2.20)-(2.22) hold. Let $W$ be as defined as above and $\mathcal{X}^{n}$ as in
(2.24). If (2.25)-(2.26) hold, that is, for all $t\in[0,T]$, and for $j=0,1$

$\displaystyle\begin{split}&{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\leq
2c_{L}\varepsilon_{1},\\\
&{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\leq
2c_{H}\varepsilon_{0}^{j}\varepsilon_{1}\langle t\rangle^{\delta},\end{split}$
(2.49)

then, for all $t\in[0,T]$,

$\displaystyle\begin{split}&{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\leq
c_{L}\varepsilon_{1},\\\
&{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\leq
c_{H}\varepsilon_{0}^{j}\varepsilon_{1}\langle t\rangle^{\delta}.\end{split}$
(2.50)

The above statement is essentially Proposition 4.1, and its proof occupies all
of Section 4. The estimates use crucially the renormalization in Proposition
4.10.

Let us give a few details of the proof of Proposition 2.15. For convenience
and ease of cross-reference, we work with the quantities defined in the flat
coordinates, and let $W$ and $V_{\omega}$ be as above, with $V$ and $\Psi$
denoting the velocity and velocity potential. $W$ and $V_{\omega}$ are the
‘rotational’ components, while $V-V_{\omega}$ and $\nabla\Psi$ (these two only
differ by a quadratic term) are the ‘rotational’ components. The rotational
components together with the height $h$ are also denoted as ‘dispersive’
variables.

The vorticity equation reads (see (4.10)-(4.11))

$\displaystyle\begin{split}&{\bf D}_{t}W=W\cdot\nabla V+\cdots\qquad{\bf
D}_{t}:=\partial_{t}+U\cdot\nabla,\qquad
U:=\nabla\Psi+V_{\omega}+\cdots\end{split}$ (2.51)

where we are denoting with “$\cdots$” lower order perturbative terms. Let us
simplify (2.51) further by replacing the material derivative just by
$\partial_{t}$, but notice that this cannot be done by trivially integrating
the Lagrangian flow since $U$ is not in $L^{1}_{t}([0,T])$ due to the non-
integrable time-decay of $\nabla\Psi$. We then arrive at the model equation

$\displaystyle\partial_{t}W=W\cdot\nabla V_{\omega}+W\cdot\nabla\nabla\Psi.$
(2.52)

One can see that the part of the quadratic stretching term that only involves
the rotational variables $W$ and $V_{\omega}$ is naturally of size
$\varepsilon_{1}^{2}$ (up to small time growing factors), and therefore is
consistent with a bound of order $\varepsilon_{1}$ for $W$ on a time scale of
order $O(\varepsilon_{1}^{-1})$. However, since
$|\nabla\Psi|\approx\varepsilon_{0}\langle t\rangle^{-1}$ (at best) the last
term in (2.52) acts as a long-range perturbation and does not allow us to
propagate bounds on $W$. Although the loss appears to be only logarithmic
here, this type of difficulty is a well-known issue when dealing with long-
term regularity for nonlinear PDE.

Before explaining our ideas to resolve the above issue, let us also mention
that the situation becomes even more delicate when looking at weighted norms
of $W$ as in (2.46). Applying vector fields to (2.51), and just concentrating
on the rotational-irrotational coupling, we obtain, schematically,

$\displaystyle\partial_{t}(\underline{\Gamma}^{k}W)=(\underline{\Gamma}^{k}W)\cdot\nabla\nabla\Psi+W\cdot(\underline{\Gamma}^{k}\nabla\nabla\Psi)+\cdots.$
(2.53)

Recall that the evolution of $\Psi$ is forced by the (time derivative) of
restriction of $V_{\omega}$, see (2.38)-(2.39), and that (in terms of norms)
we can think that $\nabla V_{\omega}\approx W$. Therefore, one should expect
that $k$ vector fields applied to $\nabla\Psi$ (or, equivalently, of
$|\nabla|^{1/2}\varphi=\operatorname{Im}u$, see (2.22)) will decay at the
sharp rate only provided that strictly more than $k$ vector fields of
$V_{\omega}$ are suitably under control. But this is then inconsistent with
the equation (2.53) where (small) polynomial losses will occur when relying on
higher order weighted $L^{2}$ norms of $\nabla\Psi$.

Our idea to resolve the issues discussed above is to introduce a “modified
vorticity” which satisfies a better equation than (2.52) where the
irrotational components only appear with quadratic or higher homogeneity. This
renormalization procedure can be thought of as a normal form for the vorticity
equation in the full three dimensional domain. The main observation is that,
up to perturbative quadratic terms,

$\displaystyle\Psi\approx e^{z|\nabla_{x}|}\partial_{t}|\nabla_{x}|^{-1}h$
(2.54)

and, therefore, $\nabla\Psi$ is approximately the time derivative of a time-
decaying component. Then, the modified vorticity defined by
$W-W\cdot\nabla\nabla e^{z|\nabla_{x}|}|\nabla_{x}|^{-1}h$ satisfies an
equation with truly perturbative nonlinear terms, and can be used to obtain
(2.50).

###  2.6. Notation

Here we give some notation used in the paper. More notation will be introduced
in the course of the proofs.

\- We use standard notations for $L^{p}$ spaces and Sobolev spaces $W^{s,p}$
and $\dot{W}^{s,p}$, with $H^{s}=W^{s,2}$.

\- With $p-$ we denote a number smaller than, but arbitrarily close to, $p$.
$\infty-$ denotes an arbitrarily large number. Similarly, $p+$ denotes a
number larger than, but arbitrarily close to, $p$.

\- We use $C$ to denoted absolute constants; these may vary from line to line
of a chain of inequalities, and may depend on the numbers in (2.10), but are
independent of the relevant quantities involved, and of $\varepsilon_{0}$ and
$\varepsilon_{1}$.

\- $A\lesssim B$ means that there exists an absolute constant $C>0$ such that
$A\leq CB$. Similarly $A\gtrsim B$ means $B\lesssim A$. $A\approx B$ means
$A\lesssim B$ and $B\lesssim A$.

\- We denote the Fourier transform over ${\mathbb{R}}^{2}$ by

$\displaystyle\widehat{f}(\xi)=\mathcal{F}(f)(\xi):=\frac{1}{2\pi}\int_{{\mathbb{R}}^{2}}e^{-ix\cdot\xi}f(x)\,dx,\qquad\mathcal{F}^{-1}(f)(x)=\frac{1}{2\pi}\int_{{\mathbb{R}}^{2}}e^{ix\cdot\xi}f(\xi)\,d\xi.$
(2.55)

\- To define frequency decomposition we fix a smooth even cutoff function
$\varphi:{\mathbb{R}}\to[0,1]$ supported in $[-8/5,8/5]$ and equal to $1$ on
$[-5/4,5/4]$. By slightly abusing notation we identify $\varphi(x)$ with its
radial extension $\varphi(|x|)$, $x\in{\mathbb{R}}^{d}$. For
$k\in{\mathbb{Z}}$ we define
$\varphi_{k}(x):=\varphi(2^{-k}x)-\varphi(2^{-k+1}x)$, so that the family
$(\varphi_{k})_{k\in{\mathbb{Z}}}$ forms a partition of unity,

$\sum_{k\in{\mathbb{Z}}}\varphi_{k}(\xi)=1,\quad\xi\in{\mathbb{R}}^{d}\smallsetminus\\{0\\}.$

We let

$\displaystyle\varphi_{I}(x):=\sum_{k\in
I\cap{\mathbb{Z}}}\varphi_{k},\quad\text{for any}\quad
I\subset{\mathbb{R}},\quad\varphi_{\leq
a}(x):=\varphi_{(-\infty,a]}(x),\quad\varphi_{>a}(x)=\varphi_{(a,\infty)}(x),$
(2.56)

with similar definitions for $\varphi_{<a},\varphi_{\geq a}$. We will also
denote $\varphi_{\sim k}$ a generic smooth cutoff function that is supported
around $|\xi|\approx 2^{k}$, e.g. $\varphi_{[k-2,k+2]}$ or
$\varphi^{\prime}_{k}$. We denote by $P_{k}$, $k\in{\mathbb{Z}}$, the
Littlewood-Paley projections defined by

$\widehat{P_{k}f}(\xi)=\varphi_{k}(\xi)\widehat{f}(\xi),\quad\widehat{P_{\leq
k}f}(\xi)=\varphi_{\leq k}(\xi)\widehat{f}(\xi),\quad\widehat{P_{\sim
k}f}(\xi)=\varphi_{\sim k}(\xi)\widehat{f}(\xi),\quad\textrm{ etc.}$ (2.57)

Note that these projections essentially commute with the vector fields:

$\displaystyle[\Omega,P_{k}]=0,\qquad[S,P_{k}]=P_{\sim k}.$ (2.58)

\- We denote by $\mathbf{1}_{\pm}$ the characteristic function of $\\{\pm
x>0\\}$.

##  3\. Estimates for the vector potential

In this section we establish bounds on the rotational part of the velocity in
the flat domain that, recall, is denoted by

$\displaystyle V_{\omega}(t,x,z)=v_{\omega}(t,x,z+h(x)),$ (3.1)

under some assumption on the vorticity in the flat domain, that is,
$W(t,x,z)=\omega(t,x,z+h(x))$. In particular, we will prove Proposition 2.14
as a combination of Proposition 3.12 and Lemma 3.7.

###  3.1. Preliminaries and flattening of the domain

We start by relating $v_{\omega}$ to $\beta$ such that
$v_{\omega}=\operatorname{curl}\beta$, and reduce matters to estimates for a
suitable elliptic system. This Hodge-type decomposition is fairly standard but
we provide some details for completeness and since we are going to need
explicit formulas for our estimates; see Appendix C.

######  Lemma 3.1 (Elliptic problem).

Let $\omega=\operatorname{curl}v$ and $v_{\omega}$ be defined as above, then
we can write

$\displaystyle v_{\omega}=-\operatorname{curl}\beta$ (3.2)

where $\beta$ is the unique solution, up to constant vector fields of the
system

$\displaystyle\Delta\beta$ $\displaystyle=\omega,\qquad\text{ in
}\mathcal{D}_{t},$ (3.3a) $\displaystyle\Pi\beta$
$\displaystyle=0,\qquad\text{ on }\partial\mathcal{D}_{t},$ (3.3b)
$\displaystyle n^{i}\partial_{i}(n^{j}\beta_{j})+H\beta_{i}n^{i}$
$\displaystyle=0,\qquad\text{ on }\partial\mathcal{D}_{t}.$ (3.3c)

Here $H:=\partial_{j}n^{j}$ and $\Pi$ denotes the projection to the tangential
components $\Pi_{i}^{j}:=\delta^{i}_{j}-n^{i}n_{j}$, with $n$ the outward unit
normal.

###### Proof.

Recall that we define the (irrotational) velocity potential $\psi$ by solving
the Neumann problem

$\displaystyle\begin{array}[]{rcll}\Delta\psi&\\!\\!=\\!\\!&0,&\qquad\text{ in
}\mathcal{D}_{t},\\\ \partial_{y}\psi-\nabla\psi\cdot\nabla
h&\\!\\!=\\!\\!&\partial_{t}h,&\qquad\text{ on
}\partial\mathcal{D}_{t}.\end{array}$

Since $v_{\omega}=v-\nabla\psi$ we have the system

$\displaystyle\begin{split}\operatorname{div}v_{\omega}&=0,\qquad\text{ in
}\mathcal{D}_{t},\\\ \operatorname{curl}v_{\omega}&=\omega,\qquad\text{ in
}\mathcal{D}_{t},\\\ v_{\omega}\cdot n&=0,\qquad\text{ on
}\partial\mathcal{D}_{t},\end{split}$ (3.4)

where $n=(1+|\nabla h|^{2})^{-1/2}(-\nabla h,1)$ denotes the outward unit
normal. In what follows we denote with the same symbol $n$ a regular extension
of the unit normal vector field $n$ defined in a neighborhood of
$\partial\mathcal{D}_{t}$ and such that $|n|=1$ close to
$\partial\mathcal{D}_{t}$.

Note that any $v_{\omega}$ decaying at infinity that solves this system is
unique since the difference of any two solutions of (3.4) is the gradient of
an harmonic function with homogeneous Neumann data, which is thus a constant.
Then, letting $\beta$ solve (3.3), we set $w:=-\operatorname{curl}\beta$ and
want to show that $w$ satisfies the system (3.4).

