# Quasiperiodic solutions of the generalized SQG equation

Javier Gómez-Serrano, Alexandru D. Ionescu, Jaemin Park

###### Abstract

Our goal in this monograph is twofold. First, we would like to develop a
robust method to construct global time-quasiperiodic solutions of large
families of quasilinear evolution equations. The method we present here
expands on earlier work of many people and combines elements from KAM theory,
normal forms, pseudo-differential calculus and a Nash-Moser scheme. One of our
main objectives is to avoid the use of any external parameters, and exploit
instead the genericity of the initial data to generate large families of
quasiperiodic solutions.

As a concrete application we prove the existence of linearly stable,
quasiperiodic patch solutions of the generalized SQG equation for all
parameters $\alpha\in(1,2)$, and for almost all initial data in a neighborhood
of the disk solution.

###### Contents

  1. 1 Introduction
    1. 1.0.1 Patch problems
    2. 1.0.2 Steady solutions and global existence of the gSQG equation
    3. 1.0.3 Quasiperiodic solutions in PDE
    4. 1.0.4 Weak solutions and finite time singularities
    5. 1.1 Main result
    6. 1.2 Strategy of the proof and the structure of the monograph
      1. 1.2.1 Sketch of the proof, Part 1: Internal parameter and the weak Birkhoff normal form
      2. 1.2.2 Sketch of the proof, Part 2: Analysis of the linearized operator
      3. 1.2.3 Summary of the sketch and the structure of the monograph
  2. 2 Preliminaries and notations
    1. 2.1 Function spaces and norms
    2. 2.2 Linear operators
      1. 2.2.1 Pseudo differential operators
      2. 2.2.2 Lipschitz-tame operators
    3. 2.3 Modified fractional Laplacians: $\Lambda^{\alpha-1}$ and $\Upsilon^{\alpha-3}$.
    4. 2.4 Hamiltonian structure in $L^{2}_{0}$
    5. 2.5 Time-reversible Hamiltonians
    6. 2.6 Translation invariance
  3. 3 Hamiltonian structure of the gSQG equations
    1. 3.1 Hamiltonian equation in the patch setting
    2. 3.2 Expansion of $\mathcal{H}$
    3. 3.3 Conservation of momentum, time-reversibility and $\mathtt{M}$-fold symmetry
      1. 3.3.1 Conservation of momentum
      2. 3.3.2 Time-reversibility
      3. 3.3.3 $\mathtt{M}$-fold symmetric patches
  4. 4 Weak Birkhoff normal form
    1. 4.1 Tangential sites and normal sites
    2. 4.2 Composition with a time-1 flow
    3. 4.3 Weak Birkhoff normal form
  5. 5 Action-angle variables
    1. 5.1 Hypotheses on the tangential sites
  6. 6 The nonlinear functional setting
    1. 6.1 Regularity of the functional $\mathcal{F}_{\omega}$
    2. 6.2 Reversible and $\frac{2\pi}{\mathtt{M}}$-translation invariant solutions
    3. 6.3 Statement of the main theorem
  7. 7 Approximate inverse
  8. 8 Linearized operator in the normal directions
    1. 8.1 Homogeneous expansion
      1. 8.1.1 Homogeneous expansion of the coefficients: Class $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$
      2. 8.1.2 Homogeneous expansion of the symbols: Class $\mathfrak{S}^{m}$
    2. 8.2 Finite dimensional operators
    3. 8.3 Linearized operator in the normal directions
    4. 8.4 Structure of the operator of size $O(\varepsilon^{2})$
  9. 9 Symplectic transformations
    1. 9.1 Properties of the flow type (1)
    2. 9.2 Properties of the flow type (2)
  10. 10 Reduction to a constant coefficients operator
    1. 10.1 Change of the space variables
    2. 10.2 Reparametrization of time
    3. 10.3 Egorov method
      1. 10.3.1 Symbolic expression for the linear operator
      2. 10.3.2 Elimination of the dependence on $x$
      3. 10.3.3 Elimination of the dependence on $\varphi$
    4. 10.4 Linear Birkhoff normal form
      1. 10.4.1 Linear Birkhoff normal form: Step 1
      2. 10.4.2 Structure of the operator of size $O(\varepsilon^{2})$
      3. 10.4.3 Linear Birkhoff normal form: Step 2
    5. 10.5 Lipschitz tame estimates for the remainders
    6. 10.6 KAM reducibility and inversion of $\mathcal{L}_{\omega}$
    7. 10.7 Invertibility of $\mathcal{L}_{\omega}$: Proof of proposition 7.14
  11. 11 Nash-Moser iteration
    1. 11.1 Measure of the frequency set
  12. 12 Proof of Theorem 6.5
  13. A Conjugation with the operators $\Lambda^{\alpha-1}_{a},\Upsilon^{\alpha-3}_{a}$
  14. B Analysis of the non-resonance condition
    1. B.1 Expressions of $\mathbb{A}$ and $\vec{D}(\xi)$
    2. B.2 Estimates for $\mathtt{B}_{1},\mathtt{B}_{2},\mathtt{B}_{3}$ and $m^{\circ}_{3,\alpha}$
    3. B.3 Asymptotics of $\mathbb{A}$ and $\vec{D}(\xi)$ for large $\mathtt{M}$
    4. B.4 Choice of tangential sites

##  1 Introduction

The generalized surface quasi-geostrophic (gSQG) equations describing the
evolution of the potential temperature $\omega$ read as

$\displaystyle\begin{cases}\omega_{t}+u\cdot\nabla\omega=0,&\text{ for
}(t,x)\in[0,\infty)\times\mathbb{R}^{2},\\\
u=-\nabla^{\perp}(-\Delta)^{-1+\alpha/2}\omega,&\alpha\in[0,2]\\\
\omega(0,\cdot)=\omega_{0}(\cdot).\end{cases}$ (1.1)

Formally, these equations interpolate between the case of the Euler equation
$(\alpha=0)$ and the case of stationary solutions $(\alpha=2)$. The case
$(\alpha=1)$ is known as the SQG equation.

The SQG equation models the evolution of the temperature from a general quasi-
geostrophic system for atmospheric and oceanic flows (see [35, 79, 111, 102]
for more details). The first rigorous mathematical study of the SQG equation
was done by Constantin–Majda–Tabak [35] where its mathematical importance due
to its analogy with the incompressible 3D Euler equations was highlighted and
the first numerical and analytical study of the equation was carried out.
Córdoba–Fontelos–Mancho–Rodrigo in [39] proposed the gSQG or (SQG)α model
(1.1) as an interpolation between the Euler and surface quasi–geostrophic
equations. Nonetheless, very little is known for this family of equations and
specifically the question of global existence versus finite time singularities
is still open, for all $\alpha>0$. In this monograph we aim to prove the
existence of a large class of initial data for which there is time quasi-
periodic behaviour and thus global existence in the more singular case
$\alpha\in(1,2)$.

####  1.0.1 Patch problems

In this monograph we will work in the _patch_ setting, where
$\omega(\cdot,t)=1_{D(t)}$ is an indicator function of a simply-connected,
bounded set that moves with the fluid. In such a situation, we parametrize
$\partial D(t)$ as $z(\theta,t),\,\theta\in[0,2\pi]$ and the evolution
equations read:

$\displaystyle\partial_{t}z(\theta,t)=\int_{0}^{2\pi}\frac{\partial_{\theta}z(\theta,t)-\partial_{\theta}z(\theta-\eta,t)}{|z(\theta,t)-z(\theta-\eta,t)|^{\alpha}}d\eta+c(\theta,t)\partial_{\theta}z(\theta,t),$
(1.2)

where $c(\theta,t)$ accounts for the reparametrization freedom of the curve.

Concerning well-posedness results for patch solutions, Rodrigo (in a
$C^{\infty}$ space) [115] and Gancedo [59] and
Chae–Constantin–Córdoba–Gancedo–Wu [30] (in a Sobolev space) proved local
existence for the case $0<\alpha\leq 1$ and $1<\alpha$ respectively. See also
[96, 60, 1].

####  1.0.2 Steady solutions and global existence of the gSQG equation

The construction of nontrivial global solutions for the generalized SQG
equations is a very challenging open problem for all parameters
$\alpha\in(0,2)$, both in the smooth case and in the patch case. For
$\alpha=0$ (the 2D Euler equations), global regularity of solutions was well-
understood a long time ago, both in the smooth case and in the patch case. See
for example the classical papers of Wolibner [125], Yudovich [126], Burbea
[21], Chemin [31], and Bertozzi-Constantin [17]. However, the construction of
global solutions in the case of $\alpha\in(0,2)$ is much more challenging than
when $\alpha=0$, since the velocity is more singular, and only partial results
have been obtained in recent years. We review some of these results below.

Most of the results around global existence of the gSQG equation have revolved
around solutions that exhibited some rigid character (steady, uniformly
rotating –V-states–, traveling). In the case where $0<\alpha<1$,
Hassainia–Hmidi [74] proved the existence of V-states with $C^{k}$ boundary
regularity. Castro–Córdoba–Gómez-Serrano then expanded upon this result in
[26] by showing that V-states also exist with $C^{\infty}$ boundary regularity
in the remaining open cases of $\alpha\in[1,2)$ for existence and
$\alpha\in(0,2)$ for regularity. This boundary regularity was later refined to
be analytic in [27]. Other notable works on rotating solutions include [44,
63, 80, 77], which discuss other families of rotating solutions or even more
steady states, [45, 113] which address the doubly connected case, and [28]
which presents a construction in the smooth setting.

In [45], de la Hoz–Hassainia–Hmidi showed that there exist non-radial patches
bifurcating from annuli at negative angular velocities and Gómez-Serrano [68]
constructed non-radial, doubly connected stationary patches. García [62]
proved the existence of a Kármán vortex street structure by desingularizing an
infinite array of point vortices in the case $\alpha\in[0,1)$. In [29] it was
ruled out by Castro–Córdoba–Gómez-Serrano–Martín Zamora that ellipses could be
rotating solutions for $\alpha>0$, as opposed to the case $\alpha=0$.
Gravejat–Smets [70], in the case $\alpha=1$, constructed smooth translating
solutions. Ao–Dávila–del Pino–Musso–Wei [2], expanded the range to
$\alpha\in(0,2)$ as well as to rotating solutions. See also [66, 67] and [22,
24, 23] for alternative constructions. In [69], Gómez-Serrano–Park–Shi–Yao
proved that any smooth, non-negative rotating solution with simply-connected
superlevel sets can only rotate with positive angular velocity, and in the
case of a patch of fixed area derived moreover a sharp upper bound on the
angular velocity.

The drawback of the aforementioned solutions is that they are special in the
sense that general solutions will not have such behavior. Concerning results
for general solutions, Córdoba–Gómez-Serrano–Ionescu [40] proved global
existence for small patch data close to a halfplane in the case
$\alpha\in(1,2)$, using a different mechanism based on dispersion and decay.
This was extended in [84, 83]. The main idea was to show that general initial
data that are small perturbations of the halfplane stationary patch solution
lead to global solutions that decay in time (at an optimal rate of
$t^{-1/2}$), thus converging back to the halfplane stationary patch.
Unfortunately, the mechanism of dispersion and decay seems to require
unbounded domains and, in particular, infinite energy solutions.

In a different direction, one could hope to use the mechanism of inviscid
damping to construct families of global-in-time solutions around explicit
stationary solutions of finite energy, such as smooth shear flows or vortices.
This has been successfully implemented in recent years in the 2D Euler case
$\alpha=0$, for perturbations of the Couette flow (by Bedrossian–Masmoudi [11]
and Ionescu–Jia [85]) and then general monotonic shear flows [86, 105]. It is
tempting to try to adapt the mechanism of inviscid damping to construct
families of nontrivial global solutions of the gSQG equations, at least for
some parameters $\alpha>0$ small. The easiest would be to perturb around the
Couette flow corresponding to $\theta(t,x,y)=-1$ on the bounded channel
$\mathcal{D}=\mathbb{T}\times[0,1]$. Unfortunately and surprisingly, recent
work of Gómez-Serrano–Ionescu–Jia (discussed in [87]) shows that this fails to
produce global solutions for any parameters $\alpha>0$, due to a forward
cascade that leads to loss of regularity in finite time.

####  1.0.3 Quasiperiodic solutions in PDE

Our main goal in this monograph is to demonstrate the existence of large
families of global solutions of the generalized SQG equations. We do this
using KAM theory, by constructing quasiperiodic solutions for almost all
initial data in a neighborhood of the unit disk (the simplest stationary patch
solution with finite energy).

The first application of KAM theory [97, 3, 108] was to prove the existence of
invariant tori that were small perturbations of finite dimensional nearly
integrable Hamiltonian systems. In order to upgrade it to the infinite
dimensional (PDE) case, the first results are due to Kuksin [98], Wayne [122],
Pöschel [112] for 1-d semilinear wave and Schrödinger equations with Dirichlet
boundary conditions and Craig–Wayne [42], Bourgain [19], Grébert–Kappeler [71]
and Chierchia–You [33] with periodic boundary conditions. See also [99]. In
the semilinear multidimensional case, we refer to the works of Bourgain [18],
Eliasson–Kuksin [49], Grébert–Paturel [72], Wang [121] and Berti–Bolle [13]
and references therein. See also De la Llave–Sire [46]. Note that all the
previous results only were able to deal with semilinear problems.

In the last decade there has been an emergence of results of quasiperiodic
solutions for quasilinear PDE, motivated by applications to the dynamics of
confined fluids, building up and polishing the techniques and the methods and
culminating with excellent theorems. Baldi–Berti–Montalto constructed
quasiperiodic solutions to the Airy equation [6] and KdV and mKdV [7, 8, 9].
See also the results of Giuliani for gKdV [65], and [51, 56, 106, 55] and
references therein for other relevant models. In the context of water waves,
Baldi–Berti–Haus–Montalto [5] (gravity case), Berti–Montalto [16] (gravity-
capillary case), Feola–Giuliani [52, 53] (infinite depth) and
Berti–Franzoi–Maspero [14] (constant non-zero vorticity) constructed
quasiperiodic solutions. Numerically, Wilkening–Zhao [123, 124] computed
quasiperiodic gravity-capillary water waves in the infinite depth case.

Berti–Hassainia–Masmoudi [15] constructed quasiperiodic solutions close to
elliptical vortex patches, introducing the angular momentum as a symplectic
variable. Hassainia–Roulley [76] constructed quasiperiodic solutions of the 2D
Euler equations in a bounded domain, Roulley [116] proved its existence for
the Euler-$\alpha$ equation and Hmidi–Roulley [81] for the QGSW equations.

Other examples of quasiperiodic solutions in the context of the incompressible
Euler and Navier-Stokes equations, even in high dimensions, were obtained by
Crouseilles–Faou, Elgindi–Jeong, Enciso–Peralta-Salas–Torres de Lizaur [43,
48, 50] for Euler, using non-KAM constructions, Baldi–Montalto [10] for forced
Euler, using a KAM construction and Franzoi–Montalto, Montalto [57, 107] for
forced Navier-Stokes, using a KAM construction. Finally, we would like to draw
the attention to the recent results by Hassainia–Hmidi–Masmoudi [75] who
proved the existence of global quasiperiodic solutions for the generalized SQG
equations, for a set of parameters $\alpha\in\left(0,1/2\right)$. The set of
acceptable parameters $\alpha$ is unknown, but of full measure in $(0,1/2)$.

We emphasize that most of these recent results in the quasilinear case (with
the notable exception of the papers [52], [65] and [8]) rely on the use of
external parameters. Quasiperiodic solutions are then constructed for all
initial data, but for an unknown set of parameters, usually generic of full
measure. The point is that the presence of external parameters improves
significantly the structure of the resonances of the system, which plays a key
role in the analysis.

The drawback is that the family of acceptable parameters is not explicit, and
one cannot guarantee that quasiperiodic solutions exist for a specific given
equation. Our broad goal in this monograph is to develop a robust and flexible
method to construct quasiperiodic solutions for certain fluid models, without
requiring the presence of external parameters. The basic idea is to replace
the genericity of the external parameters with genericity of the initial data.
This leads however to very significant difficulties at the implementation
level; see below for a more detailed discussion.

####  1.0.4 Weak solutions and finite time singularities

The generalized SQG equations have been studied extensively, by many authors.
In this subsection we discuss two other areas of active research, and provide
some references.

In his thesis [114], Resnick demonstrated the global existence of weak
solutions in $L^{2}$ through the use of the oddness of the Riesz transform to
achieve additional cancellation. Marchand [103] later extended this result to
include initial data belonging to $L^{p}$ with $p$ greater than $\frac{4}{3}$.
See also [109] for other existence results concerning weak solutions. Non-
uniqueness of weak solutions of SQG remains a difficult problem, with progress
being made through works such as Azzam–Bedrossian [4] or Isett–Vicol [89], and
most importantly, Buckmaster–Shkoller–Vicol [20], as well as alternative
proofs by Isett–Ma [88] and the investigation of the stationary problem by
Cheng–Kwon–Li [32].

One of the most significant questions in mathematical fluid mechanics is
whether the SQG and gSQG system exhibits finite time singularities or has
global existence. Kiselev–Nazarov [94] created solutions that exhibited norm
inflation, and Friedlander–Shvydkoy [58] demonstrated the presence of unstable
eigenvalues in the spectrum. He–Kiselev [78] proved an exponential in time
growth of the $C^{2}$-norm. See also the construction of singular solutions
with infinite energy by Castro–Córdoba [25] and ill-posedness results by
Córdoba–Martínez-Zoroa and Jeong–Kim [41, 91].

In order to understand the possibility of a finite time blow-up scenario,
numerical studies have been conducted. Constantin–Majda–Tabak [35] suggested
that a singularity in the form of a hyperbolic saddle may occur, closing in a
finite amount of time. However, Ohkitani–Yamada [110] and
Constantin–Nie–Schörghofer [36] proposed that the growth was actually double
exponential. Córdoba [37] bounded the growth at quadruple exponential, and
later Córdoba and Fefferman [38] proposed a double exponential bound, which
was supported by numerical simulations from Deng–Hou–Li–Yu [47].
Constantin–Lai–Sharma–Tseng–Wu [34] later reexamined the hyperbolic saddle
scenario using improved algorithms and found no evidence of blowup. Scott
[118] proposed a scenario in which filamentation occurs and blowup of
$\nabla\theta$ occurs after several cascades, starting from elliptical
configurations. This is currently the only scenario that remains valid in the
smooth setting. In [64], very recently, García–Gómez-Serrano constructed a big
class of non-trivial self-similar spiral solutions close to radial ones with a
mild singularity at the origin.

Even though the finite time singularity problem seems elusive, there exist
several numerical scenarios suggesting such a singularity. The first one,
proposed by Córdoba–Fontelos–Mancho–Rodrigo [39] initially starts as two
patches rolling onto each other and finally collapsing. At the intersection
point the curvature blows up (the curve should lose regularity due to the
results by Gancedo and Strain [61], see also [93, 90]) and the collapse is
suggested to be asymptotically self-similar. The second scenario was proposed
by Scott–Dritschel [119], taking ellipses as initial condition; starting with
an aspect ratio of $0.16$, they report a self-similar cascade of
filamentation. In [120], again taking ellipses as initial condition and
combining numerical analysis with asymptotic calculations, they conjecture a
scenario where the patch develops a corner in finite time, together with a
self-similar spiral. Finally, Kiselev–Ryzhik–Yao–Zlatoš [95] (for
$0<\alpha<\frac{1}{12}$) and later Gancedo–Patel [60] (for
$0<\alpha<\frac{1}{3}$) construct finite time singularities in the presence of
a boundary.

