# Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-
Periodic Solutions

Zineb Hassainia  and  Haroune Houamed  New York University Abu Dhabi
Abu Dhabi
United Arab Emirates<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of
two interconnected potential vorticities, each is governed by an active scalar
equation. These vorticities are linked through a distinctive combination of
their respective stream functions, which can be loosely characterized as a
parameterized blend of both Euler and shallow-water stream functions.

In this article, we study (QS2L) in two directions: First, we prove the
existence and uniqueness of global weak solutions in the class of Yudovich,
that is when the initial vorticities are only bounded and Lebesgue-integrable.
The uniqueness is obtained as a consequence of a stability analysis of the
flow-maps associated with the two vorticities. This approach replaces the
relative energy method and allows us to surmount the absence of a velocity
formulation for (QS2L). Second, we show how to construct $m$-fold time-
periodic solutions bifurcating from two arbitrary distinct initial discs
rotating with the same angular velocity. This is achieved provided that the
number of symmetry $m$ is large enough, or for any symmetry
$m\in\mathbb{N}^{*}$ as long as one of the initial radii of the discs does not
belong to some set that contains, at most, a finite number of elements. Due to
its multi-layer structure, it is essential to emphasize that the bifurcation
diagram exhibits a two-dimensional pattern. Upon analysis, it reveals some
similarities with the scheme accomplished for the doubly connected V-states of
the Euler and shallow-water equations. However, the coupling between the
equations gives rise to several difficulties in various stages of the proof
when applying Crandall-Rabinowitz’s Theorem. To address this challenge, we
conduct a careful analysis of the coupling between the kernels associated with
the Euler and shallow-water equations.

###### Key words and phrases:

Quasi-geostrophic equations, weak solutions, Lagrangian solutions, V-states,
vortex patches.

###### Contents

  1. 1 Introduction and main results
  2. 2 Functional tools and building blocks
    1. 2.1 Bessel functions and asymptotic expansions
    2. 2.2 Kernel estimates
    3. 2.3 Hölder-continuity of some singular operators
    4. 2.4 Crandall-Rabinowitz’s theorem
  3. 3 Weak and Lagrangian solutions
    1. 3.1 Existence of global solutions
      1. 3.1.1 Existence of weak-distributional solutions
      2. 3.1.2 Existence of Lagrangian solutions
    2. 3.2 Stability of the flow-maps and uniqueness
  4. 4 Uniformly rotating solutions
    1. 4.1 Contour dynamics equation
    2. 4.2 Linearization around discs
    3. 4.3 Regularity properties
    4. 4.4 Spectral analysis of the linearized operator
    5. 4.5 Applying Crandall-Rabinowitz’s theorem and proof of Theorem 1.3
  5. 5 Endnote

##  1\. Introduction and main results

In this work, we consider the quasi-geostrophic two-layer model (Phillips’
model) given by

(QS2L)
$\left\\{\begin{array}[]{ll}\partial_{t}\omega_{i}+(\nabla^{\perp}\psi_{i})\cdot\nabla\omega_{i}=0,&\qquad\qquad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{2},\vspace{2mm}\\\
\omega_{i}=\Delta\psi_{i}+(-\delta)^{i-1}\lambda^{2}(\psi_{2}-\psi_{1}),&\qquad\qquad
i\in\\{1,2\\},\vspace{2mm}\\\
\omega_{i}|_{t=0}=\omega_{i,0},\end{array}\right.$

where $\psi_{i}$ and $\omega_{i}$ stand for the stream function and potential
vorticity in the $i^{\text{the}}$ layer, respectively. The parameter
$\delta>0$ above refers to the ratio of the upper to lower layer thickness
when the fluid is at rest, whereas $\lambda\geq 0$ describes the rigidity of
the interface. In particular, when $\lambda=0$ (which corresponds to a
perfectly rigid interface), the two layers become uncoupled and behave as two
independent two-dimensional systems obeying the Euler equations.

The system of equations (QS2L) serves as a simplified model and a foundational
concept in the study of large-scale atmospheric and oceanic flows. In
atmospheric applications, these layers often represent the upper and lower
troposphere. On the other hand, in oceanic modeling, they might correspond to
the upper mixed layer and the deeper ocean. See for instance [45] for more
details.

Though it is still far less studied compared to Euler equations, from
analytical point of view, the quasi-geostrophic two-layer model has enjoyed
considerable interest in computational fluid dynamics. We refer to [1, 10, 11,
22, 44, 45] and the references therein for a series of results on the
numerical analysis of (QS2L).

In the first part of this paper, we are interested in the construction of weak
solutions to the quasi-geostrophic two-layer system (QS2L) with initial data
of Yudovich-type. These solutions solve a weak formulation of the equations in
the distribution sense, which we introduce next.