We have $\operatorname{div}w=0$ and
$\operatorname{curl}w=\Delta\beta-\nabla(\operatorname{div}\beta)=\omega-\nabla(\operatorname{div}\beta)$.
Observe that $\Delta(\operatorname{div}\beta)=0$ by (3.3a). Moreover,
decomposing into tangential and normal components, and denoting
$\beta_{n}=\beta\cdot n$, and $\partial_{n}=n\cdot\nabla$, we have, on the
surface

$\displaystyle\partial_{i}\beta^{i}$
$\displaystyle=(\Pi_{i}^{j}+n_{i}n^{j})\partial_{j}(\Pi^{i}_{k}\beta^{k}+n^{i}n_{k}\beta^{k})$
$\displaystyle=\Pi^{j}_{i}\partial_{j}\Pi^{i}_{k}\beta^{k}+n_{i}n^{j}\partial_{j}\Pi^{i}_{k}\beta^{k}+\Pi_{i}^{j}\partial_{j}(n^{i}n_{k}\beta^{k})+n_{i}n^{j}\partial_{j}(n^{i}n_{k}\beta^{k})$
$\displaystyle=n_{i}\partial_{n}(\Pi^{i}_{k}\beta^{k})+\Pi_{i}^{j}(\partial_{j}n^{i})\beta_{n}+\partial_{n}\beta_{n},$

having used the boundary condition (3.3b) to drop the first term in the second
line above, $\Pi_{i}^{j}n_{i}=0$ and $|n|=1$. Using again that the tangential
components of $\beta$ vanish we see that

$\displaystyle\left.\partial_{i}\beta^{i}\right|_{\partial\mathcal{D}_{t}}$
$\displaystyle=\partial_{n}\beta_{n}+\Pi_{i}^{j}(\partial_{j}n^{i})\beta_{n}.$

We then calculate the last term as

$\displaystyle\Pi_{i}^{j}(\partial_{j}n^{i})\beta_{n}=(\partial_{i}n^{i})\beta_{n}-n^{i}n_{j}(\partial_{j}n^{i})\beta_{n}=H\beta_{n},$

so that we eventually get

$\displaystyle\left.\partial_{i}\beta^{i}\right|_{\partial\mathcal{D}_{t}}$
$\displaystyle=\partial_{n}\beta_{n}+H\beta_{n}.$ (3.5)

Therefore, in view of (3.3c), we have $\operatorname{div}\beta=0$ on
$\partial\mathcal{D}_{t}$ and we can deduce that $\operatorname{div}\beta=0$
in $\mathcal{D}_{t}$ so that $\operatorname{curl}w=\Delta\beta=\omega$. For
the last boundary condition in (3.4) we see that since

$\displaystyle\operatorname{curl}z\cdot n=\operatorname{div}(\Pi z\times n),$
(3.6)

using again (3.3b) we get $w\cdot n=-\operatorname{curl}\beta\cdot n=0$.
Therefore $w$ solves (3.4) and (3.2) follows by uniqueness with $\beta$
solving (3.3) as desired.

Finally, observe that solutions to (3.3) (with a given divergence-free
$\omega$) are unique up to constant vector fields since any solution is
divergence-free in view of (3.5), and the curl of the difference of two
solutions solves the homogeneous system associated to (3.4). ∎

####  3.1.1. Change of coordinates

In order to obtain estimates for $\beta$ we change coordinates to a flat
domain, going from $(x,y)=(x_{1},x_{2},y)$ in $\mathcal{D}_{t}$ to $(x,z)$
with $x\in{\mathbb{R}}^{2},z<0$ with $z:=y-h(t,x)$, by defining

$\displaystyle\alpha(t,x,z):=\beta(t,x,z+h(t,x)),\qquad
W(t,x,z):=\omega(t,x,z+h(t,x)).$ (3.7)

In what follows $\nabla$ and $\Delta$ will only refer to differentiation in
$x$ unless otherwise specified.

######  Remark 3.2.

Notice that since we will be working at a lower level of regularity than the
maximal regularity available, we will not need to worry about the regularity
of the coordinate change and, in particular we can avoid paradifferential
calculus.

On the other hand, since we need to work at a level of regularity above
$N_{1}$, the top order weighted norms of $h$, which enter in the change of
coordinates, cannot be expected to be uniformly bounded in time (see the next
remark), and this creates several technical complications.

######  Remark 3.3 (A priori bounds on $h$).

Recall the a priori $L^{\infty}$ bound (2.22) and the $L^{2}$ bound (2.20) on
the height: for all $t\in[0,T]$

$\displaystyle\begin{split}&\sum_{r+k\leq
N_{1}}{\big{\|}h(t)\big{\|}}_{Z^{r,\infty}_{k}({\mathbb{R}}^{2})}\leq
c_{0}\varepsilon_{0}\langle t\rangle^{-1},\\\ &\sum_{r+k\leq
N_{0}}{\|h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\leq c_{0}\varepsilon_{0}\langle
t\rangle^{p_{0}}.\end{split}$ (3.8)

Using standard interpolation of $L^{p}$ spaces, we also deduce

$\displaystyle\sum_{r+k\leq
N_{1}}{\big{\|}h(t)\big{\|}}_{Z^{r,p}_{k}({\mathbb{R}}^{2})}\leq
c_{0}\varepsilon_{0}\langle t\rangle^{-1+(2/p)(1+p_{0})},\qquad p\geq 2.$
(3.9)

for all $t\in[0,T]$. Note that the last bound above is $\leq
c_{0}\varepsilon_{0}$ for $p\geq 11/5$, $p_{0}\leq 1/10$.

######  Remark 3.4 (A priori bounds on $\partial_{t}h$).

Using that $\partial_{t}h=G(h)\varphi$ with the second estimate in (B.38), we
have

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\big{\|}\partial_{t}h(t)\big{\|}}_{Z^{r,\infty}_{k}({\mathbb{R}}^{2})}\leq
c_{0}\varepsilon_{0}\langle t\rangle^{-1+}.$ (3.10)

Using $\partial_{t}h=v\cdot(-\nabla h,1)$ on the boundary, together with the a
priori bounds on $v$ we can also obtain, for all $t\in[0,T]$,

$\displaystyle\sum_{r+k\leq
N_{0}-5}{\|\partial_{t}h(t)\|}_{Z^{r}_{k}({\mathbb{R}}^{2})}\leq
c_{0}\varepsilon_{0}\langle t\rangle^{p_{0}}.$ (3.11)

The proof of (3.11) follows from the a priori bounds on $v$ and $h$ in (2.20),
(2.21) and (2.22), and elementary composition and product identities; we
postpone the proof until after the proof of Lemma 5.4 since it can be more
conveniently written out using some short-hand notation that will be
introduced later.

Interpolating (3.10) and (3.11) we have

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\big{\|}\partial_{t}h(t)\big{\|}}_{Z^{r,p}_{k}({\mathbb{R}}^{2})}\leq
c_{0}\varepsilon_{0}\langle t\rangle^{(2/p)(1+p_{0})-1+},\qquad p\geq 2,$
(3.12)

Note that the right-hand side of (3.12) is bounded by $\leq
c_{0}\varepsilon_{0}$ for, say, $p\geq 3$

Our goal is to establish bounds for $V_{\omega}$ and its time derivative.
Since we will do this by establishing bounds for $\alpha$, we first relate
their norms.

####  3.1.2. Basic formulas and norms

For given $f:[0,T]\times\mathcal{D}_{t}\rightarrow{\mathbb{R}}$, let us define
for $t\in[0,T]$, $x\in{\mathbb{R}}^{2}$ and $z\leq 0$ the function

$\displaystyle F(t,x,z):=f(t,x,z+h(t,x)),\qquad f(t,x,y)=F(t,x,y-h(t,x)),$

and record the basic identities

$\displaystyle\begin{split}\partial_{t}F(t,x,z)&=(\partial_{t}f)(t,x,z+h(t,x))+(\partial_{y}f)(t,x,z+h(t,x))\partial_{t}h,\\\
\partial_{x_{i}}F(t,x,z)&=(\partial_{x_{i}}f)(t,x,z+h(t,x))+(\partial_{y}f)(t,x,z+h(t,x))\partial_{x_{i}}h,\\\
\partial_{z}F(t,x,z)&=(\partial_{y}f)(t,x,z+h(t,x)),\\\ \Omega
F(t,x,z)&=(\Omega f)(t,x,z+h(t,x))+(\partial_{y}f)(t,x,z+h(t,x))\Omega h,\\\
(S+z\partial_{z})F(t,x,z)&=(\underline{S}f)(t,x,z+h(t,x))+(\partial_{y}f)(t,x,z+h(t,x))(Sh-h),\end{split}$
(3.13)

or, equivalently,

$\displaystyle\begin{split}(\partial_{t}f)(t,x,y)&=(\partial_{t}F)(t,x,y-h(x))-(\partial_{z}F)(t,x,y-h(x)\partial_{t}h(t,x),\\\
(\partial_{x_{i}}f)(t,x,y)&=(\partial_{x_{i}}F)(t,x,y-h(x))-(\partial_{z}F)(t,x,y-h(t,x))\partial_{x_{i}}h(t,x),\\\
(\partial_{y}f)(t,x,y)&=(\partial_{z}F)(t,x,y-h(x)),\\\ (\Omega
f)(t,x,y)&=(\Omega F)(t,x,y-h(x))-(\partial_{z}F)(t,x,y-h(x))\Omega h(t,x),\\\
(\underline{S}f)(t,x,y)&=((S+z\partial_{z})F)(t,x,y-h(x))-(\partial_{z}F)(t,x,y-h(t,x))(Sh(t,x)-h(t,x)).\end{split}$
(3.14)

In particular, evaluating at the boundary

$\displaystyle\begin{split}(\partial_{t}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=(\partial_{t}F)(t,x,0)-(\partial_{z}F)(t,x,0)\partial_{t}h,\\\
\big{(}\partial_{x_{i}}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=(\partial_{x_{i}}F)(t,x,0)-(\partial_{z}f)(t,x,0)\partial_{x_{i}}h,\\\
\big{(}\partial_{y}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=(\partial_{z}F)(t,x,0),\\\
\big{(}\Omega f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=(\Omega
F)(t,x,0)-(\partial_{z}F)(t,x,0)\Omega h,\\\
\big{(}\underline{S}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=(SF)(t,x,0)-(\partial_{z}F)(t,x,0)(Sh-h),\end{split}$
(3.15)

and

$\displaystyle\begin{split}\partial_{t}\big{(}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=\big{(}\partial_{t}f)|_{\partial\mathcal{D}_{t}}(t,x)+\big{(}\partial_{y}f)|_{\partial\mathcal{D}_{t}}(t,x)\partial_{t}h\\\
\partial_{x_{i}}\big{(}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=\big{(}\partial_{x_{i}}f)|_{\partial\mathcal{D}_{t}}(t,x)+(\partial_{y}f)|_{\partial\mathcal{D}_{t}}\partial_{x_{i}}h\\\
\Omega\big{(}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=\big{(}\Omega
f)|_{\partial\mathcal{D}_{t}}(t,x)+(\partial_{y}f)|_{\partial\mathcal{D}_{t}}\Omega
h\\\
S\big{(}f|_{\partial\mathcal{D}_{t}}\big{)}(t,x)&=\big{(}\underline{S}f)|_{\partial\mathcal{D}_{t}}(t,x)+(\partial_{y}f)|_{\partial\mathcal{D}_{t}}(Sh-h).\end{split}$
(3.16)

The identities (3.13), with the definitions (3.1) and (3.7), imply

$\displaystyle\begin{split}V_{\omega}(t,x,z)&=(\operatorname{curl}_{x,y}\beta)(t,x,z+h(x))\\\
&=(\operatorname{curl}_{x,z}\alpha)(t,x,z)-\partial_{z}\alpha_{3}(t,x,z)\,(\partial_{x_{2}}h(t,x)-\partial_{x_{1}}h(t,x))\\\
&-\partial_{z}\alpha_{2}(t,x,z)\partial_{x_{1}}h(t,x)+\partial_{z}\alpha_{1}(t,x,z)\partial_{x_{2}}h(t,x)\\\
&=\big{[}\operatorname{curl}\alpha+\partial_{z}\alpha\cdot(\nabla^{\perp}h,(\partial_{x_{1}}-\partial_{x_{2}})h)\big{]}(t,x,z)\end{split}$
(3.17)

and

$\displaystyle\begin{split}\partial_{t}V_{\omega}(t,x,z)&=(\operatorname{curl}_{x,z}\partial_{t}\alpha)(t,x,z)\\\
&-\partial_{z}\partial_{t}\alpha_{3}(t,x,z)\,(\partial_{x_{2}}h(t,x)-\partial_{x_{1}}h(t,x))\\\
&-\partial_{z}\alpha_{3}(t,x,z)\,(\partial_{x_{2}}\partial_{t}h(t,x)-\partial_{x_{1}}\partial_{t}h(t,x))\\\
&-\partial_{z}\partial_{t}\alpha_{2}(t,x,z)\partial_{x_{1}}h(t,x)+\partial_{z}\partial_{t}\alpha_{1}(t,x,z)\partial_{x_{2}}h(t,x)\\\
&-\partial_{z}\alpha_{2}(t,x,z)\partial_{x_{1}}\partial_{t}h(t,x)+\partial_{z}\alpha_{1}(t,x,z)\partial_{x_{2}}\partial_{t}h(t,x).\end{split}$
(3.18)

Let us record that the identities (3.17) and (3.18) schematically read as
follows:

$\displaystyle\begin{split}V_{\omega}&=\operatorname{curl}\alpha+\partial_{z}\alpha\cdot\nabla
h,\\\
\partial_{t}V_{\omega}&=\operatorname{curl}\partial_{t}\alpha+\partial_{z}\partial_{t}\alpha\cdot\nabla
h+\partial_{z}\alpha\cdot\nabla\partial_{t}h.\end{split}$ (3.19)

Then, using simple product estimates for weighted norms we will be able to
obtain bounds for $V_{\omega}$ in terms of certain norms of $\alpha$. Here are
the norms that we are going to use:

######  Definition 3.5 (Norms).

For a non-negative integer $n$, let

$\displaystyle{\|g\|}_{Y^{n}}=\sum_{|r|+|k|\leq
n}{\big{\|}g^{r,k}\big{\|}}_{Y^{0}},\qquad
g^{r,k}:=\underline{\Gamma}^{k}\nabla^{r}_{x,z}g$ (3.20)

where

$\displaystyle{\|g\|}_{Y^{0}}={\big{\|}|\nabla|^{1/2}g\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}+{\big{\|}\nabla_{x,z}g\big{\|}}_{L^{2}_{z}L^{2}_{x}}+{\|g\|}_{L^{\infty}_{z}L^{2}_{x}}.$
(3.21)

Also, let us define the “homogeneous” versions of the above spaces by

$\displaystyle{\|g\|}_{\dot{Y}^{n}}=\sum_{|r|+|k|\leq
n}{\big{\|}g^{r,k}\big{\|}}_{\dot{Y}^{0}},\qquad
g^{r,k}=\underline{\Gamma}^{k}\nabla^{r}_{x,z}g$ (3.22)

where

$\displaystyle{\|g\|}_{\dot{Y}^{0}}=\sum_{0\leq|a|\leq
1}\left({\big{\|}\nabla_{x,z}^{a}|\nabla|^{1/2}g\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}+{\big{\|}\nabla_{x,z}^{a}\nabla_{x,z}g\big{\|}}_{L^{2}_{z}L^{2}_{x}}\right)+{\|\partial_{z}g\|}_{L^{\infty}_{z}L^{2}_{x}}.$
(3.23)

Note that the above norms are defined so that

$\|\nabla_{x,z}g\|_{Y^{0}}+\||\nabla|^{1/2}g\|_{L^{\infty}_{z}L^{2}_{x}}+\|\nabla_{x,z}g\|_{L^{2}_{z}L^{2}_{x}}\lesssim\|g\|_{\dot{Y}^{0}}.$
(3.24)

In the upcoming section, we will prove estimates for $V_{\omega}$ in the
$Y^{n}$ spaces by first bounding the vector potential $\alpha$ in the
$\dot{Y}^{n}$ spaces. We will also use the bounds on $\alpha$ to obtain
estimates on $\widetilde{v}_{\omega}=V_{\omega}|_{z=0}$, hence on
$P_{\omega}$, in $Z^{r}_{k}$ spaces; see (2.48) and (2.41). The bounds for
$\alpha$ will follow from a fixed-point argument for a Poisson-type problem
for which the $\dot{Y}^{n}$ norms are well suited.

