# Periodic and quasi-periodic Euler-$\alpha$ flows close to Rankine vortices

Emeric Roulley Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France
E-mail address<EMAIL_ADDRESS>
###### Abstract

In the present contribution, we first prove the existence of $\mathbf{m}$-fold
simply-connected V-states close to the unit disc for Euler-$\alpha$ equations.
These solutions are implicitly obtained as bifurcation curves from the
circular patches. We also prove the existence of quasi-periodic in time vortex
patches close to the Rankine vortices provided that the scale parameter
$\alpha$ belongs to a suitable Cantor-like set of almost full Lebesgue
measure. The techniques used to prove this result are borrowed from the Berti-
Bolle theory in the context of KAM for PDEs.

###### Contents

  1. 1 Euler-$\alpha$ equations and main results
  2. 2 The stream function associated to the filtered velocity
  3. 3 Periodic rigid motion
    1. 3.1 Function spaces and reformulation of the vortex patch equation
    2. 3.2 Regularity aspects and structure of the linearized operator
    3. 3.3 Construction of local bifurcation branches
  4. 4 Quasi-periodic Euler-$\alpha$ patches
    1. 4.1 Topologies for functions and operators
    2. 4.2 Hamiltonian contour dynamics equation and its linearization
    3. 4.3 Transversality and linear quasi-periodic solutions
    4. 4.4 The functional of interest and associated tame estimates
    5. 4.5 Almost approximate right inverse
    6. 4.6 Construction of a non-trivial quasi-periodic solution
  5. A Properties of modified Bessel functions
  6. B Local bifurcation theorem and singular integrals

##  1 Euler-$\alpha$ equations and main results

We consider the Euler-$\alpha$ planar model with parameter $\alpha\geqslant 0$
given by

$\left\\{\begin{array}[]{ll}\partial_{t}\mathbf{u}+(\mathbf{v}\cdot\nabla)\mathbf{u}+(\nabla\mathbf{v})^{\top}\mathbf{u}+\nabla\pi=0&\textnormal{in
}\mathbb{R}_{+}\times\mathbb{R}^{2}\\\
\nabla\cdot\mathbf{v}=\nabla\cdot\mathbf{u}=0&\\\
\mathbf{u}=\mathbf{v}-\alpha^{2}\Delta\mathbf{v}&\\\
\mathbf{v}(0,\cdot)=\mathbf{v}_{0}.&\\\ \end{array}\right.$ (1.1)

This model is a regularization of Euler equations describing the flow of an
incompressible fluid on spatial scales larger than the length scale parameter
$\alpha.$ It has been introduced in the context of averaged fluid models, see
[52, 53]. In the literature, $\mathbf{v}$ and $\mathbf{u}$ are called the
filtered and unfiltered velocities, respectively. Actually, $\mathbf{u}$
corresponds to the velocity field of the fluid. Notice that
$(\nabla\mathbf{v})^{\top}$ denotes the transpose of the Jacobi matrix for
$\mathbf{v}$, namely

$(\nabla\mathbf{v})^{\top}=\left(\frac{\partial\mathbf{v}_{j}}{\partial
x_{i}}\right)_{1\leqslant i,j\leqslant 2}$

The pressure term $\pi$ is linked to the pressure field $p$ of the fluid
through the relation

$\pi\triangleq
p-\tfrac{1}{2}|\mathbf{v}|^{2}-\tfrac{\alpha^{2}}{2}|\nabla\mathbf{v}|^{2}.$

Remark that we formally recover the classical Euler model, by taking
$\alpha=0$ in the above set of equations. The rigorous justification of this
convergence can be found in [59]. In the sequel, we assume that $\alpha>0.$
Let us consider the unfiltered vorticity $\boldsymbol{\omega}$ defined by

$\boldsymbol{\omega}\triangleq\nabla^{\perp}\cdot\mathbf{u}=\nabla^{\perp}\cdot(\textnormal{Id}-\alpha^{2}\Delta)\mathbf{v},\qquad\nabla^{\perp}\triangleq\begin{pmatrix}-\partial_{2}\\\
\partial_{1}\end{pmatrix}.$ (1.2)

Applying the operator $\nabla^{\perp}\cdot$ to the first equation in (1.1), we
find, after straightforward computations using the divergence-free conditions,
that $\boldsymbol{\omega}$ is a solution to the following active scalar
equation

$\partial_{t}\boldsymbol{\omega}+\mathbf{v}\cdot\nabla\boldsymbol{\omega}=0.$
(1.3)

Such kind of nonlinear and nonlocal transport PDE has been widely studied
during the past decade in fluid dynamics, especially regarding the periodic
motions and more recently the quasi-periodic ones. We shall discuss in this
introduction two types of active scalar models in the form (1.3) which will be
of interest in the sequel. The first example is given by the classical
velocity-vorticity formulation of 2D Euler equations, namely

$\partial_{t}\boldsymbol{\omega}^{\textnormal{\tiny{E}}}+\mathbf{v}^{\textnormal{\tiny{E}}}\cdot\nabla\boldsymbol{\omega}^{\textnormal{\tiny{E}}}=0,\qquad\mathbf{v}^{\textnormal{\tiny{E}}}=\nabla^{\perp}\boldsymbol{\Psi}^{\textnormal{\tiny{E}}},\qquad\Delta\boldsymbol{\Psi}^{\textnormal{\tiny{E}}}=\boldsymbol{\omega}^{\textnormal{\tiny{E}}}.$

In this case, the potential velocity
$\boldsymbol{\Psi}^{\textnormal{\tiny{E}}}$ is obtained as the convolution
with the Green function associated to the Laplace problem set in the whole
plane, namely

$\boldsymbol{\Psi}^{\textnormal{\tiny{E}}}=\mathbf{G}^{\textnormal{\tiny{E}}}\ast\boldsymbol{\omega}^{\textnormal{\tiny{E}}},\qquad\mathbf{G}^{\textnormal{\tiny{E}}}\triangleq\tfrac{1}{2\pi}\log(|\cdot|).$
(1.4)

The second example is given by quasi-geostrophic shallow-water equations with
parameter $\lambda>0$, shorten in what follows into $(QGSW)_{\lambda}$, which
is a geophysical asymptotic model describing the circulation of the atmosphere
at large time and space scales [68, p.220]. This model is given by

$\partial_{t}\boldsymbol{q}^{\textnormal{\tiny{SW}}}+\mathbf{v}_{\lambda}^{\textnormal{\tiny{SW}}}\cdot\nabla\boldsymbol{q}^{\textnormal{\tiny{SW}}}=0,\qquad\mathbf{v}_{\lambda}^{\textnormal{\tiny{SW}}}=\nabla^{\perp}\boldsymbol{\Psi}_{\lambda}^{\textnormal{\tiny{SW}}},\qquad(\Delta-\lambda^{2})\boldsymbol{\Psi}_{\lambda}^{\textnormal{\tiny{SW}}}=\boldsymbol{q}^{\textnormal{\tiny{SW}}}.$

In this case, the potential velocity
$\boldsymbol{\Psi}_{\lambda}^{\textnormal{\tiny{SW}}}$ is obtained as the
convolution with the Green function associated to the Helmoltz problem set in
the whole plane, namely

$\boldsymbol{\Psi}_{\lambda}^{\textnormal{\tiny{SW}}}=\mathbf{G}_{\lambda}^{\textnormal{\tiny{SW}}}\ast\boldsymbol{q}^{\textnormal{\tiny{SW}}},\qquad\mathbf{G}_{\lambda}^{\textnormal{\tiny{SW}}}\triangleq-\tfrac{1}{2\pi}K_{0}(\lambda|\cdot|),$
(1.5)

where $K_{0}$ is the modified Bessel function of second kind. We refer to the
Appendix A for a presentation of modified Bessel functions and some of their
useful properties. The parameter $\lambda$ is called Rossby deformation length
or Rossby radius and quantifies the rotation/stratification balance for the
fluid. Few results are known regarding this model and we may refer to [26, 66]
for the mathematical context of interest in the sequel and to [27, 28] for
interesting numerical simulations in the physical literature.

This paper is based on the following fundamental relation, which can be found
for instance in [7, 63], giving a nice expression of the Euler-$\alpha$
velocity field $\mathbf{v}$.

$\mathbf{v}=\nabla^{\perp}\boldsymbol{\Psi},\qquad\boldsymbol{\Psi}\triangleq\mathbf{G}\ast\boldsymbol{\omega},\qquad\mathbf{G}\triangleq\mathbf{G}^{\textnormal{\tiny{E}}}-\mathbf{G}_{\frac{1}{\alpha}}^{\textnormal{\tiny{SW}}}.$
(1.6)

Due to the importance of this formula for our work, we provide in Section 2
its detailed justification. In our approach, we take advantage of this
explicit link with Euler and QGSW equations to prove the existence of periodic
and quasi-periodic vortex patch motions for the Euler-$\alpha$ equations. In
particular, we shall make appeal to the computations carried out in [14, 17,
26, 42, 49, 51] relatively to these contexts. For the Euler-$\alpha$ model,
the global existence and uniqueness of classical solutions has been obtained
in [18]. The well-posedness of weak solutions in the space of bounded Radon
measures for (1.1) has been proved in [63]. Indeed, the formula (1.6) and
(A.4) imply that the vector field $\mathbf{v}$ is less singular than the Euler
one, which allows to reach the class of measures for the well-posedness. An
other consequence is that the Yudovitch theory also applies in this context
and the weak solutions are Lagrangian. This fact is the starting point to be
able to look for vortex patches. Before presenting the notion of vortex patch
solutions together with the main results of this study, we may end this
presentation by briefly mentioning that the 2D Euler-$\alpha$ model has also
been intensively studied in the case of subset of $\mathbb{R}^{2}$ domains
with suitable boundary conditions. We may for instance refer the reader to
[19, 20, 21, 60, 72].

Now, we shall present the notion of vortex patches and discuss some historical
background about the periodic and quasi-periodic settings. The techniques used
to find these two kind of solutions are completely different. The periodic
solutions are obtained by using bifurcation theory from the stationary
solutions and the quasi-periodic ones appear in the context of KAM/Nash-Moser
theories. Considering an open bounded simply-connected domain $D_{0}$ in
$\mathbb{R}^{2}$ then the function $\mathbf{1}_{D_{0}}$ provides an initial
datum in the Yudovitch class. Therefore, we have existence and uniqueness of a
global weak solution which, according to the transport structure of (1.3),
takes the following form

$\boldsymbol{\omega}(t,\cdot)=\mathbf{1}_{D_{t}},$

where $D_{t}$ is the image of $D_{0}$ by the flow map associated to the
velocity field $\mathbf{v}$, namely

$D_{t}=\boldsymbol{\Phi}_{t}(D_{0}),\qquad\boldsymbol{\Phi}_{t}(x)=x+\int_{0}^{t}\mathbf{v}\big{(}s,\boldsymbol{\Phi}_{s}(x)\big{)}ds.$

The resulting solution $t\mapsto\mathbf{1}_{D_{t}}$ is called a vortex patch
with initial patch $\mathbf{1}_{D_{0}}.$ Given any parametrization
$z(t,\cdot):\mathbb{T}\rightarrow\partial D_{t}$ of the boundary of the patch
at time $t,$ it is well-known since the works [49, 50] that for an active
scalar equation, as in our case, the particles on the boundary move with the
flow and remain at the boundary. Hence, at least in the smooth case, one can
write the following equation called vortex patch equation.

$\Big{(}\partial_{t}z(t,\theta)-\mathbf{v}\big{(}t,z(t,\theta)\big{)}\Big{)}\cdot\mathbf{n}\big{(}t,z(t,\theta)\big{)}=0,$
(1.7)

where $\mathbf{n}(t,z(t,\theta))$ is the outward normal vector to the boundary
$\partial D_{t}$ of $D_{t}$ at the point $z(t,\theta)$. Since the dynamics is
planar, it is more convenient to use the complex notation. In particular, the
Euclidean structure of $\mathbb{R}^{2}$ is transposed into the complex world
in the following way :

$\forall u=a+{\rm i}b\in\mathbb{C},\quad\forall v=a^{\prime}+{\rm
i}b^{\prime}\in\mathbb{C},\qquad u\cdot v\triangleq\langle
u,v\rangle_{\mathbb{R}^{2}}=\mbox{Re}\left(u\overline{v}\right)=aa^{\prime}+bb^{\prime}.$

Remarking that $\mathbf{n}(t,z(t,\theta))$ is real-proportional to ${\rm
i}\partial_{\theta}z(t,\theta)$, then we get the complex form of the contour
dynamics motion (1.7),

$\mbox{Im}\Big{(}\partial_{t}z(t,\theta)\overline{\partial_{\theta}z(t,\theta)}\Big{)}=\mbox{Im}\Big{(}\mathbf{v}(t,z(t,\theta))\overline{\partial_{\theta}z(t,\theta)}\Big{)}.$
(1.8)

$\blacktriangleright$ Uniformly rotating solutions.
Uniformly rotating vortex patches, also called V-states, form a particular
subclass of vortex patch solutions taking the form

$D_{t}=e^{{\rm i}\Omega t}D_{0}.$ (1.9)

Any solution of this form is rotating with a time independent angular velocity
$\Omega\in\mathbb{R}$ around its center of mass fixed at the origine. If
$\Omega=0,$ then it is stationary, otherwise it is time periodic with period
$\tfrac{2\pi}{\Omega}.$ Historically, the first example of V-states was
provided by Kirchhoff [55] who showed that for Euler equations any ellipse
with semi-axis $a$ and $b$ uniformly rotates for an angular velocity
$\Omega=\tfrac{ab}{(a+b)^{2}}.$ Observe from (1.6), (1.4) and (1.5) that the
stream function $\boldsymbol{\Psi}$ admits a radial Green kernel. Therefore,
any radial initial profile would generate a stationary solution. In
particular, the Rankine vortices given by the discs provide such examples. The
purpose of this study is to prove the existence of periodic (and quasi-
periodic) vortex patch structures close to these equilibrium states. Due to
the invariance by dilation of the model, it is sufficient to search for these
type of solutions close to the unit disc. The analytical study of existence of
V-states close to the unit disc for Euler equations goes back to the work of
Burbea [17]. We also refer to [49] for a more recent and rigorous point of
view. Indeed, combining Crandall-Rabinowitz’s Theorem B.1 and complex
analysis, one can find the existence of branches of $\mathbf{m}$-fold V-states
bifurcating from the unit disc at the angular velocities

$\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}\triangleq\frac{\mathbf{m}-1}{2\mathbf{m}}.$
(1.10)

Later on, a lot of attention has been paid to such type of solutions for
different nonlinear transport fluid models like Euler equations in the plane
or in the unit disc, generalised surface quasi-geostrophic equations
$(SQG)_{\alpha}$ and $(QGSW)_{\lambda}$ equations. Also, several topological
and regularity settings were explored and we may refer to [22, 26, 32, 33, 34,
35, 36, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 50, 65, 66]. Nevertheless, it
seems that the Euler-$\alpha$ model has been set aside from these studies.
Hence, we propose here to study the emergence of simply-connected V-states for
this model. For this task, due to the decomposition (1.6), we may emphasize
the result in [26], where they found the following angular velocities related
to the modified Bessel functions for which the bifurcation of simply-connected
V-states from the unit disc occurs

$\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{SW}}}(\lambda)\triangleq
I_{1}(\lambda)K_{1}(\lambda)-I_{\mathbf{m}}(\lambda)K_{\mathbf{m}}(\lambda).$
(1.11)

Now, we shall present the first theorem proved in this study and dealing with
periodic simply-connected patches bifurcating from the Rankine vortices.

######  Theorem 1.1.

Let $\alpha>0$ and $\beta\in(0,1).$ For any $\mathbf{m}\in\mathbb{N}^{*},$
there exists a branch of $\mathbf{m}$-fold V-states of regularity
$C^{1+\beta}$ bifurcating from the unit disc at the angular velocity

$\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha)\triangleq\frac{\mathbf{m}-1}{2\mathbf{m}}-\Big{(}I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)-I_{\mathbf{m}}\left(\tfrac{1}{\alpha}\right)K_{\mathbf{m}}\left(\tfrac{1}{\alpha}\right)\Big{)}=\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}-\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)$
(1.12)

for Euler-$\alpha$ equations (1.1).

######  Remark 1.1.

  1. (i)

Observe that the bifurcation frequencies (1.12) are exactly the superposition
of those of Euler (1.10) and $(QGSW)_{\frac{1}{\alpha}}$ (1.11) models which
is in accordance with the remark in Section 2 and the structure of the vortex
patch equation (1.8).

  2. (ii)

The following convergences hold true.

$\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha)\underset{\mathbf{m}\rightarrow\infty}{\longrightarrow}\tfrac{1}{2}-I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)\triangleq\boldsymbol{\Omega}_{\infty}^{\textnormal{\tiny{E}}}(\alpha),\qquad\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha)\underset{\alpha\rightarrow
0}{\longrightarrow}\tfrac{\mathbf{m}-1}{2\mathbf{m}}=\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}.$

  3. (iii)

The case $\mathbf{m}=1$ corresponds to a translation of the trivial solution,
see Remark 3.1.

The Theorem 1.1 is proved by using Crandall-Rabinowitz’s Theorem B.1 in the
spirit of the previous works mentioned above. For that purpose, we reformulate
the vortex patch equation (1.8) in the uniformly rotating framework with
conformal mappings. We chose to use the conformal functional setting in order
to be able to take advantage of the computations already done in [26, 49].
Nevertheless, one could also use the polar parametrization similarly to the
next result on quasi-periodic solutions. Introducing, for an initial domain
$D_{0}$ close to the unit disc $\mathbb{D}$, the conformal parametrization
$\Phi:\mathbb{C}\setminus\overline{\mathbb{D}}\rightarrow\mathbb{C}\setminus\overline{D_{0}}$
in the form

$\Phi(z)=z+f(z),\qquad f(z)=\sum_{n=0}^{\infty}\frac{a_{n}}{z^{n}},\qquad
a_{n}\in\mathbb{R},$

we can reformulate the vortex patch equation (1.8) in the uniformly rotating
context as the following equation

$\forall w\in\mathbb{T},\quad F_{\alpha}(\Omega,f)(w)=0,\qquad
F_{\alpha}(\Omega,f)(w)\triangleq\textnormal{Im}\left\\{\Big{(}\Omega\Phi(w)+I^{\textnormal{\tiny{E}}}(f)(w)+I^{\textnormal{\tiny{SW}}}(f)(\alpha,w)\Big{)}\overline{w}\overline{\Phi^{\prime}(w)}\right\\},$

with

$I^{\textnormal{\tiny{E}}}(f)(w)\triangleq\fint_{\mathbb{T}}\Phi^{\prime}(\tau)\log\big{(}|\Phi(w)-\Phi(\tau)|\big{)}d\tau,\qquad
I^{\textnormal{\tiny{SW}}}(f)(\alpha,w)\triangleq\fint_{\mathbb{T}}\Phi^{\prime}(\tau)K_{0}\left(\tfrac{1}{\alpha}|\Phi(\tau)-\Phi(w)|\right)d\tau.$

Remarking that the disc provides a line of solutions $F_{\alpha}(\Omega,0)=0$
for $\Omega\in\mathbb{R}$, we shall apply the local bifurcation theory to find
non-trivial solutions. In Proposition 3.1, we prove that the functional
$F_{\alpha}$ is of class $C^{1}$ with respect to some Hölder regularity spaces
and its linearized operator at the equilibrium has the Fredholm property and
expresses as the following Fourier multiplier

$\forall w\in\mathbb{T},\quad
d_{f}F_{\alpha}(\Omega,0)(w)=\sum_{n=0}^{\infty}a_{n}(n+1)\Big{(}\boldsymbol{\Omega}_{n+1}^{\textnormal{\tiny{E}}}(\alpha)-\Omega\Big{)}\textnormal{Im}(w^{n+1}).$

The one dimensional kernel condition for applying bifurcation theory is
ensured by the strict monotonicity of the sequence
$\big{(}\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)\big{)}_{n\in\mathbb{N}^{*}}$
which is checked using some refined estimates on modified Bessel functions,
see Lemma 3.1. Finally, the transversality condition is a direct consequence
of the Fourier decomposition of the linearized operator and is checked in
Proposition 3.2-(iv).

$\blacktriangleright$ Quasi-periodic in time vortex patches.
This topic is rather new in vortex patch dynamics and the tools used are
borrowed from KAM/Nash-Moser theories [2, 11, 57, 61]. Recall that a function
$r:\mathbb{R}\rightarrow\mathbb{R}$ is said to be quasi-periodic if there
exists $d\in\mathbb{N}^{*},$
$\widehat{r}=\widehat{r}(\varphi):\mathbb{T}^{d}\rightarrow\mathbb{R}$
continuous and $\omega\in\mathbb{R}^{d}$ such that

$r(t)=\widehat{r}(\omega t),\qquad\forall
l\in\mathbb{Z}^{d}\setminus\\{0\\},\quad\omega\cdot l\neq 0.$ (1.13)

The existence of quasi-periodic vortex patches close to the unit disc for the
QGSW equations have been proved in [51]. These solutions are obtained by
selecting the Rossby deformation length in a massive Cantor set. Similarly, in
[40], the authors used the parameter inside the equations to generate quasi-
periodic vortex patches near the Rankine vortices for $(SQG)_{\alpha}$
equations for suitable selected values of $\alpha.$ Here our work follows the
same idea relying on the free parameter $\alpha$ to obtain these solutions. We
mention two other works recently obtained for Euler equations. The first one
concerns the quasi-periodic patches close to the Rankine vortices in the unit
disc presented in [42]. The second result, which can be found in [14], is
relative to quasi-periodic patches close to the Kirchhoff ellipses. In both
cases, this is a geometrical parameter which, when taken among a Cantor-like
set of admissible parameters, allows to find quasi-periodic solutions. Our
second result reads as follows.