###  1.1 Main result

Before we state the main result, let us first recall the definition of a
quasiperiodic function:

######  Definition 1.1.

Let $X$ be a Hilbert space and $\nu\in\mathbb{N}$ be a fixed natural number. A
function $f:\mathbb{R}\mapsto X$ is said to be quasiperiodic with frequency
${\omega}\in\mathbb{R}^{\nu}$, if there exists $i:\mathbb{T}^{\nu}\mapsto X$
such that $f(t)=i({\omega}t).$

In this monograph, we consider a patch solution to (1.2) of the form:

$\displaystyle z(x,t):=\sqrt{1+f(x,t)}(\cos x,\sin x),\text{ for some
$f(\cdot,t):\mathbb{T}\mapsto(-1,\infty)$.}$ (1.3)

Note that one of the advantages of the use of the variable $f$, instead of a
more natural parametrization $z(x,t)=R(x,t)(\cos x,\sin x)$ relies on the
conservation of the area of the patch in the dynamics in (1.1); if the patch
initially has area $|D(0)|=\pi$, then $|D(t)|=\pi$ for all $t\geq 0$,
therefore

$\displaystyle\pi=|D(t)|=\frac{1}{2}\int_{\mathbb{T}}R(x,t)^{2}dx=\pi+\int_{\mathbb{T}}f(x,t)dx.$
(1.4)

Thus, we can assume that $f$ has zero average in the variable $x$.

Plugging (1.3) into (1.2), one can find that the evolution of $f$ can be
expressed as (we refer to Subsection 3.1 for more detailed computations)

$\displaystyle\partial_{t}f(x,t)=\frac{2}{2-\alpha}\partial_{\theta}\left(\int_{\mathbb{T}}\frac{(z(x,t)-z(y,t))\cdot\partial_{x}z(y,t)^{\perp}}{|z(x,t)-z(y,t)|^{\alpha}}dy\right)=:X_{\text{gSQG}}(f(x,t)).$
(1.5)

As noted in [74, 114, 104, 117], the equation (1.5) can be seen as a
Hamiltonian system with the associated Hamiltonian

$\displaystyle\mathcal{H}(f):=\int_{D}1_{D}*\frac{1}{|\cdot|^{\alpha}}(x)dx,$
(1.6)

where $D$ is the patch determined by the parametrization $f$ as in (1.3) (see
Section 3). More precisely, the vector field $X_{\text{gSQG}}(f)$ is given by

$\displaystyle
X_{\text{gSQG}}(f)=\partial_{x}\left(\nabla_{L^{2}}\mathcal{H}(f)\right),$
(1.7)

where $\nabla_{L^{2}}\mathcal{H}(f)$ denotes the gradient vector field of
$\mathcal{H}$ at $f$ in the space $L^{2}(\mathbb{T})$.

The linearized equation of (1.5) at the unit disk ($f=0$) can be written as
(see Proposition 3.2)

$\displaystyle
f_{t}=\frac{d}{dt}X_{\text{gSQG}}(tf)\bigg{|}_{t=0}=\partial_{x}\left(-\frac{1}{2}\Lambda^{\alpha-1}f+\frac{T_{\alpha}}{4}f\right),$
(1.8)

where

$\displaystyle\Lambda^{\alpha-1}f(x):=\int_{\mathbb{T}}(2-2\cos(x-y))^{-\frac{\alpha}{2}}(f(x)-f(y))dy,\text{
and
}T_{\alpha}:=\frac{2\pi\Gamma(3-\alpha)}{\Gamma(2-\frac{\alpha}{2})\Gamma(2-\frac{\alpha}{2})}.$
(1.9)

One can also rewrite the linearized equation (1.8) as

$\displaystyle\partial_{t}f=Op(\text{i}W(j))[f](x,t),$ (1.10)

where $Op(W(j))$ denotes the pseudo differential operator associated to the
the Fourier multiplier $W(j)$, defined as

$\displaystyle
W(j):=j\left(-\frac{1}{2}\mathcal{C}_{\alpha}\left(\frac{\Gamma(|j|+\frac{\alpha}{2})}{\Gamma(1+|j|-\frac{\alpha}{2})}-\frac{\Gamma(\frac{\alpha}{2})}{\Gamma(1-\frac{\alpha}{2})}\right)+\frac{\pi(-1)^{j}\Gamma(3-\alpha)}{2\Gamma(2-\frac{\alpha}{2})\Gamma(2-\frac{\alpha}{2})}\right),$
(1.11)

where
$\mathcal{C}_{\alpha}:=-\frac{2\pi\Gamma(1-\alpha)}{\Gamma\left(\frac{\alpha}{2}\right)\Gamma\left(1-\frac{\alpha}{2}\right)}$.
A classical asymptotic analysis for the Gamma function tells us that $W(j)$
exhibits an asymptotic behavior like $j|j|^{\alpha-1}$ (e.g. [100, Theorem
2.1]), more precisely,

$W(j)=C(\alpha)j|j|^{\alpha-1}+O(1),\text{ for some constant
$C(\alpha)\in\mathbb{R}$ for $\alpha\in(0,2)\backslash\left\\{1\right\\}$}.$

Given a set of natural numbers
$S^{+}:=\left\\{j_{1},\ldots,j_{\nu}\right\\}\subset\mathbb{N}$ (also denoting
$S:=\left\\{\pm j:j\in S^{+}\right\\}$), the linear equation (1.10) possesses
time-quasiperiodic solutions of the form

$\displaystyle
f(x,t)=\sum_{j_{k}\in{S}}\sqrt{|j_{k}|\zeta_{k}}e^{\text{i}(W(j_{k})t+j_{k}x)}=\sum_{j_{k}\in
S^{+}}2\sqrt{|j_{k}|\zeta_{k}}\cos(W(j_{k})t+j_{k}x),\text{ for some
$\zeta_{1},\ldots,\zeta_{\nu}>0$,}$ (1.12)

for which the $j$-th Fourier coefficient is oscillating in time with frequency
$W(j)$. Indeed, according to Definition 1.1, the solution (1.12) to the
linearized equation can be expressed as

$\displaystyle f(t,\cdot)=f(t)=i_{\text{linear}}(\overline{\omega}t),\text{
where }i_{\text{linear}}(\varphi):=\sum_{j_{k}\in
S^{+}}2\sqrt{\zeta_{k}}\cos(\varphi_{k}+j_{k}x)\text{ and
$\overline{\omega}_{k}:=W(j_{k})$.}$ (1.13)

This naturally leads to the question whether there exists such a time-
quasiperiodic solution to the full nonlinear problem (1.5) around the steady
state $f=0$.

In our analysis, we make use of several invariance properties of the equation
(1.3). One is the so-called time-reversibility with respect to the involution
$\rho:f(x)\mapsto f(-x)$, namely,

$\mathcal{H}(\rho(f))=\mathcal{H}(f),\text{ where $\mathcal{H}$ is defined as
in \eqref{hamiltonian_intro11}.}$

We say that a solution $f(x,t)$ to (1.5) is reversible, if $\rho(f)(x,-t)$ is
also a solution. Another invariance property is the rotational invariance of
solutions. More precisely, given an integer $\mathtt{M}\in\mathbb{N}$, if the
initial data of the gSQG equation is invariant under a
$\frac{2\pi}{\mathtt{M}}$-rotation, then the solution at any time is also
invariant under a $\frac{2\pi}{\mathtt{M}}$-rotation. Such an
$\mathtt{M}$-fold symmetric patch can be associated to a parametrization $f$
in (1.3) being invariant under a $\frac{2\pi}{\mathtt{M}}$-translation of the
variable $x$:

$\displaystyle f(t,x+\frac{2\pi}{\mathtt{M}})=f(t,x).$ (1.14)

Those properties of the gSQG equation will be studied in detail in Section 3.

The main theorem we prove in this monograph is the following (stated
informally, we refer to Theorem 6.5 for a precise statement):

######  Theorem 1.2.

[= Theorem 6.5] Let $\alpha\in(1,2)$, $S^{+}\subset\mathbb{N}$ and a symmetry
class $\mathtt{M}$ satisfying some non-resonance conditions (cf. Subsection
4.1 and 5.1) be fixed. Then, for all sufficiently small $\varepsilon>0$, there
exists a set of amplitudes
$A_{\varepsilon}\subset[1,2]^{|S^{+}|}\subset\mathbb{R}^{|S^{+}|}$ such that
for each $\vec{\zeta}\in A_{\varepsilon}$, there exist a frequency vector
$\omega=\omega(\vec{\zeta})$ and a time-quasiperiodic solution to (1.5) of the
form

$\displaystyle f(\theta,t)=2\varepsilon\sum_{j_{k}\in
S^{+}}\sqrt{|j_{k}|\vec{\zeta}_{k}}\cos(\omega_{k}t+j_{k}x)+o(\varepsilon).$
(1.15)

The set $A_{\varepsilon}$ is a Cantor-like set of asymptotically full measure,
in the sense that $\lim_{\varepsilon\to
0}\frac{|A_{\varepsilon}|}{\left|[1,2]^{|S^{+}|}\right|}=1$. The solution
(1.15) is in some Sobolev space $H^{s_{0}}$ for some $s_{0}\gg 1$, and it is
reversible and invariant under $\frac{2\pi}{\mathtt{M}}$-translation in the
variable $\theta$. Lastly, the solution is linearly stable under
$\frac{2\pi}{\mathtt{M}}$-translation invariant perturbations.

Some remarks are in order:

######  Remark 1.3.

As stated above, our proof does not make use of any external parameters
($\alpha$ would be the natural candidate) as opposed to [74] and indeed this
results in needing the Diophantine constant $\gamma$ to be
$\gamma=o(\varepsilon^{2})$, which in turn requires Normal Form expansions
(cf. Section 4), and also the computation of the explicit terms of size
$O(\varepsilon)$ and $O(\varepsilon^{2})$. Relaxing this constraint would
significantly shorten the length and the complexity of this monograph.

######  Remark 1.4.

It is conceivable that our proof of Theorem 6.5 would also work in the case
$\alpha<1$, changing the relevant sections and estimates. In the case of the
SQG equation $(\alpha=1)$, the analysis in Subsection 10.3 breaks down since
the sum of pseudo differential symbols is not finite anymore and the
regularity losses coming from the Egorov method are not finite. Most of the
other parts of the reduction also hold for all cases of $\alpha$, possibly
with minimal changes.

######  Remark 1.5.

The closer $\alpha$ is to $1$, the more conjugations are required in the
reduction of the linearized operator to a constant coefficients operator. In
the adaptation of the Egorov method, inspired by [16], we use a slightly more
general flow, compared to the ones in [16, 74, 52], to avoid a large number of
iterations that might cause potential complexity. See Remark 1.7.

######  Remark 1.6.

All the amplitudes in the set of $A_{\varepsilon}$ in Theorem 1.2, which takes
asymptotically full measure, can possess quasi-periodic solutions, if the
choice of tangential sites $S^{+}$ can be made properly so that some non-
resonance conditions are satisfied. The precise conditions are stated in
Subsection 5.1. Then a natural question is whether such a set
$S^{+}=\left\\{j_{1},\ldots,j_{\nu}\right\\}\subset\mathbb{N}$ is generic or
not. The non-resonance conditions that we require can be roughly expressed as

$P(j_{1},j_{2},\ldots,j_{\nu})\neq 0,\text{ for some function
$P:\mathbb{Z}^{\nu}\mapsto\mathbb{R}$}.$

Compared to previous works (e.g. [65, 8, 9]), our $P$ involves Gamma functions
and the verification of the non-resonance condition is much more complicated.
While we expect that such conditions can be satisfied by “generic” choices of
$S$ with small $\mathtt{M}$, we will give a rigorous proof only for the case
where the frequencies are supported on multiples of sufficiently large
$\mathtt{M}$. This allows us to focus on the asymptotic behavior of the Gamma
function in the analysis.

###  1.2 Strategy of the proof and the structure of the monograph

We first outline the main ideas of the proof of Theorem 1.2 and link them with
the sections of this monograph afterwards.

In order to describe the strategy of the proof, let us fix

$\displaystyle\nu\in\mathbb{N},\quad
S^{+}:=\left\\{j_{1}<\cdots<j_{\nu}\right\\}\subset\mathbb{N},$ (1.16)

and denote

$\displaystyle S:=\left\\{\pm j:j\in S^{+}\right\\},\quad
S^{\perp}:=\mathbb{Z}\backslash(S\cup\left\\{0\right\\}).$ (1.17)

In view of (1.12), one can think of $S$ as a set of Fourier modes of the
solution at the linear level, and $S^{\perp}$ as the support of the orthogonal
correction term for the solution to the nonlinear problem, while the $0$-th
mode is excluded, since we look for a solution with zero average (see (1.4)).
We also denote the linear frequency by

$\displaystyle{\overline{\omega}}\in\mathbb{R}^{\nu},\quad(\overline{\omega})_{k}=W(j_{k}),\text{
for $k=1,\ldots,\nu$.}$ (1.18)

Using the notations in (1.5) and (1.10), we can rewrite (1.5) as

$\displaystyle
f_{t}=X_{\text{gSQG}}(f)=Op(\text{i}W(j))[f]+P_{\text{gSQG}}(f),$ (1.19)

where $P_{\text{gSQG}}(f)$ collects the nonlinear contribution of the vector
field $X_{\text{gSQG}}(f)$. Since we are interested in the solutions near
$f=0$, replacing $f$ by $\varepsilon f$ for small $\varepsilon>0$, we are led
to study the equation of the form:

$\displaystyle f_{t}=Op(\text{i}W(j))[f]+P_{\varepsilon,\text{gSQG}}(f),\text{
where
$P_{\varepsilon,\text{gSQG}}(f):=\frac{1}{\varepsilon}P_{\text{gSQG}}(\varepsilon
f)$}.$ (1.20)

As we observed in Subsection 1.1, we have an embedding $i_{\text{linear}}$
(see (1.13)) for which
$f_{\text{lin}}(t,\theta):=i_{\text{linear}}(\overline{\omega}t)$ solves the
linear equation $\partial_{t}f_{\text{lin}}=Op(\text{i}W(j))[f_{\text{lin}}]$.
The question is whether such an embedding can persist under the nonlinear
perturbation as in (1.19). Perhaps, one of the most naive attempts could be
plugging the ansatz,

$f(t,x)=i(\overline{\omega}t,x),\text{ for some
$i:\mathbb{T}^{\nu}\times\mathbb{T}\mapsto\mathbb{R}$},$

into (1.19), which leads us to find $i$ such that

$\displaystyle\mathcal{F}_{\text{gSQG}}(i):=\overline{\omega}\cdot\partial_{\varphi}i(\varphi)-Op(\text{i}W(j))[i(\varphi)]-P_{\varepsilon,\text{gSQG}}(i(\varphi))=0,\text{
for $\varphi\in\mathbb{T}^{\nu}$.}$ (1.21)

We can think of $\mathcal{F}_{\text{gSQG}}$ as a map between spaces of
functions of $(\varphi,\theta)$. Having the explicit solution
$i_{\text{linear}}$ at the linear level given in (1.13) and noting that
$P_{\varepsilon,\text{gSQG}}$ is “small” depending on $\varepsilon>0$, we
might expect the sequence of embeddings $i_{n}$, formally defined in the
spirit of Newton’s method,

$\displaystyle i_{0}=i_{\text{linear}},\quad
i_{n+1}:=i_{n}-\left(d_{i}\mathcal{F}_{\text{gSQG}}(i_{n})\right)^{-1}[\mathcal{F}_{\text{gSQG}}(i_{n})],\text{
for $n\geq 0$},$ (1.22)

where

$\displaystyle
d_{i}\mathcal{F}_{\text{gSQG}}(i)[\hat{\textbf{\i}}]:=\frac{d}{dt}\mathcal{F}_{\text{gSQG}}(i+t\hat{\textbf{\i}})\bigg{|}_{t=0}\overset{\eqref{functional_intro_1}}{=}\overline{\omega}\cdot\partial_{\varphi}\hat{\textbf{\i}}-Op(\text{i}W(j))[\hat{\textbf{\i}}]-d_{i}P_{\varepsilon,\text{gSQG}}(i)[\hat{\textbf{\i}}]$
(1.23)

to converge to a solution for (1.21). Clearly, the above argument is far less
rigorous, and we will investigate how to modify the strategy.

####  1.2.1 Sketch of the proof, Part 1: Internal parameter and the weak
Birkhoff normal form

#####  1.2.1.1  Loss of derivatives and the Nash-Moser scheme.

As a rule of thumb in usual perturbative problems, the invertibility of the
linearized operator $d_{i}\mathcal{F}_{\text{gSQG}}(i)$ in (1.22) would rely
on the invertibility of the linear part:

$\displaystyle
L[\hat{\textbf{\i}}]:=\overline{\omega}\cdot\partial_{\varphi}\hat{\textbf{\i}}-Op(\text{i}W(j))[\hat{\textbf{\i}}],$
(1.24)

assuming that the contribution of the perturbative part is negligible. While
looking for an embedding $i$ in Sobolev spaces $H^{s}(\mathbb{T}^{\nu+1})$
(for large $s\gg 1$), it is not trivial whether the operator $L$ can be
invertible between two fixed Sobolev spaces. However, the classical KAM theory
tells us that the invertibility of $L$ can be achieved depending on the
frequency vector $\overline{\omega}$. Indeed, for the frequency vectors that
satisfy the so-called “Melnikov condition” with some $\gamma,\tau>0$:

$\displaystyle\left\\{\omega:|\text{i}\omega\cdot
l-\text{i}W(j)|\geq\gamma|l|^{\tau}\text{ for all
$(l,j)\in\mathbb{Z}^{\nu}\times\mathbb{Z}$}\right\\},$ (1.25)

one can formally invert $L$ using the Fourier series, that is,

$\displaystyle
L[\hat{\textbf{\i}}](\varphi,x)=g(\varphi,x)\overset{\eqref{linearpart_1}}{\iff}\hat{\textbf{\i}}(\varphi,x)=\sum_{(l,j)\in\mathbb{Z}^{\nu}\times\mathbb{Z}}\frac{\widehat{g}(l,j)}{\text{i}(\overline{\omega}\cdot
l-W(j))}e^{\text{i}(\varphi\cdot l+jx)},$ (1.26)

where
$\hat{g}(l,j):=\frac{1}{(2\pi)^{\nu+1}}\int_{\mathbb{T}^{\nu+1}}g(\varphi,x)e^{\text{i}(\varphi\cdot
l+jx)}d\varphi dx$. Note that the expression of the inverse in (1.26) tells us
that there is a regularity mismatch between the image and the domain spaces.
If $\hat{\textbf{\i}}\in H^{s}$ for some $s>0$, we see that there is a loss of
derivatives due to the differential operators $\partial_{\varphi}$ and
$Op(W(j))$, while inverting $L$ as in (1.26) does not gain the same amount of
the regularity, and it actually causes another loss of derivatives by $\tau$;
the best estimate one can expect under the condition (1.25) is that

$\displaystyle\rVert L^{-1}[g]\rVert_{H^{s-\tau}}\lesssim\gamma^{-1}\rVert
g\rVert_{H^{s}}\text{ for $g\in H^{s}(\mathbb{T}^{\nu+1})$}.$ (1.27)

Therefore, the formal sequence of $i_{n}$ in (1.22) does not seem to be
closed, since $i_{n+1}$ must be less regular than $i_{n}$ at each iteration.
Hence, the crude iteration procedure in (1.22) needs to be replaced by the
Nash-Moser scheme, projecting each approximate solution $i_{n}$ into a finite
dimensional space so that $i_{n}$ remains in $C^{\infty}(\mathbb{T}^{\nu+1})$
for each $n\geq 0$.

#####  1.2.1.2 Internal parameter.

We have observed above that the condition on the frequency vector in (1.25) is
one of the necessary conditions to perform the iteration (1.22). Then a very
natural question is how to check whether the linear frequency
$\overline{\omega}$ satisfies such a condition. In general (for fixed
$\gamma,\tau>0$), it is very hard to determine whether a given vector
$\omega\in\mathbb{R}^{\nu}$ satisfies even a more relaxed condition (so-called
Diophantine condition):

$\displaystyle|\omega\cdot l|>\gamma|l|^{\tau},\text{ for all
$l\in\mathbb{Z}^{\nu}$.}$ (1.28)

However, it is well-known that given an open set
$\Omega\subset\mathbb{R}^{\nu}$, “almost all” $\omega\in\Omega$ satisfy
(1.28), more precisely, such non-resonance frequency vectors take
asymptotically full measure in $\Omega$ as $\gamma\to 0$. Indeed, the KAM
theory does not tell us exactly which frequency vector can possess a
quasiperiodic solution, but rather it tells us that the set of frequency
vectors that possess a quasiperiodic solution has non-zero measure in a given
set of frequencies. This is why we need parameter-dependent equations to
perform the KAM theory; if our equation depends on a parameter, and the
parameter can “properly”modulate the linear frequency $\overline{\omega}$,
then for almost all parameter values, we might expect to invert the operator
$L$. The gSQG equation (1.1) certainly involves a parameter $\alpha\in(0,2)$,
therefore one might be tempted to use $\alpha$ to modulate the linear
frequency by looking at $\overline{\omega}$ in (1.18) and (1.11) as a function
of $\alpha$, that is, $\overline{\omega}=\overline{\omega}(\alpha)$. This
attempt would enable us to obtain quasiperiodic solutions for “almost every”
$\alpha$ (without knowing precisely which $\alpha$ satisfies the condition),
while such a result cannot be, in principle, obtained for _every_ $\alpha$.
For this reason, we follow the strategy in [8, 9, 52] and derive a weak
Birkhoff normal form of the Hamiltonian $\mathcal{H}$, from which we can see a
modulation effect of the linear frequency by the amplitude. In this regard, a
bit more precise explanation will follow.