######  Definition (Weak–distributional solutions).

We say that $(\omega_{i})_{i\in\\{1,2\\}}$ is a weak distributional (or simply
weak) solution of (QS2L) with initial data $\omega_{i,0}\in
L^{p}(\mathbb{R}^{2})$, for some $p\in[1,\infty]$, if

  * (i)

$\omega_{1},\omega_{2}\in
L^{\infty}_{\text{loc}}(\mathbb{R}^{+};L^{p}(\mathbb{R}^{2})),$

  * (ii)

$\nabla^{\perp}\psi_{1},\nabla^{\perp}\psi_{2}\in
L^{\infty}_{\text{loc}}(\mathbb{R}^{+};L^{p^{\prime}}(\mathbb{R}^{2})),$ where
$p^{\prime}$ denotes the conjugate of $p$,

  * (iii)

for any test function $\varphi\in
C^{1}_{c}(\mathbb{R}^{+}\times\mathbb{R}^{2})$, it holds that

$\int_{[0,t]\times\mathbb{R}^{2}}\omega_{i}(t,x)\left(\partial_{t}\varphi+\nabla^{\perp}\psi_{i}\cdot\nabla\varphi\right)(t,x)dxdt=\int_{\mathbb{R}^{2}}\omega_{i}(t,x)\varphi(t,x)dx-\int_{\mathbb{R}^{2}}\omega_{i}(0,x)\varphi(0,x)dx,$

for all $i\in\\{1,2\\}$ and $t\in\mathbb{R}^{+}$.

Questions of global existence and uniqueness of weak solutions have been
originally addressed in the case of Euler equations by Yudovich in [41, 51,
49], where this received satisfactory answers provided that the initial
vorticity is bounded or lies in $L^{p}$ spaces, for all $p<\infty$, with
adequate assumptions on the growth of its norm as $p\to\infty$. Later on, non-
uniqueness was obtained for the forced Euler equations [4], where the
vorticity is barely bounded, and in the unforced case [8] in a weaker
functional setting.

As it is well known, the solenoidal velocity field associated with a bounded
vorticity is not Lipshitz in general. Instead, it enjoys the so-called
LogLipshitz regularity which is crucial in the original proof of uniqueness
for the Euler equations [41, 51] where the approach therein is laid out by a
stability analysis in the relative energy setting. This aproach does not seem
to be applicable in the case of quasi-geostrophic two-layer model (QS2L)
because the equations of the velocities $\nabla^{\perp}\psi_{1}$ and
$\nabla^{\perp}\psi_{2}$ are far from been compared to the velocity equation
in Euler’s system.

The alternative way to study stability aspects of models such as quasi-
geostrophic two-layer (QS2L) would be by understanding first the same question
about a different quantity. Here, this will be done for the flow-maps
associated with the velocities $\nabla^{\perp}\psi_{1}$ and
$\nabla^{\perp}\psi_{2}$. This draws insight from the recent work [16] on
nonlinear transport equations advected by a non-local velocity field. This
also motivates the consideration Lagrangian solutions that we recall next.

######  Definition (Lagrangian solutions).

We say that $(\omega_{i})_{i\in\\{1,2\\}}$ is a Lagrangian solution of (QS2L)
if $\omega_{i}(t,\cdot)$ is the push forward of the initial data
$\omega_{i,0}$ by the flow-map associated with the velocity field
$\nabla^{\perp}\psi_{i}$, for any $i\in\\{1,2\\}$. More precisely,
$(\omega_{i})_{i\in\\{1,2\\}}$ is a Lagrangian solution of (QS2L) if

$\omega_{i}(t,\cdot)=X_{i}(t,\cdot)_{\sharp}\omega_{i,0}$

and $X_{i}(t,\cdot)$ solves the ODE

(ODE)
$\left\\{\begin{array}[]{ll}\frac{d}{dt}X_{i}(t,x)=(\nabla^{\perp}\psi_{i})\left(X_{i}(t,x)\right),\vspace{2mm}\\\
X_{i}(0,x)=x,\end{array}\right.$

for all $i\in\\{1,2\\}$ and $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{2}.$

In order to construct global solutions of (QS2L) in Yudovich’s class, it is
important to understand the coupling in that system of equations. A crucial
step in our analysis below is introducing suitable new unknowns which are made
of a specific combination of the solutions $\omega_{1}$ and $\omega_{2}$. The
new unknowns allow us to recast the equations connecting the vorticities with
their stream functions in a way that is naturally compatible with the coupling
in (QS2L) and which also reveals the connexion between (QS2L), Euler and
shallow-water equations. This is fundamental in our proof of existence of
solutions and is discussed in Section 3. Afterwards, this observation inspires
introducing a similar combination of the flow–maps $X_{1}$ and $X_{2}$ which
serves in their stability analysis. The uniqueness of weak-distributional
solutions in Yudovich’s class follows subsequently. This is summarized in our
first main theorem which we state next.