######  Theorem 1.2.

Let $0<\alpha_{0}<\alpha_{1}$ and $\mathbb{S}\subset\mathbb{N}^{*}.$ There
exists $\varepsilon_{0}\in(0,1)$ small enough such that for every choice of
amplitudes
${\mathtt{a}}=(\mathtt{a}_{j})_{j\in\mathbb{S}}\in(\mathbb{R}_{+}^{*})^{|\mathbb{S}|}$
enjoying the smallness condition

$|{\mathtt{a}}|\leqslant\varepsilon_{0},$

there exists a Cantor-like set
$\mathscr{C}_{\infty}\subset(\alpha_{0},\alpha_{1})$ with asymptotically full
Lebesgue measure as ${\mathtt{a}}\rightarrow 0,$ i.e.

$\lim_{{\mathtt{a}}\rightarrow
0}|\mathscr{C}_{\infty}|=\alpha_{1}-\alpha_{0},$

such that for any $\alpha\in\mathscr{C}_{\infty}$, the equation (1.3) admits a
time quasi-periodic vortex patch solution with diophantine frequency vector
${\omega}_{\tiny{\textnormal{pe}}}(\alpha,{\mathtt{a}})\triangleq(\omega_{j}(\alpha,{\mathtt{a}}))_{j\in\mathbb{S}}\in\mathbb{R}^{|\mathbb{S}|}$
and taking the form

$\displaystyle\boldsymbol{\omega}(t,\cdot)=\mathbf{1}_{D_{t}},\qquad D_{t}$
$\displaystyle=\Big{\\{}\ell e^{{\rm i}\theta},\quad\theta\in[0,2\pi],\quad
0\leqslant\ell\leqslant R(t,\theta)\Big{\\}},\qquad
R(t,\theta)=\sqrt{1+2r(t,\theta)},$ (1.14) $\displaystyle r(t,\theta)$
$\displaystyle=\sum_{j\in\mathbb{S}}{\mathtt{a}_{j}}\cos\big{(}j\theta+\omega_{j}(\alpha,{\mathtt{a}})t\big{)}+\mathtt{p}\big{(}{\omega}_{\tiny{\textnormal{pe}}}(\alpha,\mathtt{a})t,\theta\big{)}.$

The diophantine frequency vector satisfies the following asymptotic

${\mathtt{\omega}}_{\tiny{\textnormal{pe}}}(\alpha,{\mathtt{a}})\underset{{\mathtt{a}}\rightarrow
0}{\longrightarrow}(-\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha))_{j\in\mathbb{S}},$

where $\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)$ are the equilibrium
frequencies defined by

$\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\triangleq
j\Big{(}\Omega+\boldsymbol{\Omega}_{|j|}^{\textnormal{\tiny{E}}}(\alpha)\Big{)},$

with $\boldsymbol{\Omega}_{|j|}^{\textnormal{\tiny{E}}}(\alpha)$ as in (1.12).
In addition, the perturbation
$\mathtt{p}:\mathbb{T}^{|\mathbb{S}|+1}\to\mathbb{R}$ is an even function
satisfying for some large index of regularity $s,$

$\|\mathtt{p}\|_{H^{{s}}(\mathbb{T}^{|\mathbb{S}|+1},\mathbb{R})}\underset{{\mathtt{a}}\rightarrow
0}{=}o(|{\mathtt{a}}|).$

Now we shall briefly mention the key steps of the proof of Theorem 1.2 which
are similar to the ones of [40, 42, 51]. The core of the proof of Theorem 1.2
relies on Berti-Bolle theory [11] and [40, Sec. 6]. We mention here some
results which have been obtained using this theory [3, 5, 6, 12, 13, 15, 16,
29, 31]. Plugging the ansatz (1.14) into the vortex patch equation (1.8)
provides a nonlinear Hamiltonian transport PDE for the radial deformation $r$
in the form

$\partial_{t}r=\partial_{\theta}\nabla\mathscr{H}(r),$ (1.15)

with Hamiltonian $\mathscr{H}$ related to the kinetic energy and the angular
momentum. This equation satisfies the reversibility property, namely if
$(t,\theta)\mapsto r(t,\theta)$ is a solution, then so is $(t,\theta)\mapsto
r(-t,-\theta).$ After a rescalling $r\mapsto\varepsilon r$ the equation (1.15)
can be seen as a quasilinear perturbation of its linearization at the
equilibrium ($\varepsilon=0$), see (4.42). As mentioned in Lemmata 4.5 and
4.1, this latter is an integrable Hamiltonian system, namely for any finite
set of Fourier modes $\mathbb{S}\subset\mathbb{N}^{*}$ of cardinal
$d\in\mathbb{N}^{*}$, the linearized equation at the Rankine patch admits, for
almost every $\alpha$ in $(\alpha_{0},\alpha_{1}),$ reversible quasi-periodic
solutions with $d$-dimensional frequency vector
$-\omega_{\textnormal{Eq}}(\alpha)\triangleq\big{(}-\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\big{)}$
in the form

$r(t,\theta)=\sum_{j\in\mathbb{S}}r_{j}\cos\big{(}j\theta-\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)t\big{)},\qquad
r_{j}\in\mathbb{R}^{*}.$

The property (1.13) for $\omega_{\textnormal{Eq}}(\alpha)$ holds for almost
every $\alpha$ by imposing some Diophantine conditions. The measure of the
corresponding set is a consequence of the transversality condition in Lemma
4.9-(i) together with the Rüssmann Lemma 4.6. Notice that the transversality
condition is itself a consequence of the non-degeneracy of the function
$\alpha\mapsto\omega_{\textnormal{Eq}}(\alpha)$ on $[\alpha_{0},\alpha_{1}],$
see Lemma 4.8. The introduction of the Diophantine conditions imply the
invertibility of the linearized operator at the equilibrium state but with a
fixed loss of derivatives. Therefore, to find quasi-periodic solutions for the
nonlinear model, we may apply a Nash-Moser scheme. For this aim we reformulate
the problem in terms of embedded tori. This is done by splitting the phase
space $L_{0}^{2}(\mathbb{T})$ in (4.12) associated with (1.15) into tangential
$L_{\overline{\mathbb{S}}}$ and normal $L_{\perp}^{2}$ subspaces, see (4.43).
Then we introduce in (4.44) the action-angle variables $(I,\vartheta)$ by
using a symplectic polar change of coordinates for the Fourier coefficients on
the tangential subspace. Thus, any function $r\in
L^{2}\big{(}\mathbb{T}_{\varphi}^{d},L_{0}^{2}(\mathbb{T}_{\theta})\big{)}$
can be related to an embedded torus, namely

$r(\varphi,\cdot)=\mathtt{A}\big{(}i(\varphi)\big{)},\qquad
i:\begin{array}[t]{rcl}\mathbb{T}^{d}&\rightarrow&\mathbb{T}^{d}\times\mathbb{R}^{d}\times
L_{\perp}^{2}\\\
\varphi&\mapsto&\big{(}\vartheta(\varphi),I(\varphi),z(\varphi)\big{)}\end{array},\qquad\mathtt{A}:\mathbb{T}^{d}\times\mathbb{R}^{d}\times
L_{\perp}^{2}\rightarrow L_{0}^{2}(\mathbb{T}).$

Then, the search of reversible quasi-periodic solutions to (1.15) is
equivalent to looking for reversible tori solutions to

$\mathscr{F}(i,\kappa,\alpha,\omega,\varepsilon)=0,$ (1.16)

where $\mathscr{F}$ is a nonlinear functional whose complete expression can be
found in (4.48). This provides a true solution of the original problem for the
particular value

$\kappa=-\omega_{\textnormal{Eq}}(\alpha).$ (1.17)

To find non-trivial solutions of (1.16), we apply a Nash-Moser scheme. At each
step, we shall find an approximate right inverse, with nice tame estimates, of
the linearized operator $d_{i,\kappa}\mathscr{F}(i_{0})$ at the reversible
torus $i_{0}$ of the current step. Applying the Berti-Bolle theory [11] and
[40, Sec. 6], one can conjugate the linearized operator
$d_{i,\kappa}\mathscr{F}(i_{0})$ by a suitable diffeomorphism of the toroidal
phase space $\mathbb{T}^{d}\times\mathbb{R}^{d}\times L_{\perp}^{2}$ in order
to obtain a triangular system in the action-angle-normal variables up to nice
error terms. Then to solve the triangular system, it is sufficient to find an
almost approximate right inverse for the linearized operator in the normal
directions denoted $\mathscr{L}_{\perp}$ and related to the linearized
operator $\mathcal{L}_{\varepsilon r}$ of (1.15) through the relation (4.53).
To do so, we conjugate $\mathscr{L}_{\perp}$ to a constant coefficients
operator up to error terms. This is the content of the Section 4.5. First, in
Proposition 4.3 we use a KAM reduction process with quasi-periodic symplectic
change of variables to reduce the transport part of $\mathcal{L}_{\varepsilon
r}.$ Then, in Proposition 4.4 we look at the localization effects in the
normal directions to recover the reduction of the transport part for
$\mathscr{L}_{\perp}.$ This provides a diagonal operator of order 1 plus a
remainder of order $(0,-1)$ in the variables $(\varphi,\theta).$ Finally, in
Proposition 4.5 we use a KAM reduction process in the Toeplitz operators class
to reduce the remainder term. Each KAM reduction occurs for suitable
restrictions of the parameters $(\alpha,\omega)$ to a Cantor-like set. In
addition, the inversion of the final diagonal operator also requires an
extraction of parameters. The Nash-Moser process constructs a non-trivial
solution
$(\alpha,\omega)\mapsto(i_{\infty}(\alpha,\omega),\kappa_{\infty}(\alpha,\omega))$
with reversible torus $i_{\infty}$ modulo restriction of the parameters to a
Cantor set in $(\alpha,\omega)$ obtained gathering all the restrictions of all
the steps for the contruction of the almost approximate right inverses of the
linearized operators. Then, coming back to (1.17), we rigidify the frequency
$\omega$ into $\omega(\alpha,\varepsilon)$ so that

$\kappa_{\infty}\big{(}\alpha,\omega(\alpha,\varepsilon)\big{)}=-\omega_{\textnormal{Eq}}(\alpha).$
(1.18)

We mention that the introduction of the free-parameter $\kappa$ was required
to apply the Berti-Bolle theory along the scheme, more precisely to invert the
triangular system. The condition (1.18) gives a final Cantor set in $\alpha$
only that we must check it is not empty. This latter fact is obtained in
Proposition 4.8 by estimating the Lebesgue measure of the set through the
Rüssmann Lemma 4.6 together with the perturbed transversality conditions.

##  2 The stream function associated to the filtered velocity

This short section is devoted to the justification of the formula (1.6) which
is the fundamental relation for this work. Recall that the identity (1.6) was
already observable in the literature, for instance in [7, 63]. The divergence-
free property for $\mathbf{v}$ and $\mathbf{u}$ in (1.1) implies the existence
of stream functions $\boldsymbol{\Psi}$ and $\psi$ such that

$\mathbf{v}=\nabla^{\perp}\boldsymbol{\Psi},\qquad\mathbf{u}=\nabla^{\perp}\psi$

and the goal of this section is to find a nice expression for the velocity
potential $\boldsymbol{\Psi}$. Notice that, according to the third equation in
(1.1), the stream functions are linked through the relation

$\psi=(1-\alpha^{2}\Delta)\boldsymbol{\Psi}.$

Then, (1.2) implies

$\boldsymbol{\omega}=\Delta\psi=(1-\alpha^{2}\Delta)\Delta\boldsymbol{\Psi}.$

We deduce from (1.4) and (1.5) that

$\boldsymbol{\Psi}=\tfrac{-1}{\alpha^{2}}\mathbf{G}^{\textnormal{\tiny{E}}}\ast\mathbf{G}_{\frac{1}{\alpha}}^{\textnormal{\tiny{SW}}}\ast\boldsymbol{\omega}\triangleq\mathbf{G}\ast\boldsymbol{\omega}.$

Now our goal is to compute $\mathbf{G}.$ One can write

$\displaystyle\mathbf{G}(x)$
$\displaystyle=\tfrac{1}{4\pi^{2}\alpha^{2}}\int_{\mathbb{R}^{2}}K_{0}\left(\tfrac{|y|}{\alpha}\right)\log(|x-y|)\,dA(y)$
$\displaystyle=\tfrac{1}{8\pi^{2}\alpha^{2}}\int_{\mathbb{R}^{2}}K_{0}\left(\tfrac{|y|}{\alpha}\right)\log\big{(}|x-y|^{2}\big{)}dA(y)$
$\displaystyle=\tfrac{1}{8\pi^{2}\alpha^{2}}\int_{\mathbb{R}^{2}}K_{0}\left(\tfrac{|y|}{\alpha}\right)\log\left(|x|^{2}+|y|^{2}-2x\cdot
y\right)dA(y),$

where $dA$ denotes the planar Lebesgue measure. Writing
$x=(R\cos(\theta),R\sin(\theta))$, then a polar change of variables yields

$\displaystyle\mathbf{G}(x)$
$\displaystyle=\tfrac{1}{8\pi^{2}\alpha^{2}}\int_{0}^{\infty}\int_{0}^{2\pi}rK_{0}\left(\tfrac{r}{\alpha}\right)\log\left(R^{2}+r^{2}-2Rr\cos(\theta-\eta)\right)\,drd\eta$
$\displaystyle=\tfrac{\log(R)}{2\pi\alpha}\int_{0}^{\infty}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)dr+\tfrac{1}{8\pi^{2}\alpha}\int_{0}^{\infty}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)\mathtt{I_{P}}\left(\tfrac{r}{R}\right)dr,$

where $\mathtt{I_{P}}$ denotes the Poisson integral defined by

$\forall
x\in\mathbb{R},\quad\mathtt{I_{P}}(x)=\int_{-\pi}^{\pi}\log\big{(}1+x^{2}-2x\cos(\eta)\big{)}d\eta.$

It is well-known, see for instance [23], that the Poisson integral admits the
following explicit formula

$\left\\{\begin{array}[]{ll}\mathtt{I_{P}}(x)=0&\textnormal{if }|x|\leqslant
1\\\ \mathtt{I_{P}}(x)=4\pi\log(|x|)&\textnormal{if }|x|>1.\end{array}\right.$

Therefore,

$\displaystyle\mathbf{G}(x)$
$\displaystyle=\tfrac{\log(R)}{2\pi\alpha}\int_{0}^{\infty}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)dr+\tfrac{1}{2\pi\alpha}\int_{R}^{\infty}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)\log\left(\tfrac{r}{R}\right)dr$
$\displaystyle=\tfrac{\log(R)}{2\pi\alpha}\int_{0}^{R}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)dr+\tfrac{1}{2\pi\alpha}\int_{R}^{\infty}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)\log\left(r\right)dr.$

A change of variables together with (A.3), (A.7) and (A.8) give

$\tfrac{\log(R)}{2\pi\alpha}\int_{0}^{R}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)dr=\tfrac{\log(R)}{2\pi}\int_{0}^{\frac{R}{\alpha}}uK_{0}(u)du=\tfrac{\log(R)}{2\pi}\Big{(}1-\tfrac{R}{\alpha}K_{1}\left(\tfrac{R}{\alpha}\right)\Big{)}$

and

$\displaystyle\tfrac{1}{2\pi\alpha}\int_{R}^{\infty}\tfrac{r}{\alpha}K_{0}\left(\tfrac{r}{\alpha}\right)\log\left(r\right)dr$
$\displaystyle=\tfrac{\log(\alpha)}{2\pi}\int_{\frac{R}{\alpha}}^{\infty}uK_{0}\left(u\right)du+\tfrac{1}{2\pi}\int_{\frac{R}{\alpha}}^{\infty}uK_{0}\left(u\right)\log\left(u\right)dr$
$\displaystyle=\tfrac{R\log(\alpha)}{2\pi\alpha}K_{1}\left(\tfrac{R}{\alpha}\right)+\tfrac{1}{2\pi}\int_{\frac{R}{\alpha}}^{\infty}uK_{0}\left(u\right)\log\left(u\right)du.$

An integration by parts in the last integral together with (A.8) lead to

$\displaystyle\tfrac{1}{2\pi}\int_{\frac{R}{\alpha}}^{\infty}uK_{0}\left(u\right)\log\left(u\right)du$
$\displaystyle=\tfrac{R}{2\pi\alpha}K_{1}\left(\tfrac{R}{\alpha}\right)\log\left(\tfrac{R}{\alpha}\right)+\tfrac{1}{2\pi}\int_{\frac{R}{\alpha}}^{\infty}K_{1}\left(u\right)du$
$\displaystyle=\tfrac{R}{2\pi\alpha}K_{1}\left(\tfrac{R}{\alpha}\right)\log\left(R\right)-\tfrac{R\log\left(\alpha\right)}{2\pi\alpha}K_{1}\left(\tfrac{R}{\alpha}\right)+\tfrac{1}{2\pi}K_{0}\left(\tfrac{R}{\alpha}\right).$

Gathering the foregoing computations and reminding that $R=|x|$ we obtain

$\mathbf{G}(x)=\tfrac{1}{2\pi}\log(|x|)+\tfrac{1}{2\pi}K_{0}\left(\tfrac{|x|}{\alpha}\right)=\mathbf{G}^{\textnormal{\tiny{E}}}(x)-\mathbf{G}_{\frac{1}{\alpha}}^{\textnormal{\tiny{SW}}}(x).$
(2.1)

Notice that we recover a radial function as a convolution of two radial
functions. In particular, any radial profil gives a stationary solution for
the active scalar equation (1.3) satisfied by $\boldsymbol{\omega}.$

##  3 Periodic rigid motion

This section is devoted to the proof of the Theorem 1.1 which is an
application of the Crandall-Rabinowitz’s Theorem B.1. This latter is applied
to a reformulation of the vortex patch equation (1.8) in the uniformly
rotating framework.