####  3.1.3. Consequences of bounds for $\alpha$

The next two lemmas show how bounds for $\widetilde{v}_{\omega}$ in the spaces
$Z^{r}_{k}$, and bounds for $V_{\omega}$ in the spaces $Y^{n}$, follow from
bounds for the vector potential $\alpha$ the $\dot{Y}^{n}$ spaces. Then, in
the remainder of the section, we will prove that the needed bounds on $\alpha$
stated in (3.25)-(3.26) can be obtained from our bootstrap assumptions
(3.8)-(3.9) on the dispersive variables, assumptions (2.25)-(2.26) on the
vorticity and a fixed point argument.

######  Lemma 3.6 (Bounds for $\alpha$ imply bounds for
$\widetilde{v}_{\omega}$).

Assume (3.8)-(3.9). Assume that for all $t\in[0,T]$ and for $j=0$ and $1$,
with notation as in (3.22), we have

$\displaystyle\|\partial_{t}^{j}\alpha(t)\|_{\dot{Y}^{N_{1}-10-j}}$
$\displaystyle\lesssim\varepsilon_{1},$ (3.25)
$\displaystyle\|\partial_{t}^{j}\alpha(t)\|_{\dot{Y}^{N_{1}+12-j}}$
$\displaystyle\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta}.$ (3.26)

Then, for $j=0,1$,

$\displaystyle\sum_{r+2n\leq
N_{1}+12-j}{\big{\|}\partial_{t}^{j}\widetilde{v_{\omega}}(t)\big{\|}}_{Z^{r}_{n}({\mathbb{R}}^{2})}\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta}.$ (3.27)

######  Lemma 3.7 (Bounds for $\alpha$ imply bounds for $V_{\omega}$).

Assume (3.8)-(3.9). With notation as in the previous lemma, if the bounds
(3.25)-(3.26) for the quantity $\alpha$ hold for $t\in[0,T]$, then, with the
norms $Y^{n}$ defined as in (3.20), we have

$\displaystyle\begin{split}{\|\partial_{t}^{j}V_{\omega}(t)\|}_{Y^{N_{1}-10-j}}&\lesssim\varepsilon_{1},\\\
{\|\partial_{t}^{j}V_{\omega}(t)\|}_{Y^{N_{1}+12-j}}&\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta}.\end{split}$ (3.28)

###### Proof of Lemma 3.6.

Starting from (3.17) and applying the product estimate (A.21), we estimate for
any $r+n\leq N=N_{1}+12$

$\displaystyle\begin{split}{\|\widetilde{v_{\omega}}(t,x)\|}_{Z^{r}_{n}}&\lesssim{\|(\operatorname{curl}_{x,z}\alpha)(t,x,0)\|}_{Z^{r}_{n}}\\\
&+\sum_{r+n\leq
N/2}{\|\partial_{z}\alpha(t,x,0)\|}_{Z^{r,\infty}_{n}}\,\sum_{r+n\leq
N}{\|\nabla^{\prime}h(t,x)\|}_{Z^{r}_{n}}\\\ &+\sum_{r+n\leq
N}{\|\partial_{z}\alpha(t,x,0)\|}_{Z^{r}_{n}}\,\sum_{r+n\leq
N/2}{\|\nabla^{\prime}h(t,x)\|}_{Z^{r,\infty}_{n}}\end{split}$ (3.29)

where we denoted
$\nabla^{\prime}=(\nabla^{\perp},\partial_{x_{1}}-\partial_{x_{2}})$.

The first term on the right-hand side of (3.29) is directly bounded using the
assumption (3.26): for all $r+n\leq N$, with the notation
$\alpha^{r,n}=\underline{\Gamma}^{n}\nabla^{r}_{x,z}\alpha$ as in (3.20), we
have

$\displaystyle\begin{split}{\|(\operatorname{curl}_{x,z}\alpha)(t,x,0)\|}_{Z^{r}_{n}}&\lesssim\sum_{r+n\leq
N}{\|(\nabla_{x},\partial_{z})\nabla^{r}_{x}\Gamma^{n}\alpha(t,\cdot,0)\|}_{L^{2}_{x}}\\\
&\lesssim\sum_{r+n\leq
N}{\|(\nabla_{x},\partial_{z})\alpha^{r,n}(t)\|}_{L^{\infty}_{z}L^{2}_{x}}\lesssim{\|\alpha(t)\|}_{\dot{Y}^{N}}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta}.\end{split}$ (3.30)

having also used (3.24) (recall also (3.21)) and that
$\underline{S}|_{z=0}=S$.

The second term in (3.29) is bounded by

$\displaystyle C\sum_{r+n\leq
N/2+2}{\|\partial_{z}\alpha(t,\cdot,0)\|}_{Z^{r}_{n}}\,\sum_{r+n\leq
N}{\|\nabla
h(t,\cdot)\|}_{Z^{r}_{n}}\lesssim\varepsilon_{1}\cdot\varepsilon_{0}\langle
t\rangle^{\delta}$ (3.31)

having used Sobolev’s embedding, the control the first norm in (3.25) (since
$N_{1}-10\geq N/2+2$) and the a priori assumption (3.8) (since $N+1\leq
N_{0}$).

We can instead bound the last term in (3.29) using (3.26) and (3.8) (since
$N_{1}\geq N/2+1)$, as follows:

$\displaystyle C\sum_{r+n\leq
N}{\|\partial_{z}\alpha(t,\cdot,0)\|}_{Z^{r}_{n}}\,\sum_{r+n\leq
N/2+1}{\|h(t,\cdot)\|}_{Z^{r,\infty}_{n}}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta}\cdot\varepsilon_{0}\langle t\rangle^{-1}$ (3.32)

which is more than sufficient. This proves (3.27) when $j=0$.

To obtain the estimate for the time derivative we can proceed similarly,
starting from the formula (3.18). To control the term
$\operatorname{curl}_{x,z}\partial_{t}\alpha(t,x,0)$ we can estimate as in
(3.30) replacing $\alpha$ with $\partial_{t}\alpha$ and using (3.26) with
$j=1$. All the other terms in (3.18) are of the form
$\partial_{z}\partial_{t}\alpha_{i}(t,x,0)\cdot\partial_{x_{k}}h$ or
$\partial_{z}\alpha_{i}(t,x,0)\cdot\partial_{x_{k}}\partial_{t}h$ for some
$i=1,2,3$ and $k=1,2$. We can then estimate all of these using (3.26) and
(3.25) also with $j=1$ and the bounds (3.11) and (3.12) for the terms
involving $\partial_{t}h$; these estimates are analogous to (3.31) and (3.32)
so we omit the details ∎

###### Proof of Lemma 3.7.

We argue in a similar way as in the previous lemma. For the first term on the
right-hand side of (3.17) we observe, using (3.24), that
${\|\operatorname{curl}\alpha\|}_{Y^{n}}\lesssim{\|\alpha\|}_{\dot{Y}^{n}}$,
for $n=N_{1}-10$ or $N_{1}+12$, which is consistent with the desired (3.28).
We then only need to look at the nonlinear terms on the right-hand side of
(3.17). According to the schematic version (3.19), applying vector fields and
using the notation (3.20), we have, for all $|r|+|k|\leq N$,

$\displaystyle\underline{\Gamma}^{k}\nabla_{x,z}^{r}(V_{\omega}-\operatorname{curl}\alpha)$
$\displaystyle=\sum_{\begin{subarray}{c}r_{1}+r_{2}=r,\\\
k_{1}+k_{2}=k\end{subarray}}(\partial_{z}\alpha)^{r_{1},k_{1}}\cdot(\nabla
h)^{r_{2},k_{2}}.$ (3.33)

Let us concentrate on proving the bounds in the high-norm, that is
$|r|+|k|=N:=N_{1}+12$, since the bounds in the low norm can be obtained
similarly. From (3.33) we see that it suffices to estimate the $Y^{0}$-norm of
the terms

$\displaystyle I$ $\displaystyle:=\sum_{|r_{1}|+|k_{1}|\leq
N/2}(\partial_{z}\alpha)^{r_{1},k_{1}}\cdot(\nabla h)^{r_{2},k_{2}},$ (3.34)
$\displaystyle II$ $\displaystyle:=\sum_{|r_{2}|+|k_{2}|\leq
N/2}(\partial_{z}\alpha)^{r_{1},k_{1}}\cdot(\nabla h)^{r_{2},k_{2}}.$ (3.35)

The constraints $r_{1}+r_{2}=r$ and $k_{1}+k_{2}=k$ are implicit in the above
sums.

The first component of the $Y_{0}$-norm of $I$ can be estimated using (A.24):

$\displaystyle\begin{split}{\||\nabla|^{1/2}I\|}_{L^{\infty}_{z}L^{2}_{x}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N/2}{\||\nabla|^{1/2}(\partial_{z}\alpha)^{r_{1},k_{1}}\|}_{L^{\infty}_{z}L^{2}_{x}}\cdot\sum_{|r_{2}|+|k_{2}|\leq
N}{\|(\nabla h)^{r_{2},k_{2}}\|}_{W^{1,3}}\\\
\lesssim\varepsilon_{1}\cdot\varepsilon_{0}\langle
t\rangle^{p_{0}}\end{split}$

having also used (3.25) (recall (3.23) and that $N/2\leq N_{1}-10$) and (3.8).
Note that we have also used the commutation relation (C.23).

For the second component of the $Y_{0}$-norm of $I$ we estimate using just
Hölder and the same assumptions above:

$\displaystyle\begin{split}{\|\nabla_{x,z}I\|}_{L^{2}_{z}L^{2}_{x}}\lesssim\sum_{\begin{subarray}{c}|r_{1}|+|k_{1}|\leq
N/2\\\ |a|\leq
1\end{subarray}}{\|\nabla_{x,z}^{a}(\partial_{z}\alpha)^{r_{1},k_{1}}\|}_{L^{2}_{z}L^{2}_{x}}\cdot\sum_{|r_{2}|+|k_{2}|\leq
N+1}{\|(\nabla h)^{r_{2},k_{2}}\|}_{L^{\infty}}\\\
\lesssim\varepsilon_{1}\cdot\varepsilon_{0}\langle
t\rangle^{p_{0}}.\end{split}$ (3.36)

The last $L^{\infty}_{z}L^{2}_{x}$ piece of the norm is immediate to estimate,
so we skip it.

For the term $II$, we estimate the first component of the $Y_{0}$-norm using
again the product estimate (A.24), and then the assumption (3.26) and the a
priori bound (3.9):

$\displaystyle\begin{split}{\||\nabla|^{1/2}II\|}_{L^{\infty}_{z}L^{2}_{x}}\lesssim\sum_{|r_{1}|+2|k_{1}|\leq
N}{\||\nabla|^{1/2}(\partial_{z}\alpha)^{r_{1},k_{1}}\|}_{L^{\infty}_{z}L^{2}_{x}}\cdot\sum_{|r_{2}|+2|k_{2}|\leq
N/2}{\|(\nabla h)^{r_{2},k_{2}}\|}_{W^{1,3}}\\\ \lesssim\varepsilon_{1}\langle
t\rangle^{\delta}\cdot\varepsilon_{0}.\end{split}$

The second component of the $Y_{0}$-norm is estimated just using Hölder and
the same assumptions above:

$\displaystyle\begin{split}{\|\nabla_{x,z}II\|}_{L^{2}_{z}L^{2}_{x}}\lesssim\sum_{\begin{subarray}{c}|r_{1}|+2|k_{1}|\leq
N\\\ |a|\leq
1\end{subarray}}{\|\nabla_{x,z}^{a}(\partial_{z}\alpha)^{r_{1},k_{1}}\|}_{L^{2}_{z}L^{2}_{x}}\cdot\sum_{|r_{2}|+2|k_{2}|\leq
N/2+1}{\|(\nabla h)^{r_{2},k_{2}}\|}_{L^{\infty}}\\\
\lesssim\varepsilon_{1}\langle
t\rangle^{\delta}\cdot\varepsilon_{0}.\end{split}$ (3.37)

The last piece of the norm, that is, ${\|II\|}_{L^{\infty}_{z}L^{2}_{x}}$ can
be bounded in the same way.

