The vector fields $U$ and $X$ defined respectively in (4.10) and (4.12) can be
written as

$\displaystyle U=\partial_{t}(A-h\,e_{z})+V_{\omega}-V_{\omega}\cdot\nabla
h\,e_{z}+R,$ (4.13) $\displaystyle
X=\partial_{t}A+V_{\omega}+R_{2},\qquad\mbox{with}\quad
A:=\nabla_{x,z}|\nabla|^{-1}e^{z|\nabla|}h,$ (4.14)

where $R$ is given by

$\displaystyle\begin{split}&R^{i}=R_{1}^{i}+R_{2}^{i},\\\
&R_{1}^{i}:=-\partial^{i}h\partial_{z}\Psi-(V^{\ell}-V_{\omega}^{\ell})\partial_{\ell}h\,e_{z}^{i},\\\
&R_{2}^{i}:=\partial_{i}e^{z|\nabla|}|\nabla|^{-1}\big{(}|\nabla|\varphi-G(h)\varphi\big{)}+\partial_{i}(\Psi-e^{z|\nabla|}\varphi),\qquad
i=1,2,3.\end{split}$ (4.15)

Recall that $\varphi(t,x)=\Psi(t,x,h(x))$.

More precise bounds on $U$ and $X$ are postponed for the moment (see, for
example, Lemma 4.9). The main point of the above lemma is that $U$ (and $X$)
can be written as a perfect time derivative of one of the dispersive
variables, plus terms that involve $V_{\omega}$ and other terms that will be
proven to have good time-integrability properties. In this whole section we
will distinguish and treat differently, the ‘dispersive variables’, e.g. $h$
and $\nabla\Psi$, which mainly play the role of coefficients, and the
rotational variables, e.g. $V_{\omega}$ and $W$.

###### Proof of Lemma 4.2.

From (4.10) and (4.11) we write

$\displaystyle\begin{split}&U^{i}=\partial^{i}\Psi-\partial_{t}h\,e_{z}^{i}+V^{i}_{\omega}-V_{\omega}^{\ell}\partial_{\ell}h\,e_{z}^{i}+R_{1}^{i},\\\
&R_{1}^{i}:=-\partial^{i}h\partial_{z}\Psi-(V^{\ell}-V_{\omega}^{\ell})\partial_{\ell}h\,e_{z}^{i}.\end{split}$
(4.16)

We recall that $\psi=\psi(t,x,y)$ is the harmonic extension of
$\varphi(t,x):=\psi(t,x,h(x))$ in the original domain, and that
$\partial_{t}h=G(h)\varphi=|\nabla|\varphi+\mbox{quadratic terms}$, and write,
for $z\leq 0$,

$\displaystyle\begin{split}\Psi&=e^{z|\nabla|}\varphi+(\Psi-e^{z|\nabla|}\varphi)\\\
&=e^{z|\nabla|}\partial_{t}|\nabla|^{-1}h+e^{z|\nabla|}|\nabla|^{-1}\big{(}|\nabla|\varphi-G(h)\varphi\big{)}+(\Psi-e^{z|\nabla|}\varphi).\end{split}$
(4.17)

Therefore,

$\displaystyle\begin{split}U^{i}=\partial_{t}\big{(}\partial^{i}e^{z|\nabla|}|\nabla|^{-1}h-h\,e_{z}^{i}\big{)}+\partial^{i}\big{[}e^{z|\nabla|}|\nabla|^{-1}\big{(}|\nabla|\varphi-G(h)\varphi\big{)}+(\Psi-e^{z|\nabla|}\varphi)\big{]}\\\
+V^{i}_{\omega}-V_{\omega}^{\ell}\partial_{\ell}h\,e_{z}^{i}+R_{1}^{i},\end{split}$

which is the desired conclusion (4.13)-(4.15).

The identity (4.14) is directly obtained from $X=\nabla\Psi+V_{\omega}$ and
the above formula (4.17) for $\Psi$. ∎

###  4.2. Vector fields and function classes

Our next task is to apply vector fields to the equation (4.12). To deal with
this and handle various identities and manipulations involving vector fields,
products, commutators etc. we introduce convenient shorthand notation below.
We then also define useful classes of functions satisfying linear and
quadratic bounds consistent with our energy and dispersive estimates.

####  4.2.1. Notation for vector fields

For
$\underline{\Gamma}:=(\partial_{x_{1}},\partial_{x_{2}},\partial_{z},\Omega,\underline{S})$
and for $\alpha\in{\mathbb{Z}}_{+}^{5}$, let

$\displaystyle\underline{\Gamma}^{\alpha}:=\partial_{x_{1}}^{\alpha_{1}}\cdot\partial_{x_{2}}^{\alpha_{2}}\cdot\partial_{z}^{\alpha_{3}}\cdot\Omega^{\alpha_{4}}\cdot\underline{S}^{\alpha_{5}}.$

For $n\in{\mathbb{Z}}_{+}$, we define the sets of vector fields of order $n$,
respectively, less or equal to $n$, by

$\displaystyle\mathcal{V}^{n}=\big{\\{}\underline{\Gamma}^{\alpha},\quad\alpha\in{\mathbb{Z}}_{+}^{5}\quad\mbox{with}\quad\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}=n\big{\\}},$
$\displaystyle\mathcal{V}^{\leq
n}=\big{\\{}\underline{\Gamma}^{\alpha},\quad\alpha\in{\mathbb{Z}}_{+}^{5}\quad\mbox{with}\quad\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}\leq
n\big{\\}}.$

$\mathcal{V}^{0}=\\{1\\}$, where $1$ is intended as a multiplication operator.

  * •

(Action of vector fields) Given a function $f$ we will denote by $\Gamma^{n}f$
a generic element of $\mathcal{V}^{n}f$, that is, $g=\Gamma^{n}f$ if there
exists $V\in\mathcal{V}^{n}$ such that $g=Vf$. We further denote with
$\Gamma^{\leq n}f$ a generic linear combination of elements in
$\mathcal{V}^{\leq n}f$, that is, $g=\Gamma^{\leq n}f$ if

$\displaystyle g=\sum_{V\in\mathcal{V}^{\leq n}}c_{V}\,Vf.$ (4.18)

for some real constants $c_{V}$. In particular, $\Gamma^{\leq n}f$ may also
denote a linear combination of element of $\mathcal{V}^{n}f$. We have the
following schematic identities:

$\displaystyle\Gamma^{n_{1}}(\Gamma^{n_{2}}f)=\Gamma^{n_{1}+n_{2}}f,\qquad\Gamma^{n_{1}}(\Gamma^{\leq
n_{2}}f),\,\Gamma^{\leq n_{1}}(\Gamma^{n_{2}}f)=\Gamma^{\leq n_{1}+n_{2}}f.$
(4.19)

Note that while the $3$d vector fields are denoted $\underline{\Gamma}$ using
an underline, we are not underlining the $\Gamma^{n}$ expressions for ease of
notation. This should generate no confusion since in this section we work
solely in the full $3$d (flattened) domain. When the $z$-independent function
$h$ is involved, the action of $\underline{\Gamma}$ is the obvious one, and we
adopt the same notation $\Gamma^{n}h$ and $\Gamma^{\leq n}h$.

  * •

(Products) For any $V\in\mathcal{V}^{n}$ one sees that

$\displaystyle
V(fg)=\sum_{\begin{subarray}{c}V_{1}\in\mathcal{V}^{n_{1}},\,V_{2}\in\mathcal{V}^{n_{2}}\\\
n_{1}+n_{2}=n\end{subarray}}c_{V_{1}V_{2}}\,V_{1}f\cdot V_{2}g$

for some real constants $c_{V_{1}V_{2}}=c_{V_{1}V_{2}}(V)$ determined by $V$.
We then adopt a short-hand notation for linear combinations of the form above
by omitting the constants and denoting a generic term in $\mathcal{V}^{n}(fg)$
as

$\displaystyle\Gamma^{n}(fg)=\sum_{n_{1}+n_{2}=n}\Gamma^{n_{1}}f\cdot\Gamma^{n_{2}}g.$
(4.20)

A linear combination of terms in $\mathcal{V}^{n}(fg)$ will also be denoted in
the same way. Similarly, we will write a generic linear combination of
elements of $\mathcal{V}^{\leq n}(fg)$, as

$\displaystyle\Gamma^{\leq n}(fg)=\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}f\cdot\Gamma^{n_{2}}g.$ (4.21)

We will also use a sum as in the right-hand side of (4.21) to denote a generic
linear combination (of linear combinations) of products of terms in
$\mathcal{V}^{n_{1}}$ and $\mathcal{V}^{n_{2}}$ for $n_{1}+n_{2}=0,\dots,n$.
Note that, by this convention, we can identify

$\displaystyle\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq
n_{1}}f\cdot\Gamma^{n_{2}}g=\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}f\cdot\Gamma^{n_{2}}g.$ (4.22)

  * •

(Commutators) From standard commutation relations one sees that for any
$V_{1}\in\mathcal{V}^{n_{1}}$ and $V_{2}\in\mathcal{V}^{n_{2}}$ the commutator
satisfies $[V_{1},V_{2}]=V^{\prime}$ where
$V^{\prime}\in\mathcal{V}^{n_{1}+n_{2}-1}$. Consistently with this and our
short-hand notation from above, we write

$\displaystyle[\Gamma^{\leq n_{1}},\Gamma^{\leq n_{2}}]=\Gamma^{\leq
n_{1}+n_{2}-1}$ (4.23)

to express the fact that the commutation of any linear combinations of vector
fields of order at most $n_{1}$ and $n_{2}$, gives a linear combination of
vector fields of order less or equal to $n_{1}+n_{2}-1$.

  * •

(Norms and estimates) Consistently with the above convention, for generic
terms $\Gamma^{n}f$ and $\Gamma^{\leq n}f$ we have

$\displaystyle|\Gamma^{n}f|\leq\sup_{V\in\mathcal{V}^{n}}|Vf|,\qquad|\Gamma^{\leq
n}f|\lesssim\sum_{V\in\mathcal{V}^{\leq n}}|Vf|.$ (4.24)

In particular,

$\displaystyle{\|\Gamma^{\leq
n}f\|}_{L^{q}_{z}L^{p}_{x}}\lesssim\sum_{V\in\mathcal{V}^{\leq
n}}{\|Vf\|}_{L^{q}_{z}L^{p}_{x}}\lesssim\sum_{\begin{subarray}{c}r,k\in{\mathbb{Z}}^{2}_{+}\\\
|r|+|k|\leq
n\end{subarray}}{\big{\|}(\partial_{x_{1}},\partial_{x_{2}})^{r}(\Omega,S)^{k}f\big{\|}}_{L^{q}_{z}L^{p}_{x}}.$
(4.25)

At the same time, we also have

$\displaystyle\sum_{\begin{subarray}{c}r,k\in{\mathbb{Z}}^{2}_{+}\\\
|r|+|k|\leq
n\end{subarray}}{\big{\|}(\partial_{x_{1}},\partial_{x_{2}})^{r}(\Omega,\underline{S})^{k}f\big{\|}}_{L^{q}_{z}L^{p}_{x}}\lesssim\sup_{0\leq\ell\leq
n}\sup_{V\in\mathcal{V}^{\ell}}{\|Vf\|}_{L^{q}_{z}L^{p}_{x}}.$ (4.26)

Therefore, in order to estimate the desired $\mathcal{X}^{n}$ type norms of
$W$, see Proposition 4.1, it suffices to estimate generic terms of the form
$\Gamma^{\ell}W$, $0\leq\ell\leq n$ in the appropriate $L^{p}_{x,z}$ spaces,
and under the a priori assumptions inferred from (4.25), (4.4)-(4.5) and
(4.8).

  * •

(The two dimensional case) We will adopt an analogous notation for the $2$d
vector fields, with corresponding product and commutator identities, together
with corresponding norms estimates as in (4.25). The distinction will always
be clear from context.

####  4.2.2. Classes of functions

To deal with multilinear expressions involving the vorticity and the
dispersive variables, we introduce classes of linear and quadratic functions
satisfying dispersive type bounds. We will adopt the shorthand notation from
Subsection 4.2.1. Also, recall the notation $x+$ for a real number $x$ (see
Subsection 2.6) and that $3p_{0}<\delta$.

######  Definition 4.3 ($\mathcal{O}_{i}$ classes).

We say that a function $F=F(x,z)$ defined on
${\mathbb{R}}^{2}\times{\mathbb{R}}_{-}$ is of class $\mathcal{O}_{1}$, and
write $F\in\mathcal{O}_{1}$, if

$\displaystyle{\big{\|}\Gamma^{n}F\big{\|}}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1+},$ $\displaystyle\qquad n\leq N_{1}-10,$ (4.27a)
$\displaystyle{\big{\|}\Gamma^{n}F\big{\|}}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}\langle
t\rangle^{3p_{0}},$ $\displaystyle\qquad n\leq N_{1}+12.$ (4.27b)

We say that $F$ is of class $\mathcal{O}_{2}$, and write
$F\in\mathcal{O}_{2}$, if

$\displaystyle{\big{\|}\Gamma^{n}F\big{\|}}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1-\delta},\qquad n\leq N_{1}-10,$ (4.28a)
$\displaystyle{\big{\|}\Gamma^{n}F\big{\|}}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1+\delta},\qquad n\leq N_{1}+12.$ (4.28b)

######  Remark 4.4 (About the $\mathcal{O}$ classes).

Here are a few remarks and simple consequences of the above definitions:

  1. (1)

The classes above are consistent with the bounds we have on high order
energies and low order dispersive norms for our irrotational variables.
Indeed, notice that a typical $\mathcal{O}_{1}$ function is $h$ together with
a few (up to ten) of its derivatives, in view of (3.8). The quantity $A$ in
(4.13) can also be easily seen to be in $\mathcal{O}_{1}$; see Lemma 4.6
below. Also, clearly $\mathcal{O}_{2}\subset\mathcal{O}_{1}$.

  2. (2)

If $F,G\in\mathcal{O}_{1}$, then the product $F\cdot G\in\mathcal{O}_{2}$.
This follows immediately from the definitions, distributing vector fields as
in (4.20), and using that $N_{1}$ is large. More precisely, we have

$\displaystyle{\big{\|}\Gamma^{n_{1}}F\cdot\Gamma^{n_{2}}G\big{\|}}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}^{2}\langle
t\rangle^{-2+},$ $\displaystyle\qquad n_{1}+n_{2}\leq N_{1}-10,$ (4.29)
$\displaystyle{\big{\|}\Gamma^{n_{1}}F\cdot\Gamma^{n_{2}}G\big{\|}}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}^{2}\langle
t\rangle^{-1+3p_{0}+},$ $\displaystyle\qquad n_{1}+n_{2}\leq N_{1}+12,$ (4.30)

which are sufficient since $3p_{0}<\delta$.

  3. (3)

If $F\in\mathcal{O}_{1}$, and $W$ satisfies the a priori estimates
(4.4)-(4.5), then the product $W\cdot F$ satisfies estimates that are at least
$\varepsilon_{0}$ better. More precisely

$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}F\big{\|}}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{0}\cdot\varepsilon_{1}\cdot\langle
t\rangle^{-1+},\qquad n_{1}+n_{2}\leq N_{1}-10,$ (4.31)
$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}F\big{\|}}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{0}\cdot\varepsilon_{1}\cdot\langle
t\rangle^{3p_{0}},\qquad n_{1}+n_{2}\leq N_{1}+12,$ (4.32)

which can be verified directly using Hölder’s inequality, estimating the terms
$\Gamma^{n_{2}}F$ always in $L^{\infty}_{x,z}$.

  4. (4)

The main property that we will eventually use to bound the large number of
cubic remainder terms in the renormalized vorticity equation is that the
$L^{1}_{t}([0,T_{\varepsilon_{1}}])$-norm of the product of $W$ with an
$\mathcal{O}_{2}$ function satisfies bounds consistent with the conclusions
(4.6)-(4.7). This is the content of Lemma 4.5 below.

Before proceeding with some general lemmas about the behavior of
$\mathcal{O}_{i}$ functions, let us recall here, for ease of reference, the
bounds we established on the vector potential, (see Lemma 3.7 and Definition
3.5): with the notation

$\displaystyle
V_{\omega,j}^{r,k}=\partial_{t}^{j}\underline{\Gamma}^{k}\nabla_{x,z}^{r}V_{\omega},\quad
j=0,1,$ (4.33)

we have

$\displaystyle\begin{split}\sum_{|r|+2|k|\leq
N_{1}+12-j}{\big{\|}|\nabla|^{1/2}V_{\omega,j}^{r,k}(t)\big{\|}}_{L^{\infty}_{z}L^{2}_{x}}+{\big{\|}\nabla_{x,z}V_{\omega,j}^{r,k}(t)\big{\|}}_{L^{2}_{z}L^{2}_{x}}+{\|V_{\omega,j}^{r,k}(t)\|}_{L^{\infty}_{z}L^{2}_{x}}\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta},\end{split}$ (4.34)

and

$\displaystyle\begin{split}\sum_{|r|+2|k|\leq
N_{1}-10-j}{\||\nabla|^{1/2}V_{\omega,j}^{r,k}(t)\|}_{L^{\infty}_{z}L^{2}_{x}}+{\|\nabla_{x,z}V_{\omega,j}^{r,k}(t)\|}_{L^{2}_{z}L^{2}_{x}}+{\|V_{\omega,j}^{r,k}(t)\|}_{L^{\infty}_{z}L^{2}_{x}}\lesssim\varepsilon_{1}.\end{split}$
(4.35)

In particular using the notation from 4.2.1 for vector fields, and (4.35) with
Sobolev’s embedding in $x$, we have the bounds

$\displaystyle{\|\Gamma^{n}V_{\omega}(t)\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{1},$
$\displaystyle\qquad n\leq N_{1}-12,$ (4.36)
$\displaystyle{\|\Gamma^{n}\nabla_{x,z}V_{\omega}(t)\|}_{L^{2}_{x,z}}\lesssim\varepsilon_{1}\langle
t\rangle^{\delta},$ $\displaystyle\qquad n\leq N_{1}+12,$ (4.37)

for all $t\leq T_{\varepsilon_{1}}$.

Here is a useful Lemma about products of $W$ with elements of
$\mathcal{O}_{2}$.

######  Lemma 4.5 (Bounds on trilinear expression).

Let $W$ be defined as in (4.1) and let $H\in\mathcal{O}_{2}$ as in Definition
4.3. Then,

$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}H\big{\|}}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{-1-\delta},\qquad n_{1}+n_{2}\leq N_{1}-10,$ (4.38a)
$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}H\big{\|}}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{-1+\delta},\qquad n_{1}+n_{2}\leq N_{1}+12.$ (4.38b)

Similarly, if $H,K\in\mathcal{O}_{1}$, then

$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}H\cdot\Gamma^{n_{3}}K\big{\|}}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{-1-\delta},\qquad n_{1}+n_{2}+n_{3}\leq N_{1}-10,$ (4.39a)
$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}H\cdot\Gamma^{n_{3}}K\big{\|}}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{-1+\delta},\qquad n_{1}+n_{2}+n_{3}\leq N_{1}+12.$ (4.39b)

###### Proof.

The proof is an immediate verification. We give some details for the sake of
completeness. Using Hölder’s inequality we have

$\displaystyle{\big{\|}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}H\big{\|}}_{L^{2}_{x,z}}$
$\displaystyle\lesssim\sup_{n_{1}+n_{2}=n}{\|\Gamma^{n_{1}}W\|}_{L^{2}_{x,z}}{\|\Gamma^{n_{2}}H\|}_{L^{\infty}_{x,z}}.$
(4.40)

Let us look at the case $n_{1}+n_{2}=n\leq N_{1}+12$ first. When $n_{1}\geq
n_{2}$, we use the a priori bounds (4.5) and (4.28a) to estimate

$\displaystyle{\|\Gamma^{n_{1}}W\|}_{L^{2}_{x,z}}{\|\Gamma^{n_{2}}H\|}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}+12}W\|}_{L^{2}_{x,z}}{\|\Gamma^{\leq N_{1}-10}H\|}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim\varepsilon_{1}\langle
t\rangle^{\delta}\cdot\varepsilon_{0}^{1+}\langle
t\rangle^{-1-\delta}\lesssim\varepsilon_{1}\cdot\varepsilon_{0}^{1+}\langle
t\rangle^{-1},$

which suffices. When instead $n_{2}\geq n_{1}$, we use the a priori bounds
(4.4) and (4.28b) to estimate

$\displaystyle{\|\Gamma^{n_{1}}W\|}_{L^{2}_{x,z}}{\|\Gamma^{n_{2}}H\|}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}-10}W\|}_{L^{2}_{x,z}}{\|\Gamma^{\leq N_{1}+12}H\|}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim\varepsilon_{1}\cdot\varepsilon_{0}^{1+}\langle
t\rangle^{-1+\delta}\lesssim\varepsilon_{1}\cdot\varepsilon_{0}^{1+}\langle
t\rangle^{-1+\delta}.$

Identical estimates hold with $L^{6/5}_{x,z}$ replacing $L^{2}_{x,z}$. This
gives us the desired bound (4.38b).

To obtain (4.38a) we use the bounds on low norms (4.4) and (4.28a) and see
that for all $n_{1}+n_{2}\leq N_{1}-10$,

$\displaystyle{\|\Gamma^{n_{1}}W\|}_{L^{2}\cap
L^{6/5}}{\|\Gamma^{n_{2}}H\|}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim{\|\Gamma^{\leq N_{1}-10}W\|}_{L^{2}\cap
L^{6/5}}{\|\Gamma^{\leq
N_{1}-10}H\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{1}\cdot\varepsilon_{0}^{1+}\langle
t\rangle^{-1-\delta},$

as claimed.

The proof of (4.39) is similar. It suffices to notice that terms of the form
$\Gamma^{n_{2}}\mathcal{O}_{1}\cdot\Gamma^{n_{3}}\mathcal{O}_{1}$ satisfy
better bounds than $\Gamma^{\leq n_{2}+n_{3}}\mathcal{O}_{2}$, see Remark
4.4(2). ∎

We conclude this section with a list of functions that are of class
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$.