#####  1.2.1.3 Action-angle variables.

According to the decomposition in (1.17), we denote

$\displaystyle H_{S}:=\left\\{f:f_{j}=0,\text{ if $j\notin S$}\right\\},\quad
H_{S^{\perp}}:=\left\\{f:f_{j}=0,\text{ if $j\notin S^{\perp}$}\right\\},\quad
f_{j}:=\int_{\mathbb{T}}e^{\text{i}jx}dx,$ (1.29)

and we will refer to $H_{S}$ and $H_{S^{\perp}}$ as _tangential space_ and
_normal space_ , respectively. We introduce the amplitude variable ${\zeta}$:

$\displaystyle\zeta\in[1,2]^{\nu}\subset\left(\mathbb{R}^{+}\right)^{\nu},$
(1.30)

and consider a $\zeta$-dependent change of variables,
$U_{\zeta}:\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times H_{S^{\perp}}\mapsto
L^{2}(\mathbb{T})$:

$\displaystyle U_{\zeta}(\theta,y,z):=\varepsilon\left(\sum_{j_{k}\in
S}\sqrt{|j_{k}|(\zeta_{k}+\varepsilon^{2(b-1)}y_{k})}e^{\text{i}(\theta_{k}+j_{k}x)}+\varepsilon^{(b-1)}z\right),\text{
for some $b\in(1,1+1/12)$},$ (1.31) $\displaystyle\text{ with
}\zeta_{-k}:=\zeta_{k},\quad\theta_{-k}:=-\theta_{k},\quad y_{-k}:=y_{k}.$

The variables $(\theta,y)$ are the so-called action-angle variables and the
above change of variables can be thought of as a reparametrization of
functions in $H_{S}$. Also, in order to see the motivation of the constant $b$
in (LABEL:intro_change_val), we note that in view of (1.13), the map

$\displaystyle\mathbb{T}^{\nu}\ni\varphi\mapsto U_{\zeta}(\varphi,0,0)\in
L^{2}(\mathbb{T})$ (1.32)

corresponds to $i_{\text{linear}}$ up to the rescaling factor $\varepsilon$,
therefore the terms $\varepsilon^{2(b-1)}y$ and $\varepsilon^{b-1}z$ in
(LABEL:intro_change_val) can be thought of as correction terms to solve the
nonlinear problem, which justifies the requirement $b>1$.

Now we define a $\zeta$-dependent Hamiltonian $\mathcal{H}_{\zeta}$ on
$\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times H_{S^{\perp}}$ as

$\displaystyle\mathcal{H}_{\zeta}(\theta,y,z):=\varepsilon^{-2b}\mathcal{H}\circ
U_{\zeta}(\theta,y,z).$ (1.33)

We note that the factor $\varepsilon^{-2b}$ in (1.33) naturally arises in the
rescaling of the change of variables to describe the Hamiltonian equation in
the new phase space $\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$. Indeed, the gSQG dynamics in $L^{2}(\mathbb{T})$ in (1.5) can
be easily rewritten as an evolution equation in the new phase space by pulling
back the vector field by $U_{\zeta}$, and one can obtain the equivalent
equation is given by (one can follow the same computations given in Section 5
after the proof of Proposition 5.1. Note that the factor $\frac{1}{2\pi}$ is
due to our definition for the symplectic form $\sigma$ in (2.127) and the
gradient in (2.125), but does not play a crucial role throughout the proof)

$\displaystyle\frac{d}{dt}\begin{pmatrix}\theta(t)\\\ y(t)\\\
z(t)\end{pmatrix}=\begin{pmatrix}\frac{1}{2\pi}\partial_{y}\mathcal{H}_{\zeta}(\theta,y,z)\\\
-\frac{1}{2\pi}\partial_{\theta}\mathcal{H}_{\zeta}(\theta,y,z)\\\
\partial_{x}\left(\nabla_{z}\mathcal{H}_{\zeta}(\theta,y,z)\right)\end{pmatrix},$
(1.34)

where $\nabla_{z}\mathcal{H}_{\zeta}$ is the gradient vector field of
$\mathcal{H}_{\zeta}$ restricted to the subspace $H_{S^{\perp}}$. Indeed, one
can show that if $(\theta(t),y(t),z(t))$ is a solution to (1.34), then
$f(t):=U_{\zeta}(\theta(t),y(t),z(t))$ is a solution to the gSQG equation
(1.6) (again, see Section 5).

  * $\bullet$

A toy model 1: A perturbed Airy equation.

Recall that our goal at this moment is to see whether we can extract a
dependence of the linear frequency on $\zeta$. Therefore the question becomes
whether the new system (1.34) has a linear frequency that can be modulated by
$\zeta$. Of course, the answer depends on the structure of $\mathcal{H}$. To
this end, let us consider a simpler example, where we can see that the
modulation of the linear frequency relies on the quartic homogeneous term of
the Hamiltonian.

As a toy model, let us consider a perturbed Airy equation (see [6, 8, 9, 65]
for the results of relevant but more complicated models). We define a
Hamiltonian
$\mathcal{G}:L^{2}(\mathbb{T})\mapsto\mathbb{R}\cup\left\\{\infty\right\\}$ of
the form:

$\displaystyle\mathcal{G}(f):=\mathcal{G}_{2}(f)+\mathcal{G}_{4}(f),\text{
where }$ (1.35)
$\displaystyle\mathcal{G}_{2}(f):=\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}f)^{2}(x)dx,$
$\displaystyle\mathcal{G}_{4}(f):=\sum_{\begin{subarray}{c}j_{1},j_{2},j_{3},j_{4}\in\mathbb{Z}\backslash\left\\{0\right\\}\end{subarray}}G({j_{1},j_{2},j_{3},j_{4}})f_{j_{1}}f_{j_{2}}f_{j_{3}}f_{j_{4}}\text{
for some $G:\mathbb{Z}^{4}\mapsto\mathbb{C}$}.$

The associated nonlinear Hamiltonian equation to $\mathcal{G}$ is written as
(see the comparison with (1.7))

$\displaystyle
f_{t}=\partial_{x}(\nabla_{L^{2}}\mathcal{G}(f))=-\partial_{xxx}f+\partial_{x}(\nabla_{L^{2}}\mathcal{G}_{4}(f)).$
(1.36)

The linearized Hamiltonian equation associated to $\mathcal{G}$ depends on
only the quadratic term $\mathcal{G}_{2}$ and it corresponds to the Airy
equation,

$\displaystyle f_{t}=-\partial_{xxx}f.$ (1.37)

It is trivial to see that the Airy equation possesses quasiperiodic solutions
with the linear frequency $\omega^{\text{Airy}}\in\mathbb{R}^{\nu}$ given by

$\displaystyle(\overline{\omega}^{\text{Airy}})_{k}:=j_{k}^{3},\text{ for
$S^{+}:=\left\\{j_{1}<\cdots<j_{\nu}\right\\}$.}$ (1.38)

At this point, the linearized equation does not reveal the modulation of the
linear frequency by the amplitude. To make the computations easier, let us
make the following assumptions on the quartic term $\mathcal{G}_{4}$:

    1. (A1)

${G}(j_{1},j_{2},j_{3},j_{4})$ is invariant under any permutation on
$\left\\{j_{1},j_{2},j_{3},j_{4}\right\\}$. This assumption is simply to make
the computations easier and can be assumed for general quartic Hamiltonian,
since we can take the average of the summation in (1.35) over all the
permutations.

    2. (A2)

${G}$ is supported only on modes $S$, that is,
${G}(j_{1},j_{2},j_{3},j_{4})=0$, if $j_{i}\notin S$ for some $=1,2,3,4$. This
is to focus on the contribution of $\zeta$ through this example, since $\zeta$
presents only in the modes in the set $S$ (see (LABEL:intro_change_val)).

    3. (A3)

If $G(j_{1},j_{2},j_{3},j_{4})\neq 0$, then $j_{a}=-j_{b}$ and $j_{c}=-j_{d}$
for a permutation $(a,b,c,d)$ of $(1,2,3,4)$. In other words, there is no
nontrivial resonance in $\mathcal{G}_{4}$. Therefore, using the assumption
(A1) and (A2), we can define a $\nu\times\nu$ matrix $\mathbb{G}$ as

$\displaystyle\mathbb{G}^{i}_{k}:=G(j_{i},-j_{i},j_{k},-j_{k})\text{ for
$i,k=1,\ldots,\nu$ and $j_{i},j_{k}\in S^{+}$.}$ (1.39)

To see the modulation of the frequency by $\zeta$, we compute the composition
with the change of variables $U_{\zeta}$:

$\displaystyle\mathcal{G}_{\zeta}(\theta,y,z):=\varepsilon^{-2b}\mathcal{G}\circ
U_{\zeta}(\theta,y,z).$ (1.40)

For the quadratic term, we see that

$\displaystyle\varepsilon^{-2b}\mathcal{G}_{2}\circ U_{\zeta}(\theta,y,z)$
$\displaystyle\overset{\eqref{intro_change_val}}{=}\varepsilon^{-2b}\left(2\pi\sum_{j_{k}\in
S^{+}}j_{k}^{2}\varepsilon^{2}(j_{k}\zeta_{k}+\varepsilon^{2(b-1)}j_{k}y_{k})+\frac{1}{2}\int_{\mathbb{T}}\varepsilon^{2b}(\partial_{x}z)^{2}(x)dx\right)$
$\displaystyle=C_{\varepsilon,\zeta}+2\pi\sum_{j_{k}\in
S^{+}}(j_{k})^{3}y_{k}+\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}z)^{2}(x)dx$
$\displaystyle\overset{\eqref{airy_intro}}{=}C_{\varepsilon,\zeta}+2\pi\left(\overline{\omega}^{\text{Airy}}\cdot
y\right)+\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}z)^{2}(x)dx.$ (1.41)

for some $C_{\varepsilon,\zeta}$ that does not depend on $(\theta,y,z)$. For
the quartic term, we have

$\displaystyle\mathcal{G}_{4,0}(f)$
$\displaystyle\overset{\eqref{g4_intro_1},\ref{Assumpp1_intro2}}{=}\sum_{\begin{subarray}{c}j_{1},j_{2},j_{3},j_{4}\in
S\end{subarray}}G(j_{1},j_{2},j_{3},j_{4})f_{j_{1}}f_{j_{2}}f_{j_{3}}f_{j_{4}}$
$\displaystyle\overset{\ref{Assumpp1_intro1},\ref{Assumpp1_intro3}}{=}6\sum_{j_{1}=j_{2}\in
S^{+}}G(j_{1},-j_{1},j_{2},-j_{2})|f_{j_{1}}|^{2}|f_{j_{2}}|^{2}+12\sum_{j_{1},j_{2}\in
S^{+},\ j_{1}\neq
j_{2}}G(j_{1},j_{2},-j_{1},-j_{2})|f_{j_{1}}|^{2}|f_{j_{2}}|^{2}$
$\displaystyle\overset{\eqref{G_symple}}{=}6\sum_{i=1}^{\nu}\mathbb{G}^{i}_{i}|f_{j_{i}}|^{4}+12\sum_{i,k=1,\
i\neq k}^{\nu}\mathbb{G}^{i}_{k}|f_{j_{i}}|^{2}|f_{j_{k}}|^{2}.$

Therefore, using (LABEL:intro_change_val), we can see that

$\displaystyle\varepsilon^{-2b}\mathcal{G}_{4}\circ U_{\zeta}(\theta,y,z)$
$\displaystyle\overset{\eqref{intro_change_val}}{=}6\varepsilon^{-2b}\sum_{i=1}^{\nu}\mathbb{G}^{i}_{i}\varepsilon^{4}(j_{i}\zeta_{i}+\varepsilon^{2(b-1)}j_{i}y_{i})^{2}$
$\displaystyle+12\varepsilon^{-2b}\sum_{i,k=1,\ i\neq
k}^{\nu}\mathbb{G}^{i}_{k}\varepsilon^{4}(j_{i}\zeta_{i}+\varepsilon^{2(b-1)}j_{i}y_{i})(j_{k}\zeta_{k}+\varepsilon^{2(b-1)}j_{k}y_{k})$
$\displaystyle=C_{\varepsilon,\zeta}+12\varepsilon^{2}\left(\sum_{i=1}^{\nu}\mathbb{G}^{i}_{i}j_{i}^{2}\zeta_{i}y_{i}+2\sum_{i,k=1,\
i\neq k}^{\nu}\mathbb{G}^{i}_{k}j_{i}j_{k}\zeta_{i}y_{k}\right)$
$\displaystyle\
+6\varepsilon^{2b}\left(\sum_{i=1}^{\nu}\mathbb{G}^{i}_{i}j_{i}^{2}y_{i}^{2}+2\sum_{i,k=1,\
i\neq k}^{\nu}\mathbb{G}^{i}_{k}j_{i}j_{k}y_{i}y_{k}\right)$
$\displaystyle=C_{\varepsilon,\zeta}+12\varepsilon^{2}G^{\text{mod}}\zeta\cdot
y+6\varepsilon^{2b}G^{\text{mod}}y\cdot y,$ (1.42)

where $G^{\text{mod}}$ is a $\nu\times\nu$ symmetric matrix defined as

$\displaystyle(G^{\text{mod}})^{i}_{k}:=\begin{cases}j_{i}^{2}\mathbb{G}^{i}_{i}&\text{
if $i=k$}\\\ 2j_{i}j_{k}\mathbb{G}^{i}_{k}&\text{ if $i\neq
k$}\end{cases},\text{ for $j_{i},j_{k}\in
S^{+}=\left\\{j_{1}<\cdots<j_{\nu}\right\\}$.}$ (1.43)

Plugging ($\bullet$ ‣ 1.2.1.3) and (1.41) into (1.40), we see a normal form of
the nonlinear Hamiltonian $\mathcal{G}_{\zeta}$:

$\displaystyle\mathcal{G}_{\zeta}(\theta,y,z)$
$\displaystyle=C_{\varepsilon,\zeta}+\left(2\pi\overline{\omega}^{\text{Airy}}+12\varepsilon^{2}G^{\text{mod}}\zeta\right)\cdot
y+6\varepsilon^{2b}G^{\text{mod}}y\cdot
y+\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}z)^{2}(x)dx$
$\displaystyle=C_{\varepsilon,\zeta}+2\pi\omega^{\text{Airy}}(\zeta)\cdot
y+6\varepsilon^{2b}G^{\text{mod}}y\cdot
y+\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}z)^{2}(x)dx,$ (1.44)

where

$\displaystyle\omega^{\text{Airy}}(\zeta):=\overline{\omega}^{\text{Airy}}+\frac{6}{\pi}\varepsilon^{2}G^{\text{mod}}\zeta\in\mathbb{R}^{\nu}.$
(1.45)

Recall that we can write the Hamiltonian equation (1.36) in the phase space
$L^{2}(\mathbb{T})$ as another Hamiltonian equation in the new phase space
$\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times H_{S^{\perp}}$ by pulling back
the vector field by $U_{\zeta}$, which should be written as (compare to
(1.34))

$\displaystyle\frac{d}{dt}\begin{pmatrix}\theta(t)\\\ y(t)\\\
z(t)\end{pmatrix}=\begin{pmatrix}\frac{1}{2\pi}\partial_{y}\mathcal{G}_{\zeta}(\theta,y,z)\\\
-\frac{1}{2\pi}\partial_{\theta}\mathcal{G}_{\zeta}(\theta,y,z)\\\
\partial_{x}\left(\nabla_{z}\mathcal{G}_{\zeta}(\theta,y,z)\right)\end{pmatrix}\overset{\eqref{normal_form_42}}{=}\begin{pmatrix}\omega^{\text{Airy}}(\zeta)\\\
0\\\
0\end{pmatrix}+\begin{pmatrix}\frac{6}{\pi}\varepsilon^{2b}G^{\text{mod}}y(t)\\\
0\\\ -\partial_{xxx}(z(t))\end{pmatrix}.$ (1.46)

From the above equation, we can easily see that the trivial embedding
$i_{triv}:\mathbb{T}^{\nu}\mapsto\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$, defined as

$\displaystyle i_{triv}(\varphi):=(\varphi,0,0).$ (1.47)

is invariant under the vector field in (1.46), and the solution to (1.46) can
be written as

$\displaystyle(\theta(t),y(t),z(t))=i_{triv}(\omega^{\text{Airy}}(\zeta)t)=(\omega^{\text{Airy}}(\zeta)t,0,0).$
(1.48)

Clearly, the solution (1.48) is time-quasiperiodic because each “angular
component” $\theta_{i}(t)$ oscillates with frequency
$(\omega^{\text{Airy}}(\zeta))_{i}$ for each $i=1,\ldots,\nu$ and the
frequency vector $\omega^{\text{Airy}}(\zeta)$ is modulated by the amplitude
$\zeta$ through the relation in (1.45), which cannot be observed by just
looking at the linear Airy equation (1.37). Therefore, a quasiperiodic
solution to (1.36) can be obtained as
$f(t):=U_{\zeta}(i_{triv}(\omega^{\text{Airy}}(\zeta)t))$. As shown in this
example, our strategy to study the equation (1.34) is to derive a “normal
form” where we can see a modulation of the frequency by the amplitude $\zeta$
at the linear level of the equation (that is, quadratic level of the
Hamiltonian).

#####  1.2.1.4 Weak Birkhoff normal form.

Our Hamiltonian $\mathcal{H}$ in (1.6) does not possess a simple structure as
in the toy-model example, therefore it is hopeless to expect
$\mathcal{H}_{\zeta}$ in (1.33) to have a simple form as in ($\bullet$ ‣
1.2.1.3). However, we will construct a symplectic transformation
$\Phi^{WB}:L^{2}(\mathbb{T})\mapsto L^{2}(\mathbb{T})$ so that defining
another Hamiltonian $H$ as

$\displaystyle H(f):=\mathcal{H}\circ\Phi^{WB}(f),$ (1.49)

we can rewrite $H$, under the composition with $U_{\zeta}$, as (compare below
with ($\bullet$ ‣ 1.2.1.3))

$\displaystyle
H_{\zeta}(\theta,y,z):=\varepsilon^{-2b}H\circ(U_{\zeta}(\theta,y,z))=C_{\varepsilon,\zeta}+2\pi\omega^{\text{gSQG}}(\zeta)\cdot
y+6\varepsilon^{2b}\mathbb{A}y\cdot y+\mathcal{N}(\theta)(z,z)+P(\theta,y,z),$
(1.50)

where

$\displaystyle\omega^{\text{gSQG}}(\zeta)=\overline{\omega}+\frac{6}{\pi}\varepsilon^{2}\mathbb{A}\zeta,\text{
(see \eqref{linear_fre_intro2} for the definition of $\overline{\omega}$)},$
(1.51)

for some $\nu\times\nu$ symmetric matrix $\mathbb{A}$, for some
$\theta$-dependent bilinear form $\mathcal{N}(\theta)$ on $H_{S^{\perp}}$, and
for some perturbation $P$ satisfying some smallness condition. Note that in
(1.50), the bilinear form $\mathcal{N}$ and the perturbation $P$ depend on
$\zeta$, while we do not denote its dependence to avoid notational
complication.

Certainly, the following concerns need to be taken into account:

  1. (C1)

What structure of $\omega^{\text{gSQG}}(\zeta)$ do we need? More precisely,
what do we require on the matrix $\mathbb{A}$?.