To obtain the estimates for the time derivative we can proceed in the same
way, starting from the second formula in (3.19), using (A.24) as above,
Hölder, and the assumption on $\partial_{t}\alpha$ in (3.25)-(3.26) and on
$\partial_{t}h$ in (3.10)-(3.11). ∎

For some of our applications (specifically for the estimates in Section 5), we
will need a slight variation of the above bounds for $V_{\omega}$ where we
both control the $L^{2}_{x,z}$ norms of $V_{\omega}$ directly (technically,
this is not included in the $Y^{n}$ spaces) and additionally control a higher-
order norm of $V_{\omega}$ (but with a worse bound) provided we have
additional high-order control of the vorticity. This is the lemma that we will
need:

######  Lemma 3.8 (High-order bounds for $\alpha$ imply high-order
$L^{2}_{z}L^{2}_{x}$ bounds for $V_{\omega}$).

Under the hypotheses of Lemma 3.7, and using the notation
$g^{r,k}=\underline{\Gamma}^{k}\nabla_{x,z}^{r}g$ from (3.20), we have

$\displaystyle\sum_{|r|+|k|\leq
N_{1}-10-j}\|\partial_{t}^{j}V_{\omega}^{r,k}(t)\|_{L^{2}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{1},$ (3.38) $\displaystyle\sum_{|r|+|k|\leq
N_{1}+12-j}{\|\partial_{t}^{j}V_{\omega}^{r,k}(t)\|}_{L^{2}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta}.$ (3.39)

Moreover, if

${\|\alpha(t)\|}_{\dot{Y}^{N_{0}-20}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}},$ (3.40)

then

$\sum_{|r|+|k|\leq
N_{0}-20}{\|V_{\omega}^{r,k}(t)\|}_{L^{2}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}}.$ (3.41)

In Section 5 we will see how the high order norm assumption (3.40) on $\alpha$
follows from the assumption on high-order norms of the vorticity (2.26); see
Proposition 5.2 and Lemma 5.5.

###### Proof of Lemma 3.8.

The argument is nearly identical to the proof of Lemma 3.7 using (3.19) and
simple product estimates. The only additional simple observation needed is
that
$\|\underline{\Gamma}^{k}\nabla^{r}\operatorname{curl}\alpha\|_{L^{2}_{z}L^{2}_{x}}+\|\underline{\Gamma}^{k}\nabla^{r}\nabla_{x,z}\alpha\|_{L^{2}_{z}L^{2}_{x}}\lesssim\|\alpha\|_{\dot{Y}^{n}}$
for $|k|+|r|\leq n$ by definition (note the $a=0$ term in the second term of
the definition (3.23) of the $\dot{Y}^{n}$ norms). Then, we follow the same
steps as in the above proof with $I$ and $II$ defined as in (3.34)-(3.35). In
place of (3.36) we bound

$\displaystyle\begin{split}{\|I\|}_{L^{2}_{z}L^{2}_{x}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N/2}{\|(\partial_{z}\alpha)^{r_{1},k_{1}}\|}_{L^{2}_{z}L^{2}_{x}}\cdot\sum_{|r_{2}|+|k_{2}|\leq
N+1}{\|(\nabla h)^{r_{2},k_{2}}\|}_{L^{\infty}}\\\
\lesssim\varepsilon_{1}\cdot\varepsilon_{0}\langle
t\rangle^{p_{0}};\end{split}$

for $N=N_{1}+12$, which is consistent with (3.39); with an obvious
modification when $N$ is replaced with $N_{0}-20$ this is consistent with
(3.41). Similarly, in place of (3.37) we have

$\displaystyle\begin{split}{\|II\|}_{L^{2}_{z}L^{2}_{x}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N}{\|(\partial_{z}\alpha)^{r_{1},k_{1}}\|}_{L^{2}_{z}L^{2}_{x}}\cdot\sum_{|r_{2}|+|k_{2}|\leq
N/2+1}{\|(\nabla h)^{r_{2},k_{2}}\|}_{L^{\infty}}\\\
\lesssim\varepsilon_{1}\langle
t\rangle^{\delta}\cdot\varepsilon_{0};\end{split}$

once again this is consistent with (3.39) if $N=N_{1}+12$; the obvious
modification when $N$ is replaced with $N_{0}-20$ gives (3.41).

The lower order norm in (3.38) can be estimated similarly, using the uniform
bound (3.9). The estimates for the time derivatives can also be obtain in a
completely analogous fashion, using the estimates on $\partial_{t}h$ from
Remark 3.4 ∎

###  3.2. Fixed point formulation for $\alpha$

From the system (3.3) satisfied by $\beta$ we derive a fixed point formulation
for $\alpha$. We first write out the elliptic system satisfied by $\alpha$:

######  Lemma 3.9 (The Elliptic system in the flat domain).

Let $\alpha$ and $W$ be defined as in (3.7), with $\beta$ the solution of
(3.3). Then we have

$\displaystyle(\partial_{z}^{2}+\Delta_{x})\alpha$
$\displaystyle=\partial_{z}E_{a}+|\nabla|E_{b}+F,$ $\displaystyle\qquad\text{
in }z<0,$ (3.42a) $\displaystyle\alpha_{1}$ $\displaystyle=B_{1},$
$\displaystyle\qquad\text{ on }z=0,$ (3.42b) $\displaystyle\alpha_{2}$
$\displaystyle=B_{2},$ $\displaystyle\qquad\text{ on }z=0,$ (3.42c)
$\displaystyle\partial_{z}\alpha_{3}$ $\displaystyle=B_{3},$
$\displaystyle\qquad\text{ on }z=0,$ (3.42d)

where

$E^{a}(\alpha):=\frac{\nabla}{|\nabla|}\cdot(\nabla h\partial_{z}\alpha)\qquad
E^{b}(\alpha):=-|\nabla h|^{2}\partial_{z}\alpha+\nabla
h\cdot\nabla\alpha,\qquad F=W,$ (3.43)

where

$\displaystyle B_{i}(\alpha,\nabla\alpha)$ $\displaystyle=\left((1+|\nabla
h|^{2})\partial_{i}h(\alpha_{3}-\nabla h\cdot\alpha)\right)|_{z=0},\qquad
i=1,2,$ (3.44)

and

$\displaystyle\begin{split}B_{3}(\alpha,\nabla\alpha)&=\nabla
h\cdot\partial_{z}(\alpha_{1},\alpha_{2})+\nabla\cdot\Big{[}(1+|\nabla
h|^{2})^{-1}\nabla h\,(\alpha_{3}-\nabla
h\cdot\alpha)\Big{]}\bigg{|}_{z=0}.\end{split}$ (3.45)

Notation. Note that in (3.43) we are omitting the dependence on $h$ and
implicitly on the position $(x,z)$. Later on, e.g. in (3.47), we will denote
these terms with $E_{a}(z)$ to make the dependence on the vertical variable
explicit.

The proof of the above lemma is an explicit computation. We give the details
in Appendix C. Regardless of the exact formulas, we point out that we are
dealing with an elliptic system for the vector field $\alpha$ with mixed
Dirichlet (for the first two components) and Neumann (for the third component)
boundary conditions. Note that the quantity $\alpha_{3}$, which is more
singular than $\nabla_{x}\alpha_{3}$ or $\partial_{z}\alpha_{3}$, appears in
the boundary data multiplied by a linear factor of $h$; this will create some
technical difficulties in proving bounds for $\alpha$.

Using Lemma 3.9 we write a fixed point formulation for $\alpha$, which we
record in the following:

######  Lemma 3.10 (Fixed point formulation).

Let $\alpha$ be the solution of (3.42)-(3.45). Then, it is formally a fixed
point of the map

$\displaystyle\alpha\rightarrow
L(\alpha)=(L_{1}(\alpha),L_{2}(\alpha),L_{3}(\alpha))$ (3.46)

where

$L_{i}(\alpha)(z):=e^{z|\nabla|}B_{i}(\alpha)-\frac{1}{2}\int_{-\infty}^{0}e^{(z+s)|\nabla|}(E^{a}_{i}(s)-E^{b}_{i}(s)-|\nabla|^{-1}F_{i}(s))\,ds\\\
+\frac{1}{2}\int_{-\infty}^{0}e^{-|z-s||\nabla|}(\mathrm{sign}(z-s)E^{a}_{i}(s)-E^{b}_{i}(s)-|\nabla|^{-1}F_{i}(s))\,ds,\quad
i=1,2,$ (3.47)

with (3.44), and

$\displaystyle\begin{split}L_{3}(\alpha)(z)&:=e^{z|\nabla|}B_{3,a}(\alpha)+|\nabla|^{-1}e^{z|\nabla|}B_{3,b}(\alpha)\\\
&+\frac{1}{2}\int_{-\infty}^{0}e^{(z+s)|\nabla|}(E^{a}_{3}(s)-E^{b}_{3}(s)-|\nabla|^{-1}F_{3}(s))\,ds\\\
&+\frac{1}{2}\int_{-\infty}^{0}e^{-|z-s||\nabla|}(\mathrm{sign}(z-s)E^{a}_{3}(s)-E^{b}_{3}(s)-|\nabla|^{-1}F_{3}(s))\,ds,\end{split}$
(3.48)

with

$\displaystyle\begin{split}B_{3,a}(\alpha)&=\frac{\nabla}{|\nabla|}\cdot\big{[}(1+|\nabla
h|^{2})^{-1}\nabla h\,(\alpha_{3}-\nabla h\cdot\alpha)\big{]},\end{split}$
(3.49) $\displaystyle\begin{split}B_{3,b}(\alpha,\nabla\alpha)&=\nabla
h\cdot\partial_{z}\alpha-\frac{\nabla}{|\nabla|}\cdot(\nabla
h\partial_{z}\alpha_{3}).\end{split}$ (3.50)

###### Proof of Lemma 3.10.

The fixed point formulation (3.46)-(3.50) is obtained using the solution of
Laplace’s equation given in Lemma C.1. Using (C.17) we directly obtain (3.47).

For the third component $\alpha_{3}$, an application of (C.19) gives us the
bulk integrals in (3.48), so we only need to verify the formulas for the
boundary contributions, which are given by

$\displaystyle\frac{1}{|\nabla|}e^{z|\nabla|}B_{3}-\frac{1}{|\nabla|}e^{z|\nabla|}\frac{\nabla}{|\nabla|}\cdot(\nabla
h\,\partial_{z}\alpha_{3}),$ (3.51)

with $B_{3}$ as in (3.45), which gives the result. ∎

###  3.3. Norms and main proposition

Based on the above fixed point formulation and using the a priori bounds on
$h$ from (3.9), we want to show existence and uniqueness of $\alpha$ and bound
it as in (3.25)-(3.26). As mentioned above, we will work in terms of the norms
from Definition 3.5; in particular, we will prove a contraction for the map
(3.46) in the ‘low norm’ $\dot{Y}^{N_{1}-10}$ and bounds in the ‘high norm’
$\dot{Y}^{N}$, $N:=N_{1}+12$.

######  Remark 3.11.

Directly from the definition, for all $r+2|k|\leq n$, we see that

$\displaystyle\sum_{|k^{\prime}|\leq|k|}{\big{\|}\langle\nabla\rangle^{1/2}\Gamma^{k^{\prime}}\nabla_{x,z}\alpha(0)\big{\|}}_{H^{r}({\mathbb{R}}^{2})}\lesssim\sum_{|k^{\prime}|\leq|k|}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k^{\prime}}\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}H^{r}({\mathbb{R}}^{2})}\lesssim{\|\alpha\|}_{\dot{Y}^{n}},$
(3.52)

This will be used to control some of the homogeneous boundary terms that we
will encounter.

We also define (see (2.24))

$\displaystyle\begin{split}{\|f\|}_{\mathcal{X}^{n}}:=\sum_{|r|+|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\,f\big{\|}}_{L^{2}_{z}L^{2}_{x}\cap
L^{6/5}_{x,z}}.\end{split}$ (3.53)

This is the norm that we use to measure the vorticity $W$ (see (3.7)) which
appears as a forcing term in (3.42a). Bounds on $W$ and its time derivative in
the above spaces will be bootstrapped in Section 4.

To obtain (3.25)-(3.26) it will suffice to show the following main
proposition:

######  Proposition 3.12 (Bounds for $\alpha$).

Let
$\alpha:[0,T]\times{\mathbb{R}}^{2}\times{\mathbb{R}}_{-}\mapsto{\mathbb{R}}^{3}$
be defined by $\alpha(t,x,z):=\beta(t,x,z+h(t,x))$ where $\beta$ solves the
system (3.3) in $\mathcal{D}_{t}$. Assume that $h$ satisfies (3.8)-(3.9) and
(3.11)-(3.12), and let $W$ be given so that, for $t\in[0,T]$, and for $j=0,1$

$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\lesssim\varepsilon_{1},$
(3.54)
$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\lesssim\varepsilon_{0}^{j}\varepsilon_{1}\langle
t\rangle^{\delta}.$ (3.55)

Then, there exists a unique fixed point $\alpha$ of the map in (3.46) in the
space $\dot{Y}^{N_{1}-10}$, which satisfies

$\displaystyle{\|\partial_{t}^{j}\alpha(t)\|}_{\dot{Y}^{N_{1}-10-j}}\lesssim\varepsilon_{1},$
(3.56)
$\displaystyle{\|\partial_{t}^{j}\alpha(t)\|}_{\dot{Y}^{N_{1}+12-j}}\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta}.$ (3.57)

The proof of Proposition (3.12) is carried out in the next subsection. The
desired conclusions will be a consequence of the following main estimates:

$\displaystyle{\|L(\alpha)\|}_{\dot{Y}^{N_{1}-10}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}+{\|W\|}_{\mathcal{X}^{N_{1}-10}},$
(3.58) $\displaystyle{\|L(\alpha)\|}_{\dot{Y}^{N_{1}+12}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}+{\|W\|}_{\mathcal{X}^{N_{1}+12}},$
(3.59)

and

$\displaystyle{\|\partial_{t}L(\alpha)\|}_{\dot{Y}^{N_{1}-11}}$
$\displaystyle\lesssim\varepsilon_{0}\big{(}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}+{\|\partial_{t}\alpha\|}_{\dot{Y}^{N_{1}-11}}\big{)}+{\|\partial_{t}W\|}_{\mathcal{X}^{N_{1}-11}},$
(3.60) $\displaystyle{\|\partial_{t}L(\alpha)\|}_{\dot{Y}^{N_{1}+11}}$
$\displaystyle\lesssim\varepsilon_{0}\big{(}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}+{\|\partial_{t}\alpha\|}_{\dot{Y}^{N_{1}+11}}\big{)}+\varepsilon_{0}\langle
t\rangle^{\delta}\big{(}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}+{\|\partial_{t}\alpha\|}_{\dot{Y}^{N_{1}-11}}\big{)}$
(3.61) $\displaystyle+{\|\partial_{t}W\|}_{\mathcal{X}^{N_{1}+11}}$

###  3.4. Proof of Proposition 3.12

####  3.4.1. Bounds for the Poisson kernel

We first need some bounds on the Poisson kernel.