######  Lemma 4.6 (Functions of class $\mathcal{O}_{1}$).

With the classes $\mathcal{O}_{1}$ defined as in Definition 4.3, and under our
a priori assumptions, we have

$\displaystyle
h,\,\,\partial_{t}h,\,\,A,\,\,\partial_{t}A\,\,\in\mathcal{O}_{1},$ (4.41)

recall the definition (4.14), and

$\displaystyle V-V_{\omega},\,\,\nabla_{x,z}\Psi\,\,\in\mathcal{O}_{1},$
(4.42)

see (4.1) and (4.2). The same holds true for $\Gamma^{\leq 2}$ applied to all
the quantities in (4.41)-(4.42).

###### Proof.

The fact that $h\in\mathcal{O}_{1}$ follows directly from the a priori
assumption (2.20) and (2.22). For $\partial_{t}h$ we use instead that
$\partial_{t}h=G(h)\varphi$ and the bounds in (B.38) to deduce bounds as in
(4.27a)-(4.27b).

For $A=\nabla_{x,z}|\nabla|^{-1}e^{z|\nabla|}h$ we use Sobolev’s embedding in
$x$ followed by (3.62): for $n\leq N_{1}+12$, and denoting
$\mathcal{R}=\nabla|\nabla|^{-1}$ the (vector) Riesz transform,

$\displaystyle{\big{\|}\Gamma^{n}\nabla_{x,z}|\nabla|^{-1}e^{z|\nabla|}h\big{\|}}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim{\big{\|}\Gamma^{n}(1,\mathcal{R})e^{z|\nabla|}h\big{\|}}_{L^{\infty}_{z}H^{2}_{x}}\lesssim\sup_{|r|+|k|\leq
n}{\|h\|}_{Z_{k}^{r+2}}\lesssim\varepsilon_{0}\langle t\rangle^{p_{0}}$

which is sufficient for the bound (4.27b) for this term. To verify the bound
(4.27a) we proceed similarly: for $n\leq N_{1}-10$ we have, by the maximum
principle, Sobolev’s embedding, and (3.9),

$\displaystyle{\big{\|}\Gamma^{n}\nabla_{x,z}|\nabla|^{-1}e^{z|\nabla|}h\big{\|}}_{L^{\infty}_{x,z}}\lesssim{\|(1,\mathcal{R})\Gamma^{n}h\|}_{L^{\infty}}\lesssim\sup_{|r|+|k|\leq
n}{\|h\|}_{Z_{k}^{r+1,\infty-}}\lesssim\varepsilon_{0}\langle t\rangle^{-1+}.$

We also see that applying $\Gamma^{\leq 2}$ to any of the quantities in (4.41)
we still obtain $\mathcal{O}_{1}$ functions.

Since $V-V_{\omega}=\nabla_{x,z}\Psi-\nabla h\,\partial_{z}\Psi$, in view of
Remark 4.4, we see that in order to obtain (4.42) it suffices to prove that
$\nabla_{x,z}\Psi\in\mathcal{O}_{1}$. This is a consequence of Lemma B.6.
Indeed, the property (4.27b) follows directly from the first bound in (B.49)
and Sobolev’s embedding. The bound (4.27a) follows from (B.50) which gives a
stronger bound for $\sum_{\ell}P_{\ell}\nabla_{x,z}\Psi$ with
$2^{\ell}\in[\langle t\rangle^{-5},\langle t\rangle^{5}]$, combined with the
$L^{2}$ bound (B.49) for the remaining very small and very large frequencies.
∎

######  Lemma 4.7 (Functions of class $\mathcal{O}_{2}$).

With the definitions (4.15), we have

$\displaystyle
R_{1},\quad\nabla_{x,z}(\Psi-e^{z|\nabla|}\varphi),\quad\nabla_{x,z}e^{z|\nabla|}|\nabla|^{-1}\big{(}|\nabla|\varphi-G(h)\varphi\big{)}\,\in\,\mathcal{O}_{2},$
(4.43)

The same is also true for $\Gamma^{\leq 2}$ applied to all of the above
quantities. In particular,

$\displaystyle\Gamma^{\leq 2}R_{2},\quad\Gamma^{\leq
2}R\,\in\,\mathcal{O}_{2}.$ (4.44)

###### Proof.

Since
$R_{1}^{i}=-\partial^{i}h\partial_{z}\Psi-(V^{\ell}-V^{\ell}_{\omega})\partial_{\ell}h\,e_{z}^{i}$
we see that this is in the $\mathcal{O}_{2}$ class in view of (4.41)-(4.42)
and (2) of Remark 4.4.

The second and third term in (4.43) are almost the same. For the second term
we can use directly (B.51), respectively (B.52), and Sobolev’s embedding to
deduce (4.28b) (recall $\delta>3p_{0}$), respectively, (4.28a).

For the third term in (4.43) we use the notation from (B.23), that is,
$G_{\geq 2}(h)\varphi:=G(h)\varphi-|\nabla|\varphi$, and invoke the bounds
(B.39). Using the maximum principle and Sobolev’s embedding we obtain, for
$n\leq N_{1}+12$,

$\displaystyle{\big{\|}\Gamma^{n}\nabla_{x,z}|\nabla|^{-1}e^{z|\nabla|}G_{\geq
2}(h)\varphi\big{\|}}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim\sup_{|r|+|k|\leq n}{\|(1,\mathcal{R})G_{\geq
2}(h)\varphi\|}_{Z_{k}^{r,\infty}}$ $\displaystyle\lesssim\sup_{|r|+|k|\leq
n}{\|(1,\mathcal{R})G_{\geq
2}(h)\varphi\|}_{Z_{k}^{r+2}}\lesssim\varepsilon_{0}^{2}\langle
t\rangle^{-1+3p_{0}}.$

Similarly, we can use (3.62) to obtain, for $n\leq N_{1}-10$,

$\displaystyle{\big{\|}\Gamma^{n}\nabla_{x,z}|\nabla|^{-1}e^{z|\nabla|}G_{\geq
2}(h)\varphi\big{\|}}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim\sup_{|r|+|k|\leq n}{\|(1,\mathcal{R})G_{\geq
2}(h)\varphi\|}_{Z_{k}^{r,\infty}}$ $\displaystyle\lesssim\sup_{|r|+|k|\leq
n}{\|G_{\geq
2}(h)\varphi\|}_{Z_{k}^{r+1,\infty-}}\lesssim\varepsilon_{0}^{2}\langle
t\rangle^{-6/5}.$

having used $L^{p}$ interpolation between the bounds in (B.39). ∎

###  4.3. Commutation with vector fields

We proceed to derive a transport equation for $\Gamma^{n}W$.

######  Lemma 4.8.

Let ${\bf D}_{t}:=\partial_{t}+U\cdot\nabla$ and recall the notation in 4.2.1.
We have:

  1. (1)

The following basic commutation identities hold

$\displaystyle\begin{split}[{\bf
D}_{t},\partial_{x_{i}}]&=-\partial_{x_{i}}U\cdot\nabla,\quad
i=1,2,3,\quad(x_{3}=z),\\\ [{\bf
D}_{t},\Omega]&=U_{1}\partial_{x_{2}}-U_{2}\partial_{x_{1}}-\Omega
U\cdot\nabla,\\\ [{\bf D}_{t},S]&=(1/2){\bf D}_{t}+(1/2)U\cdot\nabla-
SU\cdot\nabla.\end{split}$ (4.45)

  2. (2)

For any $\alpha\in{\mathbb{Z}}_{+}^{5}$, $|\alpha|=1$, there exists constants
$c_{1},c_{2},c_{2}^{\prime}\in{\mathbb{R}}$, such that

$\displaystyle[{\bf D}_{t},\Gamma^{\alpha}]=c_{1}{\bf
D}_{t}+c_{2}U\cdot\nabla+c_{2}^{\prime}U\cdot\nabla_{x}^{\perp}-\Gamma^{\alpha}U\cdot\nabla.$
(4.46)

Since the presence of the ⟂ and of all the constants will not play any role,
we will drop them from (4.46) and use the notation conventions from 4.2.1 to
write, for some $c\in{\mathbb{R}}$,

$\displaystyle[{\bf D}_{t},\Gamma^{1}]f=c{\bf D}_{t}f+(\Gamma^{\leq
1}U)\cdot\nabla f.$ (4.47)

  3. (3)

If ${\bf D}_{t}W=B$ then, for all $n\geq 1$,

$\displaystyle{\bf D}_{t}\Gamma^{n}W=\Gamma^{\leq n}B+\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla W;$ (4.48)

here the sum is a linear combination as per our conventions, see (4.21) and
the paragraph following that.

  4. (4)

If $W$ is the solution of (4.12), then, for any integer $n$ we have

$\displaystyle\begin{split}{\bf D}_{t}\Gamma^{n}W&=\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla W+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla X+\Gamma^{\leq n}F.\end{split}$
(4.49)

###### Proof.

The identities (4.45) follow from standard calculations. Indeed, for
$x\in{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}_{-}$ we have

$\displaystyle[{\bf D}_{t},S]$
$\displaystyle=(1/2)[\partial_{t},t\partial_{t}]+[U\cdot\nabla,(1/2)t\partial_{t}+x\cdot\nabla]$
$\displaystyle=(1/2)\partial_{t}-SU\cdot\nabla-U\cdot[S,\nabla]$
$\displaystyle=(1/2){\bf D}_{t}-(1/2)U\cdot\nabla-
SU\cdot\nabla-U\cdot(-\nabla),$

which is of the desired form. Similarly, $[{\bf
D}_{t},\Omega]=[U\cdot\nabla,x_{1}\partial_{x_{2}}-x_{2}\partial_{x_{1}}]=U_{1}\partial_{x_{2}}-U_{2}\partial_{x_{1}}-\Omega
U\cdot\nabla.$ The identity (4.46), and its shorthand version (4.47), then
follow directly from (4.45).

To prove (4.48) we proceed by induction. The case $n=1$ follows directly from
(4.47). Assuming (4.48) holds true for $n=1$ we calculate using the inductive
hypothesis and (4.47):

$\displaystyle{\bf D}_{t}\Gamma^{n}W$ $\displaystyle=\Gamma^{1}{\bf
D}_{t}\Gamma^{n-1}W+[{\bf D}_{t},\Gamma^{1}]\Gamma^{n-1}W$
$\displaystyle=\Gamma^{1}\Big{(}\Gamma^{\leq n-1}B+\sum_{n_{1}+n_{2}\leq
n-2}\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla W\Big{)}$
$\displaystyle+{\bf D}_{t}\Gamma^{n-1}W+\Gamma^{\leq
1}U\cdot\nabla\Gamma^{n-1}W;$

distributing the vector field $\Gamma$ in the first line (see (4.20)),
applying again the inductive hypothesis to ${\bf D}_{t}\Gamma^{n-1}W$, and
commuting the $\Gamma$’s and $\nabla$ in the last term, we see that the above
expression is of the desired form (4.48).

Finally, the equation (4.49) follows from applying the previous identity
(4.48) to the equation (4.12). ∎

We also have the following lemma for the transporting vector field $U$.

######  Lemma 4.9.

With the notation of 4.2.1 and under the assumptions of Proposition 4.1, for
$U$ and $X$ as in (4.13) we have:

$\displaystyle
U=\mathcal{O}_{1}+V_{\omega}+V_{\omega}\cdot\mathcal{O}_{1},\qquad\nabla
X=\mathcal{O}_{1}+\nabla V_{\omega},$ (4.50)

In particular

$\displaystyle{\|\Gamma^{n}U\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1+}+\varepsilon_{1},\qquad n\leq N_{1}-12.$ (4.51)

###### Proof.

We see from the formulas in Lemma 4.2, and using Lemmas 4.6 and 4.7, that

$\displaystyle U=\mathcal{O}_{1}+V_{\omega}-V_{\omega}\cdot\nabla
h\,e_{z}+\mathcal{O}_{2},$ (4.52)

and (4.50) follows. ∎

###  4.4. Renormalization of the vorticity equation

In this subsection we manipulate the equation for $W$, see (4.49), using the
identities from Lemma 4.2, in order to write it in a better form that allows
to propagate the desired $\mathcal{X}^{n}$ norms and prove Proposition 4.1.
These manipulations are akin to a (partial) normal form transformation on the
vorticity equation in the full three dimensional fluid domain that effectively
renormalizes the irrotational components. The next proposition is the main
result of this section.

######  Proposition 4.10 (Renormalized vorticity equation).

Under the assumptions of Proposition 4.1 we have following: for all $n\leq
N_{1}+12$, there exist corrections $G^{n}=G^{n}(t,x,z)$ such that

$\displaystyle\begin{split}{\mathbf{D}_{t}}\big{(}\Gamma^{n}W-G^{n}\big{)}=Q_{1}^{n}+Q_{2}^{n}+C_{1}^{n}+C_{2}^{n}+F^{n}\end{split}$
(4.53)

where the following holds:

  * •

The correction $G^{n}$ satisfies for all $|t|\leq T$

$\displaystyle{\|G^{n}(t)\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{0}\varepsilon_{1},\qquad n\leq N_{1}-10,$
(4.54a) $\displaystyle{\|G^{n}(t)\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta},\qquad n\leq N_{1}+12.$ (4.54b)

  * •

$Q_{1}^{n},Q_{2}^{n}$ are quadratic terms (in the rotational variables only)
given by

$\displaystyle Q_{1}^{n}$ $\displaystyle:=\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla V_{\omega},$ (4.55a) $\displaystyle
Q_{2}^{n}$ $\displaystyle:=\sum_{n_{1}+n_{2}\leq n-1}\Gamma^{\leq
n_{1}+1}V_{\omega}\cdot\Gamma^{n_{2}}\nabla W.$ (4.55b)

  * •

$C^{n}_{1},C^{n}_{2}$ are cubic terms of the form

$\displaystyle\begin{split}C_{1}^{n}&:=\sum_{n_{1}+n_{2}+n_{3}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla
V_{\omega}\cdot\Gamma^{n_{3}}\mathcal{O}_{1},\\\
C_{2}^{n}&:=\sum_{n_{1}+n_{2}+n_{3}\leq n-1}\Gamma^{\leq
n_{1}+1}V_{\omega}\cdot\Gamma^{n_{2}}\nabla
W\cdot\Gamma^{n_{3}}\mathcal{O}_{1};\end{split}$ (4.56)

recall Definition 4.3 and Lemma 4.6.

  * •

The remaining nonlinear terms satisfy

$\displaystyle{\|F^{n}\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{-1-\delta},\qquad n\leq N_{1}-10,$ (4.57a)
$\displaystyle{\|F^{n}\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{-1+\delta},\qquad n\leq N_{1}+12.$ (4.57b)

######  Remark 4.11 (“Correction” and “Acceptable Remainders”).

Here are some remarks on Proposition 4.10:

  * •

$G^{n}$ is a normal-form type “correction” of $\Gamma^{n}W$ since its norms
are $\varepsilon_{0}$ smaller than those of $\Gamma^{n}W$ (compare (4.54) and
(4.4)-(4.5)) and the equation satisfied by $\Gamma^{n}W-G^{n}$ has nonlinear
terms that are more perturbative then the ones in (4.49).

  * •

We call a (cubic) term that satisfies (4.57) an “acceptable remainder” since
such a term gives a small perturbation of the transported vector field
$\Gamma^{n}W-G^{n}$ (hence of $\Gamma^{n}W$) when integrated in over time
$t\in[0,T_{\varepsilon_{1}}]$. Many terms will be shown to be acceptable
remainders directly using the lemmas from the previous subsection.

  * •

The quadratic terms on the right-hand side of (4.53) only depend on $W$ and
$V_{\omega}$. Technically, they are not acceptable remainders in the sense
specified above and so will be estimated separately in Subsection 4.5.

###### Proof of Proposition 4.10.

We start from (4.49) and use the structure of the vector fields $U$ and $X$,
see Lemmas 4.2 and 4.9, to eventually obtain (4.53). In the course of the
proof we are going to collect several remainders denoted by
$F^{n}_{1},F^{n}_{2}$ and similar, that will eventually contribute to the
nonlinear remainder $F^{n}$.

Step 1: Renormalized equation. First, for convenience of the reader, we recall
(4.49)

$\displaystyle\begin{split}{\bf D}_{t}\Gamma^{n}W&=\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla W+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla X+\Gamma^{\leq n}F.\end{split}$
(4.58)

Using the formulas for $A$ and $X$ in (4.14), and the definition of $F$ in
(4.12), we rewrite (4.58) in the following form:

$\displaystyle{\bf D}_{t}\Gamma^{n}W$ $\displaystyle=\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}\partial_{t}(A-he_{z})\cdot\Gamma^{n_{2}}\nabla
W+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{2}}W\cdot\Gamma^{n_{1}}\nabla\partial_{t}A$ (4.59a)
$\displaystyle+\sum_{n_{1}+n_{2}\leq n-1}\Gamma^{\leq
n_{1}+1}V_{\omega}\cdot\Gamma^{n_{2}}\nabla W+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla V_{\omega}$ (4.59b)
$\displaystyle-\sum_{n_{1}+n_{2}\leq n-1}\Gamma^{\leq
n_{1}+1}(V_{\omega}\cdot\nabla he_{z})\cdot\Gamma^{n_{2}}\nabla W+\Gamma^{\leq
n}\big{(}-W\cdot\nabla h\,\partial_{z}V_{\omega}\big{)}$ (4.59c)
$\displaystyle+F^{n}_{0}+F^{n}_{1}+F^{n}_{2},$ (4.59d)

where we define

$\displaystyle F^{n}_{0}$ $\displaystyle:=\Gamma^{\leq n}\big{(}-W\cdot\nabla
h\,\partial_{z}(V-V_{\omega})-W\cdot\nabla(\nabla h\partial_{z}\Psi)\big{)},$
(4.60) $\displaystyle F^{n}_{1}$ $\displaystyle:=\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}R\cdot\Gamma^{n_{2}}\nabla W,$ (4.61) $\displaystyle
F^{n}_{2}$ $\displaystyle:=\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla R_{2}.$ (4.62)

We analyze term by term the right-hand side of (4.59). First, we see that the
terms in (4.59b) contribute to the quadratic terms on the right-hand side of
(4.53) as they match exactly (4.55), and so they are accounted for. Similarly,
the cubic terms in (4.59c) are accounted for in the terms (4.56), since up to
two derivatives of $h$ are in $\mathcal{O}_{1}$, see Lemma 4.6.

Then, since up to two derivatives of $h$, and one derivative of
$\partial_{z}\Psi$, and the term $\partial_{z}(V-V_{\omega})$ are in
$\mathcal{O}_{1}$, see Lemma 4.6, we see that (4.60) is of the form

$\Gamma^{\leq n}(W\cdot\mathcal{O}_{1}\cdot\mathcal{O}_{1})=\Gamma^{\leq
n}(W\cdot\mathcal{O}_{2}),$

see Remark 4.4; applying Lemma 4.5 and (4.20) we obtain that $F^{n}_{0}$ is an
acceptable remainder in that it satisfies the bounds (4.57).

We can easily see that the term (4.61) is an acceptable remainder using that
$R\in\mathcal{O}_{2}$, see Lemma 4.7 and (4.15), and an application of Lemma
4.5. Similarly, the term (4.62) is an acceptable remainder using that $\nabla
R_{2}\in\mathcal{O}_{2}$, see Lemma 4.7.

We now analyze the two terms in (4.59a). For the first one we use
$\Gamma^{\leq n_{1}+1}\partial_{t}=\partial_{t}\Gamma^{\leq
n_{1}+1}={\mathbf{D}_{t}}\Gamma^{\leq n_{1}+1}+U\cdot\Gamma^{\leq n_{1}+1}$ to
write

$\displaystyle\sum_{n_{1}+n_{2}\leq n-1}\Gamma^{\leq
n_{1}+1}\partial_{t}(A-he_{z})\cdot\Gamma^{n_{2}}\nabla
W=\sum_{n_{1}+n_{2}\leq n-1}{\mathbf{D}_{t}}\Gamma^{\leq
n_{1}+1}(A-he_{z})\cdot\Gamma^{n_{2}}\nabla W+F^{n}_{3},$ (4.63)
$\displaystyle\qquad F^{n}_{3}:=\sum_{n_{1}+n_{2}\leq
n-1}U\cdot\nabla\Gamma^{\leq n_{1}+1}(A-he_{z})\Gamma^{n_{2}}\nabla W.$

The first term on the right-hand side of (4.63) will be analyzed shortly
below. First, we verify that $F^{n}_{3}$ is an acceptable remainder satisfying
(4.57); indeed we can write

$\displaystyle F^{n}_{3}:=\sum_{n_{1}+n_{2}\leq n-1}U\cdot\Gamma^{\leq
n_{1}+1}\mathcal{O}_{1}\cdot\Gamma^{n_{2}}\nabla W$

and observe that $U\cdot\Gamma^{\leq
n_{1}+1}\mathcal{O}_{1}\in\Gamma^{n_{1}+1}\mathcal{O}_{2}$, since $U$
satisfies (4.51).

For the second term on the right-hand side of (4.59a) we can use again
$\Gamma^{n_{1}}\nabla\partial_{t}={\mathbf{D}_{t}}\Gamma^{\leq
n_{1}}\nabla+U\cdot\Gamma^{\leq n_{1}}\nabla$ to write

$\displaystyle\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{2}}W\cdot\Gamma^{n_{1}}\nabla\partial_{t}A$
$\displaystyle=\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{2}}W\cdot{\mathbf{D}_{t}}\Gamma^{n_{1}}\nabla A+F^{n}_{4}$ (4.64)

where

$\displaystyle F^{n}_{4}:=U\cdot\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{2}}W\cdot\Gamma^{n_{1}}\nabla A.$ (4.65)

Since $U=V_{\omega}+\mathcal{O}_{1}+V_{\omega}\cdot\mathcal{O}_{1}$, see
(4.50), it is not hard to verify that $F^{n}_{4}$ is an acceptable remainder.
Indeed, by (3) in Remark 4.4, since $\nabla A\in\mathcal{O}_{1}$, the
quadratic terms in the sum (4.65) are bounded by the right-hand side of
(4.32), respectively (4.31), when $n\leq N_{1}+12$, respectively $n\leq
N_{1}-10$; since we also have that
${\|U\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1+}+\varepsilon_{1}$ in view of (4.13), (4.41) and (4.36), we
obtain

$\displaystyle{\|F^{n}_{4}\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{-1+}\cdot(\varepsilon_{0}\langle
t\rangle^{-1+}+\varepsilon_{1}),\qquad n_{1}+n_{2}\leq N_{1}-10,$
$\displaystyle{\|F^{n}_{4}\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{3p_{0}}\cdot(\varepsilon_{0}\langle
t\rangle^{-1+}+\varepsilon_{1}),\qquad n_{1}+n_{2}\leq N_{1}+12.$

These bounds are enough for (4.57a)-(4.57b) since $3p_{0}<\delta$,
$\varepsilon_{0}\leq\varepsilon_{1}$, and $t\leq
T_{\varepsilon_{1}}=\varepsilon_{1}^{-1+\delta}$ gives
$\varepsilon_{1}\lesssim\langle t\rangle^{-1-\delta}$.