  2. (C2)

How to construct $\Phi^{WB}$ so that $H$ defined by (1.49) has the structure
in (1.50) with sufficiently small $P$.

We postpone more detailed comments on the above concerns to the next paragraph
but for now focus on how to transform the functional equation (1.21),
according to the new Hamiltonian (1.50). By requiring $\Phi^{WB}$ to be
symplectic, we have that the Hamiltonian equation associated to $\mathcal{H}$
is equivalent to the Hamiltonian equation associated to $H$, therefore, again
pulling back the Hamiltonian vector field of $H$ by $U_{\zeta}$. Thus we are
led to study the equation:

$\displaystyle\frac{d}{dt}\begin{pmatrix}\theta(t)\\\ y(t)\\\
z(t)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{1}{2\pi}\partial_{y}H_{\zeta}(\theta,y,z)\\\
-\frac{1}{2\pi}\partial_{\theta}H_{\zeta}(\theta,y,z)\\\
\partial_{x}\left(\nabla_{z}H_{\zeta}(\theta,y,z)\right)\end{pmatrix}$ (1.52)
$\displaystyle\overset{\eqref{newHamiltonian_int11ro}}{=}\begin{pmatrix}\omega^{\text{gSQG}}(\zeta)\\\
0\\\ 0\end{pmatrix}+\begin{pmatrix}\frac{6}{\pi}\varepsilon^{2b}\mathbb{A}y\\\
-\frac{1}{2\pi}\partial_{\theta}\mathcal{N}(\theta)(z,z)\\\
\partial_{x}((\mathcal{N}(\theta))^{T}[z])\end{pmatrix}+\begin{pmatrix}\frac{1}{2\pi}\partial_{y}P(\theta,y,z)\\\
-\frac{1}{2\pi}\partial_{\theta}P(\theta,y,z)\\\
\partial_{x}\left(\nabla_{z}P(\theta,y,z)\right)\end{pmatrix}$
$\displaystyle=:X_{H_{\zeta}}(\theta(t),y(t),z(t)),$

where $z\mapsto\mathcal{N}(\theta)^{T}[z]$ is the linear map on
$H_{S^{\perp}}$ such that

$\int_{\mathbb{T}}\mathcal{N}(\theta)^{T}[z](x)h(x)dx=\nabla_{z}(\mathcal{N}(\theta)(z,z))[h],\text{
for all $h\in S^{\perp}$.}$

If $t\mapsto
i_{\infty}(\omega^{\text{gSQG}}(\zeta)t)\in\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$ is a quasiperiodic solution to (1.52), then the quasiperiodic
solution to the gSQG equation (1.5) will be recovered by

$f(t)=\Phi^{WB}(U_{\zeta}(i_{\infty}(\omega^{\text{gSQG}}(\zeta)t))).$

Assuming $P\equiv 0$, the equation (1.52) indeed possesses the trivial
embedding (1.47) as a quasiperiodic solution with the frequency vector
$\omega^{\text{gSQG}}(\zeta)$. Therefore, our main goal of this monograph
becomes to study whether such an embedding can persist under a perturbation
$P$ in the system (1.52). Note that taking into account the dependence of the
frequency on $\zeta$, more precise statements to be proved are that “for
almost every $\zeta$” in (1.30), the quasiperiodic solution with frequency
$\omega^{\text{gSQG}}(\zeta)$ can survive under the perturbation. Making an
ansatz,

$\displaystyle\text{$t\mapsto i(\omega^{\text{gSQG}}(\zeta)t)$ solves
\eqref{Hamil_final_intro1}},\text{ for some
$i:\mathbb{T}^{\nu}\mapsto\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$, $i(\varphi)=:(\theta(\varphi),y(\varphi),z(\varphi))$},$
(1.53)

we are led to find $i$ such that

$\displaystyle\mathcal{F}(i):=\omega^{\text{gSQG}}(\zeta)\cdot\partial_{\varphi}i(\varphi)-X_{H_{\zeta}}(i(\varphi))=0.$
(1.54)

Now, let us discuss the concerns (C1)-(C2).

  * $\bullet$

Item (C1): Frequency vector $\omega^{\text{gSQG}}(\zeta)$ and use of $\omega$
as a parameter. Let us first make it clear why we need to care about the
structure of $\omega^{\text{gSQG}}(\zeta)$. We recall from (1.30) that our
parameter $\zeta$ lies in a fixed subset $[1,2]^{\nu}$, and we want to perform
the iteration (1.61) for sufficiently many $\zeta$ in $[1,2]^{\nu}$. Denoting

$\displaystyle\Omega:=\left\\{\omega\in\mathbb{R}^{\nu}:\omega=\omega^{\text{gSQG}}(\zeta),\
\zeta\in[1,2]^{\nu}\right\\},$ (1.55)

we can only hope that “almost every” $\omega$ in $\Omega$ satisfy all the
necessary non-resonance conditions, such as (1.28). This does not necessarily
imply that for “almost every” $\zeta\in[1,2]^{\nu}$,
$\omega^{\text{gSQG}}(\zeta)$ satisfies the necessary non-resonance
conditions, especially in case $\zeta\mapsto\omega^{\text{gSQG}}(\zeta)$ is
not one-to-one. Thus, we require that the matrix $\mathbb{A}$ in (1.51) is
invertible. The explicit form of $\mathbb{A}$ is not important at this point,
but it is important to note that $\mathbb{A}$ is completely determined by the
choice of the set $S^{+}$ in (1.16). The invertibility of $\mathbb{A}$ is one
of the “non-resonance conditions” described in the statement of Theorem 1.2.
In the proof, this condition will be verified (See Subsection 5.1 and
Proposition 5.5).

Once we have the invertibility of the map
$\zeta\mapsto\omega^{\text{gSQG}}(\zeta)$, we will use the frequency $\omega$
as a parameter of the system (1.52) and think of $\zeta$ as a quantity
determined by $\omega$. More precisely, defining

$\displaystyle\Omega_{\varepsilon}:=\left\\{\omega\in\mathbb{R}^{\nu}:\omega=\overline{\omega}+\frac{6}{\pi}\varepsilon^{2}\mathbb{A}\zeta,\text{
for }\zeta\in[1,2]^{\nu}\right\\},$ (1.56)

we think of the amplitude $\zeta$ to be a function of $\omega$ for
$\omega\in\Omega_{\varepsilon}$. With a slight abuse of notation, we will
still denote $\zeta$ as if it is an independent variable, but it is actually a
function of $\omega$, determined by

$\displaystyle\zeta=\zeta(\omega):=\left(\frac{6}{\pi}\varepsilon^{2}\mathbb{A}\right)^{-1}(\omega-\overline{\omega}),\text{
for $\omega\in\Omega_{\varepsilon}$.}$ (1.57)

Then the Hamiltonian functional (1.50), the system (1.52) and the functional
equation (1.54) become

$\displaystyle H_{\zeta}(\theta,y,z)$
$\displaystyle=C_{\varepsilon,\zeta}+2\pi\omega\cdot
y+6\varepsilon^{2b}\mathbb{A}y\cdot y+\mathcal{N}(\theta)(z,z)+P(\theta,y,z),$
(1.58) $\displaystyle\frac{d}{dt}\begin{pmatrix}\theta(t)\\\ y(t)\\\
z(t)\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\omega\\\ 0\\\
0\end{pmatrix}+{\begin{pmatrix}\frac{6}{\pi}\varepsilon^{2b}\mathbb{A}y\\\
-\frac{1}{2\pi}\partial_{\theta}\mathcal{N}(\theta)(z,z)\\\
\partial_{x}((\mathcal{N}(\theta))^{T}[z])\end{pmatrix}}+{\begin{pmatrix}\frac{1}{2\pi}\partial_{y}P(\theta,y,z)\\\
-\frac{1}{2\pi}\partial_{\theta}P(\theta,y,z)\\\
\partial_{x}\left(\nabla_{z}P(\theta,y,z)\right)\end{pmatrix}}=:X_{H_{\zeta}}(\theta(t),y(t),z(t)),$
(1.59)

and

$\displaystyle\mathcal{F}_{\omega}(i)$
$\displaystyle:=\omega\cdot\partial_{\varphi}i(\varphi)-X_{H_{\zeta}}(i(\varphi))=\begin{pmatrix}\omega\cdot\partial_{\varphi}\theta(\varphi)\\\
\omega\cdot\partial_{\varphi}y(\varphi)\\\
\omega\cdot\partial_{\varphi}z(\varphi)\end{pmatrix}-X_{H_{\zeta}}(i(\varphi))=0.$
(1.60)

Here $X_{H_{\zeta}}=X_{H_{\zeta(\omega)}}$ is now thought of as an
$\omega$-dependent vector field (instead of a $\zeta$-dependent vector field)
and the Hamiltonian $H_{\zeta}$, as well as $\mathcal{N}$ and $P$, depends on
$\omega$ implicitly through (1.57). Clearly, if an embedding $i_{\infty}$
solves (1.60) with some $\omega\in\Omega_{\varepsilon}$, then $i_{\infty}$
solves (1.54) with $\zeta(\omega)$. The reason why we use $\omega$ as a
parameter is that by doing so, it is easier to check the non-resonance
conditions such as (1.25) or (1.28) throughout the proof.

Now, we transform the initial iteration scheme given in (1.22), since our new
system (1.59) has a slightly different form, compared to (1.20). The main
scheme is quite same as described in (1.22) based on Newton’s method (more
precisely, Nash-Moser scheme as we discussed before): Noting that the trivial
embedding (1.47) solves (1.59) without the perturbation $P$, we set up a
formal sequence of approximate solutions
$i_{n}:\mathbb{T}^{\nu}\mapsto\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$,

$\displaystyle\begin{cases}i_{0}:=i_{triv},\quad
i_{n+1}:=i_{n}-(d_{i}\mathcal{F}_{\omega}(i_{n}))^{-1}[\mathcal{F}_{\omega}(i_{n})],\text{
for $n\geq 0$},\\\
d_{i}\mathcal{F}_{\omega}(i)[\hat{\textbf{\i}}]:=\frac{d}{dt}\mathcal{F}_{\omega}(i+t\hat{\textbf{\i}})\bigg{|}_{t=0}\overset{\eqref{actual_functioanleq}}{=}\omega\cdot\partial_{\varphi}\hat{\textbf{\i}}-d_{i}X_{H_{\zeta}}(i)[\hat{\textbf{\i}}],\end{cases}$
(1.61)

and study the convergence of $i_{n}$ to a solution to (1.60). In the iteration
scheme (1.61), we do not expect the inverse of the linearized operator
$d_{i}\mathcal{F}_{\omega}(i)[\hat{\textbf{\i}}]$ to be obtained for every
$\omega\in\Omega_{\varepsilon}$, but we select $\omega$ such that $\omega$
satisfies all the necessary non-resonance conditions to obtain an inverse of
the linearized operator.

Before we close our discussion concerning the use of $\omega$ as a parameter,
we emphasize that the constant $\gamma$ arising in the Diophantine condition
(1.28) needs to be small depending on $\varepsilon$. We wish to select non-
resonant frequencies from the set $\Omega_{\varepsilon}$, that is, we wish the
set (for some fixed $\tau>0$)

$\displaystyle\Omega_{0}:=\left\\{\omega\in\Omega_{\varepsilon}:|\omega\cdot
l|>\gamma|l|^{\tau},\text{ for all $l\in\mathbb{Z}^{\nu}$}\right\\}$ (1.62)

to take asymptotically full measure in $\Omega_{\varepsilon}$. However,
$\Omega_{\varepsilon}$ in (1.56) is an $\varepsilon^{2}$-neighborhood of
$\overline{\omega}$. Therefore, in case $\overline{\omega}$ is resonant, that
is, $\overline{\omega}\cdot l_{*}=0$ for some $l_{*}\in\mathbb{Z}^{\nu}$, we
have

$|\omega\cdot l_{*}|\leq|(\omega-\overline{\omega})\cdot
l_{*}|+|\overline{\omega}\cdot
l_{*}|\leq|\omega-\overline{\omega}||l_{*}|\lesssim\varepsilon^{2}|l_{*}|,\text{
for all $\omega\in\Omega_{\varepsilon}$}.$

Thus, it is not, in general, possible for $\Omega_{0}$ to obtain
asymptotically full measure in $\Omega_{\varepsilon}$, unless
$\gamma=o(\varepsilon^{2})$. For this reason, we will fix $\gamma$ to be

$\displaystyle\gamma:=\varepsilon^{2b},\text{ where $b>1$ is chosen as in
\eqref{intro_change_val}.}$ (1.63)

  * $\bullet$

Item (C2): Construction of $\Phi^{WB}$. Now, we discuss how to construct the
transformation $\Phi^{WB}$ so that we can obtain (1.50) through (1.49).

Before we start, we first fix some notation. In view of (1.29), we denote by
$v$ and $z$ the variables in spaces $H_{S}$ and $H_{S^{\perp}}$ respectively,
so that a given $f\in L^{2}(\mathbb{T})$ can be written as

$\displaystyle f=v+z,\text{ for some $v\in H_{S}$ and $z\in H_{S^{\perp}}$ in
a unique way.}$ (1.64)

The variables $v$ and $z$ will be called a “tangential variable” and a “normal
variable”. We define $v_{\zeta}:\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\mapsto
H_{S}$ by

$\displaystyle v_{\zeta}(\theta,y):=\sum_{j_{k}\in
S}\sqrt{|j_{k}|(\zeta_{k}+y_{k})}e^{\text{i}(\theta_{k}+j_{k}x)},$ (1.65)

so that $U_{\zeta}$ in (LABEL:intro_change_val) can be written as

$\displaystyle U_{\zeta}(\theta,y,z)=\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y)+\varepsilon^{b}z=:\varepsilon{v_{\varepsilon}}(\theta,y)+\varepsilon^{b}z.$
(1.66)

Let us consider a homogeneous expansion of $H$ defined by (1.49):

$\displaystyle H(f)=H_{2}(f)+H_{3}(f)+H_{4}(f)+H_{5}(f)+H_{\geq 6}(f),$ (1.67)

where $H_{i}$ is homogeneous of degree $i$ for $i=1,\ldots 5$, and $H_{\geq
6}$ collects all the terms that are homogenous of degree at least $6$. Also,
for each $H_{i}$ and $0\leq k\leq i$, we denote by $H_{i,k}$ the term in
$H_{i}$ that is homogeneous of degree $k$ in the variable $z$ . For example,
recalling the Airy equation in (1.35), we can write

$\displaystyle\mathcal{G}_{2}(f)$
$\displaystyle\overset{\eqref{g4_intro_1}}{=}\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}f)^{2}(x)dx=\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}v)^{2}(x)dx+\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}z)^{2}(x)dx=\mathcal{G}_{2,0}(f)+\mathcal{G}_{2,2}(f).$

The reason of introducing the above notation is to see the contribution of $v$
and $z$ in each homogeneous term $H_{i}$ separately.

Using the above notation, we can rearrange the expansion in (1.67) as

$\displaystyle H(f)$
$\displaystyle=\underbrace{(H_{2,0}+H_{4,0})}_{=:Z_{0}(f)}+\underbrace{(H_{2,1}+H_{3,0}+H_{3,1}+H_{4,1}+H_{5,0}+H_{5,1})}_{=:Z_{1}(f)}+\underbrace{(H_{2,2}+H_{3,2}+H_{4,2})}_{=:Z_{2}(f)}$
(1.68) $\displaystyle\
+\underbrace{(H_{3,3}+H_{4,3}+H_{4,4}+\sum_{k=2}^{5}H_{5,k}+H_{\geq
6})}_{=:Z_{3}(f)}.$

Using the change of variables from (1.64) and (1.65), and splitting
$Z_{2}(f)=Z_{2}(v_{\zeta}(\theta,y)+z)$ into
$Z_{2}(v_{\zeta}(\theta,0)+z)+(Z_{2}(v_{\zeta}(\theta,y)+z)-Z_{2}(v_{\zeta}(\theta,0)+z))$,
we can rewrite the above as

$\displaystyle H(v_{\zeta}(\theta,y)+z)$
$\displaystyle=Z_{0}(v_{\zeta}(\theta,y))+\mathcal{N}(\theta)(z,z)+\tilde{P}(v_{\zeta}(\theta,y)+z),$
(1.69)

where

$\displaystyle\mathcal{N}(\theta)(z,z)$
$\displaystyle:=Z_{2}(v_{\zeta}(\theta,0)+z),$ (1.70)
$\displaystyle\tilde{P}(v_{\zeta}(\theta,y)+z)$
$\displaystyle:=Z_{1}(v_{\zeta}(\theta,y)+z)+(Z_{2}(v_{\zeta}(\theta,y)+z)-Z_{2}(v_{\zeta}(\theta,0)+z))+Z_{3}(v_{\zeta}(\theta,y)+z).$

Therefore, using (1.66), and the homogeneity of each $H_{i,k}$ in (1.68), we
obtain

$\displaystyle H_{\zeta}(\theta,y,z)$
$\displaystyle\overset{\eqref{newHamiltonian_int11ro}}{=}\varepsilon^{-2b}H(U_{\zeta}(\theta,y,z))$
$\displaystyle\overset{\eqref{Uzeta_rewrite_intro}}{=}\varepsilon^{-2b}H(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y)+\varepsilon^{b}z)$
$\displaystyle\overset{\eqref{invandz_intro1}}{=}\varepsilon^{-2b}Z_{0}(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y))+\varepsilon^{-2b}\mathcal{N}(\theta)(\varepsilon^{b}z,\varepsilon^{b}z)+\varepsilon^{-2b}\tilde{P}(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y)+\varepsilon^{b}z)$
$\displaystyle=\varepsilon^{-2b}Z_{0}(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y))+\mathcal{N}(\theta)(z,z)+\varepsilon^{-2b}\tilde{P}(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y)+\varepsilon^{b}z),$ (1.71)

where the last equality follows from the homogeneity of
$z\mapsto\mathcal{N}(\theta)(z,z)$, which can be seen from its definition in
(1.70) and the definition of $Z_{2}$ in (1.68) noticing that $H_{2,2},H_{3,2}$
and $H_{4,2}$ are homogeneous of degree $2$ in the normal variable $z$.
Comparing the above formula to (1.50), which we aim to obtain, we now see that
in order to have a normal form as in (1.50), we need to choose $\Phi^{WB}$ in
(1.49) so that

$\displaystyle\varepsilon^{-2b}Z_{0}(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)}y))$
$\displaystyle=C_{\varepsilon,\zeta}+2\pi\omega^{\text{gSQG}}(\zeta)\cdot
y+6\varepsilon^{2b}\mathbb{A}y\cdot y.$ (1.72)

Once $\Phi^{WB}$ is chosen, the perturbation $P$ in (1.50) is given by

$\displaystyle P(\theta,y,z)=\varepsilon^{-2b}\tilde{P}(\varepsilon
v_{\zeta}(\theta,\varepsilon^{2(b-1)})+\varepsilon^{b}z),$ (1.73)

which can be seen from the comparison between ($\bullet$ ‣ 1.2.1.4) and
(1.50). In our proof, we will construct $\Phi^{WB}$ so that (see Proposition
4.4):

    1. (WB1)

(1.72) holds. Note that $Z_{0}$, that is $H_{2,0}+H_{4,0}$ (see (1.68)), is
restricted to the tangential space $H_{S}$. As we saw in the example of the
Airy equation (1.35), whether (1.72) holds or not depends on the non-existence
of nontrivial resonances (see assumption (A3) above the equation (1.39)).

    2. (WB2)

$Z_{1}(f)\equiv 0$ in (1.68), therefore, (1.68) reduces to

$\displaystyle
H=(H_{2,0}+H_{4,0})+(H_{2,2}+H_{3,2}+H_{4,2})+(H_{3,3}+H_{4,3}+H_{4,4}+\sum_{k=2}^{5}H_{5,k}+H_{\geq
6}).$ (1.74)

We do not write the exact formula here, but compare the structure above to the
formulae in Proposition 4.4. From (1.73), we see that the higher the degree of
homogeneity of $\tilde{P}$ is, the smaller $P$ becomes, thanks to the
rescaling factor $\varepsilon$. Compared to $Z_{3}$, $Z_{1}$ has the low
homogeneous contribution to $\tilde{P}$ in (1.70) and the elimination of such
low homogeneous contribution yields a sufficient smallness of $P_{\zeta}$ to
perform the iteration (1.61) (See Lemma 6.2).