###  3.1 Function spaces and reformulation of the vortex patch equation

The goal of this subsection is to set up the contour dynamics equation for the
Euler-$\alpha$ V-states near the Rankine vortices. Consider an ansatz (1.9)
with simply-connected domain $D_{0}$ and rotating uniformly with some angular
velocity $\Omega.$ At time $t\geqslant 0,$ the boundary $\partial D_{t}$ can
be parametrized by

$z(t,\theta)=e^{{\rm i}\Omega t}z(0,\theta),\qquad\theta\in[0,2\pi],\qquad
z(0,0)=z(0,2\pi).$ (3.1)

With this parametrization, the left hand side of (1.8) becomes

$\mbox{Im}\Big{(}\partial_{t}z(t,\theta)\overline{\partial_{\theta}z(t,\theta)}\Big{)}=\Omega\,\mbox{Re}\Big{(}z(0,\theta)\overline{\partial_{\theta}z(0,\theta)}\Big{)}.$
(3.2)

As for the computation of the right hand-side of (1.8), it is obtained from
the Biot-Savart law. Note that the velocity potential $\boldsymbol{\Psi}$
given by (1.6) writes in this context

$\boldsymbol{\Psi}(t,z)=\tfrac{1}{2\pi}\int_{D_{t}}\log(|z-\xi|)dA(\xi)+\tfrac{1}{2\pi}\int_{D_{t}}K_{0}\left(\tfrac{1}{\alpha}|z-\xi|\right)dA(\xi).$

Notice that the real notation $x\in\mathbb{R}^{2}$ has been replaced by the
complex notation $z\in\mathbb{C}.$ To get the Biot-Savart law we shall use
Stokes’ Theorem in complex notation

$2{\rm
i}\int_{D}\partial_{\overline{\xi}}f(\xi,\overline{\xi})dA(\xi)=\int_{\partial
D}f(\xi,\overline{\xi})d\xi.$

Hence, by making the identification $2{\rm
i}\partial_{\overline{z}}=\nabla^{\perp}$ leading to $\mathbf{v}(t,z)=2{\rm
i}\partial_{\overline{z}}\boldsymbol{\Psi}(t,z)$ one deduces, after a
regularization procedure similar to the one explained in the proof of [42,
Lem. 2.1], that

$\mathbf{v}(t,z)=-\frac{1}{2\pi}\int_{\partial
D_{t}}\log(|z-\xi|)d\xi-\frac{1}{2\pi}\int_{\partial
D_{t}}K_{0}\left(\tfrac{1}{\alpha}|z-\xi|\right)d\xi.$ (3.3)

One easily obtains from (3.1), (3.3) and change of variables,

$\displaystyle\mathbf{v}\big{(}t,z(t,\theta)\big{)}=e^{{\rm i}\Omega
t}\mathbf{v}\big{(}0,z(0,\theta)\big{)}.$ (3.4)

Thus, combining (3.4) and (3.1), we obtain

$\mbox{Im}\Big{(}\mathbf{v}\big{(}t,z(t,\theta)\big{)}\overline{\partial_{\theta}z(t,\theta)}\Big{)}=\mbox{Im}\Big{(}\mathbf{v}\big{(}0,z(0,\theta)\big{)}\overline{\partial_{\theta}z(0,\theta)}\Big{)}.$
(3.5)

Gathering (3.2) and (3.5), the equation (1.8) becomes

$\Omega\mbox{Re}\Big{(}z(0,\theta)\overline{\partial_{\theta}z(0,\theta)}\Big{)}=\mbox{Im}\Big{(}\mathbf{v}(0,z(0,\theta))\overline{\partial_{\theta}z(0,\theta)}\Big{)}.$
(3.6)

Therefore, denoting $z^{\prime}$ a tangent vector to the boundary $\partial
D_{0}$ at the point $z,$ one gets from (3.3) and (3.6) that for any
$z\in\partial D_{0},$

$\Omega\mbox{Re}\left(z\overline{z^{\prime}}\right)+\mbox{Im}\left(\left[\frac{1}{2\pi}\int_{\partial
D_{0}}\log\big{(}|z-\xi|\big{)}d\xi+\frac{1}{2\pi}\int_{\partial
D_{0}}K_{0}\big{(}\tfrac{1}{\alpha}|z-\zeta|\big{)}d\zeta\right]\overline{z^{\prime}}\right)=0.$
(3.7)

In accordance with the previous works in the field beginning with the one of
Burbea [17], we should rewrite the equation (3.7) by using conformal mappings.
For that purpose we shall now present the function spaces used throughout this
first part on periodic solutions. Notice that we shall identify
$2\pi$-periodic $g:\mathbb{R}\rightarrow\mathbb{C}$ functions with functions
$f:\mathbb{T}\rightarrow\mathbb{C}$ defined on the torus
$\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\cong\mathbb{U}$ through the relation

$f(w)=g(\theta),\quad w=e^{{\rm i}\theta}.$

In this part, we shall consider the following notation for the mean value line
integral of any continuous function $f$ defined on the torus $\mathbb{T}$

$\fint_{\mathbb{T}}f(\tau)d\tau\triangleq\frac{1}{2{\rm
i}\pi}\int_{\mathbb{T}}f(\tau)d\tau\triangleq\frac{1}{2\pi}\int_{0}^{2\pi}f\big{(}e^{{\rm
i}\theta}\big{)}e^{{\rm i}\theta}d\theta.$

Now, we introduce the Hölder spaces on the unit circle. Given $\beta\in(0,1),$
we denote by $C^{\beta}(\mathbb{T})$ the space of continuous functions $f$
such that

$\|f\|_{C^{\beta}(\mathbb{T})}\triangleq\|f\|_{L^{\infty}(\mathbb{T})}+\sup_{\underset{\tau\neq
w}{(\tau,w)\in\mathbb{T}^{2}}}\frac{|f(\tau)-f(w)|}{|\tau-w|^{\beta}}<+\infty$

and we denote by $C^{1+\beta}(\mathbb{T})$ the space of $C^{1}$ functions with
$\beta$-Hölder continuous derivative such that

$\|f\|_{C^{1+\beta}(\mathbb{T})}\triangleq\|f\|_{L^{\infty}(\mathbb{T})}+\Big{\|}\frac{df}{dw}\Big{\|}_{C^{\beta}(\mathbb{T})}<+\infty.$

For $\beta\in(0,1),$ we set

$\displaystyle X^{1+\beta}$ $\displaystyle\triangleq\left\\{f\in
C^{1+\beta}(\mathbb{T})\quad\textnormal{s.t.}\quad\forall
w\in\mathbb{T},\,f(w)=\sum_{n=0}^{+\infty}f_{n}\overline{w}^{n},\,f_{n}\in\mathbb{R}\right\\},$
$\displaystyle Y^{\beta}$ $\displaystyle\triangleq\left\\{g\in
C^{\beta}(\mathbb{T})\quad\textnormal{s.t.}\quad\forall
w\in\mathbb{T},\,g(w)=\sum_{n=1}^{+\infty}g_{n}e_{n}(w),\,g_{n}\in\mathbb{R}\right\\},\qquad
e_{n}(w)\triangleq\mbox{Im}(w^{n}).$

Remark that the $\mathbf{m}$-fold symmetry property can be translated in the
functional spaces as follows

$\displaystyle X_{\mathbf{m}}^{1+\beta}$ $\displaystyle\triangleq\left\\{f\in
X^{1+\beta}\quad\textnormal{s.t.}\quad\forall
w\in\mathbb{T},\,f(w)=\sum_{n=1}^{+\infty}f_{n\mathbf{m}-1}\overline{w}^{n\mathbf{m}-1}\right\\},$
$\displaystyle Y_{\mathbf{m}}^{\beta}$ $\displaystyle\triangleq\left\\{g\in
Y^{\beta}\quad\textnormal{s.t.}\quad\forall
w\in\mathbb{T},\,g(w)=\sum_{n=1}^{+\infty}g_{n\mathbf{m}}e_{n\mathbf{m}}(w)\right\\}.$

We shall also consider the following balls of radius $r>0$ in $X^{1+\beta}$
and $X_{\mathbf{m}}^{1+\beta}$, respectively

$B_{r}^{1+\beta}\triangleq\Big{\\{}f\in
X^{1+\beta}\quad\textnormal{s.t.}\quad\|f\|_{C^{1+\beta}(\mathbb{T})}<r\Big{\\}},\qquad
B_{r,\mathbf{m}}^{1+\beta}\triangleq B_{r}^{1+\beta}\cap
X_{\mathbf{m}}^{1+\beta}.$

Based on the Riemann mapping Theorem, we may parametrize the boundary of
$D_{0}$ by considering the conformal mapping
$\Phi:\mathbb{C}\setminus\overline{\mathbb{D}}\rightarrow\mathbb{C}\setminus\overline{D_{0}}$
given by

$\Phi(z)\triangleq z+f(z),\qquad
f(z)=\sum_{n=0}^{\infty}\frac{a_{n}}{z^{n}},\qquad a_{n}\in\mathbb{R}.$

Here, $f$ is in $B_{r}^{1+\beta}$ for small $r$. We have
$\Phi(\mathbb{T})=\partial D_{0}$ and we mention that the link between the
regularity of the mapping and of the boundary is given by Kellogg-
Warschawski’s Theorem [69, 64]. For $w\in\mathbb{T},$ a tangent vector to the
boundary $\partial D_{0}$ at the point $z=\Phi(w)$ is given by

$\overline{z^{\prime}}=-{\rm i}\overline{w}\overline{\Phi^{\prime}(w)}.$

Plugging this into (3.7) and using the change of variables $\xi=\Phi(\tau)$
give

$\forall w\in\mathbb{T},\quad F_{\alpha}(\Omega,f)(w)=0,\qquad
F_{\alpha}(\Omega,f)(w)\triangleq\textnormal{Im}\left\\{\Big{(}\Omega\Phi(w)+I^{\textnormal{\tiny{E}}}(f)(w)+I^{\textnormal{\tiny{SW}}}(f)(\alpha,w)\Big{)}\overline{w}\overline{\Phi^{\prime}(w)}\right\\},$
(3.8)

with

$I^{\textnormal{\tiny{E}}}(f)(w)\triangleq\fint_{\mathbb{T}}\Phi^{\prime}(\tau)\log\big{(}|\Phi(w)-\Phi(\tau)|\big{)}d\tau,\qquad
I^{\textnormal{\tiny{SW}}}(f)(\alpha,w)\triangleq\fint_{\mathbb{T}}\Phi^{\prime}(\tau)K_{0}\left(\tfrac{1}{\alpha}|\Phi(w)-\Phi(\tau)|\right)d\tau.$

Observe that the functional $F_{\alpha}$ makes appear a term
$I^{\textnormal{\tiny{E}}}$ associated with the Euler dynamics and a term
$I^{\textnormal{\tiny{SW}}}$ corresponding to the QGSW equations.

###  3.2 Regularity aspects and structure of the linearized operator

Our next task is to study some regularity properties for the functional
$F_{\alpha}$ defined in (3.8), look for the structure of its linearized
operator at the Rankine vortex and check some monotonicity property for its
spectrum. We first remark that the linearized operator at the equilibrium
state acts as a Fourier multiplier according to the functions spaces
introduced in Section 3.1. More precisely, we have the following result.

######  Proposition 3.1.

Let $\alpha>0.$ Then the following properties hold true.

  1. 1.

There exists $r>0$ such that for any $\beta\in(0,1)$, the following hold true.

    1. (i)

$F_{\alpha}:\mathbb{R}\times B_{r}^{1+\beta}\rightarrow Y^{\beta}$ is well-
defined and of class $C^{1}.$

    2. (ii)

For any $\mathbf{m}\in\mathbb{N}^{*},$ the restriction
$F_{\alpha}:\mathbb{R}\times B_{r,\mathbf{m}}^{1+\beta}\rightarrow
Y_{\mathbf{m}}^{\beta}$ is well-defined.

    3. (iii)

The partial derivative $\partial_{\Omega}d_{f}F_{\alpha}:\mathbb{R}\times
B_{r}^{1+\beta}\rightarrow\mathcal{L}\left(X^{1+\beta},Y^{\beta}\right)$
exists and is continuous.

    4. (iv)

For any $\Omega\in\mathbb{R},$ one has $F_{\alpha}(\Omega,0)=0.$

  2. 2.

Let
$\Omega\in\mathbb{R}\setminus\big{\\{}\tfrac{1}{2}-I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)\big{\\}}.$
Then the operator $d_{f}F_{\alpha}(\Omega,0):X^{1+\beta}\rightarrow Y^{\beta}$
is Fredholm with index $0$.
In addition, for $h\in X^{1+\beta}$ writing

$\forall w\in\mathbb{T},\quad
h(w)=\sum_{n=0}^{\infty}a_{n}\overline{w}^{n},\quad a_{n}\in\mathbb{R},$

we have

$\forall w\in\mathbb{T},\quad
d_{f}F_{\alpha}(\Omega,0)(w)=\sum_{n=0}^{\infty}a_{n}(n+1)\Big{(}\tfrac{n}{2(n+1)}-\boldsymbol{\Omega}_{n+1}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)-\Omega\Big{)}e_{n+1}(w),$
(3.9)

with $\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{SW}}}$ as in (1.11).

###### Proof.

1\. (i) The proof of the regularity is now classical. We refer the reader to
[49] for the computations associated with the Euler part and to [26] for the
computations associated with the QGSW part.
(ii) For $f\in B_{r,\mathbf{m}}^{1+\beta},$ the following identities hold

$\forall w\in\mathbb{T},\qquad\Phi\left(e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}w\right)=e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}\Phi\left(w\right),\qquad\Phi^{\prime}\left(e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}w\right)=\Phi^{\prime}\left(w\right).$

Therefore, the change of variables $\tau\mapsto e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}\tau$ implies

$I^{\textnormal{\tiny{E}}}(f)\left(e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}w\right)=e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}I^{\textnormal{\tiny{E}}}(f)(w),\qquad
I^{\textnormal{\tiny{SW}}}(f)\left(\alpha,e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}w\right)=e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}I^{\textnormal{\tiny{SW}}}(f)(\alpha,w).$

Consequently,

$\forall w\in\mathbb{T},\quad F_{\alpha}(\Omega,f)\big{(}e^{\frac{2{\rm
i}\pi}{\mathbf{m}}}w\big{)}=F_{\alpha}(\Omega,f)(w).$

This symmetry property proves the desired result.
(iii) One readily has

$\partial_{\Omega}d_{f}F_{\alpha}(\Omega,f)[h](w)=\textnormal{Im}\left\\{\overline{h^{\prime}(w)}\overline{w}\Phi(w)+h(w)\overline{w}\overline{\Phi^{\prime}(w)}\right\\}.$

Hence, for $(f_{j},g_{j})\in(B_{r}^{1+\alpha})^{2}$ and $h\in
C^{1+\alpha}(\mathbb{T})$, we get

$\Big{\|}\partial_{\Omega}d_{f}F_{\alpha}(\Omega,f_{j})[h]-\partial_{\Omega}d_{f}F_{\alpha}(\Omega,g_{j})[h]\Big{\|}_{C^{\alpha}(\mathbb{T})}\lesssim\|f_{j}-g_{j}\|_{C^{1+\alpha}(\mathbb{T})}\|h\|_{C^{1+\alpha}(\mathbb{T})}.$

This proves the continuity of
$\partial_{\Omega}d_{f}F_{\alpha}:\mathbb{R}\times
B_{r}^{1+\alpha}\rightarrow\mathcal{L}(X^{1+\alpha},Y^{\alpha}).$
(iv) For $w\in\mathbb{T},$ using the change of variable $\tau\mapsto w\tau$,
the fact that $|w|=1$ and the definition of the line integral, we get

$\displaystyle I^{\textnormal{\tiny{E}}}(0)(w)$
$\displaystyle=\fint_{\mathbb{T}}\log\big{(}|w-\tau|\big{)}d\tau$
$\displaystyle=w\fint_{\mathbb{T}}\log\big{(}|1-\tau|\big{)}d\tau$
$\displaystyle=\frac{w}{2\pi}\int_{0}^{2\pi}\log\big{(}|1-e^{{\rm
i}\theta}|\big{)}e^{{\rm i}\theta}d\theta.$

Now remark that

$|1-e^{{\rm
i}\theta}|^{2}=2\big{(}1-\cos(\theta)\big{)}=4\sin^{2}\big{(}\tfrac{\theta}{2}\big{)}.$

In particular, this latter quantity is even in $\theta.$ Hence, we infer

$I^{\textnormal{\tiny{E}}}(0)(w)=\frac{w}{4\pi}\int_{0}^{2\pi}\log\big{(}\sin^{2}\big{(}\tfrac{\theta}{2}\big{)}\big{)}\cos(\theta)d\theta=-\frac{w}{2}.$
(3.10)

The last identity is a consequence of the following formula which can be found
for instance in [22, Lem. A.3].

$\tfrac{1}{2\pi}\int_{0}^{2\pi}\log\big{(}\sin^{2}\left(\tfrac{\theta}{2}\right)\big{)}\cos(n\theta)d\theta=-\frac{1}{n}.$
(3.11)

Similarly,

$I^{\textnormal{\tiny{SW}}}(0)(w)=\frac{w}{2\pi}\int_{0}^{2\pi}K_{0}\big{(}\tfrac{2}{\alpha}\big{|}\sin\big{(}\tfrac{\theta}{2}\big{)}\big{|}\big{)}\cos(\theta)d\theta=wI_{1}\big{(}\tfrac{1}{\alpha}\big{)}K_{1}\big{(}\tfrac{1}{\alpha}\big{)}.$
(3.12)

The last identity follows from the following identity which can be found in
the proof of [51, Lem. 3.2].

$\frac{1}{2\pi}\int_{0}^{2\pi}K_{0}\big{(}\tfrac{2}{\alpha}\sin\big{(}\tfrac{\theta}{2}\big{)}\big{)}\cos(n\theta)d\theta=I_{n}\left(\tfrac{1}{\alpha}\right)K_{n}\left(\tfrac{1}{\alpha}\right).$
(3.13)

Inserting (3.10) and (3.12) into (3.8) and using $w\overline{w}=|w|^{2}=1$
gives the desired result.
2. We can write

$\displaystyle d_{f}F_{\alpha}(\Omega,0)[h](w)$
$\displaystyle=\mathcal{I}[h](w)+\mathcal{K}[h](w),$

with

$\displaystyle\mathcal{I}[h](w)$
$\displaystyle\triangleq\big{(}\Omega-\tfrac{1}{2}+I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)\big{)}\textnormal{Im}\left\\{\overline{h^{\prime}(w)}\right\\},$
$\displaystyle\mathcal{K}[h](w)$
$\displaystyle\triangleq\textnormal{Im}\left\\{\big{(}\Omega
h(w)+d_{f}I^{\textnormal{\tiny{E}}}(0)[h](w)+d_{f}I^{\textnormal{\tiny{SW}}}(0)[h](w)\big{)}\overline{w}\right\\}.$

One obviously has that $\mathcal{I}:X^{1+\beta}\rightarrow Y^{\beta}$ is an
isomorphism provided that
$\Omega\neq\tfrac{1}{2}-I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right).$
One readily has

$\displaystyle d_{f}I^{\textnormal{\tiny{E}}}(0)[h](w)$
$\displaystyle=\fint_{\mathbb{T}}h^{\prime}(\tau)\log\big{(}|w-\tau|\big{)}d\tau+\fint_{\mathbb{T}}\frac{h(w)-h(\tau)}{2\big{(}w-\tau\big{)}}d\tau+\fint_{\mathbb{T}}\frac{\overline{h(w)}-\overline{h(\tau)}}{2\big{(}\overline{w}-\overline{\tau}\big{)}}d\tau$

and

$\displaystyle d_{f}I^{\textnormal{\tiny{SW}}}(0)[h](w)$
$\displaystyle=\fint_{\mathbb{T}}h^{\prime}(\tau)K_{0}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}d\tau$
$\displaystyle\quad+\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}h(w)-h(\tau)\big{)}\big{(}\overline{w}-\overline{\tau}\big{)}}{2\alpha|w-\tau|}d\tau$
$\displaystyle\quad+\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}\overline{h(w)}-\overline{h(\tau)}\big{)}\big{(}w-\tau\big{)}}{2\alpha|w-\tau|}d\tau.$

Now, combining (A.2) and (A.4), we can write

$K_{0}(z)=-\log(z)+\mathtt{F}_{1}(z),\qquad
K_{0}^{\prime}(z)=-K_{1}(z)=-\tfrac{1}{z}+\mathtt{F}_{2}(z),\qquad\mathtt{F}_{1},\mathtt{F}_{1}^{\prime},\mathtt{F}_{2},\mathtt{F}_{2}^{\prime}\textnormal{
bounded at }0.$

Gathering the foregoing computations leads to

$\displaystyle
d_{f}I^{\textnormal{\tiny{E}}}(0)[h](w)+d_{f}I^{\textnormal{\tiny{SW}}}(0)[h](w)$
$\displaystyle=\fint_{\mathbb{T}}h^{\prime}(\tau)\mathtt{F}_{1}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}d\tau+\fint_{\mathbb{T}}\mathtt{F}_{2}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}h(w)-h(\tau)\big{)}\big{(}\overline{w}-\overline{\tau}\big{)}}{2\alpha|w-\tau|}d\tau$
$\displaystyle\quad+\fint_{\mathbb{T}}\mathtt{F}_{2}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}\overline{h(w)}-\overline{h(\tau)}\big{)}\big{(}w-\tau\big{)}}{2\alpha|w-\tau|}d\tau.$

This can be written in the form

$d_{f}I^{\textnormal{\tiny{E}}}(0)[h](w)+d_{f}I^{\textnormal{\tiny{SW}}}(0)[h](w)=\mathcal{T}_{\mathtt{K}_{1}}(h^{\prime})(w)+\mathcal{T}_{\mathtt{K_{2}}}(1)(w)+\mathcal{T}_{\overline{\mathtt{K}_{2}}}(1)(w),$

with

$\displaystyle\mathcal{T}_{\mathtt{K}}(u)(w)$
$\displaystyle\triangleq\fint_{\mathbb{T}}u(\tau)\mathtt{K}(w,\tau)d\tau,$
$\displaystyle\mathtt{K}_{1}(w,\tau)$
$\displaystyle\triangleq\mathtt{F}_{1}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)},$
$\displaystyle\mathtt{K}_{2}(w,\tau)$
$\displaystyle\triangleq\mathtt{F}_{2}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}h(w)-h(\tau)\big{)}\big{(}\overline{w}-\overline{\tau}\big{)}}{2\alpha|w-\tau|}d\tau.$

Using the boundedness of $\mathtt{F}_{1}$, $\mathtt{F}_{1}^{\prime}$,
$\mathtt{F}_{2}$ and $\mathtt{F}_{2}^{\prime}$ close to zero, we get

$\begin{array}[]{ll}|\mathtt{K}_{1}(w,\tau)|\leqslant
C,&\displaystyle|\partial_{w}\mathtt{K}_{1}(w,\tau)|\leqslant\frac{C}{|w-\tau|},\vspace{0.1cm}\\\
|\mathtt{K}_{2}(w,\tau)|\leqslant
C\|h\|_{C^{1+\beta}(\mathbb{T})},&\displaystyle|\partial_{w}\mathtt{K}_{2}(w,\tau)|\leqslant\frac{C\|h\|_{C^{1+\beta}(\mathbb{T})}}{|w-\tau|}\cdot\end{array}$

Therefore, applying Lemma B.1, one obtains the continuity of
$d_{f}I^{\textnormal{\tiny{E}}}(0)+d_{f}I^{\textnormal{\tiny{SW}}}(0):C^{1+\beta}(\mathbb{T})\rightarrow
C^{\delta}(\mathbb{T})$ for any $\beta\leqslant\delta<1.$ Coming back to the
definition of $\mathcal{K}$, we deduce that for any $\beta\leqslant\delta<1,$
the operator $\mathcal{K}:X^{1+\beta}\rightarrow Y^{\delta}$ is continuous
(the symmetry property being easily obtained by straightforward calculations
and changes of variables in the integrals). Hence, using the compact embedding
of $C^{\delta}(\mathbb{T})$ into $C^{\beta}(\mathbb{T})$ for $\beta<\delta<1,$
one deduces that the operator $\mathcal{K}:X^{1+\beta}\rightarrow Y^{\beta}$
is compact. Consequently, $d_{f}F_{\alpha}(\Omega,0):X^{1+\beta}\rightarrow
Y^{\beta}$ is a Fredholm operator with index $0$. Now, we shall compute the
Fourier representation of this operator. Fix