We now combine (4.63)-(4.65), changing the index in the sums and then pull the
${\mathbf{D}_{t}}$ out to write the first two terms on the right-hand side of
(4.59a) as

$\displaystyle\begin{split}&\sum_{n_{1}+n_{2}\leq n-1}\partial_{t}\Gamma^{\leq
n_{1}+1}(A-he_{z})\cdot\Gamma^{n_{2}}\nabla W+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{2}}W\cdot\Gamma^{n_{1}}\nabla\partial_{t}A\\\
&=\sum_{n_{1}+n_{2}\leq n}{\mathbf{D}_{t}}\big{(}\Gamma^{\leq
n_{1}+1}A+\Gamma^{\leq
n_{1}+1}he_{z}\big{)}\cdot\Gamma^{n_{2}}W+F^{n}_{3}+F^{n}_{4}\\\ &=:{\bf
D}_{t}G^{n}+F^{n}_{3}+F^{n}_{4}+F^{n}_{5},\end{split}$ (4.66)

upon defining

$\displaystyle\begin{split}G^{n}:=\sum_{n_{1}+n_{2}\leq n}\big{(}\Gamma^{\leq
n_{1}+1}A+\Gamma^{\leq n_{1}+1}he_{z}\big{)}\cdot\Gamma^{n_{2}}W\end{split}$
(4.67)

and

$\displaystyle\begin{split}F^{n}_{5}:=\sum_{n_{1}+n_{2}\leq
n}\big{(}\Gamma^{\leq n_{1}+1}A+\Gamma^{\leq
n_{1}+1}he_{z}\big{)}\cdot{\mathbf{D}_{t}}\Gamma^{n_{2}}W.\end{split}$ (4.68)

By letting

$\displaystyle F^{n}_{r}=\sum_{i=0}^{4}F^{n}_{i},$ (4.69)

we have obtained an equation of the form

$\displaystyle\begin{split}{\mathbf{D}_{t}}\Gamma^{n}W={\mathbf{D}_{t}}G^{n}+Q_{1}^{n}+Q_{2}^{n}+C_{1}^{n}+C_{2}^{n}+F^{n}_{r}+F^{n}_{5}\end{split}$
(4.70)

with quadratic and cubic terms as in (4.55) and (4.56).

Step 2: Estimates for the correction. We can directly verify that $G^{n}$
satisfies (4.54a)-(4.54b) using the definition (4.67), the fact that
$\Gamma^{\leq 1}A,\Gamma^{\leq 1}h\in\mathcal{O}_{1}$, and (3) in Remark 4.4.

Step 3: Remainder estimates. To conclude the proof of the proposition we need
to handle the remainders in (4.70). We have already proved that $F^{n}_{r}$ is
an acceptable remainder satisfying (4.57a)-(4.57b), so we only need to show
that $F^{n}_{5}$ contributes an acceptable remainder plus other contributions
that are accounted for in the cubic terms (4.56).

First, we use $\Gamma^{\leq 1}A,\Gamma^{\leq 1}h\in\mathcal{O}_{1}$ to write

$\displaystyle F^{n}_{5}$ $\displaystyle=\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot{\mathbf{D}_{t}}\Gamma^{n_{2}}W;$

then, using $\Gamma^{\leq 1}\partial_{t}A,\Gamma^{\leq
1}\partial_{t}h\in\mathcal{O}_{1}$, we express the right-hand side of (4.59a)
as

$\displaystyle\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{2}}W\cdot\Gamma^{n_{1}}\mathcal{O}_{1}.$

Therefore, using the full equation (4.59), and adopting the same notation
(4.55)-(4.56) in the statement, we have

$\displaystyle F^{n}_{5}$ $\displaystyle=\sum_{n_{1}+n_{2}+n_{3}\leq
n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\Gamma^{n_{2}}\mathcal{O}_{1}\cdot\Gamma^{n_{3}}W$
(4.71a) $\displaystyle+\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\big{(}Q^{n_{2}}_{1}+Q^{n_{2}}_{2}\big{)}$ (4.71b)
$\displaystyle+\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\big{(}C^{n_{2}}_{1}+C^{n_{2}}_{2}\big{)}$ (4.71c)
$\displaystyle+\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\big{(}F^{n_{2}}_{0}+F^{n_{2}}_{1}+F^{n_{2}}_{2}\big{)}.$
(4.71d)

The terms (4.71a) are acceptable remainders satisfying (4.57) in view of
(4.39).

From (4.55b) we see that, adopting the notation (4.21), the terms in (4.71b)
are of the form

$\displaystyle\eqref{F5q}=\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\big{(}\sum_{n_{2}+n_{3}\leq
n_{2}}\Gamma^{n_{2}}W\cdot\Gamma^{n_{3}}\nabla
V_{\omega}+\sum_{n_{2}+n_{3}\leq n_{2}-1}\Gamma^{\leq
n_{2}+1}V_{\omega}\cdot\Gamma^{n_{3}}\nabla W\big{)};$ (4.72)

we then see that these terms are actually cubic terms as in (4.56) and,
therefore, are accounted for in the main equation (4.53).

Next, we claim that also the terms (4.71c) are of the same ‘cubic’ form
(4.56). Indeed, let us look at the first of the two summands in (4.71c), that
is,

$\displaystyle\sum_{n_{1}+n_{2}\leq n}\Gamma^{\leq n_{1}}\mathcal{O}_{1}\cdot
C^{n_{2}}_{1}=\sum_{n_{1}+n_{2}+n_{3}+n_{4}\leq n}\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\Gamma^{n_{2}}\mathcal{O}_{1}\cdot\Gamma^{n_{3}}W\cdot\Gamma^{n_{4}}\nabla
V_{\omega}.$ (4.73)

Observing that $\Gamma^{\leq
n_{1}}\mathcal{O}_{1}\cdot\Gamma^{n_{2}}\mathcal{O}_{1}$ satisfies the same
bounds of $\Gamma^{n_{1}+n_{2}}\mathcal{O}_{1}$ (better ones, in fact, of
$\mathcal{O}_{2}$-type), we see that (4.73) is accounted for in (4.56). The
same reasoning applies to the term involving $C^{n_{2}}_{2}$ in (4.71c).

Finally, we look at (4.71d). As already shown above the terms $F^{n_{2}}_{i}$,
$i=0,1,2$ satisfy the acceptable remainder bounds in (4.57a)-(4.57b). Then it
is not hard to see that (4.71d) is also an acceptable remainder: if $n_{2}\geq
n_{1}$, so that $n_{1}\leq N_{1}-10$, we use the bound (4.27a) on
$\Gamma^{n_{1}}\mathcal{O}_{1}$ and the bound (4.57b) on $F^{n_{2}}_{i}$; if
instead $n_{2}\leq n_{1}$ we use the bound (4.27b) on
$\Gamma^{n_{1}}\mathcal{O}_{1}$ and the bound (4.57a) on $F^{n_{2}}_{i}$. This
concludes the proof of the proposition. ∎

###  4.5. Transport estimates and proof of Proposition 4.1

We begin with a general result about propagation of $L^{p}_{x,z}$ norms for
transport equation:

######  Lemma 4.12 (Bounds for the transport equation).

Let ${\bf D}_{t}=\partial_{t}+U\cdot\nabla$ as above, and consider
$Z=Z(t,x,z)$ a solution of

$\displaystyle{\bf D}_{t}Z=N.$ (4.74)

Then, for all $t\leq T_{\varepsilon_{1}}$, we have

$\displaystyle{\|Z(t)\|}_{L^{p}_{x,z}}\leq{\|Z(0)\|}_{L^{p}_{x,z}}+C\int_{0}^{t}{\|N(s)\|}_{L^{p}_{x,z}}\,ds.$
(4.75)

###### Proof.

We begin by proving bounds for the Lagrangian flow associated to $U$. Let
$\Phi=\Phi_{t}$ be such that $\dot{\Phi}(t)=U(t,\Phi)$ with
$\Phi(0)=\mathrm{id}$. We want to show that

$\displaystyle\sup_{t\leq
T_{\varepsilon_{1}}}|\nabla\Phi_{t}(t)-\mathrm{id}|<1/2.$ (4.76)

We do this by a bootstrap argument. Assume that (4.76) holds true and denote
$J(t):=\nabla\Phi_{t}(t)$. Since $\dot{J}(t)=J(t)\nabla U(t,\Phi_{t})$, using
(4.14) to express $U$, we have

$\displaystyle J(t)-\mathrm{id}$ $\displaystyle=\int_{0}^{t}J(s)\nabla
U(s,\Phi_{s})\,ds$
$\displaystyle=\int_{0}^{t}J(s)\nabla\big{[}\partial_{s}(A-h\,e_{z})\big{]}(s,\Phi_{s})\,ds+\int_{0}^{t}J(s)\nabla
V_{\omega}(s,\Phi_{s})\,ds+\int_{0}^{t}J(s)\nabla R(s,\Phi_{s})\,ds$
$\displaystyle=J(s)\nabla(A-h\,e_{z})(s,\Phi_{s})\Big{|}_{s=0}^{s=t}-\int_{0}^{t}J(s)\nabla
U(s,\Phi_{s})\nabla(A-h\,e_{z})(s,\Phi_{s})\,ds$
$\displaystyle+\int_{0}^{t}J(s)\nabla
V_{\omega}(s,\Phi_{s})\,ds+\int_{0}^{t}J(s)\nabla R(s,\Phi_{s})\,ds$
$\displaystyle=:J_{1}(t)-J_{1}(0)+J_{2}(t)+J_{3}(t)+J_{4}(t).$

For the first term in the above right-hand side we use $|J(t)|<3/2$ and that
$\nabla A,\nabla h\in\mathcal{O}_{1}$ (see Lemma 4.6 and (4.27a) in Definition
4.3) to estimate

$\displaystyle|J_{1}(s)|\lesssim\varepsilon_{0}\langle s\rangle^{-1+}$

For the second term we use in addition that $\nabla U\in\mathcal{O}_{1}$, and
bound

$\displaystyle|J_{2}(t)|\lesssim\int_{0}^{t}|\nabla
U(s,\Phi_{s})||\nabla(A-h\,e_{z})(s,\Phi_{s})|\,ds\lesssim\int_{0}^{t}\varepsilon_{0}\langle
s\rangle^{-1+}\cdot\varepsilon_{0}\langle
s\rangle^{-1+}\,ds\lesssim\varepsilon_{0}^{2}$

For the third term we use the bound on $V_{\omega}$ in (4.35) to estimate

$\displaystyle|J_{3}(t)|\lesssim\int_{0}^{t}|\nabla
V_{\omega}(s,\Phi_{s})|\,ds\lesssim
t\varepsilon_{1}\lesssim\varepsilon_{1}^{\delta},$

for $t\leq T_{\varepsilon_{1}}=\varepsilon_{1}^{-1+\delta}$. For the last
term, we use that $\nabla R\in\mathcal{O}_{2}$ (see Lemma 4.7 and (4.28a)) and
obtain a bound $|J_{4}(s)|\lesssim\varepsilon_{0}^{1+}$. Putting all these
together shows that, for $\varepsilon_{0},\varepsilon_{1}$ small enough,

$\displaystyle\sup_{t\leq
T_{\varepsilon_{1}}}|\nabla\Phi_{t}(t)-\mathrm{id}|<1/4,$

and therefore we obtain (4.76).

We can then use the Lagrangian map to integrate the flow (4.74),

$\displaystyle
Z(t,\Phi_{t}(x,z))=Z(0,x,z)+\int_{0}^{t}N(s,\Phi_{s}(x,z))\,ds,$

and then deduce (4.75) by Minkowski’s inequality and changing variables using
(4.76) to control the Jacobian. ∎

Next, we apply Lemma 4.12 to conclude the proof of the main Proposition 4.1.
The main task left is to obtain suitable bounds on the quadratic (and cubic)
terms on the right-hand side of (4.53).

###### Proof of Proposition 4.1.

For this proof we define

$\displaystyle\delta_{n}=\left\\{\begin{array}[]{lcl}0&\quad\mbox{if}&\quad
n\leq N_{1}-10,\\\ \delta&\quad\mbox{if}&\quad
n\in(N_{1}-10,N_{1}+12]\cap{\mathbb{Z}},\end{array}\right.$ (4.79)

and use the short-hand

$\displaystyle L:=L^{2}_{x,z}\cap L^{6/5}_{x,z}$ (4.80)

to denote the relevant Lebesgue space.

We start from (4.53), apply Lemma 4.12 to $Z=\Gamma^{n}W-G^{n}$, and use the
bound (4.54a) for $G^{n}$ to obtain

$\displaystyle\begin{split}{\|\Gamma^{n}W(t)\|}_{L}&\leq{\|\Gamma^{n}W(0)\|}_{L}+C\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta_{n}}+C\int_{0}^{t}{\|Q_{1}^{n}(s)\|}_{L}+{\|Q_{2}^{n}(s)\|}_{L}\,ds\\\
&+C\int_{0}^{t}{\|C_{1}^{n}(s)\|}_{L}+{\|C_{2}^{n}(s)\|}_{L}\,ds+C\int_{0}^{t}{\|F^{n}(s)\|}_{L}\,ds.\end{split}$
(4.81)

From (2.12) and (2.11), which give (4.82) we can bound the contribution at the
initial time

$\displaystyle{\|\Gamma^{n}W(0)\|}_{L}\leq C_{0}\varepsilon_{1};$ (4.82)

this is consistent with (4.6)-(4.7) (for $j=0$) by taking $C$ large enough.
Moreover, using (4.57a), we can bound

$\displaystyle\int_{0}^{t}{\|F^{n}(s)\|}_{L}\,ds\lesssim\int_{0}^{t}\varepsilon_{1}\varepsilon_{0}^{1+}\langle
s\rangle^{-1-\delta}\,ds\lesssim\varepsilon_{1}\varepsilon_{0}^{1+},\qquad
n\leq N_{1}-10,$

and, similarly, using (4.57b),

$\displaystyle\int_{0}^{t}{\|F^{n}(s)\|}_{L}\,ds\lesssim\int_{0}^{t}\varepsilon_{1}\varepsilon_{0}^{1+}\langle
s\rangle^{-1+\delta}\,ds\lesssim\varepsilon_{1}\varepsilon_{0}^{1+}\langle
t\rangle^{\delta}\qquad n\leq N_{1}+12.$

These last two bounds are also consistent with the desired conclusions
(4.6)-(4.7). Therefore, we see that the proof of (4.6)-(4.7) would follow with
$j=0$ if we can show that, for all $t\leq
T_{\varepsilon_{1}}:=\bar{c}\varepsilon_{1}^{-1+\delta}$ with $\bar{c}$ small
enough,

$\displaystyle\int_{0}^{t}{\|Q^{n}_{1}(s)\|}_{L}\,ds\lesssim\varepsilon_{1}^{1+}\langle
t\rangle^{\delta_{n}},$ (4.83)
$\displaystyle\int_{0}^{t}{\|Q^{n}_{2}(s)\|}_{L}\,ds\lesssim\varepsilon_{1}^{1+}\langle
t\rangle^{\delta_{n}},$ (4.84)

and

$\displaystyle\int_{0}^{t}{\|C^{n}_{1}(s)\|}_{L}+{\|C^{n}_{2}(s)\|}_{L}\,ds\lesssim\varepsilon_{1}^{1+}\langle
t\rangle^{\delta_{n}}.$ (4.85)

Proof of (4.83). Since $Q_{1}^{n}$ is given by (4.55a), we observe that in
order to obtain (4.83) it will suffice to prove the bound

$\displaystyle{\|\Gamma^{n_{1}}W(s)\cdot\Gamma^{n_{2}}\nabla
V_{\omega}(s)\|}_{L}\lesssim\varepsilon_{1}^{2}\langle
s\rangle^{\delta},\qquad n_{1}+n_{2}\leq N_{1}+12,$ (4.86)

for all $s\leq T_{\varepsilon_{1}}$; indeed, since
$T_{\varepsilon_{1}}=\bar{c}\varepsilon_{1}^{-1+\delta}$,

$\displaystyle\int_{0}^{t}\varepsilon_{1}^{2}\langle s\rangle^{\delta}\,ds\leq
2\varepsilon_{1}^{2}T_{\varepsilon_{1}}^{1+\delta}\leq
2\bar{c}\varepsilon_{1}^{1+\delta^{2}}.$

To prove (4.86) in the case $n_{2}\leq n_{1}$, with $n_{1}+n_{2}\leq
N_{1}+12$, we use the a priori estimate (4.5) and (4.35) after Sobolev’s
embedding (in $x$):

$\displaystyle{\|\Gamma^{n_{1}}W(s)\cdot\Gamma^{n_{2}}\nabla
V_{\omega}(s)\|}_{L}$ $\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}+12}W(s)\|}_{L}{\|\Gamma^{\leq N_{1}-13}\nabla
V_{\omega}(s)\|}_{L^{\infty}_{x,z}}$
$\displaystyle\lesssim\varepsilon_{1}\langle
s\rangle^{\delta}\cdot{\|\Gamma^{\leq
N_{1}-10}V_{\omega}(s)\|}_{L^{\infty}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{1}^{2}\langle s\rangle^{\delta}.$

For the case $n_{1}\leq n_{2}$, we use instead Hölder’s inequality (recall
(4.80)) with (4.37) and (4.4) after Sobolev’s embedding (in $x$):

$\displaystyle{\|\Gamma^{n_{1}}W(s)\cdot\Gamma^{n_{2}}\nabla
V_{\omega}(s)\|}_{L}$ $\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}-13}W(s)\|}_{L^{\infty}_{x,z}\cap L^{3}_{x,z}}{\|\Gamma^{\leq
N_{1}+12}\nabla V_{\omega}(s)\|}_{L^{2}_{x,z}}$
$\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}-10}W(s)\|}_{L^{2}_{x,z}}\cdot\varepsilon_{1}\langle s\rangle^{\delta}$
$\displaystyle\lesssim\varepsilon_{1}^{2}\langle s\rangle^{\delta}.$

Note that this is the place where we use the highest order estimate (4.37) for
$V_{\omega}$.

Proof of (4.84). We now look at the quadratic terms $Q_{2}^{n}$ as given by
(4.55b); these are linear combinations of terms of the form
$\Gamma^{n_{1}+1}V_{\omega}\cdot\Gamma^{n_{2}}\nabla W$ for $n_{1}+n_{2}\leq
n-1$. As before, we see that for (4.84) it suffices to show the stronger bound

$\displaystyle{\|\Gamma^{n_{1}+1}V_{\omega}(s)\cdot\Gamma^{n_{2}}\nabla
W(s)\|}_{L}\lesssim\varepsilon_{1}^{2}\langle s\rangle^{\delta},\qquad
n_{1}+n_{2}\leq N_{1}+11.$ (4.87)

In the case $n_{2}\leq n_{1}$, with $n_{1}+n_{2}\leq N_{1}+11$, we use
Hölder’s inequality with (4.34) to estimate $\Gamma^{\leq
N_{1}+12}V_{\omega}$, and the a priori estimate (4.4) for
$\Gamma^{N_{1}-10}W$:

$\displaystyle{\|\Gamma^{n_{1}+1}V_{\omega}(s)\cdot\Gamma^{n_{2}}W(s)\|}_{L}$
$\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}+12}V_{\omega}(s)\|}_{L^{\infty}_{z}L^{2}_{x}}{\|\Gamma^{\leq
N_{1}-12}\nabla W(s)\|}_{L^{2}_{z}L^{\infty}_{x}\cap L^{6/5}_{z}L^{3}_{x}}$
$\displaystyle\lesssim\varepsilon_{1}\langle
s\rangle^{\delta}\cdot{\|\Gamma^{\leq N_{1}-10}W(s)\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}$ $\displaystyle\lesssim\varepsilon_{1}^{2}\langle
s\rangle^{\delta}.$

In the case $n_{2}\geq n_{1}$, with $n_{1}+n_{2}\leq N_{1}+11$, we use instead
the bound (4.36) on low norms of $V_{\omega}$, and the a priori estimate (4.5)
on high norms of $W$:

$\displaystyle{\|\Gamma^{n_{1}+1}V_{\omega}(s)\cdot\Gamma^{n_{2}}W(s)\|}_{L}$
$\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}-12}V_{\omega}(s)\|}_{L^{\infty}_{x,z}}{\|\Gamma^{\leq N_{1}+10}\nabla
W(s)\|}_{L}\lesssim\varepsilon_{1}\cdot\varepsilon_{1}\langle
s\rangle^{\delta}.$

Proof of (4.85). Recall the form of the cubic terms from (4.56); we only
detail how to treat the first one, that is,

$\displaystyle\begin{split}C_{1}^{n}&:=\sum_{n_{1}+n_{2}+n_{3}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla
V_{\omega}\cdot\Gamma^{n_{3}}\mathcal{O}_{1},\end{split}$ (4.88)

as the other term can be dealt with in the same way.