In regards to (WB2), indeed, one can formally check whether the perturbative
term $P$, defined in (1.73) and (1.70) assuming $Z_{1}\equiv 0$, is small
enough. To see this, counting the degree of homogeneity of each term in
(1.68), one can find formally (see (6.16) for the precise estimates) that:

$\displaystyle|\mathcal{F}(i_{triv})|$ $\displaystyle=O(\varepsilon^{6-2b}),$
(1.75)
$\displaystyle|d_{i}^{2}\mathcal{F}_{\omega}(i_{triv})[\hat{\textbf{\i}},\hat{\textbf{\i}}]|$
$\displaystyle=O(\varepsilon|\hat{\textbf{\i}}|^{2}).$ (1.76)

Furthermore, for $\omega\in\Omega_{0}$ in (1.62), the best possible estimate
we can expect for $(D_{i}\mathcal{F}_{\omega}(i))^{-1}$ would be (similar to
(1.27))

$\displaystyle|d_{i}\mathcal{F}_{\omega}(i))^{-1}[\hat{\textbf{\i}}]|=O(\gamma^{-1}|\hat{\textbf{\i}}|).$
(1.77)

Therefore, the first iteration in (1.61) formally gives us that

$\displaystyle|i_{1}-i_{0}|\overset{\eqref{refined_sequence_intro2}}{=}|d_{i}\mathcal{F}_{\omega}(i_{triv}))^{-1}[\mathcal{F}_{\omega}(i_{triv})]|\overset{\eqref{formal_inverse_1intro}}{\lesssim}\gamma^{-1}|\mathcal{F}(i_{triv})|\overset{\eqref{formal_estimate_1intro1}}{\lesssim}\varepsilon^{6-2b}\gamma^{-1},$
(1.78)

while

$\displaystyle|\mathcal{F}_{\omega}(i_{1})|$
$\displaystyle\leq|\mathcal{F}_{\omega}(i_{1})-\mathcal{F}_{\omega}(i_{0})-d_{i}\mathcal{F}_{\omega}(i_{0})[i_{1}-i_{0}]|+|\mathcal{F}_{\omega}(i_{0})+d_{i}\mathcal{F}_{\omega}(i_{0})[i_{1}-i_{0}]|$
$\displaystyle\overset{\eqref{refined_sequence_intro2}}{=}|\mathcal{F}_{\omega}(i_{1})-\mathcal{F}_{\omega}(i_{0})-d_{i}\mathcal{F}_{\omega}(i_{0})[i_{1}-i_{0}]|+\underbrace{|\mathcal{F}_{\omega}(i_{0})-d_{i}\mathcal{F}_{\omega}(i_{0})[(d_{i}\mathcal{F}_{\omega}(i_{0}))^{-1}[\mathcal{F}_{\omega}(i_{0})]]}_{=0}$
$\displaystyle\overset{\eqref{formal_estimate_1intro2}}{\lesssim}\varepsilon|i_{1}-i_{0}|^{2}\overset{\eqref{first_intro_large11}}{\lesssim}\varepsilon(\varepsilon^{6-2b}\gamma^{-1})^{2}\overset{\eqref{gamma_impose_intro1}}{=}\varepsilon^{13-8b}.$
(1.79)

Since $\varepsilon^{13-8b}\leq\varepsilon^{6-2b}$ for $b\in(1,1+1/6)$ (see
(LABEL:intro_change_val) for the range of $b$), we see that the bound of
$\mathcal{F}_{\omega}(i_{1})$ in ($\bullet$ ‣ 1.2.1.4) has indeed improved
compared to $\mathcal{F}_{\omega}(i_{triv})$ in (1.75). This formal
computation can be thought of as evidence that $P$ in (1.73) is small enough
for the iteration to close. Furthermore, from the usual super-exponential
convergence rate of the Nash-Moser scheme, we expect that each $i_{n}$ stays
close to the trivial embedding,

$\displaystyle|\mathfrak{I}_{n}|=O(\varepsilon^{6-2b}\gamma^{-1})\text{ and
}|\mathcal{F}_{\omega}(i_{n})|=O(\varepsilon^{6-2b}),\text{ for each $n\geq
0$},\text{ where }\mathfrak{I}_{n}:=i_{n}-i_{triv}.$ (1.80)

Naturally, the above computations are not rigorous at all. However in our
proof, we will see that (1.80) actually holds true (see (11.59)). Therefore,
in the further analysis of the invertibility of the linearized operator
$\hat{\textbf{\i}}\mapsto d_{i}\mathcal{F}(i)[\hat{\textbf{\i}}]$, which we
will discuss in more details later, we will focus on the embeddings $i$ which
are close enough to the trivial embedding $i_{triv}$.

Lastly, we note that such a transformation $\Phi^{WB}$ will be constructed (in
Section 4) by means of time-$1$ flow maps of auxiliary Hamiltonian equations,
which guarantee that the resulting transformation is symplectic. We also note
that such a procedure only modifies a finite number of Fourier modes, which
can be deduced from the fact that $Z_{0}$ and $Z_{1}$ in (1.68) involve at
most one normal variable. Therefore, the flow maps that will be used to
construct $\Phi^{WB}$ can be constructed safely in terms of the well-posedness
of the auxiliary Hamiltonian equations.

#####  1.2.1.5 Summary of part 1.

In the above discussion, we explored how to extract an internal parameter
yielding a parameter-dependent Hamiltonian $H_{\zeta}$ in (1.58) with
sufficiently small perturbative term $P$. Afterwards, we transformed the
search for a quasiperiodic solution to the search for a zero of the functional
equation in (1.60), which can be solved by means of the iterative scheme in
(1.61).

####  1.2.2 Sketch of the proof, Part 2: Analysis of the linearized operator

In the second part of the sketch of the proof, we discuss in more details how
to “invert” the linearized operator arising in the iterative scheme (1.61):

$\displaystyle\hat{\textbf{\i}}\mapsto
d_{i}\mathcal{F}_{\omega}(i)[\hat{\textbf{\i}}]=(\omega\cdot\partial_{\varphi}-d_{i}X_{H_{\zeta}}(i))[\hat{\textbf{\i}}],$
(1.81)

for a fixed embedding
$i:\mathbb{T}^{\nu}\mapsto\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$. Denoting the embedding $i$ as

$\displaystyle i:\varphi\mapsto(\theta(\varphi),y(\varphi),z(\varphi)),$
(1.82)

we think of $\theta,y$ as smooth functions on $\mathbb{T}^{\nu}$ and $z$ as a
smooth function on $\mathbb{T}^{\nu}\times\mathbb{T}$, such that $z(\varphi)$
for each $\varphi\in\mathbb{T}^{\nu}$ is restricted to the normal space
$H_{S^{\perp}}$ (see (1.29) for $H_{S^{\perp}}$). Such functions will be
measured in the Sobolev spaces $H^{s}(\mathbb{T}^{\nu})\times
H^{s}(\mathbb{T}^{\nu})\times H^{s}(\mathbb{T}^{\nu+1})$. In the analysis of
the linearized operator $\hat{\textbf{\i}}\mapsto
d_{i}\mathcal{F}_{\omega}(i)[\hat{\textbf{\i}}]$, we will assume on $i$ that

$\displaystyle\rVert\mathfrak{I}\rVert^{\text{Lip}(\gamma,\Omega_{1})}_{s_{0}+\mu}\leq\mathtt{C}\varepsilon^{6-2b}\gamma^{-1},\text{
for some $\Omega_{1}\subset\Omega_{0}$, $s_{0},\mu,\mathtt{C}>0$},\text{ where
$\mathfrak{I}:=i-i_{triv}$.}$ (1.83)

Before we discuss the linearized operator, some comments to understand the
above assumption regarding the norm
$\rVert\cdot\rVert^{\text{Lip}(\gamma,\Omega_{1})}_{s_{0}+\mu}$, the set
$\Omega_{1}$, the constants $s_{0},\mu,\mathtt{C}>0$ and some motivation of
such an assumption will follow:

  1. $\bullet$

Constants arising in the proof. We first clarify some constants arising in the
proof. Let us denote

$\displaystyle\mathtt{p}:=\left\\{(\alpha,\nu,s_{0}):\alpha\in(1,2),\
2\leq\nu\in\mathbb{N},\ s_{0}\geq\frac{\nu+2}{2}\right\\}.$ (1.84)

The $3$-tuple $\mathtt{p}$ consists of the quantities that we can freely
choose. For instance, if we want to find a quasiperiodic solution to the gSQG
equation for $\alpha=3/2$ that can be represented by a $3$-dimensional
embedded torus $i_{\infty}$ (see Definition 1.1) satisfying the Sobolev
regularity

$\rVert i_{\infty}\rVert_{H^{10}(\mathbb{T}^{\nu})\times
H^{10}(\mathbb{T}^{\nu})\times H^{10}(\mathbb{T}^{\nu+1})}<\infty,$

then we can just choose $\mathtt{p}=(3/2,3,10)$. The requirement of
$s_{0}\geq\frac{\nu+2}{2}$ is to guarantee that the corresponding Sobolev
space becomes an algebra. Once $\mathtt{p}$ is fixed, then our previous
discussion can be summarized as follows: We choose the tangential sites
$S^{+}$ as in (1.16) (thus the sets of modes $S$ and $S^{\perp}$ are fixed
accordingly) so that we can derive a weak Birkhoff normal form as in (1.50),
and we obtain an $\omega$-dependent functional equation (1.60) for
$\omega\in\Omega_{\varepsilon}$ in (1.56). Therefore, we are led to study the
convergence of the iteration (1.61). We will also consider the following
constants fixed throughout the proof:

$\displaystyle
b\in(1,1+1/12),\quad\mathtt{M}\in\mathbb{N},\quad\tau:=\frac{2}{\alpha-1}+\nu+2.$
(1.85)

    1. $\circ$

Constant $b\in(1,1+1/12)$: This appears in the introduction of the action-
angle variables in (LABEL:intro_change_val), where we already discussed why
$b$ needs to be larger than $1$. For some technical reasons, we will consider
$b$ strictly less than $1+1/12$. Note that the constant $b$ also appears in
the Diophantine condition to measure the non-resonance of frequency vectors
(see (1.62) and (1.63)).

    2. $\circ$

Constant $\mathtt{M}\in\mathbb{N}$: As stated in our main theorem (Theorem
1.2), we will find solutions that are invariant under a
$\frac{2\pi}{\mathtt{M}}$-translation, as in (1.14) (at the patch level, this
corresponds to the invariance under a $\frac{2\pi}{\mathtt{M}}$\- rotation).
For this purpose, we will choose $S^{+}$ in (1.16) to be multiples of
$\mathtt{M}$ (see Remark 1.6 and Proposition 5.5). We note that $\mathtt{M}$
does play a crucial role in the convergence of the iteration scheme (its role
mainly lies in the rigorous analysis of the non-resonance conditions, see
Appendix B).

    3. $\circ$

Constant $\tau$: The constant $\tau$ arises in non-resonance conditions, for
example in (1.62). The motivation is that $\tau$ needs to be large enough to
obtain the measure estimate for the non-resonance frequencies (See Proposition
11.7).

  2. $\bullet$

Size assumption (1.83) on the embedding $i$. Firstly, for the definition of
the norm $\rVert\cdot\rVert_{s}^{\text{Lip}(\gamma,\Omega)}$, we refer to
(2.4) and (2.7). To understand the assumption (1.83), note that even though
the iteration (1.61) starts with $i_{triv}$, which is independent of the
parameter $\omega$, all the other approximate solutions $i_{n}$ for $n>0$ will
implicitly depend on $\omega$ since the functional $\mathcal{F}_{\omega}$ and
the linearized operator $\hat{\textbf{\i}}\mapsto
D_{i}\mathcal{F}_{\omega}(i)[\hat{\textbf{\i}}]$ depend on $\omega$. Hence,
the dependence of the embeddings on $\omega$ must be considered as well, and
we will measure it in a Lipschitz way (2.7). Also, as we noted, we do not
expect that the iteration (1.61) runs for every $\omega$. Therefore, at each
iteration, we will exclude some resonant frequencies for which the next
approximate solution may not be well-defined. Indeed, in our proof, we will
have a sequence of nested sets
$\mathcal{G}_{n+1}\subset\mathcal{G}_{n}\subset\Omega_{0}$ such that $n$-th
approximate solution $i_{n}$ is well-defined for $\omega\in\mathcal{G}_{n}$
(see item (2) of Proposition 11.1). Therefore, to analyze the linearized
operator at a given embedding $i$, we assume that $i$ is defined for
$\omega\in\Omega_{1}$ for some set $\Omega_{1}\subset\Omega_{0}$.

Another motivation that we need to keep in mind is that the approximate
solutions in our iteration are expected to stay close to the trivial
embedding, in view of our formal computations in (1.80). For this reason, we
introduce the constant $\mathtt{C}$ in the assumption (1.83), and presumably,
we expect such a constant to depend on only $\mathtt{p},{S^{+}}$ and the
constant $b$ (in (1.84), (1.16) and (1.85) respectively), since these
quantities completely determine the functional equation (1.60) (see (11.59)).

Lastly, $\mu$ in (1.83) denotes an additional necessary regularity to obtain
numerous estimates arising in the analysis of the linearized operator. Roughly
speaking, in order to obtain the invertibility or desired estimates in the
analysis of the linearized operator at $i$, we need the embedding $i$ to be
more regular than $H^{s_{0}}$, in which we wish to see a convergence of the
approximate solutions. We emphasize that as long as such an additional
regularity $\mu$ depends on only $\mathtt{p}$ in (1.84), which is fixed from
the beginning, the usual Nash-Moser scheme tells us that the sequence of
approximate solutions in the iteration can be constructed in such a way that
they converge in $H^{s_{0}}$, diverge in a finer space ($H^{\mathtt{S}}$ for
some very large $\mathtt{S}\gg s_{0}$), while stay small in the intermediate
space $H^{s_{0}+\mu}$ (interpolating the low-high norms. See item (2) of
Proposition 11.1). Thus our analysis (especially Section 7-10) should read as
“there exists $\mu(\mathtt{p})>0$ such that if (1.83) holds for
$\mu=\mu(\mathtt{p})$, then all the necessary properties of the linearized
operator hold”. Afterwards, we will see every approximate solution satisfies
such an assumption. See (11.59). The upper bound of $\mu$ is computable in
principle, but we do not do so to avoid additional complexity of the proof.
Instead, we will only focus on the non-dependence of $\mu$ on the parameters
other than $\mathtt{p}$. We will always assume $\mu$ to be as large as we
want, as long as it depends only on $\mathtt{p}$.

#####  1.2.2.1 Approximate inverse.

In practice, we will not look for the actual inverse,
$(d_{i}\mathcal{F}_{\omega}(i))^{-1}$. Instead, we will look for an
_approximate inverse_ , $T(i)$, such that

$\displaystyle|(d_{i}\mathcal{F}_{\omega}(i)\circ
T(i)-I)[g]|=O(|\mathcal{F}_{\omega}(i)|),\text{ where $I$ is the identity
operator}.$ (1.86)

(See (7.3) for the actual estimate we obtain in our proof). The estimate in
(7.3) is more complicated due to the consideration of the dependence on
$\omega$ and the contribution of the high modes when $s\gg s_{0}$, which is
split into the ones from from $i_{0},Z:=\mathcal{F}_{\omega}(i_{0})$ and $g$).
Although the operator $T(i)$ above is not the actual inverse, we expect that
as the approximate solutions in (1.61) approach to the solution $i_{\infty}$
(that is $\mathcal{F}_{\omega}(i_{n})$ gets smaller in the iteration), the
error becomes smaller and smaller, therefore eventually, $T(i_{\infty})$ is
indeed equal to the actual inverse. Thus, our iteration scheme (1.61) is now
replaced by

$\displaystyle i_{0}:=i_{triv},\quad
i_{n+1}:=i_{n}-T(i_{n})[\mathcal{F}_{\omega}(i_{n})].$ (1.87)

The error of size $O(|\mathcal{F}_{\omega}(i)|)$ in (1.86) can be managed in
the usual Nash-Moser scheme. We also note that the operator $T(i)$ depends on
$\omega$ as well, since so do $\mathcal{F}_{\omega}$ and $i$. However, we omit
the notation of such dependence to avoid notational complexity.

Now the question is how we can find such an approximate inverse $T(i)$ to run
the iteration (1.87). To this end, we follow the theory of Berti and Bolle
developed in [12], which tells us that an approximate inverse of
$\hat{\textbf{\i}}\mapsto d_{i}\mathcal{F}_{\omega}(i)[\hat{\textbf{\i}}]$ can
be obtained by an approximate inverse in the normal direction. In order to see
this sentence in more detail, let us suppose $i_{\infty}$ is a solution,
$\mathcal{F}_{\omega}(i_{\infty})=0$, and investigate what the linearized
operator $d_{i}\mathcal{F}_{\omega}(i_{\infty})$ looks like, because we can
expect that the different nature between $d_{i}\mathcal{F}_{\omega}(i)$ and
$d_{i}\mathcal{F}_{\omega}(i_{\infty})$ can be measured quantitatively by
$|\mathcal{F}_{\omega}(i)|$, which is admissible in view of (1.86).