$h(w)=\sum_{n=0}^{\infty}a_{n}\overline{w}^{n}\in X^{1+\beta}.$

First observe that

$\mathcal{I}[h](w)=-\big{(}\Omega-\tfrac{1}{2}+I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)\big{)}\sum_{n=0}^{\infty}na_{n}e_{n+1}(w)$
(3.14)

and

$\textnormal{Im}\left\\{\Omega
h(w)\overline{w}\right\\}=-\Omega\sum_{n=0}^{\infty}a_{n}e_{n+1}(w).$ (3.15)

The next task is to compute $d_{f}I^{\textnormal{\tiny{E}}}(0).$ One has

$\displaystyle\fint_{\mathbb{T}}h^{\prime}(\tau)\log\big{(}|w-\tau|\big{)}d\tau$
$\displaystyle=w\fint_{\mathbb{T}}h^{\prime}(\omega\tau)\log\big{(}|1-\tau|\big{)}d\tau$
$\displaystyle=-\sum_{n=1}^{\infty}na_{n}\overline{\omega}^{n}\fint_{\mathbb{T}}\overline{\tau}^{n+1}\log\big{(}|1-\tau|\big{)}d\tau.$

The last integral can be computed as follows by using (3.11)

$\displaystyle\fint_{\mathbb{T}}\overline{\tau}^{n+1}\log\big{(}|1-\tau|\big{)}d\tau$
$\displaystyle=\tfrac{1}{2\pi}\int_{0}^{2\pi}e^{-{\rm
i}n\theta}\log\big{(}|1-e^{{\rm i}\theta}|\big{)}d\theta$
$\displaystyle=\tfrac{1}{4\pi}\int_{0}^{2\pi}\log\big{(}\sin^{2}\left(\tfrac{\theta}{2}\right)\big{)}\cos(n\theta)d\theta$
$\displaystyle=-\tfrac{1}{2n}.$

Hence,

$\fint_{\mathbb{T}}h^{\prime}(\tau)\log\big{(}|w-\tau|\big{)}d\tau=\sum_{n=1}^{\infty}\tfrac{a_{n}}{2}\overline{w}^{n}.$

Similarly, we obtain

$\displaystyle\fint_{\mathbb{T}}\frac{h(w)-h(\tau)}{2\big{(}w-\tau\big{)}}d\tau$
$\displaystyle=\sum_{n=0}^{\infty}\tfrac{a_{n}}{2}\fint_{\mathbb{T}}\frac{\overline{w}^{n}-\overline{\tau}^{n}}{w-\tau}=-\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\tfrac{a_{n}\overline{w}^{k+1}}{2}\fint_{\mathbb{T}}\overline{\tau}^{n-k}d\tau=-\sum_{n=1}^{\infty}\tfrac{a_{n}}{2}\overline{w}^{n}$
$\displaystyle\fint_{\mathbb{T}}\frac{\overline{h(w)}-\overline{h(\tau)}}{2\big{(}\overline{w}-\overline{\tau}\big{)}}d\tau$
$\displaystyle=\sum_{n=0}^{\infty}\tfrac{a_{n}}{2}\fint_{\mathbb{T}}\frac{w^{n}-\tau^{n}}{\overline{w}-\overline{\tau}}=-\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\tfrac{a_{n}w^{k+1}}{2}\fint_{\mathbb{T}}\tau^{n-k}d\tau=0.$

Consequently,

$d_{f}I^{\textnormal{\tiny{E}}}(0)=0.$ (3.16)

Now, let us turn to the calculation of
$\textnormal{Im}\left\\{d_{f}I^{\textnormal{\tiny{SW}}}(0)[h](w)\overline{w}\right\\}.$
The formula (3.13) gives

$\displaystyle\fint_{\mathbb{T}}h^{\prime}(\tau)K_{0}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}d\tau$
$\displaystyle=-\sum_{n=1}^{\infty}na_{n}\overline{w}^{n}\fint_{\mathbb{T}}\overline{\tau}^{n+1}K_{0}\big{(}\tfrac{1}{\alpha}|1-\tau|\big{)}d\tau$
$\displaystyle=-\sum_{n=1}^{\infty}\tfrac{na_{n}\overline{w}^{n}}{2\pi}\int_{0}^{2\pi}K_{0}\big{(}\tfrac{2}{\alpha}\sin\big{(}\tfrac{\theta}{2}\big{)}\big{)}\cos(n\theta)d\theta$
$\displaystyle=-\sum_{n=1}^{\infty}nI_{n}\big{(}\tfrac{1}{\alpha}\big{)}K_{n}\big{(}\tfrac{1}{\alpha}\big{)}a_{n}\overline{w}^{n}.$

Thus

$\textnormal{Im}\left\\{\overline{w}\fint_{\mathbb{T}}h^{\prime}(\tau)K_{0}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}d\tau\right\\}=\sum_{n=1}^{\infty}nI_{n}\big{(}\tfrac{1}{\alpha}\big{)}K_{n}\big{(}\tfrac{1}{\alpha}\big{)}a_{n}e_{n+1}(w).$

On the other hand

$\displaystyle\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}h(w)-h(\tau)\big{)}\big{(}\overline{w}-\overline{\tau}\big{)}+\big{(}\overline{h(w)}-\overline{h(\tau)}\big{)}\big{(}w-\tau\big{)}}{2\alpha|w-\tau|}d\tau$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\left(\overline{w}^{n}\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|1-\tau|\big{)}\frac{\big{(}\overline{\tau}^{n}-1\big{)}\big{(}\overline{\tau}-1\big{)}}{2\alpha|1-\tau|}d\tau+w^{n}\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|1-\tau|\big{)}\frac{\big{(}\tau^{n}-1\big{)}\big{(}\tau-1\big{)}}{2\alpha|1-\tau|}d\tau\right).$

One can easily check that the above integrals are real and therefore

$\displaystyle\textnormal{Im}\left\\{\overline{w}\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|w-\tau|\big{)}\frac{\big{(}h(w)-h(\tau)\big{)}\big{(}\overline{w}-\overline{\tau}\big{)}+\big{(}\overline{h(w)}-\overline{h(\tau)}\big{)}\big{(}w-\tau\big{)}}{2\alpha|w-\tau|}d\tau\right\\}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\left(\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|1-\tau|\big{)}\frac{\big{(}\tau^{n}-1\big{)}\big{(}\tau-1\big{)}-\big{(}\overline{\tau}^{n}-1\big{)}\big{(}\overline{\tau}-1\big{)}}{2\alpha|1-\tau|}d\tau\right)e_{n+1}(w)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\left(\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|1-\tau|\big{)}\frac{\big{(}\tau^{n+1}-\overline{\tau}^{n+1}\big{)}-\big{(}\tau^{n}-\overline{\tau}^{n}\big{)}-\big{(}\tau-\overline{\tau}\big{)}}{2\alpha|1-\tau|}d\tau\right)e_{n+1}(w).$

Now, symmetry arguments together with an integration by parts and (3.13) imply
for any $k\in\mathbb{N}^{*}$

$\displaystyle\fint_{\mathbb{T}}K_{0}^{\prime}\big{(}\tfrac{1}{\alpha}|1-\tau|\big{)}\frac{\tau^{k}-\overline{\tau}^{k}}{2\alpha|1-\tau|}d\tau$
$\displaystyle=\frac{-1}{4\pi\alpha}\int_{0}^{2\pi}K_{0}^{\prime}\big{(}\tfrac{2}{\alpha}\big{|}\sin\left(\tfrac{\theta}{2}\right)\big{|}\big{)}\frac{\sin(k\theta)\sin(\theta)}{\big{|}\sin\left(\tfrac{\theta}{2}\right)\big{|}}d\theta$
$\displaystyle=\frac{-1}{2\pi\alpha}\int_{0}^{2\pi}K_{0}^{\prime}\big{(}\tfrac{2}{\alpha}\sin\left(\tfrac{\theta}{2}\right)\big{)}\sin(k\theta)\cos\left(\tfrac{\theta}{2}\right)d\theta$
$\displaystyle=\frac{k}{2\pi}\int_{0}^{2\pi}K_{0}\big{(}\tfrac{2}{\alpha}\sin\left(\tfrac{\theta}{2}\right)\big{)}\cos(k\theta)d\theta$
$\displaystyle=kI_{k}\left(\tfrac{1}{\alpha}\right)K_{k}\left(\tfrac{1}{\alpha}\right).$

Combining the foregoing computations leads to

$\textnormal{Im}\left\\{d_{f}I^{\textnormal{\tiny{SW}}}(0)[h](w)\overline{w}\right\\}=\sum_{n=0}^{\infty}a_{n}\Big{(}(n+1)I_{n+1}\left(\tfrac{1}{\alpha}\right)K_{n+1}\left(\tfrac{1}{\alpha}\right)-I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)\Big{)}e_{n+1}(w).$
(3.17)

Putting together (3.14), (3.15), (3.16) and (3.17) gives the desired result. ∎

According to Proposition 3.1 the possible values for $\Omega$ from which we
can hope to bifurcate are

$\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)\triangleq\tfrac{n-1}{2n}-\big{[}I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)-I_{n}\left(\tfrac{1}{\alpha}\right)K_{n}\left(\tfrac{1}{\alpha}\right)\big{]}=\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}-\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right),\quad
n\in\mathbb{N}^{*}.$ (3.18)

We shall now prove the monotonicity of the sequence
$\big{(}\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)\big{)}_{n\in\mathbb{N}^{*}}.$
This is given by the following result.

######  Lemma 3.1.

For any $\alpha>0,$ the sequence
$\big{(}\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)\big{)}_{n\in\mathbb{N}^{*}}$
is strictly increasing and tends to
$\boldsymbol{\Omega}_{\infty}^{\textnormal{\tiny{E}}}(\alpha),$ with

$\boldsymbol{\Omega}_{\infty}^{\textnormal{\tiny{E}}}(\alpha)\triangleq\tfrac{1}{2}-I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right).$

Moreover,

$\forall
n\in\mathbb{N}^{*},\quad\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)\underset{\alpha\rightarrow
0}{\longrightarrow}\tfrac{n-1}{2n}.$

###### Proof.

The convergences are immediate consequences of (A.9) and (A.8). We shall now
study the monotonicity. For $n\in\mathbb{N}^{*},$ we can write

$\boldsymbol{\Omega}_{n+1}^{\textnormal{\tiny{E}}}(\alpha)-\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)=\big{(}\tfrac{n}{2(n+1)}-\tfrac{n-1}{2n}\big{)}-\big{(}\boldsymbol{\Omega}_{n+1}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)-\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)\big{)}.$

We look for the monotonicity of the function $\varphi_{n}$ defined for
$n\in\mathbb{N}^{*}$ by

$\forall
x>0,\quad\varphi_{n}(x)\triangleq\boldsymbol{\Omega}_{n+1}^{\textnormal{\tiny{SW}}}\left(x\right)-\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{SW}}}\left(x\right)=I_{n}\left(x\right)K_{n}\left(x\right)-I_{n+1}\left(x\right)K_{n+1}\left(x\right).$

From the decay property of $x\mapsto I_{n}(x)K_{n}(x)$ on $(0,\infty)$ for
every $n\in\mathbb{N}^{*}$ and the asymptotic expansion (A.7) we deduce

$\forall n\in\mathbb{N}^{*},\quad\forall x>0,\quad
I_{n}(x)K_{n}(x)<\frac{1}{2n}.$ (3.19)

Using (A.5), (A.6) and (3.19), we find

$\displaystyle\varphi_{n}^{\prime}(x)$
$\displaystyle=I_{n}^{\prime}(x)K_{n}(x)+I_{n}(x)K_{n}^{\prime}(x)-I_{n+1}^{\prime}(x)K_{n+1}(x)-I_{n+1}(x)K_{n+1}^{\prime}(x)$
$\displaystyle=2\left(\frac{1}{x}+I_{n}(x)K_{n}^{\prime}(x)-I_{n+1}^{\prime}(x)K_{n+1}(x)\right)$
$\displaystyle=\frac{2}{x}\left(1+\frac{xK_{n}^{\prime}(x)}{K_{n}(x)}I_{n}(x)K_{n}(x)-\frac{xI_{n+1}^{\prime}(x)}{I_{n+1}(x)}I_{n+1}(x)K_{n+1}(x)\right)$
$\displaystyle<\frac{2}{x}\left(1-\sqrt{x^{2}+n^{2}}\big{[}I_{n}(x)K_{n}(x)+I_{n+1}(x)K_{n+1}(x)\big{]}\right)$
$\displaystyle<\frac{2}{x}\left(1-\sqrt{x^{2}+n^{2}}\right)$
$\displaystyle<0.$

We deduce that for all $n\in\mathbb{N}^{*}$, $\varphi_{n}$ is strictly
decreasing on $(0,\infty).$ In addition, from (A.7), we infer

$\forall n\in\mathbb{N}^{*},\quad\lim_{x\to 0}I_{n}(x)K_{n}(x)=\tfrac{1}{2n},$

which implies in turn

$\forall n\in\mathbb{N}^{*},\quad\lim_{x\to
0}\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{SW}}}(x)=\tfrac{n-1}{2n}.$

Therefore,

$\forall
n\in\mathbb{N}^{*},\quad\boldsymbol{\Omega}_{n+1}^{\textnormal{\tiny{E}}}(\alpha)-\boldsymbol{\Omega}_{n}^{\textnormal{\tiny{E}}}(\alpha)>\big{(}\tfrac{n}{2(n+1)}-\tfrac{n-1}{2n}\big{)}-\big{(}\tfrac{n}{2(n+1)}-\tfrac{n-1}{2n}\big{)}=0.$

This achieves the proof of Lemma 3.1. ∎

###  3.3 Construction of local bifurcation branches

We check here the hypothesis of Crandall-Rabinowitz’s Theorem B.1 which
immediately imply Theorem 1.1. Notice that the line of trivial solutions has
already been obtained in Lemma 3.1-1-(iv).

######  Proposition 3.2.

Let $\alpha>0,$ $\beta\in(0,1)$ and $\mathbf{m}\in\mathbb{N}^{*}.$ The
following assertions hold true.

  1. (i)

$F_{\alpha}:\mathbb{R}\times X_{\mathbf{m}}^{1+\beta}\rightarrow
Y_{\mathbf{m}}^{\beta}$ is well-defined and of class $C^{1}.$

  2. (ii)

The kernel
$\ker\Big{(}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}\Big{)}$
is one dimensional and generated by

$v_{0,\mathbf{m}}:\begin{array}[t]{rcl}\mathbb{T}&\rightarrow&\mathbb{C}\\\
w&\mapsto&\overline{w}^{\mathbf{m}-1}.\end{array}$

  3. (iii)

The range
$\mathcal{R}\Big{(}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}\Big{)}$
is closed and of codimension one in $Y_{\mathbf{m}}^{\beta}.$

  4. (iv)

Transversality condition :

$\partial_{\Omega}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}[v_{0,\mathbf{m}}]\not\in\mathcal{R}\Big{(}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}\Big{)}.$

###### Proof.

(i) Follows immediately from Proposition 3.1-1.
(ii) It is a direct consequence of Proposition 3.1-2 and Lemma 3.1 since for

$h(w)=\sum_{n=0}^{\infty}a_{n}\overline{w}^{n\mathbf{m}-1}\in
X_{\mathbf{m}}^{1+\beta},$

we have by (3.9)

$d_{f}F_{\alpha}\big{(}\Omega_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}[h]=\sum_{n=1}^{\infty}n\mathbf{m}a_{n}\Big{(}\Omega_{n\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha)-\Omega_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha)\Big{)}e_{n\mathbf{m}}.$
(3.20)

(iii) From Proposition 3.1-2, we know that
$d_{f}F_{\alpha}\big{(}\Omega_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}$
is a Fredholm operator with index zero. Together with the point (ii), we
conclude that the range of
$d_{f}F_{\alpha}\big{(}\Omega_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}$
is closed with codimension one. We endow $Y_{\mathbf{m}}^{\beta}$ with the
scalar product

$\big{\langle}f,g\big{\rangle}\triangleq\fint_{\mathbb{T}}f(w)\overline{g(w)}dw=\sum_{n=1}^{\infty}f_{n\mathbf{m}}g_{n\mathbf{m}},\qquad
f=\sum_{n=1}^{\infty}f_{n\mathbf{m}}e_{n\mathbf{m}},\qquad
g=\sum_{n=1}^{\infty}g_{n\mathbf{m}}e_{n\mathbf{m}}.$

The continuity of $\langle\cdot,\cdot\rangle$ on $Y_{\mathbf{m}}^{\beta}\times
Y_{\mathbf{m}}^{\beta}$ is a direct consequence of Cauchy-Schwarz inequality
and the continuous embedding $C^{\beta}(\mathbb{T})\hookrightarrow
L^{2}(\mathbb{T}).$ We claim that

$\mathcal{R}\Big{(}d_{f}F_{\alpha}\big{(}\Omega_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}\Big{)}=\langle
e_{\mathbf{m}}\rangle^{\perp},$ (3.21)

where the orthogonal is understood in the sense of the scalar product
$\langle\cdot,\cdot\rangle.$ Observe that (3.20) and the point (i) immediately
imply the first inclusion in (3.21). The inverse inclusion is deduced from the
fact that the range is of codimension one.
(iv) For any $h\in X_{\mathbf{m}}^{1+\beta},$

$\forall
w\in\mathbb{T},\quad\partial_{\Omega}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}[h](w)=\textnormal{Im}\left\\{h(w)\overline{w}+\overline{h^{\prime}(w)}\right\\}.$

Consequently, (3.21) implies

$\partial_{\Omega}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}[v_{0,\mathbf{m}}]=-\mathbf{m}e_{\mathbf{m}}\not\in\mathcal{R}\Big{(}d_{f}F_{\alpha}\big{(}\boldsymbol{\Omega}_{\mathbf{m}}^{\textnormal{\tiny{E}}}(\alpha),0\big{)}\Big{)}.$

This ends the proof of Proposition 3.2 and proves Theorem 1.1. ∎

######  Remark 3.1.

Observe that for $\mathbf{m}=1$, a similar calculation to Proposition
3.1-1-(iv), shows that for $\Phi(z)=z+a_{0},$ with $a_{0}\in\mathbb{R}$, one
has $F_{\alpha}(0,\Phi)=0.$ By uniqueness of the constructed branch of
bifurcation, this latter corresponds to a translation of the Rankine vortex.

##  4 Quasi-periodic Euler-$\alpha$ patches

This section is devoted to the proof of Theorem 1.2. It is based on KAM and
Nash-Moser techniques in a similar way to the recent works [14, 40, 42, 51].
First let us introduce the notations and topologies used along this section.

###  4.1 Topologies for functions and operators

This subsection presents the notations and topologies used along this part on
the construction of quasi-periodic vortex patches. First we fix

$0<\alpha_{0}<\alpha_{1}.$ (4.1)

The parameter $\alpha$ will live in the interval $(\alpha_{0},\alpha_{1}).$
More precisely, at the end it will belongs to a Cantor set contained in this
interval. We shall denote

$d\in\mathbb{N}^{*}$ (4.2)

the number of excited frequencies generating the quasi-periodic solutions.
Consequently, the frequency vector $\omega$ belongs to $\mathbb{R}^{d}.$ More
precisely, at the end $\omega$ should be close to the equilibrium frequency
vector obtained at the linear level at the Rankine vortex. Our solutions will
be searched in the Sobolev class constructed on $L^{2}$ in the variables
$\varphi\in\mathbb{T}^{d}$ and $\theta\in\mathbb{T}.$ Hence, we decompose any
$\rho\in L^{2}(\mathbb{T}^{d+1},\mathbb{C}),$ in Fourier series as follows

$\rho=\sum_{(l,j)\in\mathbb{Z}^{d+1}}\rho_{l,j}\,\mathbf{e}_{l,j},\qquad\rho_{l,j}\triangleq\big{\langle}\rho,\mathbf{e}_{l,j}\big{\rangle}_{L^{2}(\mathbb{T}^{d+1},\mathbb{C})},$

where $(\mathbf{e}_{l,j})_{(l,j)\in\mathbb{Z}^{d}\times\mathbb{Z}}$ denotes
the classical Hilbert basis of the complex Hilbert space
$L^{2}(\mathbb{T}^{d+1},\mathbb{C}).$ Explicitly, we have

$\mathbf{e}_{l,j}(\varphi,\theta)\triangleq e^{{\rm
i}(l\cdot\varphi+j\theta)},\qquad\mathbf{e}_{j}\triangleq\mathbf{e}_{0,j}.$

The associated Hermitian inner product is

$\big{\langle}\rho_{1},\rho_{2}\big{\rangle}_{L^{2}(\mathbb{T}^{d+1},\mathbb{C})}\triangleq\bigintssss_{\mathbb{T}^{d+1}}\rho_{1}(\varphi,\theta)\overline{\rho_{2}(\varphi,\theta)}d\varphi
d\theta,\qquad\int_{\mathbb{T}^{n}}f(x)dx\triangleq\frac{1}{(2\pi)^{n}}\bigintssss_{[0,2\pi]^{n}}f(x)dx.$

For $s\in\mathbb{R}$ the complex Sobolev space
$H^{s}(\mathbb{T}^{d+1},\mathbb{C})$ is given by

$H^{s}(\mathbb{T}^{d+1},\mathbb{C})\triangleq\Big{\\{}\rho\in
L^{2}(\mathbb{T}^{d+1},\mathbb{C})\quad\textnormal{s.t.}\quad\|\rho\|_{H^{s}}^{2}\triangleq\sum_{(l,j)\in\mathbb{Z}^{d+1}}\langle
l,j\rangle^{2s}|\rho_{l,j}|^{2}<\infty\Big{\\}},\qquad\langle
l,j\rangle\triangleq\max(1,|l|,|j|).$

The closed sub-vector space of real valued functions is denoted

$\displaystyle H^{s}\triangleq H^{s}(\mathbb{T}^{d+1},\mathbb{R})$
$\displaystyle\triangleq\left\\{\rho\in
H^{s}(\mathbb{T}^{d+1},\mathbb{C})\quad\textnormal{s.t.}\quad\forall\,(\varphi,\theta)\in\mathbb{T}^{d+1},\,\rho(\varphi,\theta)=\overline{\rho(\varphi,\theta)}\right\\}$
$\displaystyle=\Big{\\{}\rho\in
H^{s}(\mathbb{T}^{d+1},\mathbb{C})\quad\textnormal{s.t.}\quad\forall\,(l,j)\in\mathbb{Z}^{d+1},\,\rho_{-l,-j}=\overline{\rho_{l,j}}\Big{\\}}.$

In order to ensure some suitable embeddings, we shall consider the following
restrictions on the Sobolev indices. In particular $S$ is chosen large enough.