######  Lemma 3.13.

For $f:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$, any $p\in(1,\infty)$, and
$k=0,1,\dots$, we have

$\displaystyle{\big{\|}\underline{\Gamma}^{k}e^{z|\nabla|}f\big{\|}}_{L^{\infty}_{z}W^{r,p}_{x}}$
$\displaystyle\lesssim{\|f\|}_{Z^{r,p}_{k}},\qquad 1<p<\infty,$ (3.62)

and

$\displaystyle{\big{\|}\underline{\Gamma}^{k}|\nabla|^{1/2}e^{z|\nabla|}f\big{\|}}_{L^{2}_{z}H^{r}_{x}}$
$\displaystyle\lesssim{\|f\|}_{H^{r}}.$ (3.63)

Moreover, for $f:{\mathbb{R}}^{2}\times\\{z<0\\}\rightarrow{\mathbb{R}}$ we
have

$\displaystyle\begin{split}&{\Big{\|}\underline{\Gamma}^{k}|\nabla|^{1/2}\int_{-\infty}^{0}e^{-|z-s||\nabla|}\mathbf{1}_{\pm}(s-z)f(x,s)\,ds\Big{\|}}_{L^{\infty}_{z}H^{r}}\\\
&+{\Big{\|}\underline{\Gamma}^{k}|\nabla|\int_{-\infty}^{0}e^{-|z-s||\nabla|}\mathbf{1}_{\pm}(s-z)f(x,s)\,ds\Big{\|}}_{L^{2}_{z}H^{r}}\\\
&\lesssim\min\Big{(}\sum_{k^{\prime}\leq
k}{\|\underline{\Gamma}^{k^{\prime}}f\|}_{L^{2}_{z}H^{r}},\,\sum_{k^{\prime}\leq
k}{\big{\|}|\nabla|^{1/3}\underline{\Gamma}^{k^{\prime}}f\big{\|}}_{L^{6/5}_{z}H^{r}}\Big{)}.\end{split}$
(3.64)

Recall that $\mathbf{1}_{\pm}(x)$ is the indicator function of $\pm x>0$.

Lemma 3.13 follows from standard bounds for the Poisson kernel and commutation
identities for vector fields. The proof is given in C.2. Let us make a few
remarks.

######  Remark 3.14.

1\. Note that (3.64) implies the same bounds for the operators

$\displaystyle T_{1}f:=$
$\displaystyle\int_{z}^{0}e^{(z-s)|\nabla|}f(x,s)\,ds,\quad
T_{2}:=\int_{-\infty}^{z}e^{(s-z)|\nabla|}f(x,s)\,ds,\quad
T_{3}:=\int_{-\infty}^{0}e^{(z+s)|\nabla|}f(x,s)\,ds,$ (3.65)

which are those that appear in (3.46); the first two are immediate, while for
the last one we just observe that
$T_{3}=T_{1}e^{2z|\nabla|}+e^{2z|\nabla|}T_{2}$.

2\. Also note that the estimate for the second term in (3.64) implies a
similar estimate with $\partial_{z}$ replacing $|\nabla|$:

$\displaystyle\begin{split}{\Big{\|}\underline{\Gamma}^{k}\partial_{z}\int_{-\infty}^{0}e^{-|z-s||\nabla|}\mathbf{1}_{\pm}(s-z)f(x,s)\,ds\Big{\|}}_{L^{2}_{z}H^{r}}\lesssim\sum_{k^{\prime}\leq
k}{\|\underline{\Gamma}^{k^{\prime}}f\|}_{L^{2}_{z}H^{r}};\end{split}$ (3.66)

this follows since we have the identities

$\displaystyle\partial_{z}T_{1}=-\mathrm{id}+T_{1}|\nabla|,\quad\partial_{z}T_{2}=\mathrm{id}-T_{2}|\nabla|,\quad\partial_{z}T_{3}=T_{3}|\nabla|,\qquad[T_{i},|\nabla|]=0.$
(3.67)

Using these identities we can also estimate

$\displaystyle\begin{split}{\Big{\|}\underline{\Gamma}^{k}\partial_{z}\int_{-\infty}^{0}e^{-|z-s||\nabla|}\mathbf{1}_{\pm}(s-z)f(x,s)\,ds\Big{\|}}_{L^{\infty}_{z}H^{r}}\lesssim\sum_{k^{\prime}\leq
k}{\||\nabla|^{1/2}\underline{\Gamma}^{k^{\prime}}f\|}_{L^{2}_{z}H^{r}}+{\|\underline{\Gamma}^{k^{\prime}}f\|}_{L^{\infty}_{z}H^{r}}.\end{split}$
(3.68)

3\. Bounds for higher-order $z$-derivatives also hold true: for $\ell\geq 1$,

$\displaystyle\begin{split}\Big{\|}\underline{\Gamma}^{k}\partial_{z}^{\ell}\int_{-\infty}^{0}e^{-|z-s||\nabla|}\mathbf{1}_{\pm}(s-z)f(x,s)\,ds\Big{\|}_{L^{2}_{z}H^{r}}\lesssim\sum_{k^{\prime}\leq
k}\sum_{\ell_{1}+\ell_{2}\leq\ell-1}\|\underline{\Gamma}^{k^{\prime}}\partial_{z}^{\ell_{1}}f\|_{L^{2}_{z}H^{r+\ell_{2}}};\end{split}$
(3.69)

and

$\displaystyle\begin{split}&\Big{\|}\underline{\Gamma}^{k}\partial_{z}^{\ell}\int_{-\infty}^{0}e^{-|z-s||\nabla|}\mathbf{1}_{\pm}(s-z)f(x,s)\,ds\Big{\|}_{L^{\infty}_{z}H^{r}}\\\
&\lesssim\sum_{k^{\prime}\leq
k}\sum_{\ell_{1}+\ell_{2}\leq\ell-1}{\big{\|}|\nabla|^{1/2}\underline{\Gamma}^{k^{\prime}}\partial_{z}^{\ell_{1}}f\big{\|}}_{L^{2}_{z}H^{r+\ell_{2}}}+{\big{\|}\underline{\Gamma}^{k^{\prime}}\partial_{z}^{\ell_{1}}f\big{\|}}_{L^{\infty}_{z}H^{r+\ell_{2}}};\end{split}$
(3.70)

these follow by repeatedly applying (3.67) to see that for
$T\in\\{T_{1},T_{2},T_{3}\\}$ we can write $\partial_{z}^{\ell}T$ as a sum of
terms of the form

$\partial_{z}^{\ell_{1}}|\nabla|^{\ell_{2}},\qquad
T|\nabla|^{\ell},\qquad\ell_{1}+\ell_{2}\leq\ell-1,$ (3.71)

and then applying (3.68).

4\. Finally, we remark that the first norm on the right-hand side of (3.64)
will be enough to control all the terms on the right-hand sides of
(3.47)-(3.48) except the forcing term involving the (inverse gradient of the)
vorticity for which we need to use the second norm.

We now proceed to estimate the map $L(\alpha)$ in (3.46)-(3.50) in the spaces
$\dot{Y}^{n}$ defined in (3.22)-(3.23). We first prove (3.58)-(3.59) by
estimating the quantities arising from the boundary conditions (3.47)-(3.48)
in 3.4.2, and the nonlinear bulk terms in 3.4.3. In 3.4.4 we control the
forcing term. Finally, in 3.4.5 we prove (3.60)-(3.61).

####  3.4.2. Estimate for the homogeneous terms

In view of the bounds (3.62)-(3.63) for $e^{z|\nabla|}$, the fact that
$\partial_{z}e^{z|\nabla|}=|\nabla|e^{z|\nabla|}$, and the definition of the
space $\dot{Y}^{n}$, we have the estimate

$\displaystyle{\big{\|}e^{z|\nabla|}f\big{\|}}_{\dot{Y}^{n}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}r+k\leq n\\\
a=0,1\end{subarray}}{\big{\|}|\nabla|^{1/2+a}f\big{\|}}_{Z^{r}_{k}}.$ (3.72)

Using this we can bound

$\displaystyle{\big{\|}e^{z|\nabla|}B_{i}\big{\|}}_{\dot{Y}^{n}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}r+k\leq n\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}B_{i}\|}_{Z^{r}_{k}},\qquad
B\in\\{B_{1},B_{2},B_{3,a}\\},$ (3.73)
$\displaystyle{\big{\|}|\nabla|^{-1}e^{z|\nabla|}B_{3,b}\big{\|}}_{\dot{Y}^{n}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}r+k\leq n\\\
a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}B_{3,b}\|}_{Z^{r}_{k}}.$ (3.74)

To get the needed estimates for $\alpha$, we therefore want to estimate the
right-hand sides of (3.73)-(3.74) and show the following:

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}r+k\leq N_{1}-10\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}(B_{1},B_{2},B_{3,a})\|}_{Z^{r}_{k}}+\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\\\
a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}B_{3,b}\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},\end{split}$
(3.75)

and

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}r+k\leq N_{1}+12\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}(B_{1},B_{2},B_{3,a})\|}_{Z^{r}_{k}}+\sum_{\begin{subarray}{c}r+k\leq
N_{1}+12\\\ a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}B_{3,b}\|}_{Z^{r}_{k}}\\\
\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}.\end{split}$ (3.76)

Some reductions and useful estimates. From the definitions (3.44), (3.49) and
(3.50) we see that there are many terms that need to be estimated to prove
(3.75) and (3.76). However, many of them are similar and they can all be
written as linear combinations of simpler terms, as we now argue. First,
(3.44) and (3.49) are all linear combinations of terms of the form

$\displaystyle b(\nabla h,\alpha)$ $\displaystyle:=c(\nabla
h)\alpha_{j}(0),\quad j=1,2,3,$ (3.77)

with $c$ denoting a generic coefficient satisfying

$\displaystyle\sum_{r+2k\leq N_{1}-1}{\|c(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0},\qquad p\geq 3,$ (3.78)
$\displaystyle\sum_{r+k\leq N_{0}-3}{\|c(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle t\rangle^{\delta},\qquad
p\geq 2.$ (3.79)

Note that we have disregarded the Riesz transform $\nabla|\nabla|^{-1}$ in
front of (3.49) since this plays no role in the desired $L^{2}$-based
estimates. Also note that, for all practical purposes, one may think that
$c=\nabla h$.

To verify (3.77) with (3.78)-(3.79) we inspect (3.44) and see that $B_{i}$ is
a linear combination of terms as in (3.77) where the coefficients are of the
form $c(\nabla h)=(1+|\nabla h|^{2})\partial_{i}h$ and $\nabla h\,c(\nabla
h)$; using the product estimate (A.21) and the a priori assumptions (3.9), we
can verify directly that (3.78) holds: for all $p\geq 11/5$

$\displaystyle\sum_{r+k\leq N_{1}-1}{\|(1+|\nabla
h|^{2})\partial_{i}h\|}_{Z^{r,p}_{k}}$
$\displaystyle\lesssim(1+\varepsilon_{0}^{2})\sum_{r+k\leq
N_{1}-1}{\|\partial_{i}h\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}$

Similarly, we can use also (3.8) to verify (3.79):

$\displaystyle\sum_{r+k\leq N_{0}-3}{\|(1+|\nabla
h|^{2})\partial_{i}h\|}_{Z^{r,p}_{k}}$
$\displaystyle\lesssim\Big{(}1+\sum_{r+k\leq(N_{0}-3)/2}{\|\nabla
h\|}^{2}_{Z^{r,\infty}_{k}}\Big{)}\sum_{r+k\leq N_{0}-3}{\|\nabla
h\|}_{Z^{r,p}_{k}}$
$\displaystyle\lesssim(1+\varepsilon_{0}^{2})\varepsilon_{0}\langle
t\rangle^{\delta},$

having used (3.9) and $N_{1}\geq N_{0}/2$.

Again omitting the Riesz transform, the term $B_{3,b}$ in (3.50) is a linear
combination of terms of the type:

$\displaystyle b_{3}(\nabla h,\alpha)$ $\displaystyle:=c(\nabla
h)\,\nabla_{j}\alpha_{k}(0),$ (3.80)

where $c_{3}$ denotes a generic coefficient satisfying

$\displaystyle\sum_{r+k\leq N_{1}-3}{\|c_{3}(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1+(2/p)(1+\delta)},\qquad p\geq 2,$ (3.81)
$\displaystyle\sum_{r+k\leq N_{0}-5}{\|c_{3}(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle t\rangle^{\delta},\qquad
p\geq 2.$ (3.82)

In fact each $c_{3}$ we consider is just a component of $\nabla h$ and so
these bounds follow directly from (3.8)-(3.9).