We first observe that if $n_{3}\leq n_{1}+n_{2}$, then, we can use (4.86) to
estimate as follows: for all $n\leq N_{1}+12$

$\displaystyle{\|C_{1}^{n}\|}_{L}$ $\displaystyle\lesssim\sup_{n_{1}+n_{2}\leq
N_{1}+12}{\|\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla
V_{\omega}\|}_{L}\cdot{\|\Gamma^{\leq
N_{1}-10}\mathcal{O}_{1}\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{1}^{2}\langle
t\rangle^{\delta}\cdot\varepsilon_{0}\langle t\rangle^{-1+},$

which is more than sufficient. When instead $n_{3}\geq n_{1}+n_{2}$ we cannot
use directly (4.86), but using $n_{1},n_{2}\leq N_{1}-12$, together with the
bounds on low norms (4.4) and (4.36), and (4.27b), we get

$\displaystyle{\|C_{1}^{n}\|}_{L}$ $\displaystyle\lesssim{\|\Gamma^{\leq
N_{1}-12}W\|}_{L}{\|\Gamma^{\leq N_{1}-12}\nabla
V_{\omega}\|}_{L^{\infty}_{x,z}}\cdot{\|\Gamma^{\leq
N_{1}+12}\mathcal{O}_{1}\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{1}\cdot\varepsilon_{1}\cdot\varepsilon_{0}\langle
t\rangle^{3p_{0}}.$

Putting these together we get

$\displaystyle{\|C_{1}^{n}\|}_{L}+{\|C_{2}^{n}\|}_{L}$
$\displaystyle\lesssim\varepsilon_{1}^{2}\varepsilon_{0}\langle
t\rangle^{3p_{0}},\qquad n\leq N_{1}+12;$ (4.89)

upon time integration, we see that the last two bounds above are more than
sufficient for (4.85). This concludes the proof of (4.6)-(4.7) for $j=0$.

Estimates for the time derivative. We now prove (4.6)-(4.7) for $j=1$. From
(4.58) we have, for all $n\leq N_{1}+11$,

$\displaystyle\begin{split}\partial_{t}\Gamma^{n}W=-U\cdot\nabla\Gamma^{n}W&+\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla W\\\
&+\sum_{n_{1}+n_{2}\leq n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla
X+\Gamma^{\leq n}F,\end{split}$ (4.90)

where, recall, $F$ is defined in (4.12). Using (4.50), and with the notation
for quadratic and cubic terms from (4.55a)-(4.55b) and (4.56), equation (4.90)
is

$\displaystyle\begin{split}\partial_{t}\Gamma^{n}W&=\mathcal{O}_{1}\cdot\nabla\Gamma^{n}W+V_{\omega}\cdot\nabla\Gamma^{n}W+V_{\omega}\cdot\mathcal{O}_{1}\cdot\nabla\Gamma^{n}W\\\
&+\sum_{n_{1}+n_{2}\leq n-1}\Gamma^{\leq
n_{1}+1}\mathcal{O}_{1}\cdot\Gamma^{n_{2}}\nabla W+Q_{2}^{n}+C_{2}^{n}\\\
&+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\mathcal{O}_{1}+Q_{1}^{n}+\Gamma^{\leq
n}F.\end{split}$ (4.91)

To obtain (4.6), respectively (4.7), we need to show that all the terms on the
right-hand side of (4.91) are bounded by $\varepsilon_{1}$ when $n\leq
N_{1}-11$, respectively, by $\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta}$ when $n\leq N_{1}+11$.

Note that the bounds (4.86) and (4.87) already give that

$\displaystyle{\|Q^{n}_{1}(t)\|}_{L}+{\|Q^{n}_{2}(t)\|}_{L}\lesssim\varepsilon_{1}^{2}\langle
t\rangle^{\delta},\qquad n\leq N_{1}+12,$

(since the summation in the definition of $Q^{n}_{2}$ goes up to $n-1$, see
(4.55b)), which is more than sufficient for the desired bounds. Similarly, the
bounds established before on $\Gamma^{\leq n}F$ are also sufficient; indeed,
$\Gamma^{\leq n}F$ is a combination of $F^{n}_{0}$ (see (4.60)) which is an
acceptable remainder satisfying (4.57b), and of a cubic term of the form
$C_{1}^{n}$ (see the last term in (4.59c)) which satisfies (4.89). The bound
(4.89) also handles the term $C_{2}^{n}$ in (4.91).

The term $V_{\omega}\cdot\nabla\Gamma^{n}W$, for $n\leq N_{1}+11$, is one of
the terms appearing in $Q_{1}^{N_{1}+12}$ and, therefore, can be estimated as
above.

To conclude, we need to estimate the four terms involving the
$\mathcal{O}_{1}$ factors in (4.91). These can be handled directly using (3)
in Remark 4.4, with the bounds (4.31)-(4.32) giving more than what is needed.
This concludes the proof of Proposition 4.1. ∎

##  5\. Proof of Proposition 2.8

The aim of this section is to prove Proposition 2.8, that is, establish the
bound

$\displaystyle\sum_{r+k\leq
N_{0}-20}{\big{\|}|\nabla|^{1/2}\varphi(t)\big{\|}}_{Z^{r}_{k}({\mathbb{R}}^{2})}\lesssim\varepsilon_{0}\langle
t\rangle^{3p_{0}}$ (5.1)

for all $t\in[0,T]$, under the a priori energy bounds on the velocity,
vorticity and height (2.20), and the decay bound on the height (2.22) and
velocity (2.21). To prove (5.1) we will in part rely on the proof of
Proposition 3.12 without repeating most of the arguments, and on some of the
material in Section 4. We are going to use the following strategy:

  * •

We bootstrap a (weak) bound for the $L^{2}\cap L^{6/5}$ norm of $W$ with a
high number ($N_{0}-20$) of vector fields, just using the high Sobolev energy
bound (2.20); this bound is just of size $\varepsilon_{0}\langle
t\rangle^{p_{0}}$, as opposed to the much better bound $\varepsilon_{1}$ for
the low norms; see the assumptions of Proposition (3.12) and Proposition 4.1.

  * •

We input the above (weak) information into the fixed point argument used to
obtain Proposition 3.12, and obtain corresponding (weak) bounds on $\alpha$,
and therefore on the vector potential $V_{\omega}$.

  * •

Finally we obtain bounds for $|\nabla|^{1/2}\varphi$ from trace estimates,
thanks to the bounds for
$\partial^{i}\Psi=V^{i}+\partial^{i}h\partial_{z}\Psi-V^{i}_{\omega}$ that are
directly implied by the bounds on $V_{\omega}$ and $V$.

######  Remark 5.1.

Note that we get the slightly faster growth rate $3p_{0}$ in (5.1) as opposed
to the more natural $p_{0}$, as for the high energies, because in the course
of proving the above bounds we will work in the “flattened” variables
$(V,\Psi,V_{\omega},W)$ instead of the variables $(v,\psi,v_{\omega},\omega)$
in the original domain $\mathcal{D}_{t}$. When measured in low-order norms
(say, less than $N_{1}$ vector fields and gradients) all of the “flattened”
quantities are equivalent to the original ones, see (5.7). On the other hand,
for higher-order norms, one needs to control products of high-order norms of
$h$ with lower-order norms of the flattened variables in $L^{2}$. Since we do
not propagate uniform control on all of the flattened variables in
$L^{2}_{z}L^{\infty}_{x}$, this winds up generating terms which grow slightly
faster that $\langle t\rangle^{p_{0}}$, which ultimately leads to the growth
rate in (5.1). This slightly weaker bound is still sufficient for the rest of
our arguments to close, in particular those in Section 4 (see Definition 4.3
for the $\mathcal{O}_{1}$ class) and Section 6 (see (6.17)).

###  5.1. (Weak) Bounds on the vorticity

We use the notation of Section 4, see in particular Subsection 4.1, and aim to
prove the bootstrap Proposition 2.7, which we rewrite here for convenience.

######  Proposition 5.2.

Assume that the a priori bounds (2.20), (2.21), and (2.22) hold. Let $W$ be as
defined in (2.23), and assume (2.25) and (2.27). Recall the definition of the
space $\mathcal{X}^{n}$ from (2.24):

$\displaystyle\begin{split}{\|f\|}_{\mathcal{X}^{n}}:=\sum_{|r|+|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla^{r}_{x,z}\,f\big{\|}}_{L({\mathbb{R}}^{2}\times{\mathbb{R}}_{\leq
0})},\qquad L:=L^{2}_{x,z}\cap L^{6/5}_{x,z}.\end{split}$ (5.2)

Then, for all $t\in[0,T]$, $T\leq T_{\varepsilon_{1}}$, we have the bound

$\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{0}-20}}\leq
c_{W}\varepsilon_{0}\langle t\rangle^{2p_{0}}.$ (5.3)

In the proof of Proposition 5.2 and in other places, we are going to need a
basic lemma about transfer of norms from $\mathcal{D}_{t}$ to the flat domain.
As before, for a given $f:[0,T]\times\mathcal{D}_{t}\rightarrow{\mathbb{R}}$,
we use the corresponding capital letter to define, for $t\in[0,T]$,
$x\in{\mathbb{R}}^{2}$ and $z\leq 0$,

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

In what follows we use the convention about repeated applications of vector
field from Subsection 4.2 and, for additional clarity, we will underline the
$3$d vector fields, while reserving $\Gamma$ only for the $2$d vector fields.

######  Lemma 5.3.

With the above definitions, the notation from 4.2, and under the a priori
bounds on $h$ from (2.20) and (2.22), we have the following schematic
identity: if $n\leq N_{0}$,

$\displaystyle\begin{split}\underline{\Gamma}^{n}F(t,x,z)=(\underline{\Gamma}^{n}f)(t,x,z+h(t,x))&+(\underline{\Gamma}^{\leq
n}f)(t,x,z+h(t,x))\cdot O\big{(}\Gamma^{\leq n/2+1}h(t,x)\big{)}\\\
&+(\underline{\Gamma}^{\leq n/2+1}f)(t,x,z+h(t,x))\cdot O\big{(}\Gamma^{\leq
n}h(t,x)\big{)}\end{split}$ (5.4)

where the notation $G=O(\Gamma^{\leq k}h)$ here means that

$\displaystyle\big{|}\Gamma^{\ell}G\big{|}\lesssim\sum_{j=0}^{k+\ell}\big{|}\Gamma^{j}h(t,x)\big{|},$
(5.5)

with an absolute implicit constant (depending on $\ell$). Similarly, we can
write

$\displaystyle\begin{split}\underline{\Gamma}^{n}f(t,x,y)=(\underline{\Gamma}^{n}F)(t,x,y-h(t,x))&+(\underline{\Gamma}^{\leq
n}F)(t,x,y-h(t,x))\cdot O\big{(}\Gamma^{\leq n/2+1}h(t,x)\big{)}\\\
&+(\underline{\Gamma}^{\leq n/2+1}F)(t,x,y-h(t,x))\cdot O\big{(}\Gamma^{\leq
n}h(t,x)\big{)}.\end{split}$ (5.6)

In particular, for $p\in[2,\infty]$, and $n\leq N_{1}$, there exists constants
$C_{1},C_{2}>0$ such that

$\displaystyle
C_{1}\sum_{k=0}^{n}{\big{\|}\underline{\Gamma}^{k}f(t)\big{\|}}_{L^{p}(\mathcal{D}_{t})}\leq\sum_{k=0}^{n}{\big{\|}\underline{\Gamma}^{k}F(t)\big{\|}}_{L^{p}_{x,z}}\leq
C_{2}\sum_{k=0}^{n}{\big{\|}\underline{\Gamma}^{k}f(t)\big{\|}}_{L^{p}(\mathcal{D}_{t})}.$
(5.7)

For $n\leq N_{0}$ we have instead

$\displaystyle\begin{split}{\|\underline{\Gamma}^{n}F(t)\|}_{L^{2}_{x,z}}&\lesssim\sum_{k\leq
n}{\|\underline{\Gamma}^{k}f(t)\|}_{L^{2}(\mathcal{D}_{t})}+\varepsilon_{0}\langle
t\rangle^{p_{0}}\sum_{k\leq
n/2+3}{\|\underline{\Gamma}^{k}f(t)\|}_{L^{2}(\mathcal{D}_{t})}\end{split}$
(5.8)

and, similarly,

$\displaystyle\begin{split}{\|\underline{\Gamma}^{n}f(t)\|}_{L^{2}(\mathcal{D}_{t})}&\lesssim\sum_{k\leq
n}{\|\underline{\Gamma}^{k}F(t)\|}_{L^{2}_{x,z}}+\varepsilon_{0}\langle
t\rangle^{p_{0}}\sum_{k\leq
n/2+3}{\|\underline{\Gamma}^{k}F(t)\|}_{L^{2}_{x,z}}.\end{split}$ (5.9)

###### Proof.

The identities (5.4) and (5.6) follow from applying repeatedly the composition
formulas (3.13)-(3.14) and using the uniform bound on the $L^{\infty}$ norm of
$h$ from (2.22) to verify the property (5.5).

The estimates (5.7) then follow directly since
$|\Gamma^{k}h|\lesssim\varepsilon_{0}$ for all $k\leq N_{1}$. For (5.8) we
instead apply Hölder’s inequality to (5.4) by estimating in $L^{\infty}$ the
term $O(\Gamma^{\leq n/2+1}h)$ and in $L^{2}$ the term $O(\Gamma^{\leq n}h)$,
placing $\underline{\Gamma}^{\leq n/2+1}f$ in $L^{2}_{z}L^{\infty}_{x}$ and
then using Sobolev embedding. The estimate (5.9) follows similarly from (5.6).
∎

A statement similar to the one in Lemma 5.3 holds for restrictions to the
boundary:

######  Lemma 5.4.

With the same notation, definitions and a priori assumptions in Lemma 5.3, and
denoting $\widetilde{g}(t,x):=g(t,x,h(t,x))$, we have the following schematic
identity: if $n\leq N_{0}$,

$\displaystyle\begin{split}\Gamma^{n}\widetilde{f}=\widetilde{\underline{\Gamma}^{n}f}&+\widetilde{\underline{\Gamma}^{\leq
n}f}\cdot O\big{(}\Gamma^{\leq
n/2+1}h\big{)}+\widetilde{\underline{\Gamma}^{\leq n/2+1}f}\cdot
O\big{(}\Gamma^{\leq n}h\big{)}.\end{split}$ (5.10)

In particular, for $p\in[2,\infty]$, and $n\leq N_{1}$, there exists constants
$C_{1},C_{2}>0$ such that

$\displaystyle
C_{1}\sum_{k=0}^{n}{\big{\|}\Gamma^{n}\widetilde{f}(t)\big{\|}}_{L^{p}({\mathbb{R}}^{2})}\leq\sum_{k=0}^{n}{\big{\|}\widetilde{\underline{\Gamma}^{k}f}(t)\big{\|}}_{L^{p}({\mathbb{R}}^{2})}\leq
C_{2}\sum_{k=0}^{n}{\big{\|}\Gamma^{n}\widetilde{f}(t)\big{\|}}_{L^{p}({\mathbb{R}}^{2})}.$
(5.11)

For $n\leq N_{0}-2$ we have instead

$\displaystyle\begin{split}{\|\Gamma^{n}\widetilde{f}(t)\|}_{L^{p}({\mathbb{R}}^{2})}&\lesssim\sum_{k\leq
n}{\|\widetilde{\Gamma^{k}f}(t)\|}_{L^{p}({\mathbb{R}}^{2})}+\varepsilon_{0}\langle
t\rangle^{p_{0}}\sum_{k\leq
n/2+1}{\|\widetilde{\Gamma^{k}f}(t)\|}_{L^{p}({\mathbb{R}}^{2})}\end{split}$
(5.12)

###### Proof.

The proof is similar to that of Lemma 5.3. The identity (5.10) follows from
applying repeatedly (3.15) and using the uniform a priori bound on $h$ in
$L^{\infty}$. The estimate (5.11) then follows immediately, while (5.12)
follows using Hölder, estimating $\Gamma^{\leq n}h$ in $L^{\infty}$, followed
by Sobolev’s embedding and (2.20). ∎

We can now give the proof of (3.11):

###### Proof of (3.11).

Since $\partial_{t}h=\widetilde{v}\cdot(-\nabla h,1)$, distributing vector
fields we see that a schematic formula like the one in (5.10), with the
notation (5.5), holds and, in particular,

$\displaystyle\begin{split}|\Gamma^{n}\partial_{t}h|\lesssim|\widetilde{\underline{\Gamma}^{n}v}|&+|\widetilde{\underline{\Gamma}^{\leq
n}v}|\cdot|O\big{(}\Gamma^{\leq
n/2+1}h\big{)}|+|\widetilde{\underline{\Gamma}^{\leq
n/2+1}v}|\cdot|O\big{(}\Gamma^{\leq n}h\big{)}|.\end{split}$

It follows that

$\displaystyle{\|\Gamma^{n}\partial_{t}h\|}_{L^{2}}$
$\displaystyle\lesssim{\|\underline{\Gamma}^{\leq
n+1}v\|}_{L^{2}(\mathcal{D}_{t})}\big{(}1+{\|\Gamma^{\leq
n/2+1}h\|}_{L^{\infty}}\big{)}$ $\displaystyle+{\|\underline{\Gamma}^{\leq
n/2+1}v\|}_{L^{\infty}(\mathcal{D}_{t})}\cdot{\|\Gamma^{\leq
n}h\|}_{L^{2}}\lesssim\varepsilon_{0}\langle t\rangle^{p_{0}},$

where in the last inequality we have used the apriori energy bound (2.20) for
both $v$ and $h$, and the decay bounds (2.22) and (2.21) to control uniformly
the $L^{\infty}$ norms. ∎

###### Proof of Proposition 5.2.

We use the same notation from Subsection 4.1, and the conventions from
Subsection 4.2. Recall from (2.25) and (2.27) that we are assuming, for all
$t\in[0,T]$, $T\leq T_{\varepsilon_{1}}$,

$\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{1}-10}}\leq 2c_{L}\varepsilon_{1},$
(5.13) $\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{0}-20}}\leq
2c_{W}\varepsilon_{0}\langle t\rangle^{2p_{0}},$ (5.14)

for some absolute constant $c_{L},c_{W}>0$ large enough, and that the
following assumption on the initial data hold in view of (2.14):

$\displaystyle{\|W_{0}\|}_{\mathcal{X}^{N_{1}-10}}\leq
C\varepsilon_{1},\qquad{\|W_{0}\|}_{\mathcal{X}^{N_{0}-20}}\leq
C\varepsilon_{0}.$ (5.15)

We then aim to show that the improved bound (5.3) holds for all $t\in[0,T]$.

We begin by writing the vorticity equation as in (4.10):

$\displaystyle\begin{split}&{\bf D}_{t}W=W\cdot\nabla
V-W^{\ell}\partial_{\ell}h\partial_{z}V^{i},\\\ &{\bf
D}_{t}:=\partial_{t}+U\cdot\nabla,\qquad
U:=V-(\partial_{t}h+V^{\ell}\partial_{\ell}h)e_{z}.\end{split}$ (5.16)

We then apply vector fields as in Subsection 4.3 and obtain the following
equation (see Lemma 4.8):

$\displaystyle\begin{split}{\bf D}_{t}\Gamma^{n}W&=\sum_{n_{1}+n_{2}\leq
n-1}\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla W+\sum_{n_{1}+n_{2}\leq
n}\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla V+\Gamma^{\leq
n}F^{\prime},\end{split}$ (5.17)

with $F^{\prime}:=-W^{\ell}\partial_{\ell}h\partial_{z}V^{i}$. Compare this
with (4.49) and note that the only difference is that in this case we do not
separate the linear and quadratic components in $V$, nor distinguish the
rotational and irrotational components.

To obtain estimates for $W$ based on (5.16) we need energy and decay bounds
for $V$ and $U$. First, using (5.7) and the priori decay assumptions on $v$ in
(2.21) we have

$\displaystyle{\|\underline{\Gamma}^{\leq
n}V\|}_{L^{\infty}_{x,z}}\lesssim{\|\underline{\Gamma}^{\leq
n}v\|}_{L^{\infty}(\mathcal{D}_{t})}\lesssim\varepsilon_{0}\langle
t\rangle^{-1}+\varepsilon_{1}\langle t\rangle^{\delta},\qquad n\leq N_{1}-5;$
(5.18)

using (5.8) and the a priori energy control (2.20) we have

$\displaystyle{\|\Gamma^{\leq
n}V\|}_{L^{2}_{x,z}}\lesssim{\|\underline{\Gamma}^{\leq
n}v\|}_{L^{2}(\mathcal{D}_{t})}+\varepsilon_{0}\langle
t\rangle^{p_{0}}\sum_{k\leq
n/2+3}{\|\underline{\Gamma}^{k}v\|}_{L^{2}(\mathcal{D}_{t})}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}},\qquad n\leq N_{0}.$ (5.19)

Then, from the definition of $U$ in (5.16), using
$\partial_{t}h=v\cdot(-\nabla h,1)$ at the free boundary with the bounds on
$V$ just used above, and basic product estimates to handle the quadratic term
$V\cdot\nabla h$, it follows that

$\displaystyle{\|\Gamma^{\leq
n}U\|}_{L^{\infty}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}},$ $\displaystyle\qquad n\leq N_{0}-10,$ (5.20)
$\displaystyle{\|\Gamma^{\leq
n}U\|}_{L^{\infty}_{x,z}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1+}+\varepsilon_{1}\langle t\rangle^{\delta},$ $\displaystyle\qquad
n\leq N_{1}-5.$ (5.21)

Applying Lemma 4.12 to (5.17), we have

$\displaystyle{\|\Gamma^{n}W(t)\|}_{L^{p}_{x,z}}\leq{\|\Gamma^{n}W(0)\|}_{L^{p}_{x,z}}+C\int_{0}^{t}{\|{\bf
D}_{s}\Gamma^{n}W(s)\|}_{L^{p}_{x,z}}\,ds.$ (5.22)

Therefore, to obtain (5.3) it then suffices to show that

$\displaystyle{\|{\bf D}_{t}\Gamma^{n}W(t)\|}_{L^{2}_{x,z}\cap
L^{6/5}_{x,z}}\leq c\varepsilon_{0}\langle t\rangle^{-1+2p_{0}},\qquad n\leq
N_{0}-20,$ (5.23)

for all $t\leq T\leq T_{\varepsilon_{1}}$ and a sufficiently small absolute
constant $c$.