$S\geqslant s\geqslant s_{0}>\tfrac{d+1}{2}+q+2.$ (4.3)

During the scheme, we shall also keep track of the regularity of our functions
and operators with respect to the parameters $\mu\triangleq(\alpha,\omega).$
Thus, we introduce the parameters

$\gamma\in(0,1),\qquad q\in\mathbb{N}^{*}$ (4.4)

and consider the following weighted spaces

$\displaystyle W^{q,\infty,\gamma}(\mathcal{O},H^{s})$
$\displaystyle\triangleq\Big{\\{}\rho:\mathcal{O}\rightarrow
H^{s}\quad\textnormal{s.t.}\quad\|\rho\|_{q,s}^{\gamma,\mathcal{O}}<\infty\Big{\\}},\qquad\|\rho\|_{q,s}^{\gamma,\mathcal{O}}\triangleq\sum_{\underset{|\alpha|\leqslant
q}{\alpha\in\mathbb{N}^{d+1}}}\gamma^{|\alpha|}\sup_{\mu\in{\mathcal{O}}}\|\partial_{\mu}^{\alpha}\rho(\mu,\cdot)\|_{H^{s-|\alpha|}},$
$\displaystyle W^{q,\infty,\gamma}(\mathcal{O},\mathbb{C})$
$\displaystyle\triangleq\Big{\\{}\rho:\mathcal{O}\rightarrow\mathbb{C}\quad\textnormal{s.t.}\quad\|\rho\|_{q}^{\gamma,\mathcal{O}}<\infty\Big{\\}},\qquad\|\rho\|_{q}^{\gamma,\mathcal{O}}\triangleq\sum_{\underset{|\alpha|\leqslant
q}{\alpha\in\mathbb{N}^{d+1}}}\gamma^{|\alpha|}\sup_{\mu\in{\mathcal{O}}}|\partial_{\mu}^{\alpha}\rho(\mu)|.$

The set $\mathcal{O}$ is an open bounded subset of $\mathbb{R}^{d+1}$ which
will be fixed in (4.50). The parameters $\gamma$ and $q$ will be fixed in
(4.98) and (4.49), respectively. Observe that any $\rho\in
W^{q,\infty,\gamma}(\mathcal{O},H^{s})$ decomposes as follows

$\rho(\mu,\varphi,\theta)=\sum_{(l,j)\in\mathbb{Z}^{d+1}}\rho_{l,j}(\mu)\mathbf{e}_{l,j}(\varphi,\theta).$

Some classical properties of the weighted Sobolev norm are gathered in the
following lemma. We refer for instance to [16, 12, 13] for the ideas of their
proofs.

######  Lemma 4.1.

Let $(d,\gamma,q,s_{0},s)$ satisfying (4.2), (4.3) and (4.4). Take
$\rho_{1},\rho_{2}\in W^{q,\infty,\gamma}(\mathcal{O},H^{s}).$ Then the
following assertions hold true.

  1. (i)

Space translation invariance : for any $\eta\in\mathbb{T},$ we have
$\tau_{\eta}\rho_{1}:(\mu,\varphi,\theta)\mapsto\rho_{1}(\mu,\varphi,\eta+\theta)\in
W^{q,\infty,\gamma}(\mathcal{O},H^{s})$, and

$\|\tau_{\eta}\rho_{1}\|_{q,s}^{\gamma,\mathcal{O}}=\|\rho_{1}\|_{q,s}^{\gamma,\mathcal{O}}.$

  2. (ii)

Projectors properties : For all $N\in\mathbb{N}^{*}$ and for all
$t\in\mathbb{R}_{+}^{*},$

$\|\Pi_{N}\rho_{1}\|_{q,s+t}^{\gamma,\mathcal{O}}\leqslant
N^{t}\|\rho_{1}\|_{q,s}^{\gamma,\mathcal{O}}\qquad\|\Pi_{N}^{\perp}\rho_{1}\|_{q,s}^{\gamma,\mathcal{O}}\leqslant
N^{-t}\|\rho_{1}\|_{q,s+t}^{\gamma,\mathcal{O}},$

where the projectors are defined by

$\Pi_{N}\left(\sum_{j\in\mathbb{Z}}(\rho_{1})_{j}\mathbf{e}_{j}\right)\triangleq\sum_{j\in\mathbb{Z}\atop|j|\leqslant
N}(\rho_{1})_{j}\mathbf{e}_{j},\qquad\Pi_{N}^{\perp}\triangleq\textnormal{Id}-\Pi_{N}.$

  3. (iii)

Law products : $\rho_{1}\rho_{2}\in W^{q,\infty,\gamma}(\mathcal{O},H^{s})$
and

$\|\rho_{1}\rho_{2}\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\rho_{1}\|_{q,s_{0}}^{\gamma,\mathcal{O}}\|\rho_{2}\|_{q,s}^{\gamma,\mathcal{O}}+\|\rho_{1}\|_{q,s}^{\gamma,\mathcal{O}}\|\rho_{2}\|_{q,s_{0}}^{\gamma,\mathcal{O}}.$

  4. (iv)

Composition law : For $f\in
C^{\infty}(\mathcal{O}\times\mathbb{R},\mathbb{R})$, if there exists
$\mathtt{M}>0$ such that

$\|\rho_{1}\|_{q,s}^{\gamma,\mathcal{O}},\|\rho_{2}\|_{q,s}^{\gamma,\mathcal{O}}\leqslant\mathtt{M},$

then $f(\rho_{1})-f(\rho_{2})\in W^{q,\infty,\gamma}(\mathcal{O},H^{s})$ with

$\|f(\rho_{1})-f(\rho_{2})\|_{q,s}^{\gamma,\mathcal{O}}\leqslant
C(s,d,q,f,\mathtt{M})\|\rho_{1}-\rho_{2}\|_{q,s}^{\gamma,\mathcal{O}},$

where we used the notation

$\forall(\mu,\varphi,\theta)\in\mathcal{O}\times\mathbb{T}^{d+1},\quad
f(\rho)(\mu,\varphi,\theta)\triangleq f(\mu,\rho(\mu,\varphi,\theta)).$

We shall now present the operator topology used along this section. In
particular, we deal with the Toeplitz in time operator class. These notions
are based on the one introduced in [3, 12, 13, 16]. We consider parameter
dependent operators in the form

$T:\mu\in\mathcal{O}\mapsto
T(\mu)\in\mathcal{L}(H^{s}(\mathbb{T}^{d+1},\mathbb{C}))$

which can be identified with an infinite dimensional matrix
$\left(T_{l_{0},j_{0}}^{l,j}(\mu)\right)_{\underset{(j,j_{0})\in\mathbb{Z}^{2}}{(l,l_{0})\in(\mathbb{Z}^{d})^{2}}}$
through their action on the Hilbert basis
$\big{(}\mathbf{e}_{l,j}\big{)}_{(l,j)\in\mathbb{Z}^{d}\times\mathbb{Z}}$ as
follows

$T(\mu)\mathbf{e}_{l_{0},j_{0}}=\sum_{(l,j)\in\mathbb{Z}^{d+1}}T_{l_{0},j_{0}}^{l,j}(\mu)\mathbf{e}_{l,j},\qquad
T_{l_{0},j_{0}}^{l,j}(\mu)\triangleq\big{\langle}T(\mu)\mathbf{e}_{l_{0},j_{0}},\mathbf{e}_{l,j}\big{\rangle}_{L^{2}(\mathbb{T}^{d+1})}.$

More precisely, we shall see such operators acting on
$W^{q,\infty,\gamma}(\mathcal{O},H^{s}(\mathbb{T}^{d+1},\mathbb{C}))$ in the
following sense,

$\rho\in
W^{q,\infty,\gamma}\big{(}\mathcal{O},H^{s}(\mathbb{T}^{d+1},\mathbb{C})\big{)},\quad\,\quad(T\rho)(\mu,\varphi,\theta)\triangleq
T(\mu)\rho(\mu,\varphi,\theta).$

We shall now introduce the Toeplitz operators class. An operator $T(\mu)$ is
said to be Toeplitz in time iff

$T_{l_{0},j_{0}}^{l,j}(\mu)=T_{j_{0}}^{j}(\mu,l-l_{0}),\qquad
T_{j_{0}}^{j}(\mu,l)\triangleq T_{0,j_{0}}^{l,j}(\mu).$

Its action on a function
$\rho=\displaystyle\sum_{(l_{0},j_{0})\in\mathbb{Z}^{d+1}}\rho_{l_{0},j_{0}}\mathbf{e}_{l_{0},j_{0}}$
writes

$T(\mu)\rho=\sum_{(l,l_{0})\in(\mathbb{Z}^{d})^{2}\\\
\atop(j,j_{0})\in\mathbb{Z}^{2}}T_{j_{0}}^{j}(\mu,l-l_{0})\rho_{l_{0},j_{0}}\mathbf{e}_{l,j}.$
(4.5)

The Toeplitz topology is given by the following off-diagonal norm,

$\|T\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}\triangleq\sum_{\underset{|\alpha|\leqslant
q}{\alpha\in\mathbb{N}^{d+1}}}\gamma^{|\alpha|}\sup_{(b,\omega)\in{\mathcal{O}}}\|\partial_{\mu}^{\alpha}(T)(\mu)\|_{\textnormal{\tiny{O-d}},s-|\alpha|},\qquad\|T\|_{\textnormal{\tiny{O-d}},s}^{2}\triangleq\sum_{(l,m)\in\mathbb{Z}^{d+1}}\langle
l,m\rangle^{2s}\sup_{j-k=m}|T_{j}^{k}(l)|^{2}.$

We mention that we will encounter several operators acting only on the
variable $\theta$ and that can be considered as $\varphi$-dependent operators
$T(\mu,\varphi)$ taking the form of integral operators

$\displaystyle
T(\mu,\varphi)\rho(\varphi,\theta)=\int_{\mathbb{T}}K(\mu,\varphi,\theta,\eta)\rho(\varphi,\eta)d\eta.$

One can easily check that those operators are Toeplitz and therefore they
satisfy (4.5). We shall now give some classical results on the Toeplitz norm.
The proofs are very similar to those in [16] concerning pseudo-differential
operators.

######  Lemma 4.2.

Let $(d,s_{0},s,\gamma,q)$ satisfying (4.2), (4.3) and (4.4). Let $T,$ $T_{1}$
and $T_{2}$ be Toeplitz in time operators.

  1. (i)

Composition law :

$\|T_{1}T_{2}\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}\lesssim\|T_{1}\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}\|T_{2}\|_{\textnormal{\tiny{O-d}},q,s_{0}}^{\gamma,\mathcal{O}}+\|T_{1}\|_{\textnormal{\tiny{O-d}},q,s_{0}}^{\gamma,\mathcal{O}}\|T_{2}\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}.$

  2. (ii)

Link between operators and off-diagonal norms :

$\|T\rho\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|T\|_{\textnormal{\tiny{O-d}},q,s_{0}}^{\gamma,\mathcal{O}}\|\rho\|_{q,s}^{\gamma,\mathcal{O}}+\|T\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}\|\rho\|_{q,s_{0}}^{\gamma,\mathcal{O}}.$

In particular

$\|T\rho\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|T\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}\|\rho\|_{q,s}^{\gamma,\mathcal{O}}.$

Now, wee give the following definition inspired for instance from [4, Def.
2.2].

######  Definition 4.1.

We defined the following involutions

$(\mathscr{S}_{2}\rho)(\varphi,\theta)=\rho(-\varphi,-\theta),\qquad(\mathscr{S}_{c}\rho)(\varphi,\theta)=\overline{\rho(\varphi,\theta)}.$

An operator $T=T(\mu)$ is said to be

  1. •

real iff $\forall\rho\in
L^{2}(\mathbb{T}^{d+1},\mathbb{C}),\,\mathscr{S}_{c}\rho=\rho\Rightarrow\mathscr{S}_{c}(T\rho)=T\rho.$

  2. •

reversible iff $T\circ\mathscr{S}_{2}=-\mathscr{S}_{2}\circ T.$

  3. •

reversibility preserving iff $T\circ\mathscr{S}_{2}=\mathscr{S}_{2}\circ T.$

We end this subsection by recalling an important lemma whose proof can be
found in [51, Lem. 4.4].

######  Lemma 4.3.

Let $(d,s_{0},s,\gamma,q)$ satisfy (4.2), (4.3) and (4.4), then for any
integral opearator $T$ with a real-valued kernel $K$, namely

$(T\rho)(\mu,\varphi,\theta)=\int_{\mathbb{T}}\rho(\mu,\varphi,\eta)K(\mu,\varphi,\theta,\eta)d\eta,\qquad
K:(\mu,\varphi,\theta,\eta)\mapsto K(\mu,\varphi,\theta,\eta),$

the following property holds true.

  1. •

If $K$ is even in $(\varphi,\theta,\eta)$, then $T$ is a real and
reversibility preserving Toeplitz in time operator.

  2. •

If $K$ is odd in $(\varphi,\theta,\eta)$, then $T$ is a real and reversible
Toeplitz in time operator.

Moreover,

$\|T\|_{\textnormal{\tiny{O-d}},q,s}^{\gamma,\mathcal{O}}\lesssim\int_{\mathbb{T}}\|K(\ast,\cdot,\centerdot,\eta+\centerdot)\|_{q,s+s_{0}}^{\gamma,\mathcal{O}}d\eta$

and

$\|T\rho\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\rho\|_{q,s_{0}}^{\gamma,\mathcal{O}}\int_{\mathbb{T}}\|K(\ast,\cdot,\centerdot,\eta+\centerdot)\|_{q,s}^{\gamma,\mathcal{O}}d\eta+\|\rho\|_{q,s}^{\gamma,\mathcal{O}}\int_{\mathbb{T}}\|K(\ast,\cdot,\centerdot,\eta+\centerdot)\|_{q,s_{0}}^{\gamma,\mathcal{O}}d\eta,$

where the notation $\ast,\cdot,\centerdot$ denote $\mu,\varphi,\theta$,
respectively.

###  4.2 Hamiltonian contour dynamics equation and its linearization

Here, we reformulate the contour dynamics equation (1.8) as a Hamiltonian
equation on the radial deformation of the patch motion near the Rankine vortex
associated with the unit disc. We also compute the linearization of this
Hamiltonian equation both at the equilibrium and at a general state close to
it. Let us consider the following polar parametrization of the boundary of a
patch $t\mapsto\mathbf{1}_{D_{t}}$ close to the Rankine patch
$\mathbf{1}_{\mathbb{D}}.$

$z(t,\theta)\triangleq R(t,\theta)e^{{\rm i}(\theta-\Omega t)},\qquad
R(t,\theta)\triangleq\sqrt{1+2r(t,\theta)}.$ (4.6)

The radial deformation $r$ is assumed to be of small amplitudes and the
angular velocity $\Omega>0$ is introduced to avoid the resonance of the first
equilibrium frequency as in [40, 51]. Therefore, $r$ satisfies an Hamiltonian
PDE as explained in the following lemma.

######  Lemma 4.4.

The following hold true.

  1. (i)

At least for short time $T>0$, the radial deformation $r$ in (4.6) is solution
of the following equation

$\forall(t,\theta)\in[0,T]\times\mathbb{T},\quad\partial_{t}r(t,\theta)+\Omega\partial_{\theta}r(t,\theta)-F^{\textnormal{\tiny{E}}}[r](t,\theta)-F^{\textnormal{\tiny{SW}}}[r](\alpha,t,\theta)=0,$
(4.7)

where

$\displaystyle F^{\textnormal{\tiny{E}}}[r](t,\theta)$
$\displaystyle\triangleq\int_{\mathbb{T}}\log\big{(}A_{r}(t,\theta,\eta)\big{)}\partial_{\theta\eta}^{2}\Big{(}R(t,\theta)R(t,\eta)\sin(\eta-\theta)\Big{)}d\eta,$
(4.8) $\displaystyle F^{\textnormal{\tiny{SW}}}[r](\alpha,t,\theta)$
$\displaystyle\triangleq\int_{\mathbb{T}}K_{0}\left(\tfrac{1}{\alpha}A_{r}(t,\theta,\eta)\right)\partial_{\theta\eta}^{2}\Big{(}R(t,\theta)R(t,\eta)\sin(\eta-\theta)\Big{)}d\eta,$
(4.9) $\displaystyle A_{r}(t,\theta,\eta)$
$\displaystyle\triangleq\left|R(t,\theta)e^{{\rm i}\theta}-R(t,\eta)e^{{\rm
i}\eta}\right|.$ (4.10)

  2. (ii)

The nonlinear and nonlocal transport-type PDE (4.7) can be written in the
following Hamiltonian form

$\partial_{t}r=\partial_{\theta}\nabla\mathscr{H}(r),$ (4.11)

where

$\mathscr{H}(r)\triangleq\tfrac{1}{2}\big{(}\mathscr{E}(r)-\Omega\mathscr{J}(r)\big{)},\quad\mathscr{E}(r)(t)\triangleq-\tfrac{1}{2\pi}\int_{D_{t}}\mathbf{\Psi}(t,z)dA(z),\quad\mathscr{J}(r)(t)\triangleq\tfrac{1}{2\pi}\int_{D_{t}}|z|^{2}dA(z).$

The notation $\nabla$ stands for the $L_{\theta}^{2}(\mathbb{T})$-gradient
associated with the $L_{\theta}^{2}(\mathbb{T})$ normalized inner product

$\big{\langle}\rho_{1},\rho_{2}\big{\rangle}_{L^{2}(\mathbb{T})}\triangleq\int_{\mathbb{T}}\rho_{1}(\theta)\rho_{2}(\theta)d\theta.$

###### Proof.

(i) First, from the polar parametrization (4.6), it is easy to check that the
left hand-side of (1.8) writes

$\mbox{Im}\left(\partial_{t}z(t,\theta)\overline{\partial_{\theta}z(t,\theta)}\right)=-\partial_{t}r(t,\theta)-\Omega\partial_{\theta}r(t,\theta).$

Now we study the right hand-side of (1.8). Combining (3.3) and (A.1), we
deduce

$\displaystyle\mbox{Im}\left(\mathbf{v}\big{(}t,z(t,\theta)\big{)}\overline{\partial_{\theta}z(t,\theta)}\right)$
$\displaystyle=-\displaystyle\int_{\mathbb{T}}\log\big{(}|z(t,\theta)-z(t,\eta)|\big{)}\mbox{Im}\left(\partial_{\eta}z(t,\eta)\overline{\partial_{\theta}z(t,\theta)}\right)d\eta$
$\displaystyle\quad-\int_{\mathbb{T}}K_{0}\left(\tfrac{1}{\alpha}|z(t,\theta)-z(t,\eta)|\right)\mbox{Im}\left(\partial_{\eta}z(t,\eta)\overline{\partial_{\theta}z(t,\theta)}\right)d\eta.$

We conclude by remarking that

$\displaystyle\mbox{Im}\left(\partial_{\eta}z(t,\eta)\overline{\partial_{\theta}z(t,\theta)}\right)=\partial^{2}_{\theta\eta}\mbox{Im}\left(z(t,\eta)\overline{z(t,\theta)}\right)=\partial_{\theta\eta}^{2}\Big{(}R(t,\eta)R(t,\theta)\sin(\eta-\theta)\Big{)}.$

(ii) Using polar change of coordinates and (4.6), we obtain

$\mathscr{J}(r)(t)=\int_{\mathbb{T}}\int_{0}^{R(t,\theta)}\ell^{3}d\ell
d\theta=\tfrac{1}{4}\int_{\mathbb{T}}\big{(}1+2r(t,\theta)\big{)}^{2}d\theta,$

leading to

$\nabla\mathscr{J}(r)=1+2r\quad\textnormal{and}\quad\tfrac{1}{2}\Omega\partial_{\theta}\nabla\mathscr{J}(r)=\Omega\partial_{\theta}r.$

Now that we have treated the corresponding linear term in (4.7), we can assume
$\Omega=0.$ From the complex notation, we have