The key implication in [12] is the derivation of a normal form near invariant
tori. Indeed [12, Theorem 1] tells us that (roughly speaking) if

$\displaystyle\text{$i_{\infty}$ solves
$\mathcal{F}_{\omega}(i_{\infty})=0$},$ (1.88)

there exists a symplectic diffeomorphism
$G:\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}\mapsto\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$ such that $G(i_{triv}(\varphi))=i_{\infty}(\varphi)$. Moreover
the Hamiltonian $K_{\infty}$, defined as

$\displaystyle K_{\infty}(\psi,\eta,w):=H_{\zeta}\circ G(\psi,\eta,w),\text{
for $(\psi,\eta,w)\in\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$},$ (1.89)

possesses a normal form such that

$\displaystyle K_{\infty}(\psi,\eta,w)$ $\displaystyle=C+2\pi\omega\cdot\eta$
(1.90) $\displaystyle\
+\frac{1}{2}\partial_{\eta}^{2}(K_{\infty})(\psi,0,0)[\eta,\eta]+\frac{1}{2}\nabla_{w}^{2}(K_{\infty})(\psi,0,0)[w,w]+\partial_{\eta}\nabla_{w}(K_{\infty})(\psi,0,0)[\eta,w]$
$\displaystyle\ +O_{3}(\eta,w),$

where $\nabla_{w}^{2}$ denotes the Hessian operator in the variable $w$ and
$O_{3}(\eta,w)$ collects all the terms that are homogeneous of degree at least
$3$. This result implies that the linearized Hamiltonian vector field of
$K_{\infty}$ on the torus

$\left\\{(\psi,0,0)\in\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}:\psi\in\mathbb{T}^{\nu}\right\\}$

can be easily computed from (1.90) (since such a linearized vector field does
not depend on the cubic contribution of $\eta,w$). With simple computations,
one can obtain that (in the direction
$\hat{\textbf{\i}}:=(\hat{\psi},\hat{\eta},\hat{w})$)

$\displaystyle d_{i}X_{K_{\infty}}(i_{triv}(\psi))[\hat{\textbf{\i}}]$
$\displaystyle:=\frac{d}{dt}\begin{pmatrix}\frac{1}{2\pi}\partial_{\eta}K_{\infty}(i_{triv}(\psi)+t\hat{\textbf{\i}})\\\
-\frac{1}{2\pi}\partial_{\psi}K_{\infty}(i_{triv}(\psi)+t\hat{\textbf{\i}})\\\
\partial_{x}\left(\nabla_{w}K_{\infty}(i_{triv}(\psi)+t\hat{\textbf{\i}})\right)\end{pmatrix}\bigg{|}_{t=0}$
$\displaystyle=\begin{pmatrix}\frac{1}{2\pi}\left(\partial_{\eta}^{2}(K_{\infty})(i_{triv}(\psi))[\hat{\eta}]+(\partial_{\eta}(\nabla_{w}K_{\infty})(i_{triv}(\psi)))^{T}[\hat{w}]\right)\\\
0\\\
\partial_{x}(\nabla_{w}^{2}(K_{\infty})(i_{triv}(\psi))[\hat{w}])+\partial_{\eta}(\nabla_{w}K_{\infty})(i_{triv}(\psi))[\hat{\eta}]\end{pmatrix},$
(1.91)

where $(\partial_{\eta}(\nabla_{w}K_{\infty})(i_{triv}(\psi)))^{T}$ denotes
the linear map $H_{S^{\perp}}\mapsto\mathbb{R}^{\nu}$ such that for all
$\hat{w}\in H_{S^{\perp}},\hat{\eta}\in\mathbb{R}^{\nu}$,

$(\partial_{\eta}(\nabla_{w}K_{\infty})(i_{triv}(\psi)))^{T}[\hat{w}]\cdot\hat{\eta}=\frac{d}{dt}(\hat{w},\nabla_{w}K_{\infty}(\psi,t\hat{\eta},0))_{L^{2}(\mathbb{T})}|_{t=0}.$

Therefore, the linearization of
$i\mapsto\omega\cdot\partial_{\varphi}i-X_{K_{\infty}}(i)$ at $i_{triv}$ is
given by

$\displaystyle\mathbb{D}_{\infty}[\hat{\textbf{\i}}]:=\begin{pmatrix}\omega\cdot\partial_{\varphi}\hat{\psi}(\varphi)\\\
\omega\cdot\partial_{\varphi}\hat{\eta}(\varphi)\\\
\omega\cdot\partial_{\varphi}\hat{w}(\varphi)\end{pmatrix}-\begin{pmatrix}\frac{1}{2\pi}\left(\partial_{\eta}^{2}(K_{\infty})(i_{triv}(\varphi))[\hat{\eta}]+(\partial_{\eta}\nabla(K_{\infty})(i_{triv}(\varphi)))^{T}[\hat{w}]\right)\\\
0\\\
\partial_{x}(\nabla_{w}^{2}(K_{\infty})(i_{triv}(\varphi))[\hat{w}])+\partial_{\eta}(\nabla_{w}K_{\infty})(i_{triv}(\varphi))[\hat{\eta}]\end{pmatrix}.$
(1.92)

As one might have already noticed from (1.89), the linearized operator,
$\hat{\textbf{\i}}\mapsto\mathbb{D}_{\infty}[\hat{\textbf{\i}}]$, is indeed
the “pullback” of the linearized operator of $i\mapsto\mathcal{F}_{\omega}(i)$
at $i_{\infty}$. More precisely, one can obtain (we omit the computations here
but one can easily follow the proof of Proposition 7.3)

$\displaystyle
d_{i}\mathcal{F}_{\omega}(i_{\infty})[\hat{\textbf{\i}}]=DG(i_{triv})\circ\mathbb{D}_{\infty}\circ(DG(i_{triv}))^{-1}[\hat{\textbf{\i}}].$
(1.93)

Therefore, the invertibility of $\hat{\textbf{\i}}\mapsto
d_{i}\mathcal{F}_{\omega}(i_{\infty})[\hat{\textbf{\i}}]$ reduces to the
invertibility of
$\hat{\textbf{\i}}\mapsto\mathbb{D}_{\infty}[\hat{\textbf{\i}}]$.

From (1.87), recall that we need to find an approximate inverse at each
$i_{n}$, which is not necessarily a solution as in (1.88). However the
consequence of [12] even tells us that given $i$, which is not necessarily a
solution, there exists a symplectic diffeomorphism
$G_{\delta}:\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}\mapsto\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$ such that defining

$\displaystyle K(\psi,\eta,w)$ $\displaystyle:=H_{\zeta}\circ
G_{\delta}(\psi,\eta,w),$ (1.94) $\displaystyle\mathbb{D}[\hat{\textbf{\i}}]$
$\displaystyle:=\begin{pmatrix}\omega\cdot\partial_{\varphi}\hat{\psi}(\varphi)\\\
\omega\cdot\partial_{\varphi}\hat{\eta}(\varphi)\\\
\omega\cdot\partial_{\varphi}\hat{w}(\varphi)\end{pmatrix}-\begin{pmatrix}\frac{1}{2\pi}\left(\partial_{\eta}^{2}(K)(i_{triv}(\varphi))[\hat{\eta}]+(\partial_{\eta}(\nabla_{w}K)(i_{triv}(\varphi)))^{T}[\hat{w}]\right)\\\
0\\\
\partial_{x}(\nabla_{w}^{2}(K)(i_{triv}(\varphi))[\hat{w}])+\partial_{\eta}(\nabla_{w}K)(i_{triv}(\varphi))[\hat{\eta}]\end{pmatrix},$
(1.95)

it holds that (compare below with (1.93))

$\displaystyle
d_{i}\mathcal{F}_{\omega}(i)=DG_{\delta}(i_{triv})\circ\mathbb{D}\circ(DG_{\delta}(i_{triv}))^{-1}+O(|\mathcal{F}_{\omega}(i)|).$
(1.96)

(See (7.58) and (7) for the precise result of (1.96) in our proof). Hence, if
$\hat{\textbf{\i}}\mapsto\mathbb{D}[\hat{\textbf{\i}}]$ is invertible, then it
is expected to be an approximate inverse, since the difference from the actual
inverse can be quantitatively estimated by the size of
$\mathcal{F}_{\omega}(i)$. This leads us to study the invertibility of
$\mathbb{D}$.

Now, let us see how to achieve the invertibility of the linear operator
$\mathbb{D}$ defined in (1.95). To simplify the notation, we denote
$K_{ij}(\varphi):=(\partial_{\eta})^{i}(\nabla_{w})^{j}K(\varphi,0,0)$, that
is,

$\displaystyle\partial_{\eta}^{2}K(i_{triv}(\varphi))=:K_{20}(\varphi),\quad\partial_{\eta}(\nabla_{w}K)(i_{triv}(\varphi))=:K_{11}(\varphi),\quad\nabla_{w}^{2}K(i_{triv}(\varphi))=:K_{02}(\varphi).$
(1.97)

Recalling that $K$ maps $\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$ to $\mathbb{R}\cup\left\\{\infty\right\\}$, we can think of
$K_{20}(\varphi)$ as a $\varphi$-dependent $\nu\times\nu$ symmetric matrix,
$K_{11}(\varphi)$ as a $\varphi$-dependent linear operator from
$\mathbb{R}^{\nu}$ to $H_{S^{\perp}}$, and $K_{02}(\varphi)$ as a
$\varphi$-dependent linear operator from $H_{S^{\perp}}$ to $H_{S^{\perp}}$.
Therefore, given $g$, denoted as

$\displaystyle\mathbb{T}^{\nu}\ni\varphi\mapsto
g(\varphi)=(g_{1}(\varphi),g_{2}(\varphi),g_{3}(\varphi))\in\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}},$

we aim to find
$\hat{\textbf{\i}}(\varphi)=(\hat{\psi}(\varphi),\hat{\eta}(\varphi),\hat{w}(\varphi))$
such that $\mathbb{D}[\hat{\textbf{\i}}]=g$, that is,

$\displaystyle\omega\cdot\partial_{\varphi}\hat{\psi}-\frac{1}{2\pi}\left(K_{20}[\hat{\eta}]+K_{11}^{T}[\hat{w}]\right)$
$\displaystyle=g_{1},$ (1.98a)
$\displaystyle\omega\cdot\partial_{\varphi}\hat{\eta}$ $\displaystyle=g_{2},$
(1.98b)
$\displaystyle\omega\cdot\partial_{\varphi}\hat{w}-\partial_{x}(K_{02}[\hat{w}])-K_{11}[\hat{\eta}]$
$\displaystyle=g_{3}.$ (1.98c)

We start with the simplest equation, the second one. One necessary condition
for (1.98b) to have a solution $\hat{\eta}$ is that $g_{2}$ has zero-average
in $\varphi\in\mathbb{T}^{\nu}$, which can be easily seen from the elementary
Stokes theorem. In order to overcome this issue, the authors in [8, 12]
introduced an extra factor in the Hamiltonian. In our proof, we restrict the
search for an invariant embedding to the search for an invariant _reversible
embedding_ which ensures that we only need to solve (1.98b) for $g_{2}$ such
that $g_{2}(\varphi)=-g_{2}(\varphi)$ (In the statement of Proposition 7.3, we
consider the approximate inverse $T$ as a map restricted to the space $Y_{i}$,
which is defined in (2.138)). In such a case, the average of $g_{2}$ is always
zero. Therefore (1.98b) has a solution when $\omega$ is Diophantine. More
precisely, if $\omega\in\Omega_{0}$ (see (1.62)) and
$g_{2}=\sum_{l\in\mathbb{Z}^{\nu}\backslash\left\\{0\right\\}}\widehat{g_{2}}(l)e^{\text{i}l\cdot\varphi}$,
then $\hat{\eta}$ is determined by

$\displaystyle\hat{\eta}(\varphi)=\langle\hat{\eta}\rangle+\sum_{l\in\mathbb{Z}^{\nu}\backslash\left\\{0\right\\}}\frac{\widehat{g_{2}}(l)}{\omega\cdot
l}e^{\text{i}l\cdot\varphi}=:\langle\hat{\eta}\rangle+(\omega\cdot\partial_{\varphi})^{-1}g_{2},$
(1.99)

where $\langle\hat{\eta}\rangle$ is a constant that does not depend on
$\varphi$, which can be freely chosen in view of (1.98b). Now we move to the
third equation (1.98c) and let us denote

$\displaystyle\mathcal{L}_{\omega}:=\omega\cdot\partial_{\varphi}-\partial_{x}K_{02},$
(1.100)

so that we are led to solve
$\mathcal{L}_{\omega}[\hat{w}]=g_{3}+K_{11}[\hat{\eta}]$. Unlike the first two
equations (1.98a) and (1.98b), the invertibility of the operator
$\mathcal{L}_{\omega}$ is not simply guaranteed by imposing the Diophantine
condition on $\omega$, and its solvability is much more complicated, since
$H_{S^{\perp}}$ is an infinite dimensional space. The invertibility of
$\mathcal{L}_{\omega}$ is our main goal in Section 8-10 and we will assume for
the moment that (see Proposition 7.14 for the precise statement with necessary
estimates)

  1. $\star$

Invertibility in the normal direction: There exists a set of frequencies
$\Omega_{\infty}(i)\subset\Omega_{0}$ such that for
$\omega\in\Omega_{\infty}(i)$, the operator $\mathcal{L}_{\omega}$ is
invertible.

With the above assumption, we can write a solution $\hat{w}$ in (1.98c) as

$\displaystyle\hat{w}=\mathcal{L}_{\omega}^{-1}(g_{3}+K_{11}[\hat{\eta}]).$
(1.101)

Lastly, having $\hat{\eta}$ and $\hat{w}$ as in (1.99) and (1.101), the first
equation (1.98a) can be solved as

$\displaystyle\hat{\psi}=(\omega\cdot\partial_{\varphi})^{-1}\left(g_{1}+\frac{1}{2\pi}\left(K_{20}[\hat{\eta}]+K_{11}^{T}[\hat{w}]\right)\right),$
(1.102)

where $\langle\hat{\eta}\rangle$ in (1.99) must be chosen to guarantee that
the $g_{1}+\frac{1}{2\pi}\left(K_{20}[\hat{\eta}]+K_{11}^{T}[\hat{w}]\right)$
has zero-average in $\varphi$. Such a choice of
$(\hat{\psi},\hat{\eta},\hat{w})$ is a solution to (1.98).

The summary of the above discussion is that the search for an approximate
inverse leads us to the investigation of the invertibility of
$\mathcal{L}_{\omega}$ in (1.100), which is restricted to the normal
direction. In view of the invertibility assumption, we will find for each
$i_{n}$ in the sequence (1.87), a nested sequence of frequency sets
$\mathcal{G}_{n+1}\subset\mathcal{G}_{n}\subset\Omega_{\infty}(i_{n})\subset\Omega_{0}$
such that for $\omega\in\mathcal{G}_{n+1}$, the operator $T(i_{n})$ is well-
defined, therefore so is $i_{n+1}$ (over $\omega\in\mathcal{G}_{n+1}$). This
implies that we can run the iteration for
$\omega\in\cap_{n=0}^{\infty}\mathcal{G}_{n}$ towards the solution
$i_{\infty}$, which will be rigorously proved in Proposition 11.1. In the
rest, we will focus on how to invert $\mathcal{L}_{\omega}$.

#####  1.2.2.2 Structure of $\mathcal{L}_{\omega}$ at a fixed embedding
$i_{0}$.

Let us fix an embedding $i_{0}$ so that

$\displaystyle
i_{0}(\varphi):=(\theta_{0}(\varphi),y_{0}(\varphi),z_{0}(\varphi)),\text{
satisfies the assumption \eqref{size_assumption_intro1} with
$\mathfrak{I}_{0}:=i_{0}-i_{triv}$,}$ (1.103)

and investigate the invertibility of $\mathcal{L}_{\omega}$ determined by
$i_{0}$ (we use a slight abuse of notation that from now on, $i_{0}$ is not
necessarily the trivial embedding). A very natural question at this point is
what the operator $\mathcal{L}_{\omega}$ does really look like. We started
from the Hamiltonian $\mathcal{H}$ in (1.6), which is already complicated but
we have composed it with multiple changes of variables. To understand the
operator $\mathcal{L}_{\omega}$, let us trace back the compositions. From
(1.49), (1.50) and (1.94), we see that the Hamiltonian $K$ is given by

$\displaystyle K(\psi,\eta,w)=\varepsilon^{-2b}\mathcal{H}\circ\Phi^{WB}\circ
U_{\zeta}\circ
G_{\delta}(\psi,\eta,w),\quad(\psi,\eta,w)\in\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}},$ (1.104)

where $\Phi^{WB}$ is a transformation designed to eliminate some homogeneous
contributions of the tangential variables (see (WB1) and (WB2) in Part 1),
$U_{\zeta}$ is the introduction of the action-angle variables as well as the
amplitude variable (see (LABEL:intro_change_val)), and $G_{\delta}$ is
introduced in the study of the approximate inverse of the linearized operator.
The linear operator $\mathcal{L}_{\omega}$ is defined as

$\displaystyle\mathcal{L}_{\omega}[\hat{w}]:=\omega\cdot\partial_{\varphi}\hat{w}-\partial_{x}(\nabla_{w}^{2}K(\varphi,0,0))[\hat{w}]),\text{
for $\hat{w}\in C^{\infty}(\mathbb{T}^{\nu}\times\mathbb{T})$ such that
$\hat{w}(\varphi)\in H_{S^{\perp}}$ for each $\varphi\in\mathbb{T}^{\nu}$,}$
(1.105)

which follows from (1.100) and (1.97). Notice that the operator
$\hat{w}\mapsto\nabla_{w}^{2}K(\varphi,0,0))[\hat{w}]$ also depends on the
frequency $\omega$ since $K$ in (1.104) depends on $\zeta$, while we have the
one-to-one correspondence between $\omega$ and $\zeta$ through (1.57).

The earlier discussion tells us that the existence of an approximate inverse
at $i_{0}$, $T(i_{0})$, is determined by the invertibility of the operator
$\mathcal{L}_{\omega}$. Then, where is the information of $i_{0}$ encoded in
the expression (1.105)? Indeed, the symplectic diffeomorphism $G_{\delta}$ in
(1.104) is determined by $i_{0}$, while we do not explicitly write it in this
sketch level discussion. We only note that the image of $i_{triv}$ under
$G_{\delta}$ corresponds to an _isotropic embedding_ $i_{\delta}$ induced from
$i_{0}$ (see (7.5), (7.17) and Lemma 7.2), for which only the “action”
coordinate differs from $i_{0}$, that is,

$\displaystyle
i_{\delta}(\varphi):=G_{\delta}(i_{triv}(\varphi))\overset{\eqref{iso_tropic}}{=}(\theta_{0}(\varphi),y_{\delta}(\varphi),z_{0}(\varphi)),$
(1.106)

and $i_{\delta}$ is also close to the trivial embedding (see (7.16));

$\displaystyle i_{\delta}(\varphi)\text{ satisfies the assumption
\eqref{size_assumption_intro1} with
$\mathfrak{I}_{\delta}:=i_{\delta}-i_{triv}$,}$ (1.107)

In view of (1.104) and (1.105), let us take a simple example to see what a
Hessian operator looks like on the embedded torus
$G_{\delta}(i_{triv})=i_{\delta}$.

  1. $\bullet$

A toy model 2: A perturbed Airy equation. Let us denote the orthogonal
projections to $H_{S}$ and $H_{S^{\perp}}$ by (see (1.29) and (1.17) for the
notations)

$\displaystyle\Pi_{S}f=\sum_{j\in
S}f_{j}e^{\text{i}jx},\quad\Pi_{S^{\perp}}f:=\sum_{j\in
S^{\perp}}f_{j}e^{\text{i}jx},\text{ for $f\in L^{2}(\mathbb{T})$ with
$\int_{\mathbb{T}}f(x)dx=0$.}$ (1.108)

We also denote the variables in $H_{S}$ and $H_{S^{\perp}}$ by $v$ and $z$
respectively, so that given $f$, we have a unique decomposition as

$\displaystyle f=v+z,\text{ where $v\in H_{S}$ and $z\in H_{S^{\perp}}$ for
$f\in L^{2}(\mathbb{T})$ with $\int_{\mathbb{T}}f(x)dx=0$.}$ (1.109)

Let us consider a Hamiltonian
$\mathcal{G}:L^{2}(\mathbb{T})\mapsto\mathbb{R}\cup\left\\{\infty\right\\}$,

$\displaystyle\mathcal{G}(f):=\mathcal{G}_{2,2}(f)+\mathcal{G}_{3,2}(f)+\mathcal{G}_{4,2}(f)+\mathcal{G}_{5,2}(f),$
(1.110)

where

$\displaystyle\mathcal{G}_{2,2}(f):=\frac{1}{2}\int_{\mathbb{T}}(\partial_{x}z)^{2}dx,\quad\mathcal{G}_{3,2}(f):=\frac{1}{2}\int_{\mathbb{T}}vz^{2}dx,$
(1.111)
$\displaystyle\mathcal{G}_{4,2}(f):=\frac{1}{2}\int_{\mathbb{T}}vz\Pi_{S}(vz)+v^{2}z^{2}dx,\quad\mathcal{G}_{5,2}(f):=\frac{1}{2}\int_{\mathbb{T}}v^{3}z^{2}dx.$

Notice that $\mathcal{G}_{i,k}$ is homogeneous of degree $i$ in the variable
$f$ and homogeneous of degree $k$ in the variable $z$, for a fixed $v$. The
motivation for the above example is that in our case, we constructed the
symplectic transformation $\Phi^{WB}$ in (1.49) so that
$H=\mathcal{H}\circ\Phi^{WB}$ reduces to the form of (1.74). We designed our
toy model (1.110) to include quadratic contributions of the normal variable
$z$. The Hessian operator in the normal direction of $\mathcal{G}$ at $f=v+z$
can be easily computed as

$\displaystyle\nabla_{z}^{2}\mathcal{G}(f)[\hat{z}]$
$\displaystyle=-\partial_{xx}\hat{z}+\Pi_{S^{\perp}}(v\hat{z})+\Pi_{S^{\perp}}(v\Pi_{S}(v\hat{z})+v^{2}\hat{z})+\Pi_{S^{\perp}}(v^{3}\hat{z})$
$\displaystyle=\Pi_{S^{\perp}}\left(-\partial_{xx}\hat{z}+(v+v^{2}+v^{3})\hat{z}\right)+\Pi_{S^{\perp}}(v\Pi_{S}(v\hat{z})).$

Therefore, defining

$\displaystyle\mathcal{G}_{\zeta}(\theta,y,z):=\varepsilon^{-2b}\mathcal{G}\circ
U_{\zeta}(\theta,y,z),$ (1.112)

and using the rescaled variables in (1.66), we obtain the Hessian operator of
$\mathcal{G}_{\zeta}(\theta,y,z)$ in the normal direction,

$\displaystyle\nabla_{z}^{2}\mathcal{G}_{\zeta}(\theta,y,z)[\hat{z}]=\Pi_{S^{\perp}}\left(-\partial_{xx}\hat{z}+(\varepsilon
v_{\varepsilon}(\theta,y)+\varepsilon^{2}v_{\varepsilon}(\theta,y)^{2}+\varepsilon^{3}v_{\varepsilon}(\theta,y)^{3})\hat{z}\right)+\varepsilon^{2}\Pi_{S^{\perp}}(v_{\varepsilon}(\theta,y)\Pi_{S}(v_{\varepsilon}(\theta,y)\hat{z}))$
(1.113)