The estimate (5.23) can be verified directly for all the terms on the right-
hand side of (5.17) by using elementary product estimates, the a priori
assumptions (5.13)-(5.14), the bounds on $V$ and $U$ in (5.18)-(5.20), and the
usual bounds on $h$ in (2.20) and (2.22). In particular, with
$L:=L^{2}_{x,z}\cap L^{6/5}_{x,z}$ we claim that the following bounds hold:

$\displaystyle{\|\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla
W\|}_{L}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta+2p_{0}}+\varepsilon_{0}^{2}\langle t\rangle^{-1+2p_{0}},$
$\displaystyle\qquad n_{1}+n_{2}\leq N_{0}-20-1,$ (5.24a)
$\displaystyle{\|\Gamma^{n_{1}}W\cdot\Gamma^{n_{2}}\nabla
V\|}_{L}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta+2p_{0}}+\varepsilon_{0}^{2}\langle t\rangle^{-1+2p_{0}},$
$\displaystyle\qquad n_{1}+n_{2}\leq N_{0}-20,$ (5.24b)
$\displaystyle{\|\Gamma^{n}(W^{\ell}\partial_{\ell}h\partial_{z}V^{i})\|}_{L}\lesssim\varepsilon_{0}^{2}\varepsilon_{1}\langle
t\rangle^{2p_{0}}+\varepsilon_{0}^{3}\langle t\rangle^{-1+2p_{0}},$
$\displaystyle\qquad n\leq N_{0}-20.$ (5.24c)

Notice that these bounds imply the desired (5.23) since
$\varepsilon_{1}\langle t\rangle^{\delta}\ll\langle t\rangle^{-1}$ for all
$t\leq T_{\varepsilon_{1}}$.

Let us prove (5.24a). When $n_{1}\geq n_{2}$ so that, in particular $n_{2}\leq
N_{0}/2-10\leq N_{1}-10$ we can use (5.13) to estimate $\Gamma^{n_{2}}\nabla
W$, and use Sobolev embedding and (5.20) to estimate $\Gamma^{\leq n_{1}+1}U$:

$\displaystyle\begin{split}&{\|\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla
W\|}_{L}\lesssim{\|\Gamma^{\leq
n_{1}+1}U\|}_{L^{\infty}_{x,z}}{\|\Gamma^{n_{2}}\nabla
W\|}_{L}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}}\cdot\varepsilon_{1}.\end{split}$ (5.25)

When instead $n_{1}\leq n_{2}$, so that $n_{1}\leq N_{0}/2-10\leq N_{1}-10$,
we can use the a priori assumption (5.14) for $\Gamma^{n_{2}}W$, and use the
decay estimate (5.20) to estimate $\Gamma^{\leq n_{1}+1}U$:

$\displaystyle\begin{split}&{\|\Gamma^{\leq n_{1}+1}U\cdot\Gamma^{n_{2}}\nabla
W\|}_{L}\lesssim{\|\Gamma^{\leq
n_{1}+1}U\|}_{L^{\infty}_{x,z}}{\|\Gamma^{n_{2}}\nabla
W\|}_{L}\lesssim(\varepsilon_{0}\langle t\rangle^{-1}+\varepsilon_{1}\langle
t\rangle^{\delta})\cdot\varepsilon_{0}\langle t\rangle^{2p_{0}}.\end{split}$

These last two bounds are consistent with the right hand-side of (5.24a). The
other bounds in (5.24) can be proven in the same way. This concludes the proof
of (5.23) of the proposition. ∎

###  5.2. Bounds for the vector potential

We now state bounds on $V_{\omega}$ that follow from the bounds in the high
norm that we just obtained on $W$, see (5.14).

######  Proposition 5.5 (Bounds for $\alpha$ from bounds on $W$).

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 $t\in[0,T]$,

$\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{1}-10}}\lesssim\varepsilon_{1},$
(5.26)
$\displaystyle{\|W(t)\|}_{\mathcal{X}^{N_{0}-20}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}}.$ (5.27)

Then, there exists a unique fixed point $\alpha$ of the map in (3.46) in the
space $\dot{Y}^{N_{0}-20}$, which satisfies

$\displaystyle{\|\alpha(t)\|}_{\dot{Y}^{N_{1}-10}}\lesssim\varepsilon_{1},$
(5.28)
$\displaystyle{\|\alpha(t)\|}_{\dot{Y}^{N_{0}-20}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}}.$ (5.29)

This proposition is an exact analogue of Proposition 3.12 stated with high
norms that contain $N_{0}-20$ vector fields instead of $N_{1}+12$; compare
(3.55) and (5.27). The conclusion (5.29) is the one that naturally corresponds
to (3.57) with the different assumption. The proof follows verbatim the one is
Subsection 3.4. The only thing to observe is that the assumptions used on $h$
in the proof of Proposition 3.12 also suffice when working at a higher level
of derivatives. The only relevant aspect is that half of the highest number of
vector fields, that is $N_{0}/2-10$ here, needs to be (a couple of units) less
than two numbers: the number of vector fields for which we have uniform bounds
when applied to $h$, that is, $N_{1}$, and the number of vector fields in the
low norm, that is, $N_{1}-10$. These hold in view of (2.9).

To conclude the proof of (5.1) we want to use the fact that
$\varphi=\Psi|_{z=0}$ where
$\nabla^{i}\Psi=V^{i}+\nabla^{i}h\partial_{z}\Psi-V^{i}_{\omega}$; we then
need the following bound for $V_{\omega}$:

######  Lemma 5.6.

Under the same hypotheses of Proposition 5.2 we have

$\sum_{|r|+|k|\leq
N_{0}-20}{\|V_{\omega}^{r,k}\|}_{L^{2}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}}$ (5.30)

###### Proof.

By Propositions 5.2 and 4.1, the hypotheses of Proposition 5.5 hold, and the
bounds (5.28)-(5.29) for $\alpha$ follow. Since (5.29) is (3.40), the bound
(5.30) now follows from (3.41). ∎

###  5.3. Conclusion: Proof of (5.1)

To conclude the proof we first use the trace inequality (A.25) to bound the
left-hand side of (5.1)

$\displaystyle\sum_{r+|k|\leq
N_{0}-20}{\big{\|}|\nabla|^{1/2}\Gamma^{k}\varphi(t)\big{\|}}_{H^{r}({\mathbb{R}}^{2})}\lesssim$
$\displaystyle\sum_{r+|k|\leq
N_{0}-20}{\big{\|}\nabla_{x}^{r}\underline{\Gamma}^{k}\nabla_{x,z}\Psi(t)\big{\|}}_{L^{2}_{x,z}}.$
(5.31)

From the identities
$\nabla^{i}\Psi=V^{i}+\nabla^{i}h\partial_{z}\Psi-V^{i}_{\omega}$ for $i=1,2$
and $\nabla_{z}\Psi=V^{3}-V^{3}_{\omega}$, the bound (5.19) for
$\|\Gamma^{k}V\|_{L^{2}_{z}L^{2}_{x}}$, and the bound (5.30) for $V_{\omega}$,
for any $n\leq N_{0}-20$, we have

$\displaystyle\|\underline{\Gamma}^{\leq
n}\nabla_{x}\Psi\|_{L^{2}_{z}L^{2}_{x}}\lesssim\sum_{|k|\leq
n}\|\underline{\Gamma}^{k}V\|_{L^{2}_{z}L^{2}_{x}}+\sum_{|k|\leq
n}\|\underline{\Gamma}^{k}(\nabla
h\partial_{z}\Psi)\|_{L^{2}_{z}L^{2}_{x}}+\sum_{|k|\leq
n}\|\underline{\Gamma}^{k}V_{\omega}\|_{L^{2}_{z}L^{2}_{x}}$
$\displaystyle\lesssim\varepsilon_{0}\langle t\rangle^{2p_{0}}+\sum_{|k|\leq
n}\|\underline{\Gamma}^{k}(\nabla h\partial_{z}\Psi)\|_{L^{2}_{z}L^{2}_{x}},$

and

$\|\underline{\Gamma}^{\leq
n}\nabla_{z}\Psi\|_{L^{2}_{x}L^{2}_{x}}\lesssim\sum_{|k|\leq
n}\|\underline{\Gamma}^{k}V\|_{L^{2}_{z}L^{2}_{x}}+\sum_{|k|\leq
n}\|\underline{\Gamma}^{k}V_{\omega}\|_{L^{2}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}\langle
t\rangle^{2p_{0}}.$

Finally, we can estimate, for any $n\leq N_{0}-20$,

$\displaystyle\sum_{|k|\leq n}\|\underline{\Gamma}^{k}(\nabla
h\partial_{z}\Psi)\|_{L^{2}_{z}L^{2}_{x}}$ $\displaystyle\lesssim\sum_{|k|\leq
n}\|\Gamma^{k}\nabla h\|_{L^{\infty}}\sum_{k\leq
n}\|\underline{\Gamma}^{k}\partial_{z}\Psi\|_{L^{2}_{z}L^{2}_{x}}\lesssim\varepsilon_{0}^{2}\langle
t\rangle^{3p_{0}}.$

This concludes the proof of (5.1). $\Box$

###  5.4. Proofs of Lemma 2.12 and 2.13

We conclude this section by showing how to recover the a priori decay
assumption (2.21) through (2.44) and (2.45).

Proof of (2.44) First, recall that, in view of (2.43) we have

$\displaystyle\sum_{\ell\in{\mathbb{Z}}}\sum_{|k|\leq
N_{1}}{\|\Gamma^{k}P_{\ell}|\nabla|^{1/2}\varphi(t)\|}_{L^{\infty}({\mathbb{R}}^{2})}\leq
c_{B}\varepsilon_{0}\langle t\rangle^{-1}$ (5.32)

and, therefore, in view of Remark B.7, we get

$\displaystyle\sum_{|k|\leq
N_{1}-1}\sum_{\ell\in{\mathbb{Z}}}{\big{\|}P_{\ell}\underline{\Gamma}^{k}\nabla_{x,z}\Psi(t)\big{\|}}_{L^{2}_{z}L^{\infty}_{x}}\leq
Cc_{B}\varepsilon_{0}\langle t\rangle^{-1}$ (5.33)

for some generic $C>0$. We then use the composition estimate (5.7), sum over
dyadic Littlewood-Paley pieces and use (5.33) after Sobolev’s embedding in $z$
to get, with $n=N_{1}-5$,

$\displaystyle\sum_{r+k\leq
n}{\|\nabla\psi(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}$ $\displaystyle\leq
C\sum_{|k|\leq
n}{\big{\|}\underline{\Gamma}^{k}\nabla_{x,z}\Psi(t)\big{\|}}_{L^{\infty}_{x,z}}$
$\displaystyle\leq C\sum_{|k|\leq
n}\sum_{\ell\in{\mathbb{Z}}}{\big{\|}P_{\ell}\underline{\Gamma}^{k}\nabla_{x,z}\Psi(t)\big{\|}}_{L^{\infty}_{x,z}}\leq
Cc_{B}\varepsilon_{0}\langle t\rangle^{-1}.$

This gives us Lemma 2.12.

Proof of (2.45) We use the composition estimate (5.7), followed by Sobolev’s
embedding in $x$, and then apply directly the second estimate on $V_{\omega}$
from (2.47) with $j=0$ to see that

$\displaystyle\sum_{r+k\leq
N_{1}-5}{\|v_{\omega}(t)\|}_{X^{r,\infty}_{k}(\mathcal{D}_{t})}$
$\displaystyle\leq C\sum_{|k|\leq
N_{1}-3}{\|\underline{\Gamma}^{k}V_{\omega}(t)\|}_{L^{\infty}_{z}L^{2}_{x}}\leq
Cc_{H}^{\prime}\varepsilon_{1}\langle t\rangle^{\delta}.$ (5.34)

This gives Lemma 2.13 and closes the bootstrap for the norm in (2.21).

##  6\. Decay of the boundary variables

In this section we use the restriction of the free boundary Euler equations to
the boundary surface to establish time decay for the dispersive variable $u$
and prove Proposition 2.10, and, in fact, the better estimate (2.43).

###  6.1. Set-up and equations at the boundary

Recall that

$\displaystyle u=h+i\Lambda^{1/2}\varphi,\qquad u_{+}=u,\quad u_{-}=\bar{u}.$
(6.1)

With
$P_{\omega}^{i}=\widetilde{v_{\omega}}^{i}=v_{\omega}^{i}|_{\partial\mathcal{D}_{t}}$,
$i=1,2$, one can show, see (B.25)-(B.31) in Appendix B, that $u$ solves

$\displaystyle(\partial_{t}+i\Lambda^{1/2})u=B_{0}(u,u)+B_{0,1}(u+\bar{u},P_{\omega})+B_{1}(P_{\omega},P_{\omega})+L(P_{\omega})+N_{3}$
(6.2)

where:

  * •

The quadratic terms involving only $u$ are given by

$\displaystyle\begin{split}B_{0}(u,u)&=\sum_{\epsilon_{1},\epsilon_{2}\in\\{+,-\\}}B_{\epsilon_{1}\epsilon_{2}}(u_{\epsilon_{1}},u_{\epsilon_{2}}),\end{split}$
(6.3)

with the definitions (B.30) for the symbols and the notation (A.12) for the
associated operators;

  * •

The quadratic terms involving at least one copy of $P_{\omega}$ are

$\displaystyle B_{0,1}(f,g):=(i\nabla\Lambda^{-1/2}f)\cdot g,\qquad
B_{1}(f,g):=-\frac{1}{2}f\cdot g;$ (6.4)

  * •

The ‘linear forcing term’ due to the vorticity is

$\displaystyle L(P_{\omega}):=-i\Lambda^{-1/2}R\cdot\partial_{t}P_{\omega};$
(6.5)

  * •

$N_{3}$ are the cubic and higher order terms in (B.28).

####  6.1.1. Vectorfields and Duhamel’s formula

We start by applying vector fields to (6.2) and deriving an equation for

$\displaystyle
u^{n}:=\Gamma^{n}u,\qquad|n|<N_{0},\qquad\Gamma\in\\{S,\Omega\\}.$ (6.6)

Note that we are not including regular derivatives in this notation. Using
(A.3) and (B.33) to commute vector fields and quadratic symbols, we see that
there exist real constants $a_{n_{1}}$, $c_{n_{1}n_{2}}$ and $d_{n_{1}n_{2}}$
such that

$\displaystyle\begin{split}(\partial_{t}+i\Lambda^{1/2})u^{n}=\sum_{|n_{1}|+|n_{2}|\leq|n|}c_{n_{1}n_{2}}B_{0}(u^{n_{1}},u^{n_{2}})+d_{n_{1}n_{2}}B_{0,1}(u^{n_{1}}+\bar{u}^{n_{1}},P_{\omega}^{n_{2}})\\\
+\sum_{n_{1}+n_{2}=n}B_{1}(P_{\omega}^{n_{1}},P_{\omega}^{n_{2}})+\sum_{|n_{1}|\leq|n|}a_{n_{1}}L^{n_{1}}(P_{\omega}^{n_{1}})+\Gamma^{n}N_{3},\end{split}$
(6.7)

where the linear term is given by

$\displaystyle
L^{n_{1}}(P_{\omega}^{n_{1}}):=-i\Lambda^{-1/2}R^{(n_{1})}\cdot\partial_{t}P_{\omega}^{n_{1}},\qquad\mbox{with}\qquad
R^{(n_{1})}\in\mathrm{span}\\{R,R^{\perp}\\}.$ (6.8)

Duhamel’s formula for (6.7) gives

$\displaystyle u^{n}(t)=e^{-it\Lambda^{1/2}}u_{0}^{n}$
$\displaystyle+\sum_{|n_{1}|+|n_{2}|\leq|n|}c_{n_{1}n_{2}}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}B_{0}(u^{n_{1}},u^{n_{2}})\,ds$
(6.9)
$\displaystyle+\sum_{|n_{1}|+|n_{2}|\leq|n|}d_{n_{1}n_{2}}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}B_{0,1}(u^{n_{1}}+\bar{u}^{n_{1}},P_{\omega}^{n_{2}})\,ds$
(6.10)
$\displaystyle+\sum_{|n_{1}|+|n_{2}|=n}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}B_{1}(P_{\omega}^{n_{1}},P_{\omega}^{n_{2}})\,ds$
(6.11)
$\displaystyle+\sum_{|n_{1}|\leq|n|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}a_{n_{1}}L^{n_{1}}(P_{\omega}^{n_{1}})\,ds+\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}\Gamma^{n}N_{3}(s)\,ds.$
(6.12)

The linear flow $e^{-it\Lambda^{1/2}}u_{0}$ is directly handled using Lemma
A.1.

####  6.1.2. Reduction of the proof of Proposition 2.10

To prove Proposition 2.10 it then suffices to prove the following bounds:

$\displaystyle{\Big{\|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}B_{0}(u^{n_{1}},u^{n_{2}})\,ds\Big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},\qquad r+(|n_{1}|+|n_{2}|)\leq N_{1};$ (6.13)

for sufficiently small $c^{\prime}=c_{r,n}^{\prime}$

$\displaystyle\begin{split}&{\Big{\|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}B(s)\,ds\Big{\|}}_{W^{r,\infty}}\leq
c^{\prime}\varepsilon_{0}\langle t\rangle^{-1},\\\ &\mbox{with}\quad
B\in\big{\\{}B_{0,1}(u_{\pm}^{n_{1}},P_{\omega}^{n_{2}}),\,\,B_{1}(P_{\omega}^{n_{1}},P_{\omega}^{n_{2}})\big{\\}},\qquad
r+(|n_{1}|+|n_{2}|)\leq N_{1};\end{split}$ (6.14)

for sufficiently small $c^{\prime\prime}=c^{\prime\prime}_{n_{1}}$

$\displaystyle\begin{split}{\Big{\|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}L^{n_{1}}(P_{\omega}^{n_{1}})\,ds\Big{\|}}_{W^{r,\infty}}\leq
c^{\prime\prime}\varepsilon_{0}\langle t\rangle^{-1},\qquad r+|n_{1}|\leq
N_{1};\end{split}$ (6.15)

and, finally,

$\displaystyle{\Big{\|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}\Gamma^{n}N_{3}(s)\,ds\Big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{2}\langle
t\rangle^{-1},\qquad r+|n|\leq N_{1}.$ (6.16)

######  Remark 6.1.

As mentioned in Remark 2.11, we are actually going to show stronger estimates
than (6.13)-(6.16), with $\ell^{1}$ sums over frequencies, that is, we will
prove all of the estimates for the Besov $B^{r}_{\infty,1}$ instead of the
$W^{r,\infty}$ norm. These bounds are essentially automatic in view of the
following: (a) the estimates for bilinear forms (6.29) which we are going to
use to establish the bounds needed for (6.13) and (b) the linear bound (A.1)
which we are going to use (as in Lemma 6.3, for example) and which already has
the $\ell^{1}$ sum on the right-hand side.

For convenience we recall that, in what follows, we will be working under the
assumptions (2.40) and (2.22), that is, for all $t\leq T_{\varepsilon_{1}}$,

$\displaystyle\sum_{r+|k|\leq
N_{0}-20}{\|u^{k}(t)\|}_{H^{r}({\mathbb{R}}^{2})}\lesssim\varepsilon_{0}\langle
t\rangle^{3p_{0}},$ (6.17) $\displaystyle\sum_{r+|k|\leq
N_{1}}{\big{\|}u^{k}(t)\big{\|}}_{W^{r,\infty}({\mathbb{R}}^{2})}\leq
c_{0}\varepsilon_{0}\langle t\rangle^{-1}.$ (6.18)

We will often use these assumptions without referring to them explicitly. We
will also use the bound on the rotational component in (2.41):

$\displaystyle\sum_{r+k\leq
N_{1}+12-j}{\big{\|}\partial_{t}^{j}P_{\omega}(t)\big{\|}}_{Z^{r}_{k}({\mathbb{R}}^{2})}\lesssim\varepsilon_{1}\varepsilon_{0}^{j}\langle
t\rangle^{\delta},\quad j=0,1.$ (6.19)

###  6.2. Normal form transformation

For the quadratic terms which only depend on the dispersive variable we need
normal form transformations. Define the profile

$\displaystyle f^{n}(t)=e^{it\Lambda^{1/2}}u^{n}(t)$ (6.20)

and, in accordance with (6.3), write

$\displaystyle\begin{split}&\int_{0}^{t}e^{is\Lambda^{1/2}}B_{0}(u^{n_{1}},u^{n_{2}})\,ds=\frac{1}{(2\pi)^{2}}\sum_{\epsilon_{1},\epsilon_{2}\in\\{+,-\\}}\mathcal{F}^{-1}I_{\epsilon_{1}\epsilon_{2}}(t),\\\
&I_{\epsilon_{1}\epsilon_{2}}(t):=\int_{0}^{t}\\!\\!\int_{{\mathbb{R}}^{3}}e^{is\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}b_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)\widehat{f_{\epsilon_{1}}^{n_{1}}}(\xi-\eta)\widehat{f_{\epsilon_{2}}^{n_{2}}}(\eta)\,d\eta
ds,\end{split}$ (6.21)

where

$\displaystyle\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)=|\xi|^{1/2}-\epsilon_{2}|\xi-\eta|^{1/2}-\epsilon_{1}|\eta|^{1/2},$
(6.22)

and we omitted the dependence on $n_{1},n_{2}$ of
$I_{\epsilon_{1}\epsilon_{2}}$. For convenience, in what follows we will often
use the short-hand

$\displaystyle f_{j}:=f_{\epsilon_{j}}^{n_{j}},\quad
u_{j}:=u^{n_{j}}_{\epsilon_{j}}=e^{-\epsilon_{j}it\Lambda^{1/2}}f_{j},\quad
j=1,2.$ (6.23)

Define

$\displaystyle
m_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)=\frac{b_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}{i\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}$
(6.24)

where the symbols $b_{\epsilon_{1}\epsilon_{2}}$ are given by (B.30).
Integrating by parts in time we can write

$\displaystyle I_{\epsilon_{1}\epsilon_{2}}(t)$
$\displaystyle=\left.\int_{{\mathbb{R}}^{3}}e^{is\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}m_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)\widehat{f_{1}}(s,\xi-\eta)\widehat{f_{2}}(s,\eta)\,d\eta\right|^{s=t}_{s=0}$
(6.25a)
$\displaystyle-\int_{0}^{t}\\!\\!\int_{{\mathbb{R}}^{3}}e^{is\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}m_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)\partial_{s}\Big{[}\widehat{f_{1}}(s,\xi-\eta)\widehat{f_{2}}(s,\eta)\Big{]}\,d\eta
ds.$ (6.25b)

To estimate the above expressions we recall the definition in (A.11) and
observe that

$\displaystyle{\|\Phi^{k,k_{1},k_{2}}(\xi,\eta)\|}_{\mathcal{S}^{\infty}}\approx
2^{(1/2)\min(k,k_{1},k_{2})}$ (6.26)

and, therefore, in view of (B.32) and Lemma A.2,

$\displaystyle{\|m_{\epsilon_{1}\epsilon_{2}}^{k,k_{1},k_{2}}(\xi,\eta)\|}_{\mathcal{S}^{\infty}}\lesssim
2^{(1/2)k}\cdot 2^{(1/2)\max(k_{1},k_{2})}.$ (6.27)

In particular, the symbols appearing in (6.25) are not singular and, using the
estimate (A.16) of Lemma A.2 with the bound (6.27), we get, for all
$1/p=1/p_{1}+1/p_{2}$,

$\displaystyle{\|P_{k}M_{\epsilon_{1}\epsilon_{2}}(P_{k_{1}}g(t),P_{k_{2}}h(t))\|}_{W^{r,p}}\lesssim
2^{rk_{+}}2^{k/2}2^{(1/2)\max(k_{1},k_{2})}{\|P_{k_{1}}g(t)\|}_{L^{p_{2}}}{\|P_{k_{2}}h(t)\|}_{L^{p_{1}}}.$
(6.28)

With $p=p_{1}=p_{2}=\infty$, using Bernstein’s inequality, we can deduce that

$\displaystyle\begin{split}{\|M_{\epsilon_{1}\epsilon_{2}}(g,h)\|}_{L^{\infty}}\lesssim\sum_{k}{\|P_{k}M_{\epsilon_{1}\epsilon_{2}}(g,h)\|}_{L^{\infty}}\lesssim{\|g\|}_{W^{2,\infty-}}{\|h\|}_{W^{2,\infty-}}.\end{split}$
(6.29)

Using (6.28) with $p=2$ and Bernstein’s inequality we can obtain

$\displaystyle\begin{split}{\|M_{\epsilon_{1}\epsilon_{2}}(g,h)\|}_{H^{r}}\lesssim\min\big{(}{\|g\|}_{H^{r+2}}{\|h\|}_{W^{2,\infty-}},{\|g\|}_{W^{2,\infty-}}{\|h\|}_{H^{r+2}}\big{)}.\end{split}$
(6.30)

###  6.3. Proof of (6.13)

From (6.21) and (6.25a)-(6.25b) we see that the desired bound (6.13) follows
if we show

$\displaystyle{\big{\|}e^{-it\Lambda^{1/2}}\mathcal{F}^{-1}\eqref{prdec2.1}\big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},$ (6.31)
$\displaystyle{\big{\|}e^{-it\Lambda^{1/2}}\mathcal{F}^{-1}\eqref{prdec2.2}\big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},$ (6.32)

for all $r+|n_{1}|+|n_{2}|\leq N_{1}$ (recall the notation in (6.23)).