$\displaystyle\partial_{\theta}\boldsymbol{\Psi}(t,z(t,\theta))$
$\displaystyle=\nabla\boldsymbol{\Psi}(t,z(t,\theta))\cdot\partial_{\theta}z(t,\theta)$
$\displaystyle=\textnormal{Im}\left(\mathbf{v}(t,z(t,\theta))\overline{\partial_{\theta}z(t,\theta)}\right)$
$\displaystyle=-F^{\textnormal{\tiny{E}}}[r](t,\theta)-F^{\textnormal{\tiny{SW}}}[r](\alpha,t,\theta).$

From (1.6), we can write

$\boldsymbol{\Psi}(t,z)=\int_{D_{t}}\widetilde{\mathbf{G}}(z,\xi)dA(\xi),\qquad\widetilde{\mathbf{G}}(z,\xi)\triangleq\mathbf{G}(|z-\xi|).$

The kernel $\widetilde{\mathbf{G}}$ satisfies the following symmetry property

$\widetilde{\mathbf{G}}(z,\xi)=\widetilde{\mathbf{G}}(\xi,z).$

Consequently, we can apply the general [42, Prop. 2.1] giving

$\nabla\mathscr{E}(r)(t,\theta)=-2\boldsymbol{\Psi}\big{(}t,z(t,\theta)\big{)}.$

This achieves the proof of Lemma 4.4. ∎

We shall now briefly discuss the symplectic framework behind the Hamiltonian
equation (4.11). First notice that one deduces from (4.11) that the space
average is preserved along the motion so we may assume it zero and work in the
following phase space

$L_{0}^{2}(\mathbb{T})\triangleq\Big{\\{}r=\sum_{j\in\mathbb{Z}^{*}}r_{j}\mathbf{e}_{j}\quad\textnormal{s.t.}\quad
r_{-j}=\overline{r_{j}},\quad\sum_{j\in\mathbb{Z}^{*}}|r_{j}|^{2}<\infty\Big{\\}},\qquad\mathbf{e}_{j}(\theta)\triangleq
e^{{\rm i}j\theta}.$ (4.12)

The symplectic form $\mathscr{W}$ on $L_{0}^{2}(\mathbb{T})$ generated by
(4.11) writes

$\mathscr{W}(r,\widetilde{r})\triangleq\int_{\mathbb{T}}\partial_{\theta}^{-1}r(\theta)\widetilde{r}(\theta)d\theta,\qquad\partial_{\theta}^{-1}r\triangleq\sum_{j\in\mathbb{Z}^{*}}\tfrac{r_{j}}{{\rm
i}j}\mathbf{e}_{j}.$ (4.13)

The associated Hamiltonian vector-field $X_{\mathscr{H}}$ is defined by

$d\mathscr{H}(r)[\rho]=\mathscr{W}(X_{\mathscr{H}}(r),\rho),\qquad
X_{\mathscr{H}}(r)\triangleq\partial_{\theta}\nabla\mathscr{H}(r).$

We shall also present the reversibility property of the Hamiltonian
$\mathscr{H}$. For that purpose we introduce the following involution on
$L_{0}^{2}(\mathbb{T})$

$(\mathscr{S}r)(\theta)\triangleq
r(-\theta),\qquad\mathscr{S}^{2}=\textnormal{Id}.$ (4.14)

Then changes of variables give

$\mathscr{S}\circ
F^{\textnormal{\tiny{E}}}=-F^{\textnormal{\tiny{E}}}\circ\mathscr{S},\qquad\mathscr{S}\circ
F^{\textnormal{\tiny{SW}}}=-F^{\textnormal{\tiny{SW}}}\circ\mathscr{S}$

implying in turn

$\mathscr{H}\circ\mathscr{S}=\mathscr{H},\qquad
X_{\mathscr{H}}\circ\mathscr{S}=-\mathscr{S}\circ X_{\mathscr{H}}.$

Now, in view of applying a Nash-Moser scheme, we shall compute the linearized
operator both at the equilibrium and at a general state close to it. It is
proved in [42, Lem. 3.1 and 3.2] that

$-d_{r}F^{\textnormal{\tiny{E}}}(r)[\rho]=-\partial_{\theta}\big{(}V_{r}^{\textnormal{\tiny{E}}}\rho\big{)}+\partial_{\theta}\mathbf{L}_{r}^{\textnormal{\tiny{E}}}(\rho),$
(4.15)

with

$\displaystyle V_{r}^{\textnormal{\tiny{E}}}(t,\theta)$
$\displaystyle\triangleq\tfrac{1}{R(t,\theta)}\int_{\mathbb{T}}\log\big{(}A_{r}(t,\theta,\eta)\big{)}\partial_{\eta}\big{(}R(t,\eta)\sin(\eta-\theta)\big{)}d\eta,$
(4.16) $\displaystyle\mathbf{L}_{r}^{\textnormal{\tiny{E}}}(\rho)(t,\theta)$
$\displaystyle\triangleq\int_{\mathbb{T}}\rho(t,\eta)\log\left(A_{r}(t,\theta,\eta)\right)d\eta$
(4.17)

and

$-d_{r}F^{\textnormal{\tiny{E}}}(0)[\rho]=\tfrac{1}{2}\partial_{\theta}\rho+\partial_{\theta}\mathcal{K}\ast\rho={\rm
i}\sum_{j\in\mathbb{Z}}j\boldsymbol{\Omega}_{|j|}^{\textnormal{\tiny{E}}}\rho_{j}\mathbf{e}_{j},\qquad\mathcal{K}(\theta)\triangleq\tfrac{1}{2}\log\left(\sin^{2}\left(\tfrac{\theta}{2}\right)\right).$

We recall that $A_{r}$ and $R$ are defined by (4.10) and (4.6), respectively.
In addition, it has been proved in [51, Lem. 3.1 and 3.2] that

$d_{r}F^{\textnormal{\tiny{SW}}}(r)[\rho]=\partial_{\theta}\big{(}V_{r}^{\textnormal{\tiny{SW}}}\rho\big{)}-\partial_{\theta}\mathbf{L}_{r}^{\textnormal{\tiny{SW}}}(\rho),$
(4.18)

with

$\displaystyle V_{r}^{\textnormal{\tiny{SW}}}(\alpha,t,\theta)$
$\displaystyle\triangleq\tfrac{1}{R(t,\theta)}\int_{\mathbb{T}}K_{0}\big{(}\tfrac{1}{\alpha}A_{r}(t,\theta,\eta)\big{)}\partial_{\eta}\big{(}R(t,\eta)\sin(\eta-\theta)\big{)}d\eta,$
(4.19)
$\displaystyle\mathbf{L}_{r}^{\textnormal{\tiny{SW}}}(\rho)(\alpha,t,\theta)$
$\displaystyle\triangleq\int_{\mathbb{T}}\rho(t,\eta)K_{0}\left(\tfrac{1}{\alpha}A_{r}(t,\theta,\eta)\right)d\eta$
(4.20)

and

$d_{r}F^{\textnormal{\tiny{SW}}}(0)[\rho]=I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)\partial_{\theta}\rho-\partial_{\theta}\mathcal{Q}_{\alpha}\ast\rho={\rm
i}\sum_{j\in\mathbb{Z}}j\boldsymbol{\Omega}_{|j|}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)\rho_{j}\mathbf{e}_{j},\qquad\mathcal{Q}_{\alpha}(\theta)\triangleq
K_{0}\left(\tfrac{2}{\alpha}\left|\sin\left(\tfrac{\theta}{2}\right)\right|\right).$

Gathering the previous results leads to the following lemma giving the general
expression of the linearized equation and stating that the equilibrium is
given by a Fourier multiplier associated with an integrable Hamiltonian
system.

######  Lemma 4.5.

The following assertions hold true.

  1. (i)

The linearization of (4.11) at a general state $r$ writes

$\partial_{t}\rho=-\partial_{\theta}\Big{(}V_{r}\rho+\mathbf{L}_{r}(\rho)\Big{)},$
(4.21)

where $V_{r}$ and $\mathbf{L}_{r}$ are respectively defined by

$V_{r}(\alpha,t,\theta)\triangleq\Omega+V_{r}^{\textnormal{\tiny{SW}}}(\alpha,t,\theta)-V_{r}^{\textnormal{\tiny{E}}}(t,\theta),\qquad\mathbf{L}_{r}\triangleq\mathbf{L}_{r}^{\textnormal{\tiny{E}}}+\mathbf{L}_{r}^{\textnormal{\tiny{SW}}}.$
(4.22)

In addition, we have the following symmetry property

$r(-t,-\theta)=r(t,\theta)\quad\Rightarrow\quad
V_{r}(\alpha,-t,-\theta)=V_{r}(\alpha,t,\theta).$ (4.23)

  2. (ii)

    1. (a)

At $r=0$, the equation (4.21) becomes

$\partial_{t}\rho=\partial_{\theta}\mathrm{L}(\alpha)\rho=\partial_{\theta}\nabla\mathscr{H}_{\mathrm{L}}(\rho),$
(4.24)

where $\mathrm{L}(\alpha)$ is the self-adjoint operator given by

$\mathrm{L}(\alpha)\triangleq-
V_{0}(\alpha)-\mathcal{K}_{\alpha}\ast\cdot,\qquad
V_{0}(\alpha)\triangleq\Omega+\frac{1}{2}-I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right),\qquad\mathcal{K}_{\alpha}\triangleq\mathcal{K}-\mathcal{Q}_{\alpha}.$
(4.25)

The equation (4.24) is generated by the quadratic Hamiltonian

$\mathscr{H}_{\mathrm{L}}(\rho)\triangleq\tfrac{1}{2}\big{\langle}\mathrm{L}(\alpha)\rho,\rho\big{\rangle}_{L^{2}(\mathbb{T})}.$
(4.26)

    2. (b)

In Fourier expansion, the solutions of (4.24) take the form

$\rho(t,\theta)=\sum_{j\in\mathbb{Z}^{*}}\rho_{j}(0)e^{{\rm
i}\left(j\theta-\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)t\right)},$

where for all $j\in\mathbb{Z}^{*}$,

$\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\triangleq
j\Big{[}\Omega+\tfrac{|j|-1}{2|j|}-\big{[}I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)-I_{|j|}\left(\tfrac{1}{\alpha}\right)K_{|j|}\left(\tfrac{1}{\alpha}\right)\big{]}\Big{]}=j\Big{[}\Omega+\boldsymbol{\Omega}_{|j|}^{\textnormal{\tiny{E}}}(\alpha)\Big{]},$
(4.27)

with $\boldsymbol{\Omega}_{j}^{\textnormal{\tiny{E}}}(\alpha)$ are the
frequencies obtained in the periodic case and defined in (1.12). The operator
$\mathrm{L}(\alpha)$ and the Hamiltonian $\mathscr{H}_{\mathrm{L}}$ also write

$\displaystyle\mathrm{L}(\alpha)\rho=-\sum_{j\in\mathbb{Z}^{*}}\tfrac{\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)}{j}\rho_{j}\mathbf{e}_{j}\quad\mbox{
and
}\quad\mathscr{H}_{\mathrm{L}}\rho=-\sum_{j\in\mathbb{Z}^{*}}\tfrac{\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)}{2j}|\rho_{j}|^{2}.$
(4.28)

###  4.3 Transversality and linear quasi-periodic solutions

The aim of this section is to find quasi-periodic solutions for the linearized
equation (4.24), which is the basis for expecting to get them at the nonlinear
level. The result reads as follows

######  Proposition 4.1.

Let $(\alpha_{0},\alpha_{1},d)$ as in (4.1)-(4.2). Take
$\mathbb{S}\subset\mathbb{N}^{*}$ with $|\mathbb{S}|=d.$ Then, there exists a
Cantor-like set

$\mathscr{C}_{\textnormal{Eq}}\subset[\alpha_{0},\alpha_{1}],\qquad|\mathscr{C}_{\textnormal{Eq}}|=\alpha_{1}-\alpha_{0}$

such that for any $\alpha\in\mathscr{C}_{\textnormal{Eq}}$, every function in
the form

$\rho(t,\theta)=\sum_{j\in\mathbb{S}}\rho_{j}\cos\left(j\theta-\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)t\right),\qquad\rho_{j}\in\mathbb{R}^{*}$

is a time quasi-periodic reversible solution to (4.24) with frequency vector

$\omega_{\textnormal{Eq}}(\alpha)=\left(\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\right)_{j\in\mathbb{S}}.$
(4.29)

The proof of this proposition is very similar to [51, Prop. 3.1] so we refer
the reader to the corresponding paper. We mention that in the proof, to
measure the Cantor set $\mathscr{C}_{\textnormal{Eq}}$, we make appeal to the
following Rüssmann Lemma which can be found in [67, Thm. 17.1].

######  Lemma 4.6.

Let $q_{0}\in\mathbb{N}^{*},$ $a,b\in\mathbb{R}$ with $a<b$ and
$\mathtt{m},\mathtt{M}\in(0,\infty).$ Let $f\in C^{q_{0}}([a,b],\mathbb{R})$
such that

$\inf_{x\in[a,b]}\max_{q\in\llbracket
0,q_{0}\rrbracket}\big{|}f^{(q)}(x)\big{|}\geqslant\mathtt{m}.$ (4.30)

Then, there exists $C=C(a,b,q_{0},\|f\|_{C^{q_{0}}([a,b],\mathbb{R})})>0$ such
that

$\Big{|}\left\\{x\in[a,b]\quad\textnormal{s.t.}\quad|f(x)|\leqslant\mathtt{M}\right\\}\Big{|}\leqslant
C\frac{\mathtt{M}^{\frac{1}{q_{0}}}}{\mathtt{m}^{1+\frac{1}{q_{0}}}}\cdot$

To apply the previous lemma, we shall check the transversality condition
(4.30) for the equilibrium frequency vector $\omega_{\textnormal{Eq}}$ in
(4.29). It is proved in Lemma 4.9-(i). Notice that the measure of the final
Cantor set in Section 4.6 generating quasi-periodic solution for the nonlinear
model requires transversality conditions for the perturbed frequency vector.
These latter are obtained by perturbative arguments from the one for the
equilibrium frequency vector stated in Lemma 4.9 and which are themselves
deduced from the non-degeneracy of the unperturbed frequency vector proved in
Lemma 4.8. First we start by giving some properties of the frequencies (4.27).

######  Lemma 4.7.

The following properties hold true.

  1. (i)

$\forall\,\alpha>0,\quad\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\underset{j\rightarrow+\infty}{\sim}V_{0}(\alpha)j,\,$
with $\,V_{0}(\alpha)$ as in (4.25).

  2. (ii)

For all $\alpha>0,$ the sequence
$(\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha))_{j\in\mathbb{N}^{*}}$ is
strictly increasing.

  3. (iii)

For any $j\in\mathbb{Z}^{*},$ we have

$\forall\,\alpha>0,\quad\Big{|}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\Big{|}\geqslant\Omega|j|.$

  4. (iv)

For any $j,j^{\prime}\in\mathbb{Z}^{*}$, we have

$\forall\,\alpha>0,\quad\Big{|}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)-\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\Big{|}\geqslant\Omega|j-j^{\prime}|.$

  5. (v)

Given $0<\alpha_{0}<\alpha_{1}$ and $q_{0}\in\mathbb{N},$ there exists
$C_{0}>0$ such that

$\forall j,j_{0}\in\mathbb{Z}^{*},\quad\max_{q\in\llbracket
0,q_{0}\rrbracket}\sup_{\alpha\in[\alpha_{0},\alpha_{1}]}\left|\partial_{b}^{q}\Big{(}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)-\Omega^{\textnormal{\tiny{E}}}_{j_{0}}(\alpha)\Big{)}\right|\leqslant
C_{0}|j-j_{0}|.$

###### Proof.

(i) and (ii) follow immediately from Lemma 3.1 and (4.27).
(iii) Due to the symmetry (4.27), it sufficies to study the case
$j\in\mathbb{N}^{*}.$ From Lemma 3.1, we get

$\forall\alpha>0,\quad\boldsymbol{\Omega}_{j}^{\textnormal{\tiny{E}}}(\alpha)\geqslant
0.$

Consequently,

$\forall\alpha>0,\quad\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\geqslant
j\Omega.$

(iv) Due to the symmetry (4.27), it suffices to prove that

$\forall\alpha>0,\quad\forall(j,j^{\prime})\in(\mathbb{N}^{*})^{2},\quad\Big{|}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\Big{|}\geqslant\Omega|j\pm
j^{\prime}|.$

The point (ii) allows us to restrict the discussion to the case $j\geqslant
j^{\prime}.$ We can write

$\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)=\Omega(j\pm
j^{\prime})+j\boldsymbol{\Omega}_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm
j^{\prime}\boldsymbol{\Omega}_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha).$

In view of Lemma 3.1, we obtain

$j\boldsymbol{\Omega}_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm
j^{\prime}\boldsymbol{\Omega}_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\geqslant
0.$

Hence

$\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\geqslant\Omega(j\pm
j^{\prime}).$

(v) As before, by symmetry, it sufficies to prove that

$\forall(j,j_{0})\in(\mathbb{N}^{*})^{2},\quad\forall\alpha\in[\alpha_{0},\alpha_{1}],\quad\forall
q\in\llbracket
0,q_{0}\rrbracket,\quad\Big{|}\partial_{\alpha}^{q}\Big{(}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j_{0}}^{\textnormal{\tiny{E}}}(\alpha)\Big{)}\Big{|}\leqslant
C_{0}|j\pm j_{0}|.$

Let us start with the difference. We can write form (4.27)

$\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)-\Omega_{j_{0}}^{\textnormal{\tiny{E}}}(\alpha)=\Omega(j-j_{0})+\frac{j-j_{0}}{2}+\Omega_{j_{0}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)-\Omega_{j}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right),\qquad\Omega_{j}^{\textnormal{\tiny{SW}}}(\lambda)\triangleq
j\boldsymbol{\Omega}_{|j|}^{\textnormal{\tiny{SW}}}(\lambda).$

Now it has been proved in [51, Lem. 3.3-(vi)] that for some
$0<\lambda_{0}<\lambda_{1},$ there exists $C>0$ such that

$\forall(j,j_{0})\in(\mathbb{N}^{*})^{2},\quad\forall\lambda\in[\lambda_{0},\lambda_{1}],\quad\forall
q\in\llbracket
0,q_{0}\rrbracket,\quad\Big{|}\partial_{\lambda}^{q}\Big{(}\Omega_{j}^{\textnormal{\tiny{SW}}}(\lambda)\pm\Omega_{j_{0}}^{\textnormal{\tiny{SW}}}(\lambda)\Big{)}\Big{|}\leqslant
C|j\pm j_{0}|.$ (4.31)

We warn the reader about the difference of definitions of the frequencies
$\Omega_{j}^{\textnormal{\tiny{SW}}}$ between this article and [51] (with the
$\Omega$ missing). But this has no impact here. Hence, we conclude by the
triangle inequality that for some $C_{0}>0$

$\forall(j,j_{0})\in(\mathbb{N}^{*})^{2},\quad\forall\alpha\in[\alpha_{0},\alpha_{1}],\quad\forall
q\in\llbracket
0,q_{0}\rrbracket,\quad\Big{|}\partial_{\alpha}^{q}\Big{(}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)-\Omega_{j_{0}}^{\textnormal{\tiny{E}}}(\alpha)\Big{)}\Big{|}\leqslant
C_{0}|j-j_{0}|.$

We now trun to the additional case. We can write

$0<\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)+\Omega_{j_{0}}^{\textnormal{\tiny{E}}}(\alpha)=\Omega(j+j_{0})+\Big{(}j\tfrac{j-1}{2j}+j_{0}\tfrac{j_{0}-1}{2j_{0}}\Big{)}-\Omega_{j_{0}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)-\Omega_{j}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right).$

Notice that

$\forall
j\in\mathbb{N}^{*},\quad\tfrac{j-1}{2j}\leqslant\tfrac{1}{2}\quad\textnormal{and}\quad\Omega_{j}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)>0.$

Thus,

$0<\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)+\Omega_{j_{0}}^{\textnormal{\tiny{E}}}(\alpha)\leqslant\big{(}\Omega+\tfrac{1}{2}\big{)}(j+j_{0}).$

Combined with (4.31), this ends the proof of Lemma 4.7. ∎

For a fixed finite set of Fourier modes

$\mathbb{S}=\\{j_{1},\ldots,j_{d}\\}\subset\mathbb{N}^{*},\qquad
j_{1}<\ldots<j_{d},\qquad d\in\mathbb{N}^{*},$ (4.32)

we define the equilibrium frequency vector

$\omega_{\textnormal{Eq}}(\alpha)=\big{(}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\big{)}_{j\in\mathbb{S}}.$
(4.33)

Then, in view of the measure of the final Cantor set, we may check the
Rüssmann conditions for the unperturped frequency vector (4.33). They are
obtained by using the following non-degeneracy conditions.