On the embedded torus
$i_{\delta}(\mathbb{T}^{\nu})\subset\mathbb{T}^{\nu}\times\mathbb{R}^{\nu}\times
H_{S^{\perp}}$, we obtain a $\varphi$-dependent operator in $H_{S^{\perp}}$:

$\displaystyle\nabla_{z}^{2}\mathcal{G}_{\zeta}(i_{\delta}(\varphi))[\hat{z}]$
$\displaystyle\overset{\eqref{idlsotrophi_intro_3}}{=}\nabla_{z}^{2}\mathcal{G}_{\zeta}(\theta_{0}(\varphi),y_{\delta}(\varphi),z_{0}(\varphi))[\hat{z}]$
(1.114)
$\displaystyle\overset{\eqref{normal_driec1nont1}}{=}\Pi_{S^{\perp}}\left(a^{\text{Airy}}_{2}(\varphi,x)\partial_{xx}\hat{z}+a^{\text{Airy}}_{0}(\varphi,x)\hat{z}\right)+R^{\text{Airy}}(\varphi)[\hat{z}]=:\Pi_{S^{\perp}}M^{\text{Airy}}(\varphi)+R^{\text{Airy}}(\varphi)[\hat{z}],$

where the coefficients $a_{2},a_{0}$ are given by

$\displaystyle a^{\text{Airy}}_{2}(\varphi,x):=-1,\quad
a^{\text{Airy}}_{0}(\varphi,x):=\sum_{i=1}^{3}\varepsilon^{i}(v_{\varepsilon}(\theta_{0}(\varphi),y_{\delta}(\varphi)))^{i},$
(1.115)

and the operator $R^{\text{Airy}}(\varphi)$ is defined as

$\displaystyle
R^{\text{Airy}}(\varphi)[\hat{z}]:=\varepsilon^{2}\Pi_{S^{\perp}}(v_{\varepsilon}(\theta_{0}(\varphi),y_{\delta}(\varphi))\Pi_{S}(v_{\varepsilon}(\theta_{0}(\varphi),y_{\delta}(\varphi))\hat{z})).$
(1.116)

Note that for each fixed $\varphi$, the operator
$\nabla_{z}^{2}\mathcal{G}_{\zeta}(i_{\delta}(\varphi))$ consists of two types
of operators; the operator $\hat{z}\mapsto M^{\text{Airy}}(\varphi)[\hat{z}]$
is a pseudo differential operator of order $2$, and $\hat{z}\mapsto
R^{\text{Airy}}(\varphi)[\hat{z}]$ is a smoothing operator. Indeed, since
$v_{\varepsilon}(\theta_{0}(\varphi),y_{\delta}(\varphi))$ is supported on a
finite number of Fourier modes, $S$ (see (1.65) and (1.66)), only a finite
number of modes of $\hat{z}(\varphi)$ is involved in (1.116), therefore
$R^{\text{Airy}}(\varphi):H_{S^{\perp}}\mapsto H_{S^{\perp}}$ is a smoothing
operator.

We first analyze the coefficients $a_{2}^{\text{Airy}},a_{0}^{\text{Airy}}$,
and then study the operator $R^{\text{Airy}}$. From the definition of
$v_{\varepsilon}$ in (1.66) and (1.65), we have

$\displaystyle v_{\varepsilon}(\theta_{0}(\varphi),y_{\delta}(\varphi))(x)$
$\displaystyle=\sum_{j_{k}\in
S}\sqrt{|j_{k}|(\zeta_{k}+\varepsilon^{2(b-1)}(y_{\delta}(\varphi))_{k})}e^{\text{i}(\theta_{0}(\varphi)_{k}+j_{k}x)},$
(1.117)

while each summand can be split as

$\displaystyle\sqrt{|j_{k}|(\zeta_{k}+\varepsilon^{2(b-1)}(y_{\delta}(\varphi))_{k})}e^{\text{i}(\theta_{0}(\varphi)_{k}+j_{k}x)}=\underbrace{\sqrt{|j_{k}|\zeta_{k}}e^{\text{i}(\varphi\cdot\mathtt{l}(j_{k})+j_{k}x)}}_{=:\overline{v}_{j_{k}}(\varphi,x)}+O(|\mathfrak{I}_{\delta}|).$
(1.118)

where $\mathtt{l}(j_{k})$ is defind as

$\displaystyle\mathtt{l}(j_{k}):=\mathtt{e}_{k},\text{ and
}\mathtt{l}(-j_{k}):=-\mathtt{l}(j_{k})\text{ for $j_{k}\in
S^{+}=\left\\{j_{1},\ldots,j_{\nu}\right\\}$}.$ (1.119)

Writing

$\displaystyle
v_{\varepsilon}:=v_{\varepsilon}(\theta_{0}(\varphi),y_{\delta}(\varphi)),\quad\overline{v}:=\sum_{j_{k}\in
S}\overline{v}_{j_{k}}(\varphi,x),$ (1.120)

for simplicity, the decomposition (1.118) shows that the tangential variable
$v_{\varepsilon}$ can be decomposed into a sum of the $i_{\delta}$-independent
monomial ($\overline{v}_{j_{k}}$) and a small contribution of the size of the
embedding
$O(|\mathfrak{I}_{\delta}|)\overset{\eqref{isotropic_123intro231},\eqref{size_assumption_intro1}}{=}O(\varepsilon^{6-2b}\gamma^{-1})$.
Hence, from (1.115), we see that the coefficient $a_{0}^{\text{Airy}}$
consists of at most cubic interactions of the monomials $\overline{v}_{j_{k}}$
and the small contribution of size $O(\varepsilon|\mathfrak{I}_{0}|)$, that
is,

$\displaystyle
a_{0}^{\text{Airy}}(\varphi,x)=\sum_{i=1}^{3}\varepsilon^{i}\left(\sum_{j_{k_{1}},\ldots,j_{k_{i}}\in
S}\overline{v}_{j_{k_{1}}}(\varphi,x)\cdots\overline{v}_{j_{k_{i}}}(\varphi,x)\right)+O(\varepsilon|\mathfrak{I}_{\delta}|).$
(1.121)

Now, we consider the operator $R^{\text{Airy}}$. As above, we can decompose it
into

$\displaystyle R^{\text{Airy}}(\varphi)[\hat{z}]$
$\displaystyle=\varepsilon^{2}\Pi_{S^{\perp}}\left(\overline{v}\Pi_{S}(\overline{v}\hat{z})\right)+R_{2}^{\text{Airy}},$
(1.122) $\displaystyle R^{\text{Airy}}_{2}(\varphi)[\hat{z}]$
$\displaystyle:=\varepsilon^{2}\left(\Pi_{S^{\perp}}(v_{\varepsilon}-\overline{v})\Pi_{S}(v_{\varepsilon}\hat{z})\right)+\varepsilon^{2}\left(\overline{v}\Pi_{S}((v_{\varepsilon}-\overline{v})\hat{z})\right).$

This shows that $R^{\text{Airy}}$ can be also decomposed into a sum of an
operator that does not depend on the embedding $i_{\delta}$ and a operator
$R^{\text{Airy}}_{2}$ whose size is
$O(\varepsilon^{2}|v_{\varepsilon}-\overline{v}|)=O(\varepsilon^{2}|\mathfrak{I}_{\delta}|)$,
which is much smaller than $O(\varepsilon^{2})$.

The summary of this example is that while the Hessian operator of
$\mathcal{G}$ at $f=0$ is simply

$\nabla_{z}^{2}\mathcal{G}(0)[\hat{z}]=-\partial_{xx}\hat{z},$

the Hessian operator of rescaled Hamiltonian $\mathcal{G}_{\zeta}$ on the
embedded torus $i_{\delta}$,
$\nabla_{z}^{2}\mathcal{G}_{\zeta}(i_{\delta}(\varphi))[\hat{z}]$ in (1.114)
reveals perturbations in the coefficients and a smoothing operator
$R^{\text{Airy}}$. Moreover, the deviation between these two Hessian operators
is composed of the contribution of simple monomials $\overline{v}$ in (1.120),
and smaller contributions from the embedding $i_{\delta}$.

Comparing $K$ in (1.104) and $\mathcal{G}_{\zeta}$ in (1.112), we can deduce
from the above example that in the case of gSQG the Hessian operator,
$\nabla_{w}^{2}K(i_{triv}(\varphi))$, will be expressed as a perturbation from
the $\nabla_{z}^{2}\mathcal{H}(0)$,

$\displaystyle\nabla_{z}^{2}\mathcal{H}(0)[\hat{z}]\overset{\eqref{linear_intro_operator_language1}}{=}-\frac{1}{2}\Lambda^{\alpha-1}\hat{z}+\frac{T_{\alpha}}{4}\hat{z}.$
(1.123)

Indeed, in Section 8, we will find that $\nabla_{w}^{2}K(i_{triv}(\varphi))$
has the form (compare below to (1.114)):

$\displaystyle\nabla_{w}^{2}K(i_{triv}(\varphi))[\hat{w}]=\Pi_{S^{\perp}}M(\varphi)[\hat{w}]+R(\varphi)[\hat{w}],$
(1.124)

where $M(\varphi)$ is an operator with perturbed coefficients and and
$R(\varphi)$ is a smoothing operator. The operator $M(\varphi)$ is of the
form:

$\displaystyle
M(\varphi)[\hat{w}]:=\Lambda^{\alpha-1}_{-\frac{1}{2}+a_{1}(\varphi)}\hat{w}+\left(\frac{T_{\alpha}}{4}+a_{2}(\varphi)\right)\hat{w}+\Upsilon^{\alpha-3}_{a_{3}(\varphi)}\hat{w},$
(1.125)

where

$\displaystyle\Lambda^{\alpha-1}_{a}h:=\int_{\mathbb{T}}(2-2\cos(x-y))^{-\frac{\alpha}{2}}a(x,y)(h(x)-h(y))dy,$
(1.126)
$\displaystyle\Upsilon^{\alpha-3}_{a}h:=\int_{\mathbb{T}}(2-2\cos(x-y))^{1-\frac{\alpha}{2}}a(x,y)h(y)dy.$

The $\varphi$-dependent “coefficients”
$a_{1}(\varphi)=a_{1}(\varphi,x,y),a_{2}(\varphi)=a_{2}(\varphi,x),a_{3}(\varphi)=a_{3}(\varphi,x,y)$
and the operator $R(\varphi)$ are small;

$\displaystyle a_{i}=O(\varepsilon)\text{ for $i=1,2,3$},\quad
R=O(\varepsilon).$ (1.127)

With (1.124), the definition of $\mathcal{L}_{\omega}$ in (1.105) tells us
that the linear operator $\mathcal{L}_{\omega}$ is of the form :

$\displaystyle\mathcal{L}_{\omega}=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}M(\varphi)+\mathcal{R}(\varphi),\text{
where $\mathcal{R}:=-\partial_{x}R$.}$ (1.128)

The derivation of the from of $\mathcal{L}_{\omega}$ is the main result of
Section 8 and Proposition 8.18.

Actually, Proposition 8.18 tells us something more. As we observed in the toy
example (see (1.121) and (1.122)), we can decompose the contribution to the
coefficients and $R$ into the ones from the monomials $\overline{v}$ and the
ones from $i_{\delta}$. This is the motivation for the definition of the class
of coefficients $\mathfrak{C}_{1}(i_{0})$ and $\mathfrak{C}_{2}(i_{0})$ in
Definition 8.2 and the class of a “finite dimensional operator”
$\mathfrak{R}(i_{0})$ in Definition 8.11. In Proposition 8.18, we prove that

$\displaystyle a_{1},a_{3}\in\mathfrak{C}_{2}(i_{0}),\quad
a_{2}\in\mathfrak{C}_{1}(i_{0}),\quad\mathcal{R}\in\mathfrak{R}(i_{0}).$
(1.129)

Indeed, we will see that the coefficients/operators of size
$O(\varepsilon^{2})$ are not perturbative (see (1.141)). Therefore, in the
reduction procedure, which we will discuss later, we need more precise
information about coefficients/operators of sizes
$O(\varepsilon),O(\varepsilon^{2})$. The inclusions (1.129) tell us that the
eigenvalues of the Hessian $\nabla_{w}^{2}K(i_{triv}(\varphi))$ are
independent of $i_{0},i_{\delta}$ up to $o(\varepsilon^{2})$, since the
contributions of size $O(\varepsilon),O(\varepsilon^{2})$ are solely
determined by the monomials $\overline{v}$ in (1.120).

#####  1.2.2.3 KAM reducibility and homological equation

The invertibility of $\mathcal{L}_{\omega}$ is achieved by diagonalizing the
operator. Roughly speaking, we aim to find $\varphi$-dependent invertible
transformations
$\Phi_{1,\infty}(\varphi),\Phi_{2,\infty}(\varphi):H_{S^{\perp}}\mapsto
H_{S^{\perp}}$ and Fourier multipliers $j\mapsto d_{\infty}(j)$ such that

$\displaystyle\Phi_{1,\infty}\circ\mathcal{L}_{\omega}\circ\Phi_{2,\infty}=\mathcal{L}_{\infty}:=\omega\cdot\partial_{\varphi}-\text{diag}_{j\in
S^{\perp}}d_{\infty}(j).$ (1.130)

Note that since the multipliers $d_{\infty}(j)$ are the eigenvalues of
$\hat{w}\mapsto\partial_{x}(\nabla_{w}^{2}K(\varphi,0,0))[\hat{w}])$ (we
remove the dependence on $\varphi$ by reparametrizing in the variable
$\varphi$) and $K$ depends on $\omega$, so does $d_{\infty}$ as well,
therefore, we denote

$\displaystyle d_{\infty}(j)=d_{\infty}(\omega,j),\text{ for $j\in
S^{\perp}$}.$ (1.131)

Then, for the frequencies satisfying the so-called first order Melnikov
condition (as we saw in (1.26)):

$\displaystyle\left\\{\omega:|\text{i}\omega\cdot
l-d_{\infty}(\omega,j)|\geq\gamma|l|^{-\tau}|j|^{\alpha},\text{ for all
$(l,j)\in\mathbb{Z}^{\nu}\times S^{\perp}$}\right\\},$ (1.132)

we can invert the operator in (1.130).

In order to achieve (1.130), let us suppose we have an operator $L$ of the
form:

$\displaystyle L:=\omega\cdot\partial_{\varphi}-\text{diag}_{j\in
S^{\perp}}d(\omega,j)+\mathcal{R}(\varphi)=:\omega\cdot\partial_{\varphi}-D+\mathcal{R}(\varphi),$
(1.133)

for some Fourier multipliers $j\mapsto d(\omega,j)$ and for some
$\varphi$-dependent operator $\mathcal{R}(\varphi):H_{S^{\perp}}\mapsto
H_{S^{\perp}}$ and investigate how to obtain, from $L$, a fully diagonalized
operator such as (1.130). We consider a transformation $\phi(\varphi)$ such
that

$\displaystyle\phi(\varphi)=I+\psi(\varphi),\text{ for some small operator
$\psi(\varphi):H_{S^{\perp}}\mapsto H_{S^{\perp}}$, so that
$\phi(\varphi)^{-1}=:I+\tilde{\psi}(\varphi)$ exists.}$ (1.134)

We compute the conjugation of $L$ with $\phi$, that is $\phi^{-1}\circ
L\circ\phi$. First, we compute

$\displaystyle L\circ\phi=L\circ(I+\psi)$
$\displaystyle\overset{\eqref{l_0inftronotext123}}{=}L+(\omega\cdot\partial_{\varphi}\circ\psi-D\circ\psi+\mathcal{R}\circ\psi)$
$\displaystyle=L+\omega\cdot\partial_{\varphi}(\psi)+\psi\circ(\omega\cdot\partial_{\varphi})-D\circ\psi+\mathcal{R}\circ\psi$
$\displaystyle=(\omega\cdot\partial_{\varphi}-D)+\mathcal{R}+\psi\circ(\omega\cdot\partial_{\varphi})-\psi\circ
D+\omega\cdot\partial_{\varphi}(\psi)+(\psi\circ
D-D\circ\psi)+\mathcal{R}\circ\psi$
$\displaystyle=(I+\psi)\circ(\omega\cdot\partial_{\varphi}-D)+(\omega\cdot\partial_{\varphi}(\psi)+[\psi,D]+\mathcal{R})+\mathcal{R}\circ\psi.$
(1.135)

Denoting

$\displaystyle r(j):=\mathcal{R}^{j}_{j}(0),\quad D_{r}:=\text{diag}_{j\in
S^{\perp}}r(j),$ (1.136)

where $\mathcal{R}^{j_{1}}_{j_{2}}(l)$ is the matrix representation of the
operator $\mathcal{R}(\varphi)$ such that

$\mathcal{R}(\varphi)[f]=\sum_{(l,j_{1})\in\mathbb{Z}^{\nu}\times\mathbb{Z}}\mathcal{R}^{j_{1}}_{j_{2}}(l)f_{j_{1}}e^{\text{i}(\varphi\cdot
l+j_{2}x)},$

we solve a homological equation for the operator $\psi$:

$\displaystyle\omega\cdot\partial_{\varphi}(\psi)+[\psi,D]+\mathcal{R}=\text{diag}_{j\in
S^{\perp}}r(j)=:D_{r}.$ (1.137)

For the solution $\psi$, it follows from (1.2.2.3) that

$L\circ\phi=\phi\circ(\omega\cdot\partial_{\varphi}-D)+D_{r}+\mathcal{R}\circ\psi=\phi\circ(\omega\cdot\partial_{\varphi}-(D+D_{r}))-\psi\circ
D_{r}+\mathcal{R}\circ\psi.$

Therefore, we see that

$\displaystyle\phi^{-1}\circ
L\circ\phi=\omega\cdot\partial_{\varphi}-\underbrace{(D-D_{r})}_{=:D_{\text{new}}}+\underbrace{\phi^{-1}\circ\left(-\psi\circ
D_{r}+\mathcal{R}\circ\psi\right)}_{=:\mathcal{R}_{\text{new}}}.$ (1.138)

In the above formal computations, we observe that given a linear operator of
the form (1.133), a solution to the homological equation (1.137) leads us to
an updated linear operator with a new diagonal part and a new remainder part.
If the size of the new remainder $\mathcal{R}_{new}$ is shown to be
significantly smaller than the previous remainder $\mathcal{R}$, we can deduce
that an infinite number of such conjugations converge to a fully diagonalized
operator (1.130). The first thing we can observe from the homological equation
(1.137) is that the size of $\psi$ will depend on $\mathcal{R}$, while we do
not expect the solution to exist for every frequency $\omega$ and we need to
select $\omega$ that satisfies sufficient non-resonance conditions. From the
Diophantine condition (1.28), heuristically we can deduce that the size of
$\psi$ must be

$\displaystyle\psi=O(\gamma^{-1}|\mathcal{R}|).$ (1.139)

Therefore, the new remainder (1.138) can be inferred to be

$\displaystyle\mathcal{R}_{\text{new}}=O(\gamma^{-1}|\mathcal{R}|^{2}).$
(1.140)

In order for this new remainder to be smaller than the previous one,
$|\mathcal{R}|\gtrsim\gamma^{-1}|\mathcal{R}|^{2}$, we must have

$\displaystyle\mathcal{R}=o(\gamma^{-1})\overset{\eqref{gamma_impose_intro1}}{=}o(\varepsilon^{2b}).$
(1.141)

Furthermore, since $\mathcal{R}_{\text{new}}$ is quadratic in $\mathcal{R}$,
we need $\mathcal{R}$ to be bounded in the sense that the composition
$\mathcal{R}\circ\mathcal{R}$ does not accumulate the derivatives (for
example, compositions of an operator such as $\varepsilon^{3}\partial_{x}$ do
not get smaller after the compositions, since its contribution to high modes
are increasing because of the accumulation of the derivatives).