####  6.3.1. Proof of (6.31)

We only look at the terms in (6.25a) with $s=t$ since the $s=0$ contribution
is easier to estimate. We write these as

$\displaystyle\int_{{\mathbb{R}}^{3}}e^{it\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}\frac{b_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}{i\Phi_{\epsilon_{1}\epsilon_{2}}(\xi,\eta)}\widehat{f_{1}}(t,\xi-\eta)\widehat{f_{2}}(t,\eta)\,d\eta=\mathcal{F}\big{[}e^{it\Lambda^{1/2}}M_{\epsilon_{1}\epsilon_{2}}(u_{1}(t),u_{2}(t))\big{]},$

and, according to (6.31), aim to show that

$\displaystyle{\|M_{\epsilon_{1}\epsilon_{2}}(\nabla^{r_{1}}u_{1}(t),\nabla^{r_{2}}u_{2}(t))\|}_{L^{\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},\qquad r_{1}+r_{2}+(|n_{1}|+|n_{2}|)\leq N_{1}.$ (6.33)

Using (6.29) we can estimate

$\displaystyle\begin{split}&{\|M_{\epsilon_{1}\epsilon_{2}}(\nabla^{r_{1}}u_{1}(t),\nabla^{r_{2}}u_{2}(t))\|}_{L^{\infty}}\lesssim{\|\nabla^{r_{1}}u_{1}(t)\|}_{W^{2,\infty-}}{\|\nabla^{r_{2}}u_{2}(t)\|}_{W^{2,\infty-}}.\end{split}$
(6.34)

Then, recall from (6.17)-(6.18) that since $u_{j}=u^{n_{j}}_{\epsilon_{j}}$,
we have in particular

$\displaystyle\langle t\rangle{\|u_{j}\|}_{W^{r_{j},\infty}}+\langle
t\rangle^{-3p_{0}}{\|u_{j}\|}_{H^{r_{j}+(N_{0}-N_{1}-20)}}\lesssim\varepsilon_{0},\qquad
r_{j}+|n_{j}|\leq N_{1},$ (6.35)

so that Sobolev-Gagliardo-Nirenberg interpolation gives

$\displaystyle{\|u_{j}\|}_{W^{r_{j}+2,\infty-}}\lesssim\varepsilon_{0}\langle
t\rangle^{-2/3},\qquad r_{j}+|n_{j}|\leq N_{1}.$ (6.36)

Using this inequality in (6.34) gives (6.33).

####  6.3.2. Proof of (6.32)

To estimate bulk terms (6.25b) we need estimates for the time derivative of
the profile $f_{j}$. First, from the definition (6.23) we have
$\partial_{t}f_{j}=e^{\epsilon_{j}it\Lambda^{1/2}}(\partial_{t}+\epsilon_{j}i\Lambda^{1/2})u_{j}$.
Assuming $\epsilon_{j}=+$ (the other case is obtained by conjugation), with
the notation (6.6) and using the equation (6.7) we have

$\displaystyle\partial_{t}f^{n}=e^{it\Lambda^{1/2}}\sum_{|n_{1}|\leq|n|}a_{n_{1}}L^{n_{1}}(P_{\omega}^{n_{1}})+e^{it\Lambda^{1/2}}Q^{n}$
(6.37)

with

$\displaystyle\begin{split}Q^{n}:=\sum_{|n_{1}|+|n_{2}|\leq|n|}c_{n_{1}n_{2}}B_{0}(u^{n_{1}},u^{n_{2}})+d_{n_{1}n_{2}}B_{0,1}(u^{n_{1}}+\bar{u}^{n_{1}},P_{\omega}^{n_{2}})\\\
+\sum_{|n_{1}|+|n_{2}|\leq|n|}B_{1}(P_{\omega}^{n_{1}},P_{\omega}^{n_{2}})+\Gamma^{n}N_{3}.\end{split}$
(6.38)

We first establish some estimates for $Q^{n}$ and will then rely on (6.19) to
estimate the contribution from the operator $L(P_{\omega})$.

######  Lemma 6.2.

Under the a priori assumptions, for any $t\leq T$, we have

$\displaystyle{\big{\|}Q^{n}\big{\|}}_{H^{r}}\lesssim\varepsilon_{0}\langle
t\rangle^{-2/3},\qquad r+|n|\leq N_{1}+11,$ (6.39)

and

$\displaystyle{\big{\|}Q^{n}\big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}\langle
t\rangle^{-5/4},\qquad r+|n|\leq N_{1}-5.$ (6.40)

Notice that the estimates in the above Lemma are not optimal in terms of decay
rates, since $Q^{n}$ is effectively quadratic in $(u,P_{\omega})$, but they
will suffice for our purposes.

###### Proof of Lemma 6.2.

We first estimate separately all the terms on the right-hand side of (6.38) in
$H^{r}$ with the claimed number of vector fields. The $L^{\infty}$-type
estimate will follow similarly.

Proof of (6.39). In view of (B.32), $B_{0}(u,u)$ satisfies standard product
estimates up to a small loss of derivatives:

$\displaystyle{\|B_{0}(g,h)\|}_{L^{p}}\lesssim\min\big{(}{\|g\|}_{W^{2,p}}{\|h\|}_{W^{2,\infty}},{\|g\|}_{W^{2,\infty}}{\|h\|}_{W^{2,p}}\big{)}.$
(6.41)

The desired bound on the $B_{0}$ terms is implied by

$\displaystyle{\|B_{0}(\nabla^{r_{1}}u^{n_{1}},\nabla^{r_{2}}u^{n_{2}})\|}_{L^{2}}\lesssim\varepsilon_{0}\langle
t\rangle^{-2/3},\qquad r_{1}+r_{2}+(|n_{1}|+|n_{2}|)\leq N_{0}-10.$ (6.42)

To show (6.42), without loss of generality, let us assume that
$r_{1}+|n_{1}|\leq(N_{0}-10)/2\leq N_{1}-2$, see (2.9), and estimate using
(6.41),

$\displaystyle{\|B_{0}(\nabla^{r_{1}}u^{n_{1}},\nabla^{r_{2}}u^{n_{2}})\|}_{L^{2}}$
$\displaystyle\lesssim{\|\nabla^{r_{1}}u^{n_{1}}\|}_{W^{2,\infty}}{\|\nabla^{r_{2}}u^{n_{2}}\|}_{H^{2}}\lesssim\varepsilon_{0}\langle
t\rangle^{-1}\cdot\varepsilon_{0}\langle t\rangle^{3p_{0}},$

which is more than sufficient; we have used (6.17) (since $r_{2}+2+|n_{2}|\leq
N_{0}-8$) and (6.18) for the last inequality .

The terms $B_{0,1}(u_{\pm}^{n_{1}},P_{\omega}^{n_{2}})$ and
$B_{1}(P_{\omega}^{n_{1}},P_{\omega}^{n_{2}})$ are easier to treat, using the
estimates for $P_{\omega}^{n}$ in (6.19). From the definition in (6.4),
Hölder’s inequality and Sobolev’s embedding we have, for
$r_{1}+r_{2}+(|n_{1}|+|n_{2}|)\leq N_{1}+11$,

$\displaystyle{\|B_{0,1}(u_{\pm}^{n_{1}},P_{\omega}^{n_{2}})\|}_{L^{2}}\lesssim{\|u_{\pm}^{n_{1}}\|}_{H^{3}}{\|P_{\omega}^{n_{2}}\|}_{L^{2}}\lesssim\varepsilon_{0}\langle
t\rangle^{3p_{0}}\cdot\varepsilon_{1}\langle t\rangle^{\delta},$

having used again (6.17); this is sufficient since
$\varepsilon_{1}\lesssim\langle t\rangle^{-1}$. Similarly, assuming without
loss of generality that $r_{1}+|n_{1}|\leq(N_{1}+11)/2$, we can estimate,
using again (6.19),

$\displaystyle{\|B_{1}(P_{\omega}^{n_{1}},P_{\omega}^{n_{2}})\|}_{L^{2}}\lesssim{\|P_{\omega}^{n_{1}}\|}_{H^{2}}{\|P_{\omega}^{n_{2}}\|}_{L^{2}}\lesssim\varepsilon_{1}^{2}\langle
t\rangle^{2\delta}$

which clearly suffices. Since the bound for $\Gamma^{n}N_{3}$ follows directly
from the stronger estimate (B.37), the proof of (6.39) is concluded.

Proof of (6.40). The $L^{\infty}$ type bound (6.40) can be obtained similarly,
by estimating in $W^{r,\infty}$ all the terms on the right-hand side of (6.7)
by means of the product estimate (6.41) with $p=\infty$, and using (B.37) for
$N_{3}$. ∎

We now go back to the proof of (6.32). First observe that by symmetry it
suffices to consider the case when $\partial_{s}$ hits the first profile in
the formulas for (6.25b), and show

$\displaystyle\begin{split}{\Big{\|}e^{-it\Lambda^{1/2}}\int_{0}^{t}e^{is\Lambda^{1/2}}M_{\epsilon_{1}\epsilon_{2}}\big{(}e^{-\epsilon_{1}is\Lambda^{1/2}}\partial_{s}f^{n_{1}}_{\epsilon_{1}}(s),u^{n_{2}}_{\epsilon_{2}}(s)\big{)}\,ds\Big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},\\\ \qquad r+(|n_{1}|+|n_{2}|)\leq N_{1}.\end{split}$ (6.43)

In what follows we drop the $\epsilon_{1},\epsilon_{2}$ signs since they do
not play any role. Using (6.37)-(6.38) we see that (6.43) reduces to showing
the two following estimates:

$\displaystyle\begin{split}{\Big{\|}e^{-it\Lambda^{1/2}}\int_{0}^{t}e^{is\Lambda^{1/2}}M\big{(}Q^{n_{1}}(s),u^{n_{2}}(s)\big{)}\,ds\Big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},\qquad r+(|n_{1}|+|n_{2}|)\leq N_{1}\end{split}$ (6.44)

and

$\displaystyle\begin{split}{\Big{\|}e^{-it\Lambda^{1/2}}\int_{0}^{t}e^{is\Lambda^{1/2}}M\big{(}L^{n_{1}}(P_{\omega}^{n_{1}}),u^{n_{2}}(s)\big{)}\,ds\Big{\|}}_{W^{r,\infty}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1},\qquad r+(|n_{1}|+|n_{2}|)\leq N_{1}.\end{split}$ (6.45)

In what follows we are going to use the following lemma, which is a
consequence of the linear decay estimate in Lemma A.1.

######  Lemma 6.3.

Let $F=F(t,x)$ be such that, for all $k=0,\dots,3$ and $n=0,1$,

$\displaystyle\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\|S^{n}\Omega^{k}P_{\ell}F(t)\|}_{L^{2}}\leq
A_{n}(t).$ (6.46)

Then we have the non-homogeneous decay bound

$\displaystyle{\Big{\|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}F(s)\,ds\Big{\|}}_{L^{\infty}}\lesssim
A_{0}(t)+\langle t\rangle^{-1}\int_{0}^{t}(A_{0}(s)+A_{1}(s))\,ds.$ (6.47)

###### Proof.

We first apply the linear estimate A.1 to obtain

$\displaystyle\langle
t\rangle\,{\Big{\|}e^{-it\Lambda^{1/2}}\int_{0}^{t}e^{is\Lambda^{1/2}}F(s)\,ds\Big{\|}}_{L^{\infty}}$
$\displaystyle\lesssim\sup_{k=0,\dots,3}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\Big{\|}\Sigma\Omega^{k}\int_{0}^{t}e^{is\Lambda^{1/2}}P_{\ell}F(s)\,ds\Big{\|}}_{L^{2}}$
$\displaystyle\lesssim\sup_{k=0,\dots,3,\,n=0,1}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\Big{\|}\int_{0}^{t}e^{is\Lambda^{1/2}}S^{n}\Omega^{k}P_{\ell}F(s)\,ds\Big{\|}}_{L^{2}}$
$\displaystyle+\sup_{k=0,\dots,3}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\Big{\|}\int_{0}^{t}s\partial_{s}\Big{[}e^{is\Lambda^{1/2}}\Omega^{k}P_{\ell}F(s)\Big{]}\,ds\Big{\|}}_{L^{2}}.$

We have used that $\Sigma:=x\cdot\nabla=S-(1/2)s\partial_{s}$, and
$[S,e^{is\Lambda^{1/2}}]=-1$ and $[\Omega,e^{is\Lambda^{1/2}}]=0$. The first
of the two terms on the above right-hand side is already accounted for in the
bound (6.47). For the last term we integrate by parts in $s$ and use the
assumption (6.46) to conclude. ∎

###### Proof of (6.44).

Using Lemma 6.3 above, and commuting the scaling and rotation vector fields
and derivatives, we see that in order to obtain (6.44) it suffices to show

$\displaystyle\begin{split}{\big{\|}M\big{(}\nabla^{r_{1}}Q^{n_{1}}(t),\nabla^{r_{2}}u^{n_{2}}(t)\big{)}\big{\|}}_{L^{2}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1-},\\\ \qquad|r_{1}|+|r_{2}|+|n_{1}|+|n_{2}|\leq
N_{1}+5.\end{split}$ (6.48)

The number $N_{1}+5$ is coming from the presence of four vector fields in the
definition of $A_{1}$ in (6.46) and taking the $H^{1}$ norm instead of the
Besov norm $\dot{B}^{1/2}_{2,1}$.

Case $|r_{1}|+|n_{1}|\geq N_{1}/2+6$. Using (6.30) we can estimate the left-
hand side of (6.48) by

$\displaystyle\begin{split}C{\|Q^{n_{1}}(t)\|}_{H^{|r_{1}|+2}}{\|u^{n_{2}}(t)\|}_{W^{|r_{2}|+2,\infty-}}.\end{split}$

We can then use (6.39) (since $|r_{1}|+2+|n_{1}|\leq N_{1}+11$) to estimate
the first term, and (6.35) (since $|r_{2}|+|n_{2}|\leq N_{1}/2+3<N_{1}$) to
estimate the second; this gives and upper bound of $C\varepsilon_{0}\langle
t\rangle^{-2/3}\cdot\varepsilon_{0}\langle t\rangle^{-1+}$ which suffices.

Case $|r_{1}|+|n_{1}|\leq N_{1}/2+5$. In this case we use (6.30) to estimate
the left-hand side of (6.48) by

$\displaystyle\begin{split}&C{\|Q^{n_{1}}(t)\|}_{W^{|r_{1}|+2,\infty-}}{\|u^{n_{2}}(t)\|}_{H^{|r_{2}|+2}}\lesssim\varepsilon_{0}\langle
t\rangle^{-11/10}\cdot\varepsilon_{0}\langle t\rangle^{3p_{0}},\end{split}$

having used (6.40) and Sobolev-Gagliardo-Nirenberg interpolation with (6.39),
and the a priori bound (6.17). This concludes the proof of (6.48), hence of
(6.44). ∎

###### Proof of (6.45).

To prove (6.45) we are going to use the following lemma, which gives a (non-
optimal) interpolation-type estimate for $u$ when more than $N_{1}$ vector
fields and derivatives are applied to it.

######  Lemma 6.4.

Under the assumptions (6.17)-(6.18) and (6.19), we have

$\displaystyle\sup_{2^{k}\geq\langle
t\rangle^{-2/3}}{\|P_{k}\nabla^{r}u^{n}(t)\|}_{L^{\infty}}\lesssim\varepsilon_{0}\langle
t\rangle^{-2/3},\qquad|r|+|n|\leq N_{1}+7\quad(=N-5).$ (6.49)

A lower bound on the frequencies in (6.49) is needed to avoid dealing with
very small frequencies in the arguments below, but the exact restriction
$2^{k}\gtrsim\langle t\rangle^{2/3}$ is rather arbitrary and it is unrelated
to the decay rate on the right-hand side.

###### Proof of Lemma 6.4.

We begin by using (A.2) to see that, for all $|r|+|n|\leq N_{1}+7$,

$\displaystyle{\|P_{k}\nabla^{r}u^{n}\|}_{L^{\infty}}$
$\displaystyle\lesssim\langle t\rangle^{-1}\sup_{|r|+|n|\leq
N_{1}+11}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\|\nabla^{r}P_{\ell}u^{n}\|}_{L^{2}({\mathbb{R}}^{2})}$
(6.50) $\displaystyle+\sup_{|r|+|n|\leq N_{1}+10}\sum_{2^{\ell}\gtrsim\langle
t\rangle^{-2/3}}2^{\ell/2}{\|\nabla^{r}P_{\ell}\partial_{t}f^{n}\|}_{L^{2}({\mathbb{R}}^{2})}.$
(6.51)

Notice how we kept the restriction on not-too-low frequencies in (6.51).

The term on the right-hand side of (6.50) is estimated directly using the
$L^{2}$ based a priori assumption (6.17), which gives a bound for (6.50) by
$C\langle t\rangle^{-1}\cdot\varepsilon_{0}\langle t\rangle^{3p_{0}}$; this is
more than enough for the desired conclusion (6.49).