######  Theorem 1.1 (Existence and uniqueness of weak solutions).

For any initial data satisfying

$(\omega_{1,0},\omega_{2,0})\in L^{1}\cap L^{\infty}(\mathbb{R}^{2}),$

there is a unique global weak solution $(\omega_{1},\omega_{2})$ to (QS2L)
enjoying the bounds

$(\omega_{1},\omega_{2})\in L^{\infty}(\mathbb{R}^{+},L^{1}\cap
L^{\infty}(\mathbb{R}^{2})),$

as well as the conservation of norms

$\left\|\omega_{j}(t,\cdot)\right\|_{L^{q}(\mathbb{R}^{2})}=\left\|\omega_{j,0}\right\|_{L^{q}(\mathbb{R}^{2})},$

for any $i\in\\{1,2\\}$, $q\in[1,\infty]$ and $t\geq 0$. Moreover, the
solution is continuous in time in the sense that

$(\omega_{1},\omega_{2})\in C(\mathbb{R}^{+},L^{p}(\mathbb{R}^{2})),$

for all $p\in[1,\infty)$.

Before we move on to our second main result of this paper, allow us to
establish a remarkable consequence of Theorem 1.1. In particular, the next
corollary exhibits the connexion between solutions of the system of equation
(QS2L) in the case $\delta=1$ and the Euler equations.

######  Corollary 1.2.

The solution $(\omega_{1},\omega_{2})$ of the system of equations (QS2L) with
$\delta=1$ and initial data

$\omega_{0,1}=\omega_{0,1}\overset{\rm def}{=}\omega_{0}\in L^{1}\cap
L^{\infty}(\mathbb{R}^{2})$

is given by

$\omega_{1}=\omega_{2}=\omega,$

where $\omega$ is the unique Yudovich solution of the Euler’s equation

(E) $\left\\{\begin{array}[]{ll}\partial_{t}\omega+v\cdot\nabla\omega=0,\\\
v=-\nabla^{\perp}(-\Delta)^{-1}\omega,\\\
\omega_{|t=0}=\omega_{0}.\end{array}\right.$

###### Proof.

The proof is straightforward once we notice that the system of equations
(QS2L) is symmetric when $\delta=1$ in the sense that, if
$(\omega_{1},\omega_{2})$ is the solution associated with the initial data
$(\omega_{0,1},\omega_{0,2})$, then $(\omega_{2},\omega_{1})$ is the solution
associated with the initial data $(\omega_{0,2},\omega_{0,1})$. Accordingly,
if we set initially

$\omega_{0,1}=\omega_{0,1},$

then it immediately follows, by uniqueness of solutions from Theorem 1.1, that

$(\omega_{2},\omega_{1})=(\omega_{1},\omega_{2}).$

Consequently, we obtain from the second equation in (QS2L) that

$\left(-\Delta+\lambda^{2}\right)(\psi_{1}-\psi_{2})=0,$

whereby, we deduce that

$\psi_{1}=\psi_{2}.$

Therefore, by inserting the latter identity in (QS2L), we conclude that
$\omega\overset{\rm def}{=}\omega_{1}=\omega_{2}$ solves the Euler equation
(E), thereby completing the proof of the corollary. ∎

Typical examples of solutions covered by Theorem 1.1 are the so-called vortex-
patches. These are vorticities uniformly distributed in a bounded region $D$
and are generated from given initial data of the format

$(\omega_{1,0},\omega_{2,0})=(\mathrm{1}_{D_{1}},\mathrm{1}_{D_{2}}),$

where $D_{i}$, for $i\in\\{1,2\\}$, is a bounded domain in $\mathbb{R}^{2}$
with smooth boundary. These initial data fall in the setting of Theorem 1.1
which ensures the existence of a unique global solution associated with them.
Moreover, the patch structure is preserved by the evolution and, at each time
$t\geq 0$, the potential vorticities are given by

$(\omega_{1},\omega_{2})=(\mathrm{1}_{D_{1,t}},\mathrm{1}_{D_{2,t}}),$

with, for all $i\in\\{1,2\\}$, the transported domain

$D_{i,t}\overset{\rm def}{=}X_{i}(t,D_{i,t})$

denotes the image of $D_{i}$ by the flow $X_{i}$ defined as the solution of
(ODE). Note that, when $D_{1}$ and $D_{2}$ are discs, then, the potential
vorticities are stationary solutions to (QS2L). Thus, it is natural to look
for periodic solutions close to these

The second aim of this paper is to study the existence of time-periodic
solutions of (QS2L) given by rigid body rotating vortex patches around the
origin which are described by