In our proof, we will measure the size of the remainder using the notion of
“modulo-tame constant, Lip-modulo-tame constant” (see Definition 2.20, 2.22).
Such notions were introduced in [16]. In regard to the above discussion,
compositions of $0$-modulo-tame operators remain as a $0$-modulo-tame
operator, which implies that compositions of such operators do not accumulate
the derivatives (Lemma 2.23). We also note that in order to rigorously
construct the infinite sequence of conjugations (1.138) and (1.134), we need
to regularize the remainder $\mathcal{R}(\varphi)$ at each step (regularize
with respect to the variable $\varphi$), which requires the estimates of not
only the remainder $\mathcal{R}$ but also the derivatives of it with respect
to $\varphi$, that is $\partial_{\varphi}^{\mathtt{b}_{0}}(\mathcal{R}_{0})$
for sufficiently large $\mathtt{b}_{0}>0$. A question how large
$\mathtt{b}_{0}$ needs to be could be answered by solving (1.137) rigorously
(in our paper, our choice is given in (10.70)).

Our operator $\mathcal{L}_{\omega}$ in (1.141) does not satisfy such
conditions. Therefore, inspired by [6], we will conjugate
$\mathcal{L}_{\omega}$ a finite number of times into the form with a
sufficiently small remainder. That is, we will construct invertible
transformations $\Phi_{1-6,1},\Phi_{1-6,2}$ such that the conjugated operator
has the form:

$\displaystyle
L_{0}:=\Phi_{1-6,1}\circ\mathcal{L}_{\omega}\circ\Phi_{1-6,2}=\omega\cdot\partial_{\varphi}-D_{0}+\mathcal{R}_{0},$
(1.142)

with

  1. (R1)

$D_{0}$ is a Fourier multiplier, that is, $D_{0}$ does not depend on
$\varphi,x$.

  2. (R2)

$\partial_{\varphi}^{\mathtt{b}_{0}}\mathcal{R}_{0}(\varphi),\
\mathcal{R}_{0}$ satisfies the smallness condition
$\mathcal{R}_{0}=o(\varepsilon^{2b})$ (as a $0$-Lip-modulo-tame operator) for
sufficiently large $\mathtt{b}_{0}>0$.

This is the main task in Subsections 10.1-10.5 yielding Proposition 10.76.
After achieving (1.142), the rigorous iteration to construct $\phi$ in (1.138)
towards the full diagonal will be proved rigorously throughout Subsection
10.6.

#####  1.2.2.4 Reduction to a constant coefficient operator: Subsections
10.1-10.5

In view of the above discussion, we are left to study how to construct
transformations $\Phi_{1-6,1},\Phi_{1-6,2}$ with which the conjugation of
$\mathcal{L}_{\omega}$ is reduced to the form of (1.142) satisfying (R1) and
(R2).

Recall from (1.128) and (1.125) that $\mathcal{L}_{\omega}$ has the form

$\displaystyle\mathcal{L}_{\omega}=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}\left(\Lambda^{\alpha-1}_{-\frac{1}{2}+a_{1}(\varphi)}+\left(\frac{T_{\alpha}}{4}+a_{2}(\varphi)\right)+\Upsilon^{\alpha-3}_{a_{3}(\varphi)}\right)+\mathcal{R}(\varphi),$
(1.143)

for some $a_{1}(\varphi)=a_{1}(\varphi,x,y),\ a_{2}(\varphi)=a_{2}(\varphi,x)$
and $a_{3}(\varphi)=a_{3}(\varphi,x,y)$ such that
$a_{1},a_{2},a_{3}=O(\varepsilon)$ and for some smoothing operator
$\mathcal{R}$ of size $O(\varepsilon^{2})$. We reduce the coefficients to
constants (remove the dependence on $x$ and $\varphi$) from the highest order
term to the lower ones, using the conjugation techniques motivated from [6, 8,
16].

  1. $\bullet$

Change of the space variables (Subsection 10.1): We conjugate
$\mathcal{L}_{\omega}$ with a $\varphi$-dependent time-$1$ flow map $\Phi_{1}$
determined by the PDE,
$\partial_{t}u=\Pi_{S^{\perp}}\partial_{x}\left(\frac{\beta(\varphi,x)}{1+t\beta_{x}(\varphi,x)}u(t,\varphi,x)\right),\text{
for some function $\beta(\varphi,x)$}$. The function $\beta$ is chosen so that
the conjugated operator is of the form:

$\mathcal{L}^{1}:=(\Phi_{1})^{-1}\circ\mathcal{L}_{\omega}\circ\Phi_{1}=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}\left(b_{1}(\varphi)\Lambda^{\alpha-1}+b_{2}(\varphi,x)+\Upsilon^{\alpha-3}_{b_{3}(\varphi)}\right)+R_{1}(\varphi),$

for some $b_{1}(\varphi),b_{2}(\varphi,x),b_{3}(\varphi,x,y)$, where $b_{1}$,
does not depend on $x$ anymore and
$b_{2}-\frac{T_{\alpha}}{4},b_{3}=O(\varepsilon)$. ${R}_{1}(\varphi)$ is a
smoothing operator of size $O(\varepsilon^{2})$. A more precise statement can
be found in Proposition 10.1.

  2. $\bullet$

Reparametrization of time (Subsection 10.2): We conjugate $\mathcal{L}^{1}$
with a transformation $\Phi_{2}:[h](\varphi,x):=h(\varphi+\omega
p_{1}(\varphi),x)$ for some function $p_{1}(\varphi)$. The function $p_{1}$ is
chosen so that the conjugated operator is of the form:

$\displaystyle\mathcal{L}^{2}:=\frac{1}{\rho(\varphi)}(\Phi_{2})^{-1}\circ\mathcal{L}^{1}\circ\Phi_{2}=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}\left(\mathtt{m}_{\alpha}\Lambda^{\alpha-1}+d_{2}(\varphi,x)+\Upsilon^{\alpha-3}_{d_{3}(\varphi)}\right)+R_{2}(\varphi),$
(1.144)

for some functions $\rho(\varphi),d_{2}(\varphi,x),d_{3}(\varphi,x,y)$ and for
a constant $\mathtt{m}_{\alpha}$ that does not depend on any of $\varphi,x$,
and $d_{2}-\frac{T_{\alpha}}{4},d_{3}=O(\varepsilon)$. $R_{2}(\varphi)$ is a
smoothing operator of size $O(\varepsilon^{2})$. A more precise statement can
be found in Proposition 10.10.

  3. $\bullet$

Symbolic representation of the operator (Subsection 10.3.1): We rewrite the
operator
$\mathtt{m}_{\alpha}\Lambda^{\alpha-1}+d_{2}(\varphi,x)+\Upsilon^{\alpha-3}_{d_{3}(\varphi)}$
in (1.144) as a pseudo differential operator with associated symbols. Using
the Fourier multiplier $m_{1,\alpha}(\xi)$ associated to the operator
$\Lambda^{\alpha-1}$, that is, $Op^{W}(m_{1,\alpha}(\xi))=\Lambda^{\alpha-1}$
(see Proposition 2.34), we rewrite $\mathcal{L}^{2}$ as

$\displaystyle\mathcal{L}^{2}=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}Op^{W}\left(\mathtt{m}_{\alpha}m_{1,\alpha}(\xi)+\frac{T_{\alpha}}{4}+\mathfrak{d}_{0}(\varphi)\right)+R_{2}(\varphi),$
(1.145)

for some $\varphi$-dependent classical symbol
$\mathfrak{d}_{0}(\varphi,x,\xi)$ of order $0$ such that
$\mathfrak{d}_{0}=O(\varepsilon)$, where $Op^{W}$ denotes the Weyl
quantization of the symbol (2.13). A more detailed statement can be found in
(10.28).

  4. $\bullet$

Egorov method (step 1) (Subsection 10.3.2): We conjugate $\mathcal{L}^{2}$
with $\varphi$-dependent time-$1$ flow map $\Phi_{3}$ determined by the PDE,

$\displaystyle\partial_{t}u=\Pi_{S^{\perp}}\partial_{x}Op^{W}(\mathfrak{a}(\varphi))[u],$
(1.146)

for some $\varphi$-dependent symbol function
$\mathfrak{a}(\varphi)=\mathfrak{a}(\varphi,x,\xi)$ of order $1-\alpha$. The
symbol $\mathfrak{a}$ chosen so that the conjugated operator is of the form:

$\displaystyle\mathcal{L}^{3}$
$\displaystyle:=(\Phi_{3})^{-1}\circ\mathcal{L}^{2}\circ\Phi_{3}$
$\displaystyle=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}Op^{W}\left(\mathtt{m}_{\alpha}m_{1,\alpha}(\xi)+\frac{T_{\alpha}}{4}+\sum_{k=0}^{\mathtt{N}_{\alpha}}M_{x}(\mathfrak{d}_{k}(\varphi))+\mathfrak{r}_{-2}(\varphi)\right)+\partial_{x}\Pi_{S^{\perp}}W_{0}(\varphi)+R_{3}(\varphi).$

where

    1. $\circ$

$\mathfrak{d}_{k}(\varphi)=\mathfrak{d}_{k}(\varphi,x,\xi)$ are
$\varphi$-dependent symbols of order $k(1-\alpha)$ for
$k=1\ldots,\mathtt{N}_{\alpha}$. $\mathtt{N}_{\alpha}$ is a constant that
depends on only $\mathtt{p}$ (see (1.84) and (10.70)). The motivation of
$\mathtt{N}_{\alpha}$ is that this constant is arising in the Taylor expansion
of the flow map of (1.146) in the time variable $t$ to obtain necessary
estimates.

    2. $\circ$

$M_{x}(\mathfrak{d}_{k}(\varphi))$ is the average of the symbol
$\mathfrak{d}_{k}(\varphi,x,\xi)$, therefore the symbol
$\sum_{k=0}^{\mathtt{N}_{\alpha}}M_{x}(\mathfrak{d}_{k}(\varphi))$ in the
expression of $\mathcal{L}^{3}$ does not depend on the variable $x$.

    3. $\circ$

$\mathfrak{r}_{-2}(\varphi)=\mathfrak{r}_{-2}(\varphi,x,\xi)$ is a
$\varphi$-dependent symbol of order $-2$ such that
$\mathfrak{r}_{-2}=O(\varepsilon)$.

    4. $\circ$

$\partial_{x}\Pi_{S^{\perp}}W_{0}(\varphi)$ is a remainder satisfying the
requirement (R2) for (1.142).

    5. $\circ$

$R_{3}$ is a smoothing operator of size $O(\varepsilon^{2})$.

A more detailed statement can be found in Proposition 10.30.

######  Remark 1.7.

As opposed to the earlier works [16, 75], we choose a slightly more general
symbol $\mathfrak{a}(\varphi)$ in (1.146) to eliminate the dependence of the
lower order symbols on the variable $x$. This is why we use the Weyl
quantization in the reduction procedure. The PDE in (1.146) may not be well-
posed in general, unless the operator $Op^{W}(\mathfrak{a}(\varphi))$ is a
symmetric operator. The use of the Weyl quantization makes it easier to verify
the symmetry of the operator, since $Op^{W}(\mathfrak{a}(\varphi))$ is
symmetric if and only if $\mathfrak{a}(\varphi)$ is a real-valued symbol.
Furthermore, compared to [16, 75], the choice of the symbol
$\mathfrak{a}(\varphi)$ is more delicate, since the highest order of our
equation is “weak”. We overcome this issue by writing $\mathfrak{a}(\varphi)$
as a finite sum of lower order symbols,
$\mathfrak{a}(\varphi)=\sum_{k=0}^{\mathtt{N}_{\alpha}}\mathfrak{a}_{k}(\varphi)$
for some $\mathfrak{a}_{k}\in\mathcal{S}^{(k+1)(1-\alpha)}$ and search for
$\mathfrak{a}_{k}$ to obtain the desired conjugated operator of the form
(1.146).

  5. $\bullet$

Egorov method (step 2) (Subsection 10.3.3): We conjugate $\mathcal{L}^{3}$
with a $\varphi$-dependent time-$1$ flow map $\Phi_{4}$ determined by the PDEs
$\partial_{t}u=\Pi_{S^{\perp}}\partial_{x}Op^{W}(\mathfrak{p}_{2}(\varphi))[u],$
for some $\varphi$-dependent Fourier multiplier
$\mathfrak{p}_{2}(\varphi)=\mathfrak{p}_{2}(\varphi,\xi)$ of order $1-\alpha$
that does not depend on $x$. The function $\mathfrak{p}_{2}$ is chosen so that
the conjugated operator is of the form:

$\displaystyle\mathcal{L}^{4}$
$\displaystyle:=(\Phi_{4})^{-1}\circ\mathcal{L}^{3}\circ\Phi_{4}$ (1.147)
$\displaystyle=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}Op^{W}\left(\mathtt{m}_{\alpha}m_{1,\alpha}(\xi)+\frac{T_{\alpha}}{4}+\mathfrak{m}_{\leq
0}(\xi)+\mathfrak{r}_{-2,1}(\varphi)\right)+\partial_{x}\Pi_{S^{\perp}}W_{1}(\varphi)+R_{4}(\varphi),$

where

    1. $\circ$

$\mathfrak{m}_{\leq 0}(\xi)$ is a Fourier multiplier of order $0$ that is
independent of $\varphi,x$.

    2. $\circ$

$\mathfrak{r}_{-2,1}(\varphi)=\mathfrak{r}_{-2,1}(\varphi,x,\xi)$ is a
$\varphi$-dependent symbol of order $-2$ such that
$\mathfrak{r}_{-2,1}=O(\varepsilon)$.

    3. $\circ$

$\partial_{x}\Pi_{S^{\perp}}W_{1}(\varphi)$ is a remainder satisfying the
requirement (R2) for (1.142).

    4. $\circ$

$R_{4}$ is a smoothing operator of size $O(\varepsilon^{2})$.

A more detailed statement can be found in Proposition 10.41.

  6. $\bullet$

Linear Birkhoff normal form (step 1) (Subsection 10.4.1): Comparing (1.147) to
(1.142) and recalling the requirement for the remainder
$\mathcal{R}_{0}=o(\varepsilon^{2b})$ from (R2), the operators
$Op^{W}(\mathfrak{r}_{-2,1}(\varphi))$ and $R_{4}(\varphi)$ are not
perturbative, that is, their size is not $o(\varepsilon^{2b})$ since $b>1$
(see (1.85)). In order to remove terms of size $O(\varepsilon)$, we conjugate
$\mathcal{L}^{4}$ with a $\varphi$-dependent time-$1$ flow map $\Phi_{5}$
determined by the PDEs $\frac{d}{dt}u=\partial_{x}\Pi_{S^{\perp}}\varepsilon
Op^{W}(\rho_{1}(\varphi))[u]$ for some $\varphi$-dependent symbol
$\rho_{1}(\varphi)=\rho_{1}(\varphi,x,\xi)$ of order $-1-\alpha$. The function
$\rho_{1}(\varphi,x,\xi)$ is chosen so that the conjugated operator has the
form:

$\displaystyle\mathcal{L}^{5}$
$\displaystyle=(\Phi_{5})^{-1}\circ\mathcal{L}^{4}\circ\Phi_{5}$
$\displaystyle=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}Op^{W}\left(\mathtt{m}_{\alpha}m_{1,\alpha}(\xi)+\left(\frac{T_{\alpha}}{4}+\mathfrak{m}_{\leq
0}(\xi)\right)+\varepsilon^{2}\tilde{\mathfrak{b}}_{2}(\varphi)+{\mathfrak{r}}_{-2,\leq
3,*}(\varphi)\right)+\partial_{x}\Pi_{S^{\perp}}W_{2}(\varphi)+R_{5}(\varphi),$

where

    1. $\circ$

$\varepsilon^{2}\tilde{\mathfrak{b}}_{2}(\varphi)=\varepsilon^{2}\tilde{\mathfrak{b}}_{2}(\varphi,x,\xi)$
is a $\varphi$-dependent symbol of order $-2$ such that
$\varepsilon^{2}\tilde{\mathfrak{b}}_{2}=O(\varepsilon^{2})$.

    2. $\circ$

$\mathfrak{r}_{-2,\leq 3,*}(\varphi)=\mathfrak{r}_{-2,\leq
3,*}(\varphi,x,\xi)$ is a $\varphi$-dependent symbol of order $-2$ such that
$\mathfrak{r}_{-2,\leq 3,*}=o(\varepsilon^{2b})$.

    3. $\circ$

$\partial_{x}\Pi_{S^{\perp}}W_{2}(\varphi)$ is a remainder satisfying the
requirement (R2) for (1.142).

    4. $\circ$

$R_{5}$ is a smoothing operator of size $O(\varepsilon^{2})$.

A more detailed statement can be found in Proposition 10.51.

  7. $\bullet$

Linear Birkhoff normal form (step 2) (Subsection 10.4.3): Comparing (1.147) to
(1.142) and recalling the requirement for the remainder
$\mathcal{R}_{0}=o(\varepsilon^{2b})$ from (R2), the operators
$Op^{W}(\varepsilon^{2}\tilde{\mathfrak{b}}_{2}(\varphi))$ and
$R_{5}(\varphi)$ are not perturbative, that is, their size is not
$o(\varepsilon^{2b})$ since $b>1$ (see (1.85)). To eliminate the terms of size
$O(\varepsilon^{2})$, we conjugate $\mathcal{L}^{5}$ with a
$\varphi$-dependent time-$1$ flow map $\Phi_{6}$ determined by the PDEs
$\frac{d}{dt}u=\partial_{x}\Pi_{S^{\perp}}\varepsilon
Op^{W}(\rho_{2}(\varphi))[u]$ for some $\varphi$-dependent symbol
$\rho_{2}(\varphi)=\rho_{2}(\varphi,x,\xi)$ of order $-1-\alpha$. The function
$\rho_{2}(\varphi,x,\xi)$ is chosen so that the conjugated operator has the
form:

$\displaystyle\mathcal{L}^{6}$
$\displaystyle:=(\Phi_{6})^{-1}\circ\mathcal{L}_{\omega}\circ\Phi_{6}$ (1.148)
$\displaystyle=\omega\cdot\partial_{\varphi}-\Pi_{S^{\perp}}\partial_{x}Op^{W}\left(\mathtt{m}_{\alpha}m_{1,\alpha}(\xi)+\left(\frac{T_{\alpha}}{4}+\mathfrak{m}_{\leq
0}(\xi)+\varepsilon^{2}\mathfrak{m}_{\mathfrak{b}}(\xi)\right)\right)$
$\displaystyle\ +\Pi_{S^{\perp}}\partial_{x}Op^{W}(\mathfrak{r}_{-2,\leq
3,\sharp}(\varphi))+\partial_{x}\Pi_{S^{\perp}}W_{3}(\varphi)+R_{6}(\varphi),$

where

    1. $\circ$

$\mathfrak{m}_{\mathfrak{b}}(\xi)$ is a Fourier multiplier of order $-2$ that
is independent of $\varphi,x$.

    2. $\circ$

$\mathfrak{r}_{-2,\leq 3,\sharp}(\varphi)=\mathfrak{r}_{-2,\leq
3,\sharp}(\varphi,x,\xi)$ is a $\varphi$-dependent symbol of order $-2$ such
that $\mathfrak{r}_{-2,\leq 3,\sharp}=o(\varepsilon^{2b})$.

    3. $\circ$

$\partial_{x}\Pi_{S^{\perp}}W_{3}(\varphi)$ is a remainder satisfying the
requirement (R2) for (1.142).

    4. $\circ$

$R_{6}$ is a smoothing operator of size $o(\varepsilon^{2b})$.

A more detailed statement can be found in Proposition 10.66.

Finally, denoting

$\displaystyle\mathcal{R}_{0}$
$\displaystyle:=\Pi_{S^{\perp}}\partial_{x}Op^{W}(\mathfrak{r}_{-2,\leq
3,\sharp}(\varphi))+\partial_{x}\Pi_{S^{\perp}}W_{3}(\varphi)+R_{6}(\varphi),$
$\displaystyle D_{0}$
$\displaystyle:=\Pi_{S^{\perp}}\partial_{x}Op^{W}\left(\mathtt{m}_{\alpha}m_{1,\alpha}(\xi)+\left(\frac{T_{\alpha}}{4}+\mathfrak{m}_{\leq
0}(\xi)+\varepsilon^{2}\mathfrak{m}_{\mathfrak{b}}(\xi)\right)\right),$
$\displaystyle L_{0}$