To handle the terms in (6.51) we use the equation (6.37). The contribution to
$\partial_{t}f^{n}$ from the terms $Q^{n}$ can be estimated directly using
(6.39), which is consistent with the bound (6.49). For the contribution of the
linear forcing term instead, we recall the definition (6.5) and estimate for
any $|r|+|n|\leq N_{1}+10$,

$\displaystyle\sum_{2^{\ell}\gtrsim\langle
t\rangle^{-2/3}}2^{\ell/2}{\|\nabla^{r}P_{\ell}L^{n}(P_{\omega}^{n})\|}_{L^{2}({\mathbb{R}}^{2})}\lesssim\sum_{2^{\ell}\gtrsim\langle
t\rangle^{-2/3}}{\|\nabla^{r}P_{\ell}\,\partial_{t}P_{\omega}^{n}\|}_{L^{2}({\mathbb{R}}^{2})}$
$\displaystyle\lesssim\log(2+t)\,{\|\nabla^{r}\partial_{t}P_{\omega}^{n}\|}_{L^{2}({\mathbb{R}}^{2})}+\sum_{2^{\ell}\gtrsim
1}{\|\nabla^{r}P_{\ell}\,\partial_{t}P_{\omega}^{n}\|}_{L^{2}({\mathbb{R}}^{2})}$
$\displaystyle\lesssim\log(2+t){\|\nabla^{r}\partial_{t}P_{\omega}^{n}\|}_{H^{1}({\mathbb{R}}^{2})}\lesssim\log(2+t)\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta},$

having used (6.19); since $\varepsilon_{1}\lesssim\langle t\rangle^{-1}$ this
concludes the proof of the lemma. ∎

We now proceed with the proof of (6.45). Using again Lemma 6.3 as in the proof
of (6.44) above, we reduce matters to an estimate analogous to (6.48), that
is,

$\displaystyle\begin{split}{\big{\|}M\big{(}\nabla^{r_{1}}L^{n_{1}}(P_{\omega}^{n_{1}})(t),\nabla^{r_{2}}u^{n_{2}}(t)\big{)}\big{\|}}_{L^{2}}\lesssim\varepsilon_{0}^{1+}\langle
t\rangle^{-1-},\\\ \qquad|r_{1}|+|r_{2}|+(|n_{1}|+|n_{2}|)\leq
N_{1}+5.\end{split}$ (6.52)

We first prove that

$\displaystyle\sum_{k\in{\mathbb{Z}}}{\|P_{k}\nabla^{r}L^{n}(P_{\omega}^{n})\|}_{L^{4+}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta},\qquad r+|n|\leq N_{1}+10,$ (6.53)

and

$\displaystyle\sum_{k\in{\mathbb{Z}}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{4-}}$
$\displaystyle\lesssim\varepsilon_{0}\langle t\rangle^{-1/4},\qquad r+|n|\leq
N_{1}+7.$ (6.54)

From the definition (6.8), the estimate (6.19) and Bernstein’s inequality, we
have, for all $r+|n|\leq N_{1}+10$,

$\displaystyle\sum_{k\in{\mathbb{Z}}}{\|P_{k}\nabla^{r}L^{n}(P_{\omega}^{n})\|}_{L^{4+}}$
$\displaystyle\lesssim\sum_{k\leq
0}2^{-k/2}{\|P_{k}\nabla^{r}\partial_{t}P_{\omega}^{n}\|}_{L^{4+}}+\sum_{k\geq
1}2^{-k/2}{\|P_{k}\nabla^{r}\partial_{t}P_{\omega}^{n}\|}_{L^{4+}}$
$\displaystyle\lesssim\sum_{k\leq
0}2^{(0+)k}{\|\nabla^{r}\partial_{t}P_{\omega}^{n}\|}_{L^{2}}+{\|\nabla^{r}\partial_{t}P_{\omega}^{n}\|}_{H^{1}}\lesssim{\|\partial_{t}P_{\omega}^{n}\|}_{H^{|r|+1}}\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta}.$

Using Bernstein’s inequality and interpolation with the bounds (6.49) and
(6.17) we have, for all $|r|+|n|\leq N_{1}+7$,

$\displaystyle\sum_{k\in{\mathbb{Z}}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{4-}}\lesssim\sum_{2^{k}\leq\langle
t\rangle^{-2/3}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{4-}}+\sum_{2^{k}\geq\langle
t\rangle^{-2/3}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{4-}}$
$\displaystyle\lesssim\langle
t\rangle^{(-1/3)+}{\big{\|}\nabla^{r}u^{n}(t)\big{\|}}_{L^{2}}+\sum_{2^{k}\geq\langle
t\rangle^{-2/3}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{2}}^{(1/2)+}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{\infty}}^{(1/2)-}$
$\displaystyle\lesssim\langle t\rangle^{(-1/3)+}\varepsilon_{0}\langle
t\rangle^{3p_{0}}+(\varepsilon_{0}\langle
t\rangle^{-2/3})^{(1/2)-}\cdot\sum_{2^{k}\geq\langle
t\rangle^{-2/3}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{2}}^{(1/2)+}$
$\displaystyle\lesssim\langle t\rangle^{(-1/3)+}\varepsilon_{0}\langle
t\rangle^{3p_{0}}+(\varepsilon_{0}\langle
t\rangle^{-2/3})^{(1/2)-}\log(2+t)\sup_{2^{k}\geq\langle
t\rangle^{-2/3}}\big{(}2^{k_{+}}{\big{\|}P_{k}\nabla^{r}u^{n}(t)\big{\|}}_{L^{2}}\big{)}^{(1/2)+}$
$\displaystyle\lesssim\varepsilon_{0}\langle t\rangle^{-1/4}.$

Note how we included a small loss in the second to last inequality coming from
the summation of $k$ with $\langle t\rangle^{-2/3}\lesssim 2^{k}\lesssim 1$,
and how we used the validity of (6.49) for frequencies $2^{k}\gtrsim\langle
t\rangle^{-2/3}$ in the third inequality.

Using the Hölder-type bound (6.28) with the fact that
$k\leq\max(k_{1},k_{2})+5$, and the estimates (6.53) and (6.54) above, we can
bound the left-hand side of (6.52) for all
$|r_{1}|+|r_{2}|+|n_{1}|+|n_{2}|\leq N_{1}+5$ as follows:

$\displaystyle\begin{split}&{\big{\|}M\big{(}\nabla^{r_{1}}L^{n_{1}}(P_{\omega}^{n_{1}})(t),\nabla^{r_{2}}u^{n_{2}}(t)\big{)}\big{\|}}_{L^{2}}\\\
&\lesssim\sum_{k_{1},k_{2}}2^{\max(k_{1},k_{2})}{\big{\|}P_{k_{1}}\nabla^{r_{1}}L^{n_{1}}(P_{\omega}^{n_{1}})(t)\|}_{L^{4+}}{\big{\|}P_{k_{2}}\nabla^{r_{2}}u^{n_{2}}(t)\big{\|}}_{L^{4-}}\\\
&\lesssim\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta}\cdot\varepsilon_{0}\langle t\rangle^{-1/4};\end{split}$

since $\varepsilon_{1}\leq\langle t\rangle^{-1}$, this concludes the proof of
(6.45).

∎

With (6.44)-(6.45) we have obtained the desired estimates for the cubic bulk
terms, and the proof of (6.32) is concluded. The bound (6.13) follows.

###  6.4. Proof of (6.14)

The proof of (6.14) in the case of the $B_{1}$ terms is easier than for the
$B_{0,1}$ terms so we can just focus on these latter. From Lemma 6.3 we see
that it suffices to show (we drop the $\pm$)

$\displaystyle\begin{split}{\big{\|}B_{0,1}(\nabla^{r_{1}}u^{n_{1}}(t),\nabla^{r_{2}}P_{\omega}^{n_{2}}(t))\big{\|}}_{L^{2}}\leq
c^{\prime}\varepsilon_{0}\langle t\rangle^{-1-},\\\
|r_{1}|+|r_{2}|+|n_{1}|+|n_{2}|\leq N_{1}+5.\end{split}$ (6.55)

Let us consider the case $|r_{1}|+|n_{1}|\geq N_{1}/2$, and disregard the
complementary case which is easier since we can estimate
$\nabla^{r_{1}}u^{n_{1}}$ in $L^{\infty}$ and obtain a bound of the form
$C\varepsilon_{0}\langle t\rangle^{-1}\cdot\varepsilon_{1}\langle
t\rangle^{\delta}$. We then recall the definition (6.4) of $B_{0,1}$ and
estimate using the bounds (6.19) for $P_{\omega}$, Bernstein, the estimate
(6.49) for $P_{k}\nabla^{r}u^{n}$ and the a priori assumption (6.17) on the
energy: for all $|r_{1}|+|n_{1}|\geq N_{1}/2$, $|r_{2}|+|n_{2}|\leq N_{1}/2+5$
we have

$\displaystyle{\big{\|}B_{0,1}(\nabla^{r_{1}}u^{n_{1}},\nabla^{r_{2}}P_{\omega}^{n_{2}})\big{\|}}_{L^{2}}\lesssim{\big{\|}\Lambda^{-1/2}\nabla\nabla^{r_{1}}u^{n_{1}}\|}_{L^{\infty}}{\|\nabla^{r_{2}}P_{\omega}^{n_{2}}\big{\|}}_{L^{2}}$
$\displaystyle\lesssim\Big{(}\sum_{2^{k_{1}}\leq\langle
t\rangle^{-2/3}}2^{k_{1}/2}{\|P_{k_{1}}\nabla^{r_{1}}u^{n_{1}}\|}_{L^{\infty}}+\sum_{2^{k_{1}}\geq\langle
t\rangle^{-2/3}}2^{k_{1}/2}{\|P_{k_{1}}\nabla^{r_{1}}u^{n_{1}}\|}_{L^{\infty}}\Big{)}\cdot\varepsilon_{1}\langle
t\rangle^{\delta}$ $\displaystyle\lesssim\Big{(}\langle
t\rangle^{-1}{\|\nabla^{r_{1}}u^{n_{1}}\|}_{L^{2}}+\varepsilon_{0}\langle
t\rangle^{-2/3}\Big{)}\cdot\varepsilon_{1}\langle t\rangle^{\delta}$
$\displaystyle\lesssim\varepsilon_{0}\langle
t\rangle^{-2/3}\cdot\varepsilon_{1}\langle t\rangle^{\delta}.$

This is a more than sufficient bound for (6.55) since
$\varepsilon_{1}\lesssim\langle t\rangle^{-1}$.

###  6.5. Proof of (6.15)

Because of the $\Lambda^{-1/2}$ factor in (6.8) we need to be careful once
again about the handling of low frequencies and summations over dyadic
indexes. With the notation in (6.8), we denote the quantity on the left-hand
side of (6.15) as

$\displaystyle
I(t):=\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}\Lambda^{-1/2}R^{(n_{1})}\cdot\partial_{s}P_{\omega}^{n_{1}}(s)\,ds;$
(6.56)

we drop the dependence on $r,n_{1}$ with $|r|+|n_{1}|\leq N_{1}$, and recall
that $R^{(n_{1})}\in\mathrm{span}\\{R,R^{\perp}\\}$. We then write

$\displaystyle\begin{split}I&=I_{l}+I_{m}+I_{h},\\\ I_{l}&:=P_{<L}I,\qquad
I_{m}:=P_{[L,H]}I,\qquad I_{h}:=P_{>H}I,\\\
&L:=\log_{2}\varepsilon_{0}^{2},\quad
H:=\log_{2}\varepsilon_{0}^{-2},\end{split}$ (6.57)

with the notation (2.56)-(2.57); note that here $\varepsilon_{0}$ is the same
quantity in (6.15). $I_{l}$ is a low-frequency contribution with frequencies
of size less than $\varepsilon_{0}^{2}$; $I_{h}$ is a high-frequency
contribution with frequencies of size larger than $\varepsilon_{0}^{-2}$; and
$I_{m}$ is the remaining ‘medium-frequencies’ contribution. We estimate
separately the three contributions in (6.57).

We first look at the medium frequencies contribution and begin by estimating

$\displaystyle{\big{\|}I_{m}\big{\|}}_{W^{r,\infty}}\leq
C|\log(\varepsilon_{0})|\sup_{\ell^{\prime}\in[\varepsilon_{0}^{2},\varepsilon_{0}^{-2}]}{\|P_{\ell^{\prime}}I\|}_{W^{r,\infty}}.$
(6.58)

We then want to apply Lemma 6.3 with
$F=P_{\ell^{\prime}}\Lambda^{-1/2}R^{(n_{1})}\cdot\partial_{s}P_{\omega}^{n_{1}}$.
We note that, for fixed $\ell^{\prime}$, and $k=0,\dots 3$, $n=0,1$, we have

$\displaystyle\begin{split}&\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\Big{\|}S^{n}\Omega^{k}P_{\ell}P_{\ell^{\prime}}\Lambda^{-1/2}R^{(n_{1})}\cdot\partial_{s}P_{\omega}^{n_{1}}(s)\Big{\|}}_{H^{r}}\\\
&\leq C\sum_{n_{1}^{\prime}\leq
n_{1}+4}{\big{\|}\partial_{s}P_{\omega}^{n_{1}^{\prime}}(s)\big{\|}}_{H^{r}}\leq
C\varepsilon_{1}\varepsilon_{0}\langle s\rangle^{\delta},\end{split}$ (6.59)

having used (6.19) for the last inequality. Then (6.58) and the conclusion of
Lemma 6.3 give

$\displaystyle{\big{\|}P_{[\varepsilon_{0}^{2},\varepsilon_{0}^{-2}]}I\big{\|}}_{W^{r,\infty}}$
$\displaystyle\leq
C|\log(\varepsilon_{0})|\Big{(}\varepsilon_{1}\varepsilon_{0}\langle
t\rangle^{\delta}+\langle
t\rangle^{-1}\int_{0}^{t}\varepsilon_{1}\varepsilon_{0}\langle
s\rangle^{\delta}\,ds\Big{)}$ $\displaystyle\leq
C|\log(\varepsilon_{0})|\varepsilon_{1}\varepsilon_{0}\langle
t\rangle^{\delta};$

the last quantity above is bounded by the right-hand side of (6.15) as
desired, since $|\log\varepsilon_{0}|\leq|\log\varepsilon_{1}|$ and we have
$t\leq T:=c\varepsilon_{1}^{-1+\delta}$ for $c$ sufficiently small, so that
$C|\log(\varepsilon_{0})|\varepsilon_{1}\varepsilon_{0}\langle
t\rangle^{\delta}\leq c^{\prime\prime}\varepsilon_{0}\langle t\rangle^{-1}$
for all $t\in[0,T]$.

To handle the low-frequency contributions from $I_{l}$, and the high-frequency
contributions from $I_{h}$, we first rewrite (6.56) in a different way
integrating by parts in $s$:

$\displaystyle I(t)$ $\displaystyle=\Lambda^{-1/2}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(t)$ (6.60)
$\displaystyle-e^{-it\Lambda^{1/2}}\Lambda^{-1/2}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(0)$ (6.61)
$\displaystyle-i\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(s)\,ds.$ (6.62)

Let us first look at the contribution from (6.60). For the low frequencies,
using Bernstein’s inequality and the bound (6.19), we get

$\displaystyle{\|P_{<L}\Lambda^{-1/2}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(t)\|}_{W^{r,\infty}}\lesssim
2^{L/2}{\|P_{\omega}^{n_{1}}(t)\|}_{L^{2}}\lesssim\varepsilon_{0}\cdot\varepsilon_{1}\langle
t\rangle^{\delta}.$ (6.63)

As before this is sufficient for the desired bound by the right-hand side of
(6.15). For the high frequencies we instead estimate using Sobolev’s embedding
and (6.19),

$\displaystyle{\|P_{>H}\Lambda^{-1/2}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(t)\|}_{W^{r,\infty}}\lesssim
2^{-H/2}{\|P_{\omega}^{n_{1}}(t)\|}_{W^{r,\infty}}\lesssim\varepsilon_{0}\cdot\varepsilon_{1}\langle
t\rangle^{\delta}.$ (6.64)

The term (6.61) can be handled in the same way, relying on the bounds at the
initial time:

$\displaystyle{\|P_{<L}e^{-it\Lambda^{1/2}}\Lambda^{-1/2}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(0)\|}_{W^{r,\infty}}\leq
2^{L/2}{\|P_{\omega}^{n_{1}}(0)\|}_{L^{2}}\leq
C\varepsilon_{0}\varepsilon_{1},$
$\displaystyle{\|P_{>H}e^{-it\Lambda^{1/2}}\Lambda^{-1/2}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(0)\|}_{W^{r,\infty}}\leq
2^{-H/2}{\|P_{\omega}^{n_{1}}(0)\|}_{H^{r+1+}}\leq
C\varepsilon_{0}\varepsilon_{1}.$

Finally, we estimate the small and high frequencies contributions from the
term (6.62). We want to apply again Lemma 6.3 in a suitable way. First, we
look at $P_{<L}\eqref{decest'ibp3}$, let $F=P_{<L}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(s)$ and, for all $k=0,\dots 3$, $n=0,1$, estimate

$\displaystyle\begin{split}&\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\Big{\|}S^{n}\Omega^{k}P_{<L}P_{\ell}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(s)\Big{\|}}_{H^{r}}\leq C2^{L/2}\sum_{n_{1}^{\prime}\leq
n_{1}+4}{\big{\|}P_{\omega}^{n_{1}^{\prime}}(s)\big{\|}}_{L^{2}}\leq
C\varepsilon_{0}\varepsilon_{1}\langle s\rangle^{\delta},\end{split}$

having used (6.19). Then, applying the conclusion (6.47) from Lemma 6.3 we get

$\displaystyle\begin{split}{\Big{\|}\int_{0}^{t}e^{i(s-t)\Lambda^{1/2}}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(s)\,ds\Big{\|}}&\leq C\varepsilon_{0}\varepsilon_{1}\langle
t\rangle^{\delta},\end{split}$ (6.65)

which again is consistent with (6.15). For the high frequency contribution we
can proceed similarly using again Lemma 6.3 with the bound

$\displaystyle\begin{split}&\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\Big{\|}S^{n}\Omega^{k}P_{>H}P_{\ell}R^{(n_{1})}\cdot
P_{\omega}^{n_{1}}(s)\Big{\|}}_{H^{r}}\leq C2^{-H/2}\sum_{n_{1}^{\prime}\leq
n_{1}+4}{\big{\|}P_{\omega}^{n_{1}^{\prime}}(s)\big{\|}}_{H^{r+1}}\leq
C\varepsilon_{0}\varepsilon_{1}\langle s\rangle^{\delta}.\end{split}$

The proof of (6.15) is completed.

###  6.6. Proof of (6.16)

The last estimate (6.16) follows similarly using again the linear decay
estimate in Lemma 6.3 and then the bound (B.37) for $N_{3}$.

The proof of Proposition 2.10 is concluded. $\hfill\Box$

##  Appendix A Supporting material

###  A.1. Linear decay estimate

Here is the linear estimate which we use to prove decay in Section 6.

######  Lemma A.1 (Linear estimate).

With the definitions (2.3) and $\Sigma:=x\cdot\nabla_{x}$,
$x\in{\mathbb{R}}^{2}$, we have

$\displaystyle{\big{\|}e^{-it\Lambda^{\frac{1}{2}}}f\big{\|}}_{L^{\infty}_{x}({\mathbb{R}}^{2})}\lesssim|t|^{-1}\sum_{k\leq
3}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\|\Sigma\Omega^{k}P_{\ell}f\|}_{L^{2}}.$
(A.1)

As a consequence

$\displaystyle\begin{split}{\|u\|}_{L^{\infty}_{x}({\mathbb{R}}^{2})}&\lesssim\langle
t\rangle^{-1}\sum_{|I|\leq
3}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\|S(\Omega,\nabla)^{I}P_{\ell}u\|}_{L^{2}({\mathbb{R}}^{2})}\\\
&+\sum_{|I|\leq
3}\sum_{\ell\in{\mathbb{Z}}}2^{\ell/2}{\|(\Omega,\nabla)^{I}(\partial_{t}+i\Lambda^{\frac{1}{2}})P_{\ell}u\|}_{L^{2}({\mathbb{R}}^{2})}.\end{split}$
(A.2)

Note the importance of just one scaling vector field appearing in (A.1) and
(A.2). We will often just use the less precise estimate

$\displaystyle\begin{split}{\|u\|}_{L^{\infty}_{x}({\mathbb{R}}^{2})}&\lesssim\langle
t\rangle^{-1}\sum_{|I|\leq
4}{\|S(\Omega,\nabla)^{I}u\|}_{L^{2}({\mathbb{R}}^{2})}\\\ &+\sum_{|I|\leq
4}{\|(\Omega,\nabla)^{I}(\partial_{t}+i\Lambda^{\frac{1}{2}})u\|}_{L^{2}({\mathbb{R}}^{2})}.\end{split}$
(A.3)

which dispenses of the summation over frequencies. The presence of the
$2^{\ell/2}$ factor at small frequencies is important when estimating the
contribution from the vector potential (6.15) with (6.8).

Similar estimates were proved in [20] for the propagator
$e^{it\Lambda^{3/2}}$, and used in [49] as well.

###### Proof of Lemma A.1.

We begin by writing

$\displaystyle(e^{-it|\nabla|^{1/2}}f)(x)=\sum_{\ell\in{\mathbb{Z}}}(e^{-it|\nabla|^{1/2}}P_{\ell}f)(x)=\sum_{\ell\in{\mathbb{Z}}}\int_{{\mathbb{R}}^{2}}e^{i(x\cdot\xi-t|\xi|^{1/2})}\varphi_{\ell}(\xi)\widehat{f}(\xi)\,d\xi,$

and aim to prove that for all $g=P_{[\ell-2,\ell+2]}g$ with $\sum_{|I|\leq
3}{\|\Sigma(\Omega,\nabla)^{I}g\|}_{L^{2}({\mathbb{R}}^{2})}=1$, we have

$\displaystyle{\|e^{it|\nabla|^{1/2}}g\|}_{L^{\infty}_{x}}\lesssim
2^{\ell/2}\langle t\rangle^{-1}.$ (A.4)

We use polar coordinates $\xi=\rho\theta$, $\rho\geq 0$ and
$\theta\in\mathbb{S}^{1}$ and expand $\widehat{g}(\xi)$ in Fourier series in
the angular variable:

$\displaystyle(e^{it|\nabla|^{1/2}}g)(x)=\sum_{m\in{\mathbb{Z}}}\int_{0}^{\infty}e^{i(|x|\rho\cos\theta-t\rho^{1/2})}e^{i\theta
m}\widehat{g}_{m}(\rho)\,\varphi_{\ell}(\rho)\,d\theta\rho d\rho,$ (A.5)
$\displaystyle\widehat{g}_{m}(\rho):=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-i\theta
m}\widehat{g}(\rho(\cos\theta,\sin\theta))d\theta,$

having assumed without loss of generality that $x=(|x|,0)$. Then we can
rewrite (A.5) as

$\displaystyle(e^{it|\nabla|^{1/2}}g)(x)=\sum_{m\in{\mathbb{Z}}}\int_{0}^{\infty}e^{-it\rho^{1/2}}J_{m}(|x|\rho)\widehat{g}_{m}(\rho)\,\varphi_{\ell}(\rho)\,\rho
d\rho,$ (A.6)

where $J_{m}$ is the Bessel function of order $m$ and satisfies (see Stein
[38])

$\displaystyle\begin{split}&J_{m}(y):=\int_{0}^{2\pi}\,e^{i(y\cos\theta+m\theta)}\,d\theta=e^{iy}J_{m,+}(y)+e^{-iy}J_{m,-}(y),\\\
&\mbox{with}\qquad\langle y\rangle^{1/2}\big{|}J_{m,\pm}(y)\big{|}+\langle
y\rangle^{3/2}\big{|}J_{m,\pm}^{\prime}(y)\big{|}\lesssim m^{2},\end{split}$
(A.7)

where the implicit constant is independent of $m$. In what follows we only
consider the contribution from $J_{m,+}$ since the one involving $J_{m,-}$ is
similar. The term to bound in the sum in (A.6) is

$\displaystyle
I_{m}(t,x):=\int_{0}^{\infty}e^{i(|x|\rho-t\rho^{1/2})}J_{m,+}(|x|\rho)\widehat{g}_{m}(\rho)\,\varphi_{\ell}(\rho)\,\rho
d\rho.$ (A.8)

Notice how this term resembles a $1$ dimensional evolution of the form
$(e^{-it|\partial_{x}|^{1/2}}G)(|x|)$, with
$\widehat{G}(\rho)\sim\widehat{g}_{m}(\rho)\,\varphi_{\ell}(\rho)\,\rho$. For
such a $1$d evolution one can use a stationary phase argument and integration
by parts to see that

$\displaystyle{\|e^{-it|\partial_{x}|^{1/2}}P_{\ell}G\|}_{L^{\infty}}\lesssim
2^{\ell/4}\langle
t\rangle^{-1/2}{\|x\partial_{x}P_{\ell}G\|}_{L^{2}}^{1/2}{\|P_{\ell}G\|}_{L^{2}}^{1/2}.$
(A.9)

See for example [29] in the case of $e^{-it|\partial_{x}|^{3/2}}$. Applying a
similar argument to (A.8), and taking into account

(1) the presence of the factor $J_{m,+}$ which decays (see (A.7)) in the
quantity $|x|\rho\approx t2^{\ell/2}$ (at the stationary point of the phase
$|x|\rho-t\rho^{1/2}$), and

(2) the extra factor of $\rho\approx 2^{\ell}$, we deduce

$\displaystyle{\|I_{m}(t)\|}_{L^{\infty}}\lesssim\langle
t\rangle^{-1}2^{\ell/2}\Big{[}{\|\rho\partial_{\rho}\widehat{g_{m}}\|}_{L^{2}(\rho
d\rho)}+{\|\widehat{g_{m}}\|}_{L^{2}(\rho d\rho)}\Big{]}.$ (A.10)

This implies the desired bound after summation over $m$, using the bound (A.7)
and the control on three angular derivatives of $g$.