In view of the above reductions, we see that in order to prove the desired
bounds (3.75) and (3.76), it suffices to show that for coefficients $c,c_{3}$
satisfying the above bounds, we have

$\displaystyle\sum_{\begin{subarray}{c}r+k\leq N_{1}-10\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}c(\nabla
h)\alpha(0)\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{W^{N_{1}-10}},$
(3.83) $\displaystyle\sum_{\begin{subarray}{c}r+k\leq N_{1}-10\\\
a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}c_{3}(\nabla
h)\nabla_{x,z}\alpha(0)\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{W^{N_{1}-10}},$
(3.84)

and

$\displaystyle\sum_{\begin{subarray}{c}r+k\leq N_{1}+12\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}c(\nabla
h)\alpha(0)\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{W^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{W^{N_{1}-10}},$ (3.85)
$\displaystyle\sum_{\begin{subarray}{c}r+k\leq N_{1}+12\\\
a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}c_{3}(\nabla
h)\nabla_{x,z}\alpha(0)\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{W^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{W^{N_{1}-10}}.$ (3.86)

Before proving the above estimates, we record a simple but useful product
estimate that we are going to use repeatedly below:

$\displaystyle{\big{\|}|\nabla|^{1/2}(fg)\big{\|}}_{L^{2}}\lesssim{\|f\|}_{W^{1,3}}{\big{\|}|\nabla|^{1/2}g\big{\|}}_{L^{2}},$
(3.87)

see Lemma A.5. In what follows $g$ will essentially play the role of
$\alpha(0)$, and $f$ will be nonlinear expressions in $h$ and its derivatives.

Proof of (3.83). Distributing vector fields using also (C.23), and applying
the estimate (3.87), we can bound

$\displaystyle\sum_{\begin{subarray}{c}r+k\leq N_{1}-10\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}c(\nabla h)\alpha(0)\|}_{Z^{r}_{k}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}r+k\leq
N_{1}-3\end{subarray}}{\|c(\nabla
h)\|}_{Z^{r,3}_{k}}\sum_{\begin{subarray}{c}r+k\leq N_{1}-10\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}\alpha(0)\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$

having used (3.78) to control the coefficient.

Proof of (3.84). Due to the possibly singular factor of $|\nabla|^{-1/2}$,
here we distinguish the cases $a=0$ and $a=1$. If $a=0$ we first apply
fractional integration followed by (A.21):

$\displaystyle\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\end{subarray}}{\||\nabla|^{-1/2}c_{3}(\nabla
h)\nabla_{x,z}\alpha(0)\|}_{Z^{r}_{k}}\lesssim\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\end{subarray}}{\|c_{3}(\nabla
h)\nabla_{x,z}\alpha(0)\|}_{Z^{r,4/3}_{k}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\end{subarray}}{\|c_{3}(\nabla
h)\|}_{Z^{r,4}_{k}}\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\end{subarray}}{\|\nabla_{x,z}\alpha(0)\|}_{Z^{r}_{k}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$

having used (3.81) for the coefficient, and (3.52). When $a=1$ we estimate
using (3.87):

$\displaystyle\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\end{subarray}}{\||\nabla|^{1/2}c_{3}(\nabla
h)\nabla_{x,z}\alpha(0)\|}_{Z^{r}_{k}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}r+k\leq
N_{1}-4\end{subarray}}{\|c_{3}(\nabla
h)\|}_{Z^{r,3}_{k}}\sum_{\begin{subarray}{c}r+k\leq
N_{1}-10\end{subarray}}{\||\nabla|^{1/2}\nabla_{x,z}\alpha(0)\|}_{Z^{r}_{k}}\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}.$

Proof of (3.85). Distributing vector fields we can estimate

$\displaystyle\sum_{r+k\leq N_{1}+12}{\||\nabla|^{1/2+a}c(\nabla
h)\alpha(0)\|}_{Z^{r}_{k}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|r_{1}|+|k_{1}|\leq n_{1}\\\
|r_{2}|+|k_{2}|\leq
n_{2}\end{subarray}}{\big{\|}|\nabla|^{1/2+a}\big{(}\nabla^{r_{1}}\Gamma^{k_{1}}c(\nabla
h)\,\nabla^{r_{2}}\Gamma^{k_{2}}\alpha(0)\big{)}\big{\|}}_{L^{2}}:=M_{n_{1},n_{2}},$

where $n_{1}+n_{2}=N_{1}+12$, and we do not make explicit the dependence on
$a=0,1$ which is unimportant here. We distinguish two cases depending which of
the indexes $n_{1}$ and $n_{2}$ is smaller. If $n_{1}\leq N_{1}-15$ we use
(3.87) to estimate

$\displaystyle M_{n_{1},n_{2}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N_{1}-13}{\|\nabla^{r_{1}}\Gamma^{k_{1}}c(\nabla
h)\|}_{L^{3}}\sum_{\begin{subarray}{c}|r_{2}|+|k_{2}|\leq N_{1}+12\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}\nabla^{r_{2}}\Gamma^{k_{2}}\alpha(0)\|}_{L^{2}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}$

having used (3.81) to estimate the coefficient. If instead $n_{2}\leq
N_{1}-15$, using again (3.87), followed by (3.82), we get

$\displaystyle M_{n_{1},n_{2}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N_{1}+13}{\|\nabla^{r_{1}}\Gamma^{k_{1}}c(\nabla
h)\|}_{L^{3}}\sum_{\begin{subarray}{c}|r_{2}|+|k_{2}|\leq N_{1}-15\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}\nabla^{r_{2}}\Gamma^{k_{2}}\alpha(0)\|}_{L^{2}}$
$\displaystyle\lesssim\varepsilon_{0}\langle
t\rangle^{\delta}\cdot{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}.$

These last two bounds above give (3.85).

Proof of (3.86). Distributing vector fields we have, for $a=0,1$,

$\displaystyle\sum_{r+k\leq
N_{1}+12}{\big{\|}|\nabla|^{-1/2+a}\big{(}c_{3}(\nabla
h)\nabla_{x,z}\alpha(0)\big{)}\big{\|}}_{Z^{r}_{k}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|r_{1}|+|k_{1}|\leq n_{1}\\\
|r_{2}|+|k_{2}|\leq
n_{2}\end{subarray}}{\big{\|}|\nabla|^{-1/2+a}\big{(}\nabla^{r_{1}}\Gamma^{k_{1}}c_{3}(\nabla
h)\cdot\nabla^{r_{2}}\Gamma^{k_{2}}\nabla_{x,z}\alpha(0)\big{)}\big{\|}}_{L^{2}}:=M_{n_{1},n_{2}}^{a},$

where $n_{1}+n_{2}=N_{1}+12$.

We look at the case $a=0$ first, apply fractional integration as before and
then Hölder to bound first

$\displaystyle M_{n_{1},n_{2}}^{0}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
n_{1}}{\|\nabla^{r_{1}}\Gamma^{k_{1}}c_{3}(\nabla
h)\|}_{L^{4}}\sum_{|r_{2}|+|k_{2}|\leq
n_{2}}{\|\nabla^{r_{2}}\Gamma^{k_{2}}\nabla_{x,z}\alpha(0)\|}_{L^{2}};$

then, when $n_{1}\leq N_{1}-15$ we use (3.81) and (3.52) to obtain
$M_{n_{1},n_{2}}^{0}\lesssim\varepsilon_{0}{\|\alpha\|}_{W^{N_{1}+12}}$; when,
instead, $n_{2}\leq N_{1}-15$ we use (3.82) to obtain
$M_{n_{1},n_{2}}^{0}\lesssim\varepsilon_{0}\langle
t\rangle^{\delta}\cdot{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}$.

In the case $a=1$ we can use the product estimate (3.87) to see that

$\displaystyle M_{n_{1},n_{2}}^{1}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
n_{1}+1}{\|\nabla^{r_{1}}\Gamma^{k_{1}}c_{3}(\nabla
h)\|}_{L^{3}}\sum_{|r_{2}|+|k_{2}|\leq
n_{2}}{\||\nabla|^{1/2}\nabla^{r_{2}}\Gamma^{k_{2}}\nabla_{x,z}\alpha(0)\|}_{L^{2}};$

then, for $n_{1}\leq N_{1}-15$ we use (3.81) and (3.52) to bound
$M_{n_{1},n_{2}}^{1}\lesssim\varepsilon_{0}{\|\alpha\|}_{W^{N_{1}+12}}$, and
for $n_{2}\leq N_{1}-15$ we use (3.82) and (3.52) to get
$M_{n_{1},n_{2}}^{1}\lesssim\varepsilon_{0}\langle
t\rangle^{\delta}\cdot{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}$.

This concludes the proof the bounds (3.75) and (3.76).

####  3.4.3. Bounds for the nonlinear bulk terms

To estimate the nonlinear expressions in the bulk integrals on the right-hand
side of (3.47)-(3.48) we proceed similarly to the above, this time using the
bounds in Lemma 3.13 and Remark 3.14 first, and then product estimates in
weighted spaces. Let us denote

$\displaystyle\begin{split}N_{i}^{a}(\alpha)(z)&:=\int_{-\infty}^{0}\big{(}e^{(z+s)|\nabla|}-e^{-|z-s||\nabla|}\mathrm{sign}(z-s)\big{)}E^{a}_{i}(s)\,ds,\\\
N_{i}^{b}(\alpha)(z)&:=\int_{-\infty}^{0}\big{(}e^{(z+s)|\nabla|}-e^{-|z-s||\nabla|}\big{)}E^{b}_{i}(s)\,ds,\qquad
i=1,2,3,\end{split}$ (3.88)

with $E^{a}$ and $E^{b}$ defined in (3.43). We then want to show, for
$i=1,2,3$,

$\displaystyle{\|N_{i}^{\ast}(\alpha)\|}_{\dot{Y}^{N_{1}-10}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$
(3.89) $\displaystyle{\|N_{i}^{\ast}(\alpha)\|}_{\dot{Y}^{N_{1}+12}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},\qquad\ast\in\\{a,b\\},$
(3.90)

consistently with (3.58) and (3.59).

We start by noting that the $N_{i}^{\ast}$ can be written in terms of the
operators $T_{1},T_{2},T_{3}$ in (3.65):

$N_{i}^{a}(\alpha)=T_{1}(E^{a})-T_{2}(E^{a})+T_{3}(E^{a}),\qquad
N_{i}^{b}(\alpha)=T_{1}(E^{b})+T_{2}(E^{b})+T_{3}(E^{b}).$

Then, from the definition of the $\dot{Y}^{n}$ norm in (3.22)-(3.23) the
estimates (3.64), and (3.69)-(3.70), for $\ast\in\\{a,b\\}$, we have

$\displaystyle{\big{\|}N_{i}^{\ast}(\alpha)\|}_{\dot{Y}^{n}}$
$\displaystyle=\sum_{\begin{subarray}{c}|r|+|k|\leq n\\\ 0\leq a\leq
1\end{subarray}}{\big{\|}\nabla_{x,z}^{a}|\nabla|^{1/2}\underline{\Gamma}^{k}\nabla^{r}_{x,z}N_{i}^{\ast}(\alpha)\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}+{\big{\|}\nabla_{x,z}^{a}\nabla_{x,z}\underline{\Gamma}^{k}\nabla^{r}_{x,z}N_{i}^{\ast}(\alpha)\big{\|}}_{L^{2}_{z}L^{2}_{x}}$
$\displaystyle+{\big{\|}\partial_{z}\underline{\Gamma}^{k}\nabla_{x,z}^{r}N_{i}^{\ast}(\alpha)\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|r|+|k|\leq n\\\
a=0,1\end{subarray}}{\big{\|}|\nabla|^{a}\,\underline{\Gamma}^{k}\nabla^{r}_{x,z}E^{\ast}_{i}\big{\|}}_{L^{2}_{z}L^{2}_{x}}+\sum_{|r|+|k|\leq
n}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k}\nabla^{r}_{x,z}E^{\ast}_{i}\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}.$
(3.91)

Therefore, in view of the definitions (3.43), and the commutation identity
(C.23) to handle the Riesz transform in front of $E^{a}$, for (3.89)-(3.90) it
suffices to prove the following bounds

$\displaystyle\sum_{\begin{subarray}{c}|r|+|k|\leq N_{1}-10\\\ |\ell|\leq
1\end{subarray}}{\big{\|}|\nabla|^{\ell}\underline{\Gamma}^{k}\nabla_{x,z}^{r}c(\nabla
h)\nabla_{x,z}\alpha\big{\|}}_{L^{2}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$
(3.92) $\displaystyle\sum_{\begin{subarray}{c}|r|+|k|\leq N_{1}+12\\\
|\ell|\leq
1\end{subarray}}{\big{\|}|\nabla|^{\ell}\underline{\Gamma}^{k}\nabla_{x,z}^{r}c(\nabla
h)\nabla_{x,z}\alpha\big{\|}}_{L^{2}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$ (3.93)

and

$\displaystyle\sum_{|r|+|k|\leq
N_{1}-10}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k}\nabla_{x,z}^{r}c(\nabla
h)\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$
(3.94) $\displaystyle\sum_{|r|+|k|\leq
N_{1}+12}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k}\nabla_{x,z}^{r}c(\nabla
h)\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$ (3.95)

where $c(\nabla h)$ is a component of $\nabla h$ or is $|\nabla h|^{2}$ so
that, in particular, it satisfies

$\displaystyle\sum_{r+k\leq N_{1}-1}{\|c(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1+(2/p)(1+\delta)},$ (3.96) $\displaystyle\sum_{r+k\leq
N_{0}-3}{\|c(\nabla h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle
t\rangle^{\delta},\qquad p\geq 2,$ (3.97)

in view of (3.8)-(3.9).