######  Lemma 4.8.

Let $(\alpha_{0},\alpha_{1})$ as in (4.1). The equilibrium frequency vector
$\omega_{\textnormal{Eq}}$ and the vector-valued functions
$(\omega_{\textnormal{Eq}},V_{0})$ and $(\omega_{\textnormal{Eq}},V_{0},1)$
are non-degenerate on $[\alpha_{0},\alpha_{1}]$, namely the curves

$\begin{array}[]{l}\alpha\in[\alpha_{0},\alpha_{1}]\mapsto\omega_{\textnormal{Eq}}(\alpha),\\\
\alpha\in[\alpha_{0},\alpha_{1}]\mapsto(\omega_{\textnormal{Eq}}(\alpha),V_{0}(\alpha)),\\\
\alpha\in[\alpha_{0},\alpha_{1}]\mapsto(\omega_{\textnormal{Eq}}(\alpha),V_{0}(\alpha),1)\end{array}$

are not contained in an hyperplane of $\mathbb{R}^{d}$, $\mathbb{R}^{d+1}$ and
$\mathbb{R}^{d+2}$, respectively.

###### Proof.

$\blacktriangleright$ Assume that there exists
$(c_{1},\ldots,c_{d})\in\mathbb{R}^{d}$ such that

$\forall\alpha\in[\alpha_{0},\alpha_{1}],\quad\sum_{k=1}^{d}c_{k}\Omega_{j_{k}}^{\textnormal{\tiny{E}}}(\alpha)=0,$

which is equivalent to

$\displaystyle\forall\alpha\in[\alpha_{0},\alpha_{1}],\quad\sum_{k=1}^{d}c_{k}j_{k}\Big{(}\Omega+\tfrac{j_{k}-1}{2j_{k}}\Big{)}=\sum_{k=1}^{d}c_{k}\Omega_{j_{k}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right).$
(4.34)

By analyticity of the product $I_{n}K_{n}$ on
$\\{z\in\mathbb{C}\quad\textnormal{s.t.}\quad\textnormal{Re}(z)>0\\}$ for any
$n\in\mathbb{N}^{*}$, then by continuation principle, the previous identity is
still true for $\alpha>0.$ Taking the limit $\alpha\to 0$ in the previous
relation implies from (A.8)

$\sum_{k=1}^{d}c_{k}j_{k}\Big{(}\Omega+\tfrac{j_{k}-1}{2j_{k}}\Big{)}=0.$

The equation (4.34) is reduced to

$\sum_{k=1}^{d}c_{k}\Omega_{j_{k}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)=0.$

Then, proceeding as in [51, Lem. 3.4], the asymptotic expansion of large
argument for $I_{j}K_{j}$ provides an invertible Vandermonde system leading to
$\forall k\in\llbracket 1,d\rrbracket,\,c_{k}=0.$
$\blacktriangleright$ Assume that there exists
$(c_{1},\ldots,c_{d},c_{d+1})\in\mathbb{R}^{d+1}$ (resp.
$(c_{1},\ldots,c_{d},c_{d+1},c_{d+2})\in\mathbb{R}^{d+2}$) such that

$\forall\alpha\in[\alpha_{0},\alpha_{1}],\qquad(\textnormal{resp.
}c_{d+2}+)\quad
c_{d+1}V_{0}(\alpha)+\sum_{k=1}^{d}c_{k}\Omega_{j_{k}}^{\textnormal{\tiny{E}}}(\alpha)=0,$

which is equivalent to the fact that for any
$\alpha\in[\alpha_{0},\alpha_{1}],$

$\displaystyle(\textnormal{resp. }c_{d+2}+)\quad
c_{d+1}\big{(}\Omega+\tfrac{1}{2}\big{)}+\sum_{k=1}^{d}c_{k}j_{k}\Big{(}\Omega+\tfrac{j_{k}-1}{2j_{k}}\Big{)}=c_{d+1}I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)+\sum_{k=1}^{d}c_{k}\Omega_{j_{k}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right).$
(4.35)

As in the previous point, this identity can be extended to $(0,\infty)$ and
taking the limit $\alpha\to 0$, one gets by (A.8)

$(\textnormal{resp. }c_{d+2}+)\quad
c_{d+1}\big{(}\Omega+\tfrac{1}{2}\big{)}+\sum_{k=1}^{d}c_{k}j_{k}\Big{(}\Omega+\tfrac{j_{k}-1}{2j_{k}}\Big{)}=0.$

Inserting this information into (4.35) yields

$\forall\alpha>0,\quad
c_{d+1}I_{1}\left(\tfrac{1}{\alpha}\right)K_{1}\left(\tfrac{1}{\alpha}\right)+\sum_{k=1}^{d}c_{k}\Omega_{j_{k}}^{\textnormal{\tiny{SW}}}\left(\tfrac{1}{\alpha}\right)=0.$

This equation has also been studied in [51, Lem. 3.4] leading to
$c_{1}=\ldots=c_{d}=c_{d+1}=0$ (resp. supplemented by $c_{d+2}=0$). This
achieves the proof of Lemma 4.8. ∎

Now we shall prove the transversality conditions for the equilibrium frequency
vector.

######  Lemma 4.9.

[Transversality] Let $(\alpha_{0},\alpha_{1})$ as in (4.1). Then, there exist
$q_{0}\in\mathbb{N}$ and $\rho_{0}>0$ such that the following results hold
true. Recall that $\omega_{\textnormal{Eq}}$ and
$\Omega_{j}^{\textnormal{\tiny{E}}}$ are defined in (4.33) and (4.27)
respectively.

  1. (i)

For any $l\in\mathbb{Z}^{d}\setminus\\{0\\},$ we have

$\inf_{\alpha\in[\alpha_{0},\alpha_{1}]}\max_{q\in\llbracket
0,q_{0}\rrbracket}\Big{|}\partial_{\alpha}^{q}\omega_{\textnormal{Eq}}(\alpha)\cdot
l\Big{|}\geqslant\rho_{0}\langle l\rangle.$

  2. (ii)

For any $(l,j)\in\mathbb{Z}^{d}\times(\mathbb{N}^{*}\setminus\mathbb{S})$

$\quad\inf_{\alpha\in[\alpha_{0},\alpha_{1}]}\max_{q\in\llbracket
0,q_{0}\rrbracket}\Big{|}\partial_{\alpha}^{q}\Big{(}\omega_{\textnormal{Eq}}(\alpha)\cdot
l\pm jV_{0}(\alpha)\Big{)}\Big{|}\geqslant\rho_{0}\langle l\rangle.$

  3. (iii)

For any $(l,j)\in\mathbb{Z}^{d}\times(\mathbb{N}^{*}\setminus\mathbb{S})$

$\quad\inf_{\alpha\in[\alpha_{0},\alpha_{1}]}\max_{q\in\llbracket
0,q_{0}\rrbracket}\Big{|}\partial_{\alpha}^{q}\Big{(}\omega_{\textnormal{Eq}}(\alpha)\cdot
l\pm\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\Big{)}\Big{|}\geqslant\rho_{0}\langle
l\rangle.$

  4. (iv)

For any $l\in\mathbb{Z}^{d},j,j^{\prime}\in\mathbb{N}^{*}\setminus\mathbb{S}$
with $(l,j)\neq(0,j^{\prime}),$ we have

$\,\quad\inf_{\alpha\in[\alpha_{0},\alpha_{1}]}\max_{q\in\llbracket
0,q_{0}\rrbracket}\Big{|}\partial_{\alpha}^{q}\Big{(}\omega_{\textnormal{Eq}}(\alpha)\cdot
l+\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\Big{)}\Big{|}\geqslant\rho_{0}\langle
l\rangle.$

###### Proof.

We shall prove the point (iv) which is the most difficult one. The arguments
are similar in the other cases using the corresponding non-degeneracy
conditions provided by Lemma 4.8. Fix some $l\in\mathbb{Z}^{d}$ and
$j,j^{\prime}\in\mathbb{N}^{*}\setminus\mathbb{S}$ with
$(l,j)\neq(0,j^{\prime}).$ If for some $c_{0}>0$

$|j\pm j^{\prime}|\geqslant c_{0}\langle l\rangle,$

then applying the triangle inequality together with Lemma 4.7-(iv), we get

$\Big{|}\omega_{\textnormal{Eq}}(\alpha)\cdot
l+\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\Big{|}\geqslant\Big{|}\Omega_{j}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\Big{|}-|\omega_{\textnormal{Eq}}(\alpha)\cdot
l|\geqslant\Omega|j\pm j^{\prime}|-C|l|\geqslant\langle l\rangle.$

Therefore it remains to check the proof for indices satisfying

$|j\pm j^{\prime}|<c_{0}\langle l\rangle,\qquad
l\in\mathbb{Z}^{d}\setminus\\{0\\},\qquad
j,j^{\prime}\in\mathbb{N}^{*}\setminus\mathbb{S}.$ (4.36)

We assume in view of a contradiction that for all $m\in\mathbb{N}$, there
exist real numbers $l_{m}\in\mathbb{Z}^{d}\setminus\\{0\\}$,
$j_{m},j^{\prime}_{m}\in\mathbb{N}^{*}\setminus\mathbb{S}$ satisfying (4.36)
and $\alpha_{m}\in[\alpha_{0},\alpha_{1}]$ such that

$\max_{q\in\llbracket
0,m\rrbracket}\left|\partial_{\alpha}^{q}\left(\omega_{\textnormal{Eq}}(\alpha)\cdot\tfrac{l_{m}}{|l_{m}|}+\tfrac{\Omega_{j_{m}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}_{m}}^{\textnormal{\tiny{E}}}(\alpha)}{|l_{m}|}\right)_{|\alpha=\alpha_{m}}\right|<\tfrac{1}{m+1}.$

This implies that

$\forall q\in\mathbb{N},\quad\forall m\geqslant
q,\quad\left|\partial_{\alpha}^{q}\left(\omega_{\textnormal{Eq}}(\alpha)\cdot\tfrac{l_{m}}{|l_{m}|}+\tfrac{\Omega_{j_{m}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}_{m}}^{\textnormal{\tiny{E}}}(\alpha)}{|l_{m}|}\right)_{|\alpha=\alpha_{m}}\right|<\tfrac{1}{m+1}\cdot$
(4.37)

By compactness and (4.36), up to considering a subsequence, we can assume that

$\lim_{m\to\infty}\tfrac{l_{m}}{|l_{m}|}=\bar{c}\neq
0,\qquad\lim_{m\to\infty}\tfrac{j_{m}\pm
j^{\prime}_{m}}{|l_{m}|}=\bar{d},\qquad\lim_{m\to\infty}\alpha_{m}=\bar{\alpha}.$
(4.38)

Now we shall study separetely the cases whether the sequence $(l_{m})_{m}$ is
bounded or not.
$\blacktriangleright$ Here we assume that the sequence $(l_{m})_{m}$ is
bounded. Then, by compactness, we can assume, up to an extraction, that we
have the following convergence

$\displaystyle\lim_{m\to\infty}l_{m}=\bar{l}\neq 0.$

Now according to (4.36) we have two sub-cases to discuss depending whether the
sequences $(j_{m})_{m}$ and $(j^{\prime}_{m})_{m}$ are simultaneously bounded
or unbounded.
$\bullet$ We first study the case where the sequences $(j_{m})_{m}$ and
$(j^{\prime}_{m})_{m}$ are bounded. Observe that the is the only case to
consider if we work with the sign $"+"$ in (4.36). Since they are sequences of
integers, then by compactness we may assume, up to considering an extraction,
that they are constant, namely

$\exists\,\bar{j},\bar{j^{\prime}}\in\mathbb{N}^{*}\setminus\mathbb{S}\qquad\textnormal{s.t.}\qquad\forall
m\in\mathbb{N},\quad j_{m}=\bar{j}\quad\textnormal{and}\quad
j^{\prime}_{m}=\bar{j^{\prime}}.$

Hence taking the limit as $m\rightarrow\infty$ in (4.37), we obtain

$\forall
q\in\mathbb{N},\quad\partial_{\alpha}^{q}\left(\omega_{\textnormal{Eq}}({\alpha})\cdot\bar{l}+\Omega_{\bar{j}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{\bar{j^{\prime}}}^{\textnormal{\tiny{E}}}(\alpha)\right)_{|\alpha=\overline{\alpha}}=0.$

Thus, the analytic function
$\alpha\mapsto\omega_{\textnormal{Eq}}(\alpha)\cdot\bar{l}+\Omega_{\bar{j}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{\bar{j^{\prime}}}^{\textnormal{\tiny{E}}}(\alpha)$
is identically zero which enters in contradiction with Lemma 4.8 up to
replacing $\omega_{\textnormal{Eq}}$ by
$(\omega_{\textnormal{Eq}},\Omega_{\bar{j}}^{\textnormal{\tiny{E}}})$ or
$(\omega_{\textnormal{Eq}},\Omega_{\bar{j}}^{\textnormal{\tiny{E}}},\Omega_{\bar{j^{\prime}}}^{\textnormal{\tiny{E}}})$.
$\bullet$ Now we study the case where $(j_{m})_{m}$ and $(j^{\prime}_{m})_{m}$
are both unbounded and without loss of generality we can assume that

$\lim_{m\to\infty}j_{m}=\lim_{m\to\infty}j^{\prime}_{m}=\infty.$ (4.39)

Notice that here this case only concerns the situation with a sign $"-"$ in
(4.36). Nevertheless, for later purposes, we may treat both sign situations.
From the expression (4.27) we can write

$\displaystyle\Omega_{j_{m}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j_{m}^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)=$
$\displaystyle(j_{m}\pm
j^{\prime}_{m})V_{0}(\alpha)-\big{(}\tfrac{1}{2}\pm\tfrac{1}{2}\big{)}$
$\displaystyle+(j_{m}\pm
j^{\prime}_{m})(I_{j_{m}}K_{j_{m}})\big{(}\tfrac{1}{\alpha}\big{)}\pm
j^{\prime}_{m}\Big{(}(I_{j^{\prime}_{m}}K_{j^{\prime}_{m}})\big{(}\tfrac{1}{\alpha}\big{)}-(I_{j_{m}}K_{j_{m}})\big{(}\tfrac{1}{\alpha}\big{)}\Big{)},$

with $V_{0}(\alpha)$ as in (4.25). It has been proved in [51, Lem. 3.5] that
for any $q\in\mathbb{N},$ and for any $0<\lambda_{0}<\lambda_{1},$

$\displaystyle\lim_{m\to\infty}\sup_{\lambda\in[\lambda_{0},\lambda_{1}]}\Big{|}j_{m}^{\prime}\,\partial_{\lambda}^{q}(I_{j_{m}}K_{j_{m}}-I_{j_{m}^{\prime}}K_{j_{m}^{\prime}})(\lambda)\Big{|}=0,\qquad\lim_{m\to\infty}\sup_{\lambda\in[\lambda_{0},\lambda_{1}]}\Big{|}\partial_{\lambda}^{q}(I_{j_{m}}K_{j_{m}})(\lambda)\Big{|}=0.$
(4.40)

Therefore (4.38) and (4.40) imply for any $q\in\mathbb{N},$

$\lim_{m\to\infty}|l_{m}|^{-1}\partial_{\alpha}^{q}\left(\Omega_{j_{m}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j^{\prime}_{m}}^{\textnormal{\tiny{E}}}(\alpha)\right)_{|\alpha=\alpha_{m}}=\partial_{\alpha}^{q}\,\Big{(}\bar{d}V_{0}(\alpha)+|\bar{l}|^{-1}\big{(}\tfrac{1}{2}\pm\tfrac{1}{2}\big{)}\Big{)}_{|\alpha=\overline{\alpha}}.$

By taking the limit as $m\rightarrow\infty$ in (4.37), we find

$\forall
q\in\mathbb{N},\quad\partial_{\alpha}^{q}\Big{(}{\omega}_{\textnormal{Eq}}({\alpha})\cdot\bar{c}+\bar{d}V_{0}(\alpha)+|\bar{l}|^{-1}\big{(}\tfrac{1}{2}\pm\tfrac{1}{2}\big{)}\Big{)}_{|\alpha=\overline{\alpha}}=0.$

Thus, the analytic function
$\alpha\mapsto{\omega}_{\textnormal{Eq}}(\alpha)\cdot\bar{c}+\bar{d}V_{0}(\alpha)+|\bar{l}|^{-1}\big{(}\tfrac{1}{2}\pm\tfrac{1}{2}\big{)}$
with $(\bar{c},\bar{d})\neq 0$ and
$\big{(}\bar{c},\bar{d},|\bar{l}|^{-1}\big{)}\neq 0$ is vanishing which
contradicts Lemma 4.8.
$\blacktriangleright$ Now we treat the case where the sequence $(l_{m})_{m}$
is unbounded. Up to an extraction we can assume that

$\lim_{m\to\infty}|l_{m}|=\infty.$

We shall distinguish three sub-cases.
$\bullet$ We first assume that the sequences $(j_{m})_{m}$ and
$(j^{\prime}_{m})_{m}$ are bounded. Hence we have convergences of the type
(4.39). Then taking the limit in (4.37) yields,

$\forall
q\in\mathbb{N},\quad\partial_{\alpha}^{q}{\omega}_{\textnormal{Eq}}(\bar{\alpha})\cdot\bar{c}=0.$

which leads to a contradiction as before.
$\bullet$ The case where the sequences $(j_{m})_{m}$ and
$(j^{\prime}_{m})_{m}$ are both unbounded is similar to what has been done
previously.
$\bullet$ Now we assume that the sequence $(j_{m})_{m}$ is unbounded and
$(j^{\prime}_{m})_{m}$ is bounded (the symmetric case is similar). Without
loss of generality we can assume that

$\lim_{m\to\infty}j_{m}=\infty,\qquad j_{m}^{\prime}=\overline{j}.$

One obtains from (4.40) and (4.38)

$\displaystyle\forall
q\in\mathbb{N},\quad\lim_{m\to\infty}|l_{m}|^{-1}\partial_{\alpha}^{q}\Big{(}\Omega_{j_{m}}^{\textnormal{\tiny{E}}}(\alpha)\pm\Omega_{j_{m}^{\prime}}^{\textnormal{\tiny{E}}}(\alpha)\Big{)}_{|\alpha=\alpha_{m}}=\bar{d}\partial_{\alpha}^{q}V_{0}(\bar{\alpha}).$

Consequently, taking the limit $m\to\infty$ in (4.37) gives

$\forall
q\in\mathbb{N},\quad\partial_{\alpha}^{q}\Big{(}{\omega}_{\textnormal{Eq}}({\alpha})\cdot\bar{c}+\bar{d}V_{0}(\alpha)\Big{)}_{\alpha=\overline{\alpha}}=0.$

Thus, the analytic function
$\alpha\mapsto{\omega}_{\textnormal{Eq}}(\alpha)\cdot\bar{c}+\bar{d}V_{0}(\alpha)$
is identically zero with $(\bar{c},\bar{d})\neq 0$ which contradicts Lemma
4.8. This completes the proof of Lemma 4.9. ∎

###  4.4 The functional of interest and associated tame estimates

In this subsection, we shall reformulate the problem in terms of embedded tori
through the introduction of action-angle variables. This leads to look for the
zeros of a nonlinear functional. Observe that the equation (4.11) can be seen
as a quasilinear perturbation of its linearization at the equilibrium state,
namely

$\partial_{t}r=\partial_{\theta}\mathrm{L}(\alpha)(r)+X_{\mathscr{P}}(r),\qquad
X_{\mathscr{P}}(r)\triangleq\tfrac{1}{2}\partial_{\theta}r+\partial_{\theta}\mathcal{K}_{\alpha}\ast
r+F^{\textnormal{\tiny{E}}}[r]+F^{\textnormal{\tiny{SW}}}[r],$ (4.41)

with $F^{\textnormal{\tiny{E}}},$ $F^{\textnormal{\tiny{SW}}}$ and
$\mathrm{L}(\alpha)$ as in (4.8), (4.9) and (4.25). The smallness property is
encoded by the introduction of a small parameter $\varepsilon.$ Then we
consider the rescalling $r\mapsto\varepsilon r$ with $r$ bounded. Therefore
(4.11) becomes

$\partial_{t}r=\partial_{\theta}\mathrm{L}(\alpha)(r)+\varepsilon
X_{\mathscr{P}_{\varepsilon}}(r),\qquad
X_{\mathscr{P}_{\varepsilon}}(r)\triangleq\varepsilon^{-2}X_{\mathscr{P}}(\varepsilon
r).$ (4.42)

Remark that (4.42) can be written in the Hamiltonian form

$\partial_{t}r=\partial_{\theta}\nabla\mathcal{H}_{\varepsilon}(r),\qquad\mathcal{H}_{\varepsilon}(r)\triangleq\varepsilon^{-2}\mathscr{H}(\varepsilon
r)\triangleq\mathscr{H}_{\mathrm{L}}(r)+\varepsilon\mathscr{P}_{\varepsilon}(r),$

with $\mathscr{H}_{\mathrm{L}}$ as in (4.26) and
$\varepsilon\mathscr{P}_{\varepsilon}(r)$ containing the higher order terms
more than cubic. We shall now reformulate the problem in terms of embedded
tori. For that purpose, we introduce the action-angle variables. This is done
in the following way. Introducing the symmetrized tangential sets
$\overline{\mathbb{S}}$ and $\mathbb{S}_{0}$ associated to $\mathbb{S}$ in
(4.32)

$\overline{\mathbb{S}}\triangleq\mathbb{S}\cup(-\mathbb{S})=\\{\pm
j,\,\,j\in\mathbb{S}\\},\qquad\mathbb{S}_{0}\triangleq\overline{\mathbb{S}}\cup\\{0\\},$

we can split the phase space $L^{2}_{0}(\mathbb{T})$ into tangential and
normal subspaces