The estimate (A.2) follows from (A.1) by letting $u:=e^{-it\Lambda^{1/2}}f$,
writing $\Sigma=S-\frac{1}{2}t\partial_{t}$ and using
$\partial_{t}f=(\partial_{t}+i\Lambda^{\frac{1}{2}})u$. ∎

###  A.2. Bilinear operators and estimates

We consider the class of symbols

$\displaystyle\mathcal{S}^{\infty}:=\big{\\{}m:{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}\rightarrow{\mathbb{C}}\,:\,{\|m\|}_{\mathcal{S}^{\infty}}:={\|\mathcal{F}^{-1}(m)\|}_{L^{1}}<\infty\big{\\}}.$
(A.11)

Given a symbol
$b:{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}\rightarrow{\mathbb{C}}$ we define
the corresponding bilinear operator

$\displaystyle
B(f,g)=\frac{1}{(2\pi)^{2}}\mathcal{F}^{-1}\Big{(}\int_{{\mathbb{R}}^{2}}b(\xi,\eta)\widehat{f}(\xi-\eta)\widehat{g}\,d\eta\Big{)}.$
(A.12)

We use the following notation for the localized symbols/operator:

$\displaystyle
b^{k,k_{1},k_{2}}(\xi,\eta):=b(\xi,\eta)\varphi_{k}(\xi)\varphi_{k_{1}}(\xi-\eta)\varphi_{k_{2}}(\eta).$
(A.13)

We have the following basic lemma:

######  Lemma A.2.

(i) We have $\mathcal{S}^{\infty}\hookrightarrow
L^{\infty}({\mathbb{R}}\times{\mathbb{R}})$. If
$m,m^{\prime}\in\mathcal{S}^{\infty}$ then $m\cdot
m^{\prime}\in\mathcal{S}^{\infty}$ and

${\|m\cdot
m^{\prime}\|}_{S^{\infty}}\lesssim{\|m\|}_{S^{\infty}}{\|m^{\prime}\|}_{S^{\infty}}.$
(A.14)

Moreover, if $m\in\mathcal{S}^{\infty}$,
$A:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ is a linear transformation,
$v\in{\mathbb{R}}^{2}$, and $m_{A,v}(\xi,\eta):=m(A(\xi,\eta)+v)$, then

${\|m_{A,v}\|}_{\mathcal{S}^{\infty}}={\|m\|}_{\mathcal{S}^{\infty}}.$ (A.15)

(ii) Assume $p,q,r\in[1,\infty]$ satisfy $1/p+1/q=1/r$, and
$m\in\mathcal{S}^{\infty}$. Then, for any $f,g\in L^{2}({\mathbb{R}})$,

${\|B(f,g)\|}_{L^{r}}\lesssim{\|m\|}_{\mathcal{S}^{\infty}}{\|f\|}_{L^{p}}{\|g\|}_{L^{q}}.$
(A.16)

We also need a lemma which describes the commutation with our vector fields:

######  Lemma A.3.

Given a bilinear operator as in (A.12), define the bilinear commutator with
$\Gamma=\Omega$ or $S$ as

$\displaystyle[\Gamma,B(f,g)]:=\Gamma B(f,g)-B(\Gamma f,g)-B(f,\Gamma g).$
(A.17)

We have

$\displaystyle[\Gamma,B(f,g)]=B^{\Gamma}(f,g)$ (A.18)

with symbols

$\displaystyle\begin{split}b^{S}(f,g)&:=-(\xi\cdot\nabla_{\xi}+\eta\cdot\nabla_{\eta})b(\xi,\eta),\\\
b^{\Omega}(f,g)&:=(\xi\wedge\nabla_{\xi}+\eta\wedge\nabla_{\eta})b(\xi,\eta),\end{split}$
(A.19)

###### Proof.

These formulas can be checked by a direct calculation. ∎

Here is basic lemma to handle product and pseudo-product in our weighted
spaces.

######  Lemma A.4.

Recall the definition (2.8) with (2.3). For all $N\geq 0$

$\displaystyle\sum_{r+k\leq N}{\|fg\|}_{Z^{r}_{k}}$
$\displaystyle\lesssim\sum_{r+2k\leq
N/2}{\|f\|}_{Z^{r,\infty}_{k}}\sum_{r+k\leq
N}{\|g\|}_{Z^{r}_{k}}+\sum_{r+k\leq N}{\|f\|}_{Z^{r}_{k}}\sum_{r+k\leq
N/2}{\|g\|}_{Z^{r,\infty}_{k}},$ (A.20)

and, more in general, for $1/p=1/p_{1}+1/p_{2}=1/q_{1}+1/q_{2}$

$\displaystyle\sum_{r+k\leq N}{\|fg\|}_{Z^{r,p}_{k}}$
$\displaystyle\lesssim\sum_{r+k\leq N/2}{\|f\|}_{Z^{r,p_{1}}_{k}}\sum_{r+k\leq
N}{\|g\|}_{Z^{r,p_{2}}_{k}}+\sum_{r+k\leq
N}{\|f\|}_{Z^{r,q_{1}}_{k}}\sum_{r+k\leq N/2}{\|g\|}_{Z^{r,q_{2}}_{k}}.$
(A.21)

The same bounds hold if we replace the product $fg$ by a pseudo-product
$B(f,g)$ as in (A.12) with a symbol $b$ satisfying

$\displaystyle{\big{\|}b^{\Gamma^{k}}\big{\|}}_{S^{\infty}}\lesssim 1,\quad
k=0,\dots,N,$ (A.22)

where $b^{\Gamma}$, $\Gamma\in\\{S,\Omega\\}$ is defined as in (A.19) and
$b^{\Gamma^{k}}$ is defined inductively by
$b^{\Gamma^{k}}=(b^{\Gamma^{k-1}})^{\Gamma}$ with $b_{0}=b$.

###### Proof.

We use the notation from 4.2.1 for repeated applications of vector fields
(just $2$d ones, here). For any $|r|+|k|\leq N$ with $r=r_{1}+r_{2}$ and
$k=k_{1}+k_{2}$ we have

$\displaystyle\begin{split}\nabla_{x}^{r}\Gamma^{k}(fg)&=\big{(}\nabla_{x}^{r_{1}}\Gamma^{k_{1}}f\big{)}\,\big{(}\nabla_{x}^{r_{2}}\Gamma^{k_{2}}g\big{)}.\end{split}$
(A.23)

Without loss of generality, by the symmetry of the right-hand side of (A.20),
we may assume that $|r_{1}|+|k_{1}|\leq N/2$ and estimate

$\displaystyle{\big{\|}\big{(}\nabla_{x}^{r_{1}}\Gamma^{k_{1}}f\big{)}\,\big{(}\nabla_{x}^{r_{2}}\Gamma^{k_{2}}g\big{)}\big{\|}}_{L^{2}}$
$\displaystyle\lesssim{\big{\|}\nabla_{x}^{r_{1}}\Gamma^{k_{1}}f\big{\|}}_{L^{\infty}}{\big{\|}\nabla_{x}^{r_{2}}\Gamma^{k_{2}}g\big{\|}}_{L^{2}}$
$\displaystyle\lesssim\sum_{r_{1}+k_{1}\leq
N/2}{\big{\|}f\big{\|}}_{Z^{r_{1},\infty}_{k_{1}}}\sum_{r_{2}+k_{2}\leq
N}{\big{\|}g\big{\|}}_{Z^{r_{2}}_{k_{2}}}.$

The estimate (A.21) follows identically. For the same estimate with $f\,g$
replaced by $B(f,g)$ it suffices to use Lemma A.3 to commute vector fields,
followed by an application (A.16) using the assumption (A.22). ∎

We also use the following standard product estimate:

######  Lemma A.5.

for $f,g:{\mathbb{R}}^{2}\rightarrow{\mathbb{C}}$, the following estimate
holds:

$\displaystyle{\big{\|}|\nabla|^{1/2}(fg)\big{\|}}_{L^{2}}\lesssim{\|f\|}_{W^{1,3}}{\big{\|}|\nabla|^{1/2}g\big{\|}}_{L^{2}},$
(A.24)

###  A.3. A basic trace inequality

We use the following basic trace estimate:

######  Lemma A.6 (Trace inequalities).

Let $f:\mathcal{D}_{t}\to\mathbb{R}$, define
$F:\mathbb{R}^{2}\times(-\infty,0]$ by $F(x,z)=f(x,z+h(x))$, and assume that
$\lim_{z\rightarrow-\infty}F=0$. Then,

$\displaystyle\||\nabla|^{1/2}F\|_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\|\nabla_{x,z}F\|_{L^{2}_{z}L^{2}_{z}},$ (A.25)
$\displaystyle\|F\|_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\|F\|_{L^{2}_{z}L^{2}_{x}}+\|\nabla_{x,z}F\|_{L^{2}_{z}L^{2}_{z}}.$
(A.26)

In particular, if $\nabla h\in L^{\infty}$,

$\displaystyle\||\nabla|^{1/2}F\|_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\|\nabla_{x,y}f\|_{L^{2}(\mathcal{D}_{t})},$ (A.27)
$\displaystyle\|F\|_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\|f\|_{L^{2}(\mathcal{D}_{t})}+\|\nabla_{x,y}f\|_{L^{2}(\mathcal{D}_{t})}.$
(A.28)

###### Proof.

We have

$\displaystyle\int_{\mathbb{R}^{2}}||\nabla|^{1/2}F(x,0)|^{2}\,dx$
$\displaystyle=2\int_{-\infty}^{0}\int_{\mathbb{R}^{2}}\partial_{z}|\nabla|^{1/2}F(x,z)|\nabla|^{1/2}F(x,z)\,dxdz$
$\displaystyle=2\int_{-\infty}^{0}\int_{\mathbb{R}^{2}}\partial_{z}F(x,z)|\nabla|F(x,z)\,dxdz$
$\displaystyle\lesssim\|\nabla_{x,z}F\|_{L^{2}_{z}L^{2}_{x}}^{2}\lesssim(1+\|\nabla
h\|_{L^{\infty}(\mathbb{R}^{2})})^{2}\|\nabla_{x,y}f\|_{L^{2}(\mathcal{D}_{t})}^{2},$

which gives (A.25) and (A.27). The bounds (A.26) and (A.28) are proved in
exactly the same way without including the factor $|\nabla|^{1/2}$. ∎

##  Appendix B The boundary equations

Recall the definitions

$\displaystyle
v=\nabla\psi+v_{\omega},\qquad\varphi=\psi|_{\partial\mathcal{D}_{t}}$ (B.1)

with $\Delta\psi=0$, and $v_{\omega}\cdot n=0$. Define the restrictions of the
horizontal and vertical components to the boundary as follows:

$\displaystyle
v|_{\partial\mathcal{D}_{t}}=(P,B),\qquad\nabla\psi|_{\partial\mathcal{D}_{t}}=(P_{ir},B_{ir}),\qquad
v_{\omega}|_{\partial\mathcal{D}_{t}}=\widetilde{v_{\omega}}=(P_{\omega},B_{\omega}).$
(B.2)

We have

$\displaystyle P_{ir}=\nabla\varphi-B_{ir}\nabla h,\qquad
B_{ir}=\frac{G(h)\varphi+\nabla h\cdot\nabla\varphi}{1+|\nabla h|^{2}},$ (B.3)

Recall that

$\displaystyle G(h)\varphi=(P_{ir},B_{ir})\cdot(-\nabla
h,1)=(P,B)\cdot(-\nabla h,1),$ (B.4)

where the last identity follows since $v_{\omega}\cdot n=0$ and, therefore, we
have

$\displaystyle B_{\omega}=\nabla h\cdot P_{\omega},$ (B.5)

that is, the rotational part of the velocity does not move the boundary.
Moreover, one can show that there exists $a_{\omega}$ such that

$\displaystyle\nabla a_{\omega}=U_{\omega}:=P_{\omega}+\nabla hB_{\omega},$
(B.6)

since
$\partial_{2}U_{\omega}^{1}-\partial_{1}U_{\omega}^{2}=\omega|_{\partial\mathcal{D}_{t}}\cdot(-\nabla
h,1)=0$ in view of our assumption that $\omega$ vanishes on the boundary.

Notice that we can also express $a_{\omega}$ just in terms of $h$ and
$P_{\omega}$:

$\displaystyle a_{\omega}=\Lambda^{-1}R\cdot\nabla
a_{\omega}=\Lambda^{-1}R\cdot\big{(}P_{\omega}+\nabla h(\nabla h\cdot
P_{\omega})\big{)}.$ (B.7)

###  B.1. Evolution equations at the boundary

The equation for the motion of the interface is the standard
$\partial_{t}h=G(h)\varphi$. The equation for the evolution of the potential
is more involved and given by the following:

######  Lemma B.1 (Boundary evolution equations).

Assume that $\omega\cdot n=0$. With the notation (B.1)-(B.6) we have

$\displaystyle\left\\{\begin{array}[]{rl}\partial_{t}h&=G(h)\varphi\\\
\partial_{t}\varphi&=-h-\dfrac{1}{2}|\nabla\varphi|^{2}+\dfrac{1}{2}\dfrac{(G(h)\varphi+\nabla
h\cdot\nabla\varphi)^{2}}{1+|\nabla
h|^{2}}-\Lambda^{-1}R\cdot\partial_{t}P_{\omega}-\nabla\varphi\cdot
P_{\omega}-\dfrac{1}{2}|P_{\omega}|^{2}+R_{\omega}\end{array}\right.$ (B.10)

where

$\displaystyle\begin{split}R_{\omega}=-\Lambda^{-1}R\cdot\partial_{t}\big{(}\nabla
h(\nabla h\cdot P_{\omega})\big{)}-\frac{1}{2}(P_{\omega}\cdot\nabla
h)^{2}+(G(h)\varphi)P_{\omega}\cdot\nabla h.\end{split}$ (B.11)

######  Remark B.2.

In the irrotational case the Zakharov formulation in terms of $(P,B)$ reads

$\displaystyle\begin{split}\partial_{t}\varphi&=-h-\frac{1}{2}|\nabla\varphi|^{2}+\frac{1}{2}(1+|\nabla
h|^{2})B^{2}=-h-\frac{1}{2}P^{2}+\frac{1}{2}B^{2}-B\nabla h\cdot
P.\end{split}$ (B.12)

One can check (and we will do this in the proof below) that the same equation
extends to the rotational case, in the sense that

$\displaystyle\begin{split}\partial_{t}(P+\nabla
hB)&=\nabla\big{(}-h-\frac{1}{2}P^{2}+\frac{1}{2}B^{2}-B\nabla h\cdot
P\big{)},\end{split}$ (B.13)

where $(P,B)$ are now defined as in (B.2). Then, by expanding
$P=P_{ir}+P_{\omega}$ and $B=B_{ir}+B_{\omega}$, using (B.3) and (B.5), (B.6)
and (B.7), one can arrive at (B.10)-(B.11).

Equation (B.13) is also equivalent to the formulation used by Castro-Lannes
[8] which has the form

$\displaystyle\begin{split}\partial_{t}U_{\parallel}&=-\nabla
h-\frac{1}{2}\nabla|U_{\parallel}|^{2}+\frac{1}{2}\nabla\big{(}(1+|\nabla
h|^{2})B^{2}\big{)}\end{split}$ (B.14)

where $U_{\parallel}=P+B\nabla h=\nabla(\varphi+a_{\omega})$.

For completeness we give here a derivation of the equation for
$\partial_{t}\varphi$ in (B.10), which slightly differs from the one in [21].

###### Proof of Lemma B.1.

Restricting Euler’s equations (1.1) to the boundary $\partial\mathcal{D}_{t}$,
using that $p=0$ on the boundary, we have ($g=1$)

$\displaystyle(\partial_{t}+P\cdot\nabla)P=a\nabla h,\qquad
a:=-\partial_{3}p,$ (B.15) $\displaystyle(\partial_{t}+P\cdot\nabla)B=a-1.$
(B.16)

From these we get an evolution equations for $\nabla\varphi+\nabla
a_{\omega}=P+\nabla hB$, that is, the tangential component of the velocity
field restricted to the boundary:

$\displaystyle\begin{split}\partial_{t}(\nabla\varphi+\nabla
a_{\omega})&=-P\cdot\nabla P-a\nabla h+\nabla h\big{(}-P\cdot\nabla
B+a-1\big{)}+(\nabla\partial_{t}h)B\\\ &=-\nabla h-P\cdot\nabla P-\nabla
h(P\cdot\nabla B)+B\nabla(-\nabla h\cdot P+B),\end{split}$ (B.17)

having used the kinematic boundary condition $\partial_{t}h=-\nabla h\cdot
P+B$. We then rewrite (B.17) as

$\displaystyle\begin{split}\partial_{t}(\nabla\varphi+\nabla
a_{\omega})&=\nabla\big{(}-h+(1/2)B^{2}-B\nabla h\cdot
P-(1/2)|P|^{2}\big{)}\\\ &+(1/2)\nabla|P|^{2}-P\cdot\nabla P-\nabla
h(P\cdot\nabla B)+\nabla B(\nabla h\cdot P).\end{split}$ (B.18)

To conclude, we observe that the last line above vanishes in view of the
condition $\omega\cdot n=0$. In fact, using that, for $i=1,2$,
$(\partial_{i}F)|_{\partial\mathcal{D}_{t}}=\partial_{i}(F|_{\partial\mathcal{D}_{t}})-(\partial_{3}F)|_{\partial\mathcal{D}_{t}}\partial_{i}h$,
we can calculate

$\displaystyle\begin{split}\nabla\times v\cdot
n&=-\partial_{1}h\big{(}\partial_{2}v_{3}-\partial_{3}v_{2}\big{)}|_{\partial\mathcal{D}_{t}}-\partial_{2}h\big{(}\partial_{3}v_{1}-\partial_{1}v_{3}\big{)}|_{\partial\mathcal{D}_{t}}+(\partial_{1}v_{2}-\partial_{2}v_{1})|_{\partial\mathcal{D}_{t}}\\\
&=-\partial_{1}h\big{(}\partial_{2}B-\partial_{3}v_{3}|_{\partial\mathcal{D}_{t}}\partial_{2}h-\partial_{3}v_{2}|_{\partial\mathcal{D}_{t}}\big{)}-\partial_{2}h\big{(}\partial_{3}v_{1}|_{\partial\mathcal{D}_{t}}-\partial_{1}B+\partial_{3}v_{3}|_{\partial\mathcal{D}_{t}}\partial_{1}h\big{)}\\\
&+(\partial_{1}P_{2}-\partial_{3}v_{2}|_{\partial\mathcal{D}_{t}}\partial_{1}h-\partial_{2}P_{1}+\partial_{3}v_{1}|_{\partial\mathcal{D}_{t}}\partial_{2}h)\\\
&=-\partial_{1}h\partial_{2}B+\partial_{2}h\partial_{1}B+\partial_{1}P_{2}-\partial_{2}P_{1}=\operatorname{curl}(P+B\nabla
h).\end{split}$

The $\operatorname{curl}$ above naturally denotes the scalar operator in $2$d.
Note how the same calculation with $v_{\omega},P_{\omega},B_{\omega}$ instead
of $v,P,B$, shows (B.6) since $\nabla\times v_{\omega}\cdot
n=\operatorname{curl}(P_{\omega}+\nabla hB_{\omega})$. Using also vector
identities, it follows that

$\displaystyle\begin{split}&(1/2)\nabla|P|^{2}-P\cdot\nabla P-\nabla
h(P\cdot\nabla B)+\nabla B(\nabla h\cdot P)\\\ &=\big{(}-\nabla
B\cdot\nabla^{\perp}h+(\operatorname{curl}P)\big{)}P^{\perp}=0\end{split}$
(B.19)

From (B.18) we have thus arrived at

$\displaystyle\begin{split}\partial_{t}(\varphi+a_{\omega})&=-h+\frac{1}{2}B^{2}-B\nabla
h\cdot P-\frac{1}{2}|P|^{2}.\end{split}$ (B.20)

To finally obtain (B.10) we rewrite this as

$\displaystyle\begin{split}\partial_{t}\varphi&=-h-\frac{1}{2}|P_{ir}|^{2}+\frac{1}{2}B_{ir}^{2}-B_{ir}\nabla
h\cdot P_{ir}-\partial_{t}a_{\omega}\\\
&-\frac{1}{2}|P_{\omega}|^{2}+\frac{1}{2}B_{\omega}^{2}-B_{\omega}\nabla
h\cdot P_{\omega}-P_{ir}P_{\omega}+B_{ir}B_{\omega}-B_{ir}\nabla h\cdot
P_{\omega}-B_{\omega}\nabla h\cdot P_{ir}.\end{split}$ (B.21)

The first line of (B.21) matches the first three terms on the right-hand side
of the equation (B.10) for $\partial_{t}\varphi$ (see (B.3)) plus the terms
that contain a $\partial_{t}$ (see (B.7)).