Proof of (3.92). Distributing vector fields and using Hölder we can simply
bound the left-hand side of (3.92) by

$\displaystyle\sum_{|r|+|k|\leq
N_{1}-5}{\big{\|}\langle\nabla\rangle\Gamma^{k}\nabla^{r}c(\nabla
h)\big{\|}}_{L^{\infty}_{x}}\sum_{|r|+|k|\leq
N_{1}-5}{\big{\|}\langle\nabla\rangle\underline{\Gamma}^{k}\nabla^{r}\nabla_{x,z}\alpha\big{\|}}_{L^{2}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-5}}$

in view of (3.96) and the definition of the $\dot{Y}^{n}$ norm.

Proof of (3.93). Distributing vector fields we see that the left-hand side of
(3.93) is bounded by the terms

$\displaystyle\sum_{\begin{subarray}{c}|r_{1}|+|k_{1}|\leq n_{1}\\\
|r_{2}|+|k_{2}|\leq
n_{2}\end{subarray}}{\big{\|}\nabla_{x}^{\ell}\big{(}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\cdot\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{)}\big{\|}}_{L^{2}_{x,z}}:=B_{n_{1},n_{2}}^{\ell},$
(3.98)

with $n_{1}+n_{2}=N_{1}+12$ and $\ell=0,1$. In the case $n_{1}\leq N_{1}-15$
we can bound (3.98) by

$\displaystyle B_{n_{1},n_{2}}^{\ell}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N_{1}-14}{\big{\|}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\big{\|}}_{L^{\infty}_{x}}\sum_{|r_{2}|+|k_{2}|\leq
N_{1}+12}{\big{\|}\langle\nabla\rangle\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{\|}}_{L^{2}_{x,z}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}},$

having used (3.96). When instead $n_{2}\leq N_{1}-15$ we can bound similarly

$\displaystyle B_{n_{1},n_{2}}^{\ell}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N_{1}+12}{\big{\|}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\big{\|}}_{L^{\infty}_{x}}\sum_{|r_{2}|+|k_{2}|\leq
N_{1}-10}{\big{\|}\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{\|}}_{L^{2}_{x,z}}$
$\displaystyle\lesssim\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$

having used (3.97).

Proof of (3.94). Distributing vector fields and using (3.87) we can bound the
left-hand side of (3.94) by

$\displaystyle\sum_{|r|+|k|\leq N_{1}-9}{\big{\|}\Gamma^{k}\nabla^{r}c(\nabla
h)\big{\|}}_{L^{3}_{x}}\sum_{|r|+|k|\leq
N_{1}-10}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}$

where we have used (3.52) for the last inequality.

Proof of (3.95). Finally, we examine the left-hand side of (3.95), distribute
the vector fields and estimate it, using again (3.87), by a linear combination
of the terms

$\displaystyle\begin{split}&\sum_{\begin{subarray}{c}r_{1}+|k_{1}|\leq
n_{1}\\\ r_{2}+|k_{2}|\leq
n_{2}\end{subarray}}{\big{\|}\langle\nabla\rangle^{1/2}\big{(}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\cdot\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{)}\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}\\\
&\lesssim\sum_{r_{1}+|k_{1}|\leq
n_{1}+1}{\big{\|}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\big{\|}}_{L^{3}_{x}}\sum_{r_{2}+|k_{2}|\leq
n_{2}}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}:=C_{n_{1},n_{2}}.\end{split}$
(3.99)

Then, in the case $n_{1}\leq N_{1}-15$ we can bound (3.99) as follows:

$\displaystyle C_{n_{1},n_{2}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N_{1}-14}{\big{\|}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\big{\|}}_{L^{3}_{x}}\sum_{|r_{2}|+|k_{2}|\leq
N_{1}+12}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+12}},$

having used (3.96). When instead $n_{2}\leq N_{1}-15$ we estimate

$\displaystyle C_{n_{1},n_{2}}\lesssim\sum_{|r_{1}|+|k_{1}|\leq
N_{1}+12}{\big{\|}\Gamma^{k_{1}}\nabla^{r_{1}}c(\nabla
h)\big{\|}}_{L^{3}_{x}}\sum_{|r_{2}|+|k_{2}|\leq
N_{1}-10}{\big{\|}\langle\nabla\rangle^{1/2}\underline{\Gamma}^{k_{2}}\nabla^{r_{2}}_{x,z}\nabla_{x,z}\alpha\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}},$

having used (3.97) and (3.52).

####  3.4.4. Bounds for the linear forcing terms

We now estimate the forcing term involving the vorticity on the right-hand
side of (3.47)-(3.48). These are (up to constants) given by

$\displaystyle
T_{3}F=\int_{-\infty}^{0}e^{(z+s)|\nabla|}|\nabla|^{-1}F(s)\,ds,\quad\mbox{or}\quad(T_{1}+T_{2})F=\int_{-\infty}^{0}e^{-|z-s||\nabla|}|\nabla|^{-1}F(s)\,ds,$
(3.100)

recall $F=W$ and the notation in (3.65). From the identities (3.67) we have

$\partial_{z}T_{3}=|\nabla|T_{3},\qquad\partial_{z}(T_{1}+T_{2})=|\nabla|(T_{1}-T_{2}).$
(3.101)

Let $P_{>0}$ and $P_{\leq 0}$ denote the standard Littlewood-Paley projections
(in the $x$ variable) defined according to (2.56)-(2.57), and note that they
commute with the operators in (3.100) above. Let us denote by $G$ any of the
two expressions in (3.100). Using the bound in (3.64) by the first argument on
the right-hand side, recalling the definition (3.53) and (3.22)-(3.23), and
using Remark 3.14 and (3.101) to handle the $z$-derivatives, we have

$\displaystyle\begin{split}{\big{\|}P_{>0}G\|}_{\dot{Y}^{n}}\lesssim\sum_{|r|+2|k|\leq
n}{\big{\|}\langle\nabla\rangle\underline{\Gamma}^{k}\nabla_{x,z}^{r}P_{>0}(|\nabla|^{-1}F(s))\big{\|}}_{L^{2}_{z}L^{2}_{x}}\lesssim{\|W\|}_{\mathcal{X}^{n}}.\end{split}$
(3.102)

For small frequencies, we instead use the bound in (3.64) by the second
argument on the right-hand side, followed by fractional integration, and
obtain

$\displaystyle\begin{split}{\big{\|}P_{\leq
0}G\|}_{\dot{Y}^{n}}&\lesssim\sum_{|r|+2|k|\leq
n}{\big{\|}\langle\nabla\rangle\underline{\Gamma}^{k}\nabla^{r}_{x,z}\,|\nabla|^{1/3}P_{\leq
0}(|\nabla|^{-1}F(s))\big{\|}}_{L^{6/5}_{z}L^{2}_{x}}\\\
&\lesssim\sum_{|r|+2|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\,|\nabla|^{-2/3}W\big{\|}}_{L^{6/5}_{z}L^{2}_{x}}\\\
&\lesssim\sum_{|r|+2|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}W\big{\|}}_{L^{6/5}_{x,z}}\lesssim{\|W\|}_{\mathcal{X}^{n}}.\end{split}$
(3.103)

Using (3.102) and (3.103) with $n=N_{1}-10$ and $n=N_{1}+12$, we get bounds
consistent with the desired inequalities (3.58) and (3.59). The proof of
(3.58)-(3.59) is thus concluded.

####  3.4.5. Proof of the bounds (3.60)-(3.61)

We now prove the bounds for the time derivatives of the map $L(\alpha)$ in
Lemma 3.9. These can be obtained in the same way as the bounds (3.58)-(3.59)
proved above, using in addition the bounds on $\partial_{t}h$ from (3.11) and
(3.12). We give some details for completeness. Let us define the map $\dot{L}$
through the identity

$\displaystyle\partial_{t}L(\alpha)=L(\partial_{t}\alpha)+\dot{L}(\alpha);$
(3.104)

Under the assumptions (3.26)-(3.25) the same exact arguments above give that

$\displaystyle\begin{split}{\|L(\partial_{t}\alpha)\|}_{\dot{Y}^{N_{1}-11}}&\lesssim\varepsilon_{0}{\|\partial_{t}\alpha\|}_{\dot{Y}^{N_{1}-11}}+{\|\partial_{t}W\|}_{\mathcal{X}^{N_{1}-11}},\\\
{\|L(\partial_{t}\alpha)\|}_{\dot{Y}^{N_{1}+11}}&\lesssim\varepsilon_{0}{\|\partial_{t}\alpha\|}_{\dot{Y}^{N_{1}+11}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\partial_{t}\alpha\|}_{\dot{Y}^{N_{1}-11}}+{\|\partial_{t}W\|}_{\mathcal{X}^{N_{1}+11}}.\end{split}$
(3.105)

Therefore, it suffices to prove that

$\displaystyle\begin{split}{\|\dot{L}(\alpha)\|}_{\dot{Y}^{N_{1}-11}}&\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-11}},\\\
{\|\dot{L}(\alpha)\|}_{\dot{Y}^{N_{1}+11}}&\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+11}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-10}}.\end{split}$ (3.106)

By definition the map $\dot{L}$ is given by the right-hand side of (3.47) and
(3.48) with $F=0$, and where we replace $(E_{i}^{a},E_{i}^{b})$ and
$(B_{1},B_{2},B_{3,a},B_{3,b})$ by new quantities
$(\dot{E}_{i}^{a},\dot{E}_{i}^{b})$ and
$(\dot{B}_{1},\dot{B}_{2},\dot{B}_{3,a},\dot{B}_{3,b})$ defined by
differentiating the coefficients that multiply $\alpha$; in other words, for
$X(\alpha)=E_{i}^{a}(\alpha),E_{i}^{b}(\alpha),B_{1}(\alpha)$ and so on, we
define

$\displaystyle\dot{X}(\alpha)=\partial_{t}X(\alpha)-X(\partial_{t}\alpha).$

More explicitly, we have

$\displaystyle\begin{split}\dot{E}^{a}(\alpha)&:=\frac{\nabla}{|\nabla|}\cdot(\partial_{t}\nabla
h)\,\partial_{z}\alpha\qquad\dot{E}^{b}(\alpha):=\big{(}-\partial_{t}|\nabla
h|^{2}\big{)}\,\partial_{z}\alpha+(\partial_{t}\nabla
h)\cdot\nabla\alpha,\end{split}$ (3.107)

and

$\displaystyle\begin{split}\dot{B}_{i}(\alpha)&:=\partial_{t}\big{[}(1+|\nabla
h|^{2})\partial_{i}h\big{]}\alpha_{3}(0)-\partial_{t}\big{[}(1+|\nabla
h|^{2})\partial_{i}h\nabla h\big{]}\cdot\alpha(0),\qquad i=1,2,\\\
\dot{B}_{3,a}(\alpha)&:=\frac{\nabla}{|\nabla|}\cdot\partial_{t}\big{(}\nabla
h(1+|\nabla h|^{2})^{-1/2}\big{)}\,\alpha_{3}(0).\end{split}$ (3.108)

For the last boundary term, that is,
$\dot{B}_{3,b}(\alpha)=\partial_{t}B_{3,b}(\alpha)-B_{3,b}(\partial_{t}\alpha)$,
we use its schematic representation from (3.80) to write it as a linear
combination of terms of the form

$\displaystyle b_{3}(\nabla h,\nabla^{2}h,\alpha)$
$\displaystyle:=\partial_{t}c_{3}(\nabla h)\,\nabla_{x,z}\alpha(0),$ (3.109)

with the natural definitions of the coefficients $c_{3}$ according to the
formula (3.50). In particular, we can verify that the following analogues of
(3.77)-(3.82) hold:

$-$ The terms $\dot{B}_{1},\dot{B}_{2}$ and $\dot{B}_{3,a}$ are linear
combinations of terms of the form

$\displaystyle\partial_{t}c(\nabla h)\,\alpha_{j}(0),\quad j=1,2,3,$ (3.110)

with

$\displaystyle\sum_{r+k\leq N_{1}-6}{\|\partial_{t}c(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0},\qquad p\geq 3,$ (3.111)
$\displaystyle\sum_{r+k\leq N_{0}-8}{\|\partial_{t}c(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle t\rangle^{\delta},\qquad
p\geq 2;$ (3.112)

to check the above bounds one can just proceed as in the proofs of
(3.78)-(3.79), using Lemma A.4, and also the bounds for $\partial_{t}h$ in
(3.11)-(3.12) besides the usual (3.8)-(3.9).

$-$ The term $\dot{B_{3,b}}(\alpha)$ is a linear combination of terms of the
form (3.109) with

$\displaystyle\sum_{r+k\leq N_{1}-8}{\|\partial_{t}c_{3}(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle
t\rangle^{-3/4+(2/p)(1+\delta)},\qquad p\geq 3,$ (3.113)
$\displaystyle\sum_{r+k\leq N_{0}-10}{\|\partial_{t}c_{3}(\nabla
h)\|}_{Z^{r,p}_{k}}\lesssim\varepsilon_{0}\langle t\rangle^{\delta},\qquad
p\geq 2;$ (3.114)

the above bounds are analogous to (3.81) and (3.82) and can be obtained in the
same way using in addition (3.11)-(3.12). Similarly to before, for all
practical purposes one may think that $c$ and $c_{3}$ are both just $\nabla
h$.