$L^{2}_{0}(\mathbb{T})=L_{\overline{\mathbb{S}}}\overset{\perp}{\oplus}L^{2}_{\bot},\qquad
L_{\overline{\mathbb{S}}}\triangleq\Big{\\{}v=\sum_{j\in\overline{\mathbb{S}}}v_{j}\mathbf{e}_{j},\,\,v_{-j}=\overline{v_{j}}\Big{\\}},\qquad
L^{2}_{\bot}\triangleq\Big{\\{}z=\sum_{j\in\mathbb{Z}\setminus\mathbb{S}_{0}}z_{j}\mathbf{e}_{j},\,\,z_{-j}=\overline{z_{j}}\Big{\\}}\,.$
(4.43)

Therefore, we can decompose any $r\in L_{0}^{2}(\mathbb{T})$ as follows

$r=v+z,\qquad
v=\Pi_{\mathbb{S}_{0}}r\triangleq\sum_{j\in\overline{\mathbb{S}}}r_{j}\mathbf{e}_{j}\in
L_{\overline{\mathbb{S}}},\qquad
z=\Pi_{\mathbb{S}_{0}}^{\perp}r\triangleq\sum_{j\in\mathbb{Z}\setminus\mathbb{S}_{0}}r_{j}\mathbf{e}_{j}\in
L_{\perp}^{2}.$

We consider small amplitudes

$(\mathtt{a}_{j})_{j\in\overline{\mathbb{S}}}\in(\mathbb{R}_{+}^{*})^{d},\qquad\mathtt{a}_{-j}=\mathtt{a}_{j}$

and introduce the action-angle variables

$(I,\vartheta)=\Big{(}\big{(}I_{j}\big{)}_{j\in\overline{\mathbb{S}}},\big{(}\vartheta_{j}\big{)}_{j\in\overline{\mathbb{S}}}\Big{)},\qquad
I_{-j}=I_{j}\in\mathbb{R},\qquad\vartheta_{-j}=-\vartheta_{j}\in\mathbb{T}$

such that on the tangential set $L_{\overline{\mathbb{S}}}$ we have

$\forall r\in L_{\overline{\mathbb{S}}},\quad
r=\sum_{j\in\overline{\mathbb{S}}}\sqrt{\mathtt{a}_{j}^{2}+|j|I_{j}}\,e^{{\rm
i}\vartheta_{j}}\mathbf{e}_{j}\triangleq v(\vartheta,I).$ (4.44)

This defines an application

$\mathtt{A}:\begin{array}[t]{rcl}\mathbb{T}^{d}\times\mathbb{R}^{d}\times
L_{\perp}^{2}(\mathbb{T})&\rightarrow&L_{0}^{2}(\mathbb{T})\\\
(\vartheta,I,z)&\mapsto&r=v(\vartheta,I)+z\end{array}$

which is symplectic with respect to the symplectic structure $\mathscr{W}$
defined in (4.13). We refer for instance to [42] for a proof of this result.
In these coordinates the new Hamiltonian system is generated by the
Hamiltonian

$\mathscr{H}_{\varepsilon}\triangleq\mathcal{H}_{\varepsilon}\circ\mathtt{A}=\mathscr{N}+\varepsilon\mathcal{P}_{\varepsilon},\qquad\mathscr{N}\triangleq-{\omega}_{\textnormal{Eq}}(\alpha)\cdot
I+\frac{1}{2}\big{\langle}\mathrm{L}(\alpha)\,z,z\big{\rangle}_{L^{2}(\mathbb{T})},\qquad\mathcal{P}_{\varepsilon}\triangleq\mathscr{P}_{\varepsilon}\circ\mathtt{A}.$
(4.45)

We look for an embedded invariant torus

$i:\begin{array}[t]{rcl}\mathbb{T}^{d}&\rightarrow&\mathbb{T}^{d}\times\mathbb{R}^{d}\times
L^{2}_{\bot}\\\
\varphi&\mapsto&i(\varphi)\triangleq(\vartheta(\varphi),I(\varphi),z(\varphi))\end{array}$

of the Hamiltonian vector field

$X_{\mathscr{H}_{\varepsilon}}\triangleq(\partial_{I}\mathscr{H}_{\varepsilon},-\partial_{\vartheta}\mathscr{H}_{\varepsilon},\Pi_{\mathbb{S}_{0}}^{\bot}\partial_{\theta}\nabla_{z}\mathscr{H}_{\varepsilon})$
(4.46)

filled by quasi-periodic solutions with Diophantine frequency vector $\omega$.
Note that for value $\varepsilon=0$, the Hamiltonian system

$\omega\cdot\partial_{\varphi}i(\varphi)=X_{\mathscr{H}_{0}}(i(\varphi))$

possesses, for any value of the parameter $\alpha\in(\alpha_{0},\alpha_{1})$,
the flat invariant torus $(\varphi,0,0).$ Similarly to [11, 40, 42, 51], we
shall introduce a free parameter $\kappa\in\mathbb{R}^{d}$ to deal with zero
$\varphi$-average conditions in the construction of an almost approximate
right inverse for the linearized operator and therefore consider the following
family of modified Hamiltonians,

$\displaystyle\mathscr{H}_{\varepsilon}^{\kappa}\triangleq\mathscr{N}_{\kappa}+\varepsilon\mathcal{P}_{\varepsilon},\qquad\mathscr{N}_{\kappa}\triangleq\kappa\cdot
I+\frac{1}{2}\big{\langle}\mathrm{L}(\alpha)\,z,z\big{\rangle}_{L^{2}(\mathbb{T})}.$
(4.47)

We mention that the original problem is recovered by taking
$\kappa=-{\omega}_{\textnormal{Eq}}(\alpha).$ Now we are interested in finding
non-trivial zeros of the nonlinear functional

$\begin{array}[]{l}\mathscr{F}(i,\kappa,(\alpha,\omega),\varepsilon)\triangleq\omega\cdot\partial_{\varphi}i(\varphi)-X_{\mathscr{H}_{\varepsilon}^{\kappa}}(i(\varphi))=\left(\begin{array}[]{c}\omega\cdot\partial_{\varphi}\vartheta(\varphi)-\kappa-\varepsilon\partial_{I}\mathcal{P}_{\varepsilon}(i(\varphi))\\\
\omega\cdot\partial_{\varphi}I(\varphi)+\varepsilon\partial_{\vartheta}\mathcal{P}_{\varepsilon}(i(\varphi))\\\
\omega\cdot\partial_{\varphi}z(\varphi)-\partial_{\theta}\big{[}\mathrm{L}(\alpha)z(\varphi)+\varepsilon\nabla_{z}\mathcal{P}_{\varepsilon}\big{(}i(\varphi)\big{)}\big{]}\end{array}\right).\end{array}$
(4.48)

We point out that we can easily check that the Hamiltonian
$\mathscr{H}_{\varepsilon}^{\kappa}$ is reversible in the sense of the
Definition 4.1, that is,

$\mathscr{H}_{\varepsilon}^{\kappa}\circ\mathbf{S}=\mathscr{H}_{\varepsilon}^{\kappa},\qquad\mathbf{S}(\vartheta,I,z)\triangleq(-\vartheta,I,\mathscr{S}z),$

with $\mathscr{S}$ as in (4.14). Thus, we look for reversible tori solutions
of $\mathscr{F}(i,\kappa,(\alpha,\omega),\varepsilon)=0,$ that is satisfying

$\mathbf{S}i(\varphi)=i(-\varphi),\qquad\textnormal{i.e.}\qquad\begin{array}[]{ccc}\vartheta(-\varphi)=-\vartheta(\varphi),&I(-\varphi)=I(\varphi),&z(-\varphi)=(\mathscr{S}z)(\varphi).\end{array}$

Now we define the periodic component $\mathfrak{I}$ of the torus $i$ together
with its Sobolev norm by

$\mathfrak{I}(\varphi)\triangleq
i(\varphi)-(\varphi,0,0)=(\Theta(\varphi),I(\varphi),z(\varphi)),\qquad\Theta(\varphi)\triangleq\vartheta(\varphi)-\varphi,\qquad\|\mathfrak{I}\|_{q,s}^{\gamma,\mathcal{O}}\triangleq\|\Theta\|_{q,s}^{\gamma,\mathcal{O}}+\|I\|_{q,s}^{\gamma,\mathcal{O}}+\|z\|_{q,s}^{\gamma,\mathcal{O}}.$

The norm $\|\cdot\|_{q,s}^{\gamma,\mathcal{O}}$ is defined in Section 4.1. We
shall fix $q$ as follows

$q\triangleq q_{0}+1,$ (4.49)

with the $q_{0}$ appearing in Lemma 4.9. This particular choice is relevant in
the final subsection when checking the perturbed Rüssmann conditions. We also
define the set of parameters $\mathcal{O}$ as follows

$\mathcal{O}\triangleq(\alpha_{0},\alpha_{1})\times
B(0,R_{0})\subset\mathbb{R}^{d+1},$ (4.50)

where the open ball $B(0,R_{0})\subset\mathbb{R}^{d}$ with $R_{0}>0$ is chosen
such that

$\omega_{\textnormal{Eq}}\big{(}(\alpha_{0},\alpha_{1})\big{)}\subset
B\big{(}0,\tfrac{R_{0}}{2}\big{)}.$

This is well-defined by continuity of the application
$\omega_{\textnormal{Eq}}$ in (4.33). This particular structure is chosen so
that the perturbed frequency vector in Section 4.6 is included in the
corresponding component of $\mathcal{O}$ in order to check trivial inclusions.
We shall now prove tame estimates for $X_{\mathscr{P}}$.

######  Lemma 4.10.

Let $(\gamma,s_{0},s,q)$ satisfying (4.4), (4.3) and (4.49). There exists
$\varepsilon_{0}\in(0,1]$ such that if

$\|r\|_{q,s_{0}+2}^{\gamma,\mathcal{O}}\leqslant\varepsilon_{0},$

then we have the following estimates for the vector field $X_{\mathscr{P}}$ in
(4.41)

  1. (i)

$\|X_{\mathscr{P}}(r)\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|r\|_{q,s+2}^{\gamma,\mathcal{O}}\|r\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}.$

  2. (ii)

$\|d_{r}X_{\mathscr{P}}(r)[\rho]\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\rho\|_{q,s+2}^{\gamma,\mathcal{O}}\|r\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}+\|r\|_{q,s+2}^{\gamma,\mathcal{O}}\|\rho\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}.$

  3. (iii)

$\|d_{r}^{2}X_{\mathscr{P}}(r)[\rho_{1},\rho_{2}]\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\rho_{1}\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}\|\rho_{2}\|_{q,s+2}^{\gamma,\mathcal{O}}+\|\rho_{1}\|_{q,s+2}^{\gamma,\mathcal{O}}\|\rho_{2}\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}+\|r\|_{q,s+2}^{\gamma,\mathcal{O}}\|\rho_{1}\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}\|\rho_{2}\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}$.

###### Proof.

It suffices to prove the estimate (iii). Indeed, the estimates (i) and (ii)
are consequences of (iii) by a direct application of Taylor formula since
$X_{\mathscr{P}}(0)=0$ and $d_{r}X_{\mathscr{P}}(0)=0.$ Recall from (4.15) and
(4.18) that

$\displaystyle d_{r}X_{\mathscr{H}}(r)[\rho]$
$\displaystyle=d_{r}F^{\textnormal{\tiny{E}}}(r)[\rho]+d_{r}F^{\textnormal{\tiny{SW}}}(r)[\rho]$
$\displaystyle=\partial_{\theta}\big{(}V_{r}^{\textnormal{\tiny{E}}}\rho\big{)}+\partial_{\theta}\mathbf{L}_{r}^{\textnormal{\tiny{E}}}(\rho)-\partial_{\theta}\big{(}V_{r}^{\textnormal{\tiny{SW}}}\rho\big{)}-\partial_{\theta}\mathbf{L}_{r}^{\textnormal{\tiny{SW}}}(\rho).$

Differentiating the last expression with respect to $r$ yields

$d_{r}^{2}X_{\mathscr{P}}(r)[\rho_{1},\rho_{2}]=\partial_{\theta}\big{(}(d_{r}V_{r}^{\textnormal{\tiny{E}}}[\rho_{2}])\rho_{1}\big{)}-\partial_{\theta}(d_{r}\mathbf{L}_{r}^{\textnormal{\tiny{E}}}[\rho_{2}]\rho_{1})+\partial_{\theta}\big{(}(d_{r}V_{r}^{\textnormal{\tiny{SW}}}[\rho_{2}])\rho_{1}\big{)}-\partial_{\theta}(d_{r}\mathbf{L}_{r}^{\textnormal{\tiny{SW}}}[\rho_{2}]\rho_{1}).$

But it has been proved in [42, Lem. 5.2] and [51, Lem. 5.2] that each term in
the right hand-side of the previous equality satisfy an estimate as in (iii)
in the statement of this lemma which concludes the proof of this latter. ∎

Consequently proceeding as for [51, Lem. 5.3], the previous lemma implies the
following one stating tame estimates for the perturbed Hamiltonian vector
field in the action-angle-normal variables.

######  Lemma 4.11.

Let $(\gamma,s_{0},s,q)$ satisfy (4.4), (4.3) and (4.49). There exists
$\varepsilon_{0}\in(0,1)$ such that if

$\varepsilon\leqslant\varepsilon_{0},\qquad\|\mathfrak{I}\|_{q,s_{0}+2}^{\gamma,\mathcal{O}}\leqslant
1,$

then the perturbed Hamiltonian vector field in the new variables

$X_{\mathcal{P}_{\varepsilon}}=(\partial_{I}\mathcal{P}_{\varepsilon},-\partial_{\vartheta}\mathcal{P}_{\varepsilon},\Pi_{\mathbb{S}}^{\perp}\partial_{\theta}\nabla_{z}\mathcal{P}_{\varepsilon})$

defined through (4.45) and (4.46) satisfies the following tame estimates

  1. (i)

$\|X_{\mathcal{P}_{\varepsilon}}(i)\|_{q,s}^{\gamma,\mathcal{O}}\lesssim
1+\|\mathfrak{I}\|_{q,s+2}^{\gamma,\mathcal{O}}.$

  2. (ii)

$\big{\|}d_{i}X_{\mathcal{P}_{\varepsilon}}(i)[\,\widehat{i}\,]\big{\|}_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\,\widehat{i}\,\|_{q,s+2}^{\gamma,\mathcal{O}}+\|\mathfrak{I}\|_{q,s+2}^{\gamma,\mathcal{O}}\|\,\widehat{i}\,\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}.$

  3. (iii)

$\big{\|}d_{i}^{2}X_{\mathcal{P}_{\varepsilon}}(i)[\,\widehat{i},\widehat{i}\,]\big{\|}_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\,\widehat{i}\,\|_{q,s+2}^{\gamma,\mathcal{O}}\|\,\widehat{i}\,\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}+\|\mathfrak{I}\|_{q,s+2}^{\gamma,\mathcal{O}}\left(\|\,\widehat{i}\,\|_{q,s_{0}+1}^{\gamma,\mathcal{O}}\right)^{2}.$

###  4.5 Almost approximate right inverse

The aim of this section is to construct for any vector-valued function
$\kappa_{0}:\mathcal{O}\to\mathbb{R}^{d}$ and any reversible torus
$i_{0}=(\vartheta_{0},I_{0},z_{0})$ close to the flat one an almost
approximate right inverse for the linearized operator

$d_{(i,\kappa)}\mathscr{F}(i_{0},\kappa_{0})[\widehat{\imath}\,,\widehat{\kappa}]=\omega\cdot\partial_{\varphi}\widehat{\imath}-d_{i}X_{\mathscr{H}_{\varepsilon}^{\kappa_{0}}}(i_{0}(\varphi))[\widehat{\imath}]-(\widehat{\kappa},0,0)$
(4.51)

associated with the functional $\mathscr{F}$ defined in (4.48). For this
purpose, we use the Berti-Bolle theory developed in [11] and [40, Sec. 6].
Namely, we can find a linear diffeomorphism of the toroidal phase space
$\mathbb{T}^{d}\times\mathbb{R}^{d}\times L_{\perp}^{2}$ such that the
conjugation of (4.51) by this application is a triangular system in the
action-angles-normal variables up to error terms either vanishing at an exact
solution or small fast decaying. The key point to solve the triangular system
is that it is sufficient to almost invert the linearized operator in the
normal directions. According to the computations done in [11, 40], this latter
admits the following form

$\mathscr{L}_{\perp}\triangleq\mathscr{L}_{\perp}(i_{0})\triangleq\Pi_{\mathbb{S}_{0}}^{\perp}\Big{(}\omega\cdot\partial_{\varphi}-\partial_{\theta}\,\partial_{z}\nabla_{z}\mathscr{H}_{\varepsilon}^{\kappa_{0}}(i_{0}(\varphi))-\varepsilon\partial_{\theta}\mathcal{R}(\varphi)\Big{)}\Pi_{\mathbb{S}_{0}}^{\perp},$
(4.52)

where $\mathscr{H}_{\varepsilon}^{\kappa_{0}}$ is as in (4.47) and
$\mathcal{R}(\varphi)$ is a remainder operator coming from a coupling with the
tangential part and given by

$\mathcal{R}(\varphi)\triangleq
L_{2}^{\top}(\varphi)\partial_{I}\nabla_{I}\mathcal{P}_{\varepsilon}(i_{0}(\varphi))L_{2}(\varphi)+L_{2}^{\top}(\varphi)\partial_{z}\nabla_{I}\mathcal{P}_{\varepsilon}(i_{0}(\varphi))+\partial_{I}\nabla_{z}\mathcal{P}_{\varepsilon}(i_{0}(\varphi))L_{2}(\varphi),$

with $\mathcal{P}_{\varepsilon}$ as in (4.45) and

$L_{2}:\mathbb{R}^{d}\rightarrow L^{2}_{\perp},\qquad
L_{2}(\phi)\triangleq-[(\partial_{\vartheta}\widetilde{z}_{0})(\vartheta_{0}(\phi))]^{\top}\partial_{\theta}^{-1},\qquad\widetilde{z}_{0}(\vartheta)\triangleq
z_{0}(\vartheta_{0}^{-1}(\vartheta)).$

We used the following definition by duality for the transposed operator
$L_{2}^{\top}:L^{2}_{\perp}\to\mathbb{R}^{d}$

$\forall\,u\in
L^{2}_{\perp}\,,\quad\forall\,v\in\mathbb{R}^{d},\quad\big{\langle}L_{2}^{\top}(\varphi)u,v\big{\rangle}_{\mathbb{R}^{d}}=\big{\langle}u,L_{2}(\varphi)v\big{\rangle}_{L^{2}(\mathbb{T}^{d})}.$

Furthermore, we can have a more explicit decomposition of the operator
$\mathscr{L}_{\perp}$. The result is described in the following proposition
whose proof is similar to [51, Prop. 6.1] or [42, Prop. 6.1].

######  Proposition 4.2.

Let $(d,\gamma,s_{0},q)$ satisfy (4.2), (4.4), (4.3) and (4.49). Then, the
operator $\mathscr{L}_{\perp}$ in (4.52) writes

$\mathscr{L}_{\perp}=\Pi_{\mathbb{S}_{0}}^{\perp}\Big{(}\mathcal{L}_{\varepsilon
r}-\varepsilon\partial_{\theta}\mathcal{R}\Big{)}\Pi_{\mathbb{S}_{0}}^{\perp},$
(4.53)

where the operator $\mathcal{L}_{\varepsilon r}$ is defined as follows with
$V_{\varepsilon r}$ and $\mathbf{L}_{\varepsilon r}$ obtained from (4.22),

$\mathcal{L}_{\varepsilon
r}\triangleq\omega\cdot\partial_{\varphi}+\partial_{\theta}\big{(}V_{\varepsilon
r}\cdot\big{)}+\partial_{\theta}\mathbf{L}_{\varepsilon r}.$ (4.54)

The function $r$ is linked to the reversible torus $i_{0}$ in the following
way

$r(\varphi,\cdot)\triangleq\mathtt{A}\big{(}i_{0}(\varphi)\big{)},\qquad
r(-\varphi,-\theta)=r(\varphi,\theta)$ (4.55)

and satisfies the following estimates

$\|r\|_{q,s}^{\gamma,\mathcal{O}}\lesssim
1+\|\mathfrak{I}_{0}\|_{q,s}^{\gamma,\mathcal{O}},\qquad\|\Delta_{12}r\|_{q,s}^{\gamma,\mathcal{O}}\lesssim\|\Delta_{12}i\|_{q,s}^{\gamma,\mathcal{O}}+\|\Delta_{12}i\|_{q,s_{0}}^{\gamma,\mathcal{O}}\max_{j\in\\{1,2\\}}\|\mathfrak{I}_{j}\|_{q,s}^{\gamma,\mathcal{O}}.$
(4.56)

Finally, the operator $\mathcal{R}$ is an integral operator with kernel
$\mathcal{J}$ satisfying the symmetry property

$\mathcal{J}(-\varphi,-\theta,-\eta)=\mathcal{J}(\varphi,\theta,\eta)$ (4.57)

and the following estimates for all $\ell\in\mathbb{N}$,

$\sup_{\eta\in\mathbb{T}}\|(\partial_{\theta}^{\ell}\mathcal{J})(\ast,\cdot,\centerdot,\eta+\centerdot)\|_{q,s}^{\gamma,\mathcal{O}}\lesssim
1+\|\mathfrak{I}_{0}\|_{q,s+3+\ell}^{\gamma,\mathcal{O}}$ (4.58)

and