With the formulas (3.107), (3.110) and (3.109), and the estimate
(3.111)-(3.112) and (3.113)-(3.114) we can then proceed in a way completely
analogous to Subsections 3.4.2 and 3.4.3. More precisely, as in Subsection
3.4.2 we can use (3.62), and reduce matters to showing the analogues of
(3.75)-(3.76) for the dotted quantities, that is, we want to show

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}r+k\leq N_{1}-11\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}(\dot{B}_{1},\dot{B}_{2},\dot{B}_{3,a})\|}_{Z^{r}_{k}}+\sum_{\begin{subarray}{c}r+k\leq
N_{1}-11\\\
a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}\dot{B}_{3,b}\|}_{Z^{r}_{k}}\\\
\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-11}},\end{split}$ (3.115)

and

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}r+k\leq N_{1}+11\\\
a=0,1\end{subarray}}{\||\nabla|^{1/2+a}(\dot{B}_{1},\dot{B}_{2},\dot{B}_{3,a})\|}_{Z^{r}_{k}}+\sum_{\begin{subarray}{c}r+k\leq
N_{1}+11\\\
a=0,1\end{subarray}}{\||\nabla|^{-1/2+a}\dot{B}_{3,b}\|}_{Z^{r}_{k}}\\\
\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+11}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-11}}.\end{split}$ (3.116)

The proofs of (3.115) and (3.116) can then be obtained in the same exact way
as the proof of the bounds (3.75) and (3.76), through analogues of
(3.83)-(3.86) where the coefficients $c$ and $c_{3}$ are replaced by
$\partial_{t}c$ and $\partial_{t}c_{3}$, and using (3.111)-(3.112) and
(3.113)-(3.114); we omit the details.

Proceeding as in Subsection 3.4.3 we can let

$\displaystyle\begin{split}\dot{N}_{i}(\alpha)(z)&:=\int_{-\infty}^{0}e^{(z+s)|\nabla|}(\dot{E}^{a}_{i}(s)-\dot{E}^{b}_{i}(s))\,ds\\\
&+\int_{-\infty}^{0}e^{-|z-s||\nabla|}(\mathrm{sign}(s-z)\dot{E}^{a}_{i}(s)-\dot{E}^{b}_{i}(s))\,ds,\qquad
i=1,2,3,\end{split}$ (3.117)

and obtain the analogues of (3.89) and (3.90) using the estimates in Lemma
3.13 and Remark 3.14, the product estimate from Lemma A.4, (3.87), and the
bounds for $h$ and $\partial_{t}h$ in (3.9) and (3.12):

$\displaystyle{\|\dot{N}_{i}(\alpha)\|}_{\dot{Y}^{N_{1}-11}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}-11}},$
$\displaystyle{\|\dot{N}_{i}(\alpha)\|}_{\dot{Y}^{N_{1}+11}}$
$\displaystyle\lesssim\varepsilon_{0}{\|\alpha\|}_{\dot{Y}^{N_{1}+11}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha\|}_{\dot{Y}^{N_{1}-11}}.$

These are consistent with the desired (3.106). and conclude the proof of
(3.60)-(3.61).

####  3.4.6. Conclusion

Finally, we show how the main bounds (3.58)-(3.59) and (3.60)-(3.61) imply
Proposition 3.12. Consider the sequence

$\alpha^{(0)}=0,\qquad\alpha^{(m+1)}=L(\alpha^{(m)}),\qquad m\geq 0.$

so that, under the assumptions (3.54)-(3.55), the bounds (3.58)–(3.59) read

$\displaystyle\begin{split}{\|\alpha^{(m+1)}\|}_{\dot{Y}^{N_{1}-10}}&\lesssim\varepsilon_{0}{\|\alpha^{(m)}\|}_{\dot{Y}^{N_{1}-10}}+\varepsilon_{1}\\\
{\|\alpha^{(m+1)}\|}_{\dot{Y}^{N_{1}+12}}&\lesssim\varepsilon_{0}{\|\alpha^{(m)}\|}_{\dot{Y}^{N_{1}+12}}+\varepsilon_{0}\langle
t\rangle^{\delta}{\|\alpha^{(m)}\|}_{\dot{Y}^{N_{1}-10}}+\varepsilon_{1}\langle
t\rangle^{\delta},\end{split}$ (3.118)

and (3.60)–(3.61) read

$\displaystyle\begin{split}{\|\partial_{t}\alpha^{(m+1)}\|}_{\dot{Y}^{N_{1}-11}}&\lesssim\varepsilon_{0}\big{(}{\|\alpha^{(m)}\|}_{\dot{Y}^{N_{1}-10}}+{\|\partial_{t}\alpha^{(m)}\|}_{\dot{Y}^{N_{1}-11}}\big{)}+\varepsilon_{1}\\\
{\|\partial_{t}\alpha^{(m+1)}\|}_{\dot{Y}^{N_{1}+11}}&\lesssim\varepsilon_{0}{\|\partial_{t}\alpha^{(m)}\|}_{\dot{Y}^{N_{1}+11}}+\varepsilon_{0}\langle
t\rangle^{\delta}\big{(}{\|\alpha^{(m)}\|}_{\dot{Y}^{N_{1}-10}}+{\|\partial_{t}^{j}\alpha^{(m)}\|}_{\dot{Y}^{N_{1}-11}}\big{)}+\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta}.\end{split}$ (3.119)

From these we see that, for $\varepsilon_{0}$ small enough, and all $t\leq T$,

$\displaystyle\begin{split}{\|\alpha^{(m)}(t)\|}_{\dot{Y}^{N_{1}-10}}&\lesssim\varepsilon_{1},\qquad{\|\alpha^{(m)}(t)\|}_{\dot{Y}^{N_{1}+12}}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta},\\\
{\|\partial_{t}\alpha^{(m)}(t)\|}_{\dot{Y}^{N_{1}-11}}&\lesssim\varepsilon_{1},\qquad{\|\partial_{t}\alpha^{(m)}(t)\|}_{\dot{Y}^{N_{1}+11}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta},\end{split}$ (3.120)

Moreover, since $L$ is linear in $\alpha$, we also have that, for $j=0,1$

$\displaystyle{\big{\|}\partial_{t}^{j}\big{(}L(\alpha_{1})-L(\alpha_{2})\big{)}\big{\|}}_{\dot{Y}^{N_{1}-10-j}}\lesssim\varepsilon_{0}\big{(}{\|\alpha_{1}-\alpha_{2}\|}_{\dot{Y}^{N_{1}-10}}+{\|\partial_{t}(\alpha_{1}-\alpha_{2})\|}_{\dot{Y}^{N_{1}-11}}\big{)},$
(3.121)

so that $L$ is a contraction in a ball of radius $C\varepsilon_{1}$, with some
absolute constant $C$, in the space $C^{0}([0,T],\dot{Y}^{N_{1}-10})\cap
C^{1}([0,T],\dot{Y}^{N_{1}-11})$. Let us denote by $\alpha$ the unique fixed
point of $L$ in this space; we have for $j=0,1$,

$\displaystyle{\|\partial_{t}^{j}\alpha(t)\|}_{\dot{Y}^{N_{1}-10-j}}\lesssim\varepsilon_{1}$
(3.122)

for all $t\leq T$. In addition, from (3.120), we have
${\|\partial_{t}^{j}\alpha^{(m)}(t)\|}_{\dot{Y}^{N_{1}+12-j}}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta}\varepsilon_{0}^{j}$ and therefore, up to passing to a sub-
sequence, we get that $\partial_{t}^{j}\alpha^{(m)}(t)$ converges weak-$\ast$
in $\dot{Y}^{N_{1}+12-j}$ to a limit $\alpha_{j}^{\prime}(t)$, $j=0,1$. Then
we have $\partial_{t}\alpha(t)=\alpha_{j}^{\prime}(t)$ for all $t\leq T$, and
by lower semi-continuity

$\displaystyle{\|\alpha(t)\|}_{\dot{Y}^{N_{1}+12}}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta},\qquad{\|\partial_{t}\alpha(t)\|}_{\dot{Y}^{N_{1}+11}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta}.$ (3.123)

With (3.122) and (3.123) we conclude the proof of Proposition 3.12
$\hfill\Box$

##  4\. Estimates for the vorticity

In this section we bootstrap weighted bounds for the vorticity in the three
dimensional (flat) domain proving the main Proposition 2.15. For the
convenience of the reader, and ease of reference, we restate this result below
as Proposition 4.1. In particular this will prove the validity of the
assumptions (3.54)-(3.55) used in Proposition 3.12 to obtain (3.56)-(3.57),
which in turn give us bounds on the vector potential $V_{\omega}$ and its
restriction to the boundary $\widetilde{v_{\omega}}$ (see Lemma 3.7 and the
conclusions of Proposition 2.14).

Let us recall here some of our notation: with $v$ the velocity field,
$\omega=\operatorname{curl}v$, we let, for $x\in{\mathbb{R}}^{2},z\leq 0$

$\displaystyle V(t,x,z):=v(t,x,z+h(t,x)),\qquad
W(t,x,z):=\omega(t,x,z+h(t,x)),$ (4.1)

and will generally use capital letters for quantities defined in the
transformed domain $\mathbb{R}^{2}\times\\{z\leq 0\\}$; accordingly, given the
Hodge decomposition (2.1) we let

$\displaystyle\Psi(t,x,z):=\psi(t,x,z+h(t,x)),\qquad
V_{\omega}(t,x,z):=v_{\omega}(t,x,z+h(t,x)).$ (4.2)

######  Proposition 4.1.

Assume that $h$ satisfies (3.8)-(3.9) and (3.11)-(3.12), and let $W$ be as
defined in (4.1). Let $\mathcal{X}^{n}$ be the space defined in (2.24), which
for convenience we recall here:

$\displaystyle\begin{split}{\|f\|}_{\mathcal{X}^{n}}:=\sum_{|r|+2|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\,f\big{\|}}_{L^{2}_{z}L^{2}_{x}\cap
L^{6/5}_{x,z}}.\end{split}$ (4.3)

Assume that, for all $t\in[0,T]$ for some $T\leq T_{\varepsilon_{1}}$, and for
$j=0,1$

$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\leq
2C\varepsilon_{1},$ (4.4)
$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\leq
2C\varepsilon_{0}^{j}\varepsilon_{1}\langle t\rangle^{\delta},$ (4.5)

for some absolute constant $C>0$ large enough, and where $\delta$ satisfies
(2.9). Then, for all $t\in[0,T_{\varepsilon_{1}}]$, we have the improved
bounds

$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}-10-j}}\leq
C\varepsilon_{1},$ (4.6)
$\displaystyle{\|\partial_{t}^{j}W(t)\|}_{\mathcal{X}^{N_{1}+12-j}}\leq
C\varepsilon_{0}^{j}\varepsilon_{1}\langle t\rangle^{\delta}.$ (4.7)

Notice that the a priori assumptions (4.4) and Sobolev’s embedding in $x$ and
$z$, imply

$\displaystyle\begin{split}\sum_{|r|+|k|\leq
N_{1}-15}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\partial_{t}^{j}W(t)\big{\|}}_{L^{\infty}_{x,z}}\leq
2C\varepsilon_{1}.\end{split}$ (4.8)

The proof of Proposition 4.1 will follow from estimates on the transport
equation satisfied by the vorticity in the transformed flat domain. Several
difficulties arise in trying to control this flow, in particular the fact that
the transport velocity is not integrable in time. We will explain in more
detail below how to resolve this by a normal form type argument in the full
three dimensional (transformed) fluid domain.

###  4.1. Vorticity equation and transport velocity

As a first step we write the transport of vorticity equation in the
transformed flat domain. We think of $\omega$ as a vector field letting
$\omega^{i}=\epsilon^{ijk}\partial_{j}v_{k}$ and from the vorticity equation
$(\partial_{t}+v^{\ell}\partial_{\ell})\omega^{i}=\omega^{\ell}\partial_{\ell}v^{i}$
and (4.1) we get

$\displaystyle\partial_{t}W^{i}-\partial_{t}h\partial_{z}W^{i}+V^{\ell}\partial_{\ell}W^{i}-(V^{\ell}\partial_{\ell}h)\partial_{z}W^{i}=W^{\ell}(\partial_{\ell}-\partial_{\ell}h\partial_{z})V^{i}.$
(4.9)

Above, and in what follows, we adopt the natural convention that
$\partial_{3}h=0$. We then write (4.9) as

$\displaystyle\begin{split}&{\bf D}_{t}W=W\cdot\nabla
V-W^{\ell}\partial_{\ell}h\partial_{z}V^{i}\\\ &{\bf
D}_{t}:=\partial_{t}+U\cdot\nabla,\qquad
U:=V-(\partial_{t}h+V^{\ell}\partial_{\ell}h)e_{z}\end{split}$ (4.10)

where $\nabla=\nabla_{x,z}$ and $e_{z}=(0,0,1)$.

Next, from (4.2) we write, for $x\in{\mathbb{R}}^{2},z\leq 0$,

$\displaystyle
V^{i}=\partial^{i}\Psi-\partial^{i}h\partial_{z}\Psi+V^{i}_{\omega}$ (4.11)

and then rewrite (4.10) as follows:

$\displaystyle\begin{split}{\bf D}_{t}W=W\cdot\nabla X+F,\qquad
X&:=V_{\omega}+\nabla\Psi,\\\
F^{i}&:=-W^{\ell}\partial_{\ell}h\partial_{z}V^{i}-W\cdot\nabla(\partial^{i}h\partial_{z}\Psi).\end{split}$
(4.12)

The system (4.12) is a transport equation for $W$ with a quadratic stretching
term $W\cdot\nabla X$ and an additional nonlinearity $F$ that contains only
cubic terms. Note that $W$ is transported by the vector field $U$ which has
some components that are not integrable in time, e.g. because of the presence
of the ‘dispersive’ components $\nabla\Psi$ and $\partial_{t}h$; at the same
time, some of the quadratic terms are also weakly decaying because of the
presence of $\nabla\Psi$ in $X$. In particular we cannot close a bootstrap
argument for norms of $W$ using (4.12) directly. Instead, we will need to use
the fact that $U$ and $X$ have additional structure. This is the content of
the next lemma.

######  Lemma 4.2.