$(D_{1,t},D_{2,t})=(e^{{\rm i}\Omega t}D_{1},e^{{\rm i}\Omega t}D_{2}),$

for some time-independent parameter $\Omega$ referring to as the angular
velocity. The potential vorticities corresponding to the preceding domains are
then stationary along the rotating frame with angular velocity $\Omega$ and
the boundaries $\partial D_{1,t}$ and $\partial D_{1,t}$ evolve according to
the non-linear, non-local, system

(1.1) $\left(\nabla^{\perp}\psi_{i}(t,x)-\Omega
x^{\perp}\right)\cdot\vec{n}_{i}(x)=0,$

for all $x\in\partial D_{i}$ and $i\in\\{1,2\\}$, where $\vec{n}_{i}$ is the
outward normal unit vector associated with the initial boundary $\partial
D_{i}$. Such solutions are called V-states and have been explored, first,
numerically in the case of Euler’s equations by Deem and Zabusky [18]. The
analytical proof of their existence is due to Burbea [9] and relies on the
contour dynamics equation, the conformal mapping parametrization and the
celebrated local bifurcation theorem of Crandall and Rabinowitz [15]. The
outcome is the existence of a countable family of local curves of V-states
with $m$-fold symmetries (i.e. invariant by $\frac{2\pi}{m}$ angular
rotation), with $m\geq 2$, bifurcating from the disc at the angular velocities
$\Omega_{m}=\tfrac{m-1}{2}$. A global continuation of these local curves were
constructed in [32], where the global curves limit to a vanishing of the
angular fluid velocity.

Other uniformly rotating vortex patch solutions to Euler’s equations were
constructed, using Burbea’s techniques, close to the annulus in [17] and close
to the Kirchoff ellipses in [12, 36]. We also point out that, in the last few
years, there have been several investigations on the V-states in different
settings, such as the study of the boundary regularity of the V-states [38],
the existence of multipole vortex patches [23, 24, 29, 37, 34] and the radial
symmetry properties of stationary and uniformly-rotating solutions [28, 35,
27]. Very recently quasi-periodic patch solutions to Euler’s equations were
constructed using the Nash-Moser scheme and KAM theory, see [7, 31, 33].

Similar results to the ones mentioned above were also obtained for other
active scalar equations such as the generalized surface quasi-geostrophic
equation, the quasi-geostrophic shallow-water equations and Euler equations on
the rotating unit 2-sphere. We refer to [26, 30, 46, 39, 25] and the
references therein for a series of relevant results.

In the second part of this paper, we adapt the techniques from [9, 36] and
show how to extend their validity to build time-periodic solutions of (QS2L).
In particular, our second result consists in establishing the existence of
$m$-fold symmetric V-states for (QS2L).

We point out in passing that there have been some numerical simulations
suggesting the existence of V-state solutions for (QS2L) in some specific
cases, see [44, 45] for instance. In the present study, we provide an
analytical proof of their existence through the application of local
bifurcation theory. Informally stated, our second main result is outlined in
the following theorem. A slightly more detailed and precise version will be
discussed in Section 4, later on.

######  Theorem 1.3 (Periodic solutions bifurcating from simple eigenvalues).

Let $m\in\mathbb{N}^{*}$, $\delta\geq 1$, $\lambda>0$ and $0<b_{2}\leq b_{1}$
be a set of real numbers. Then, there exist two curves of $m$-fold symmetric
pairs of simply connected V-states solving (QS2L) and bifurcating from the
stationary states

(1.2)
$\omega_{1}=\mathds{1}_{\\{x\in\mathbb{R}^{2}:|x|<b_{1}\\}}\quad\text{and}\quad\omega_{2}=\mathds{1}_{\\{x\in\mathbb{R}^{2}:|x|<b_{2}\\}}$

if one of the following two conditions is fulfilled:

  * •

either $b_{1}\neq b_{2}$ and the number of symmetry $m$ is large enough, i.e.,
$m\gg 1$,

  * •

or $m\in\mathbb{N}^{*}$, $b_{1}\in(0,\infty)$ and $b_{2}\not\in S_{m,b_{1}}$
for some set $S_{m,b_{1}}\subset(0,b_{1}]$ containing, at most, a finite
number of elements.

######  Remark 1.1.

It is to be emphasize that periodic solutions with $m$-fold symmetry, for any
$m\in\mathbb{N}^{*}$, bifurcating from the two discs (1.2), exist for almost
all values of $b_{1},b_{2}\in(0,\infty)$. However, there are some values of
these radii (for instance some values of $b_{1}=b_{2}$, as we will justify
later on) where the celebrated Crandall-Rabinowitz’s Theorem does not directly
apply due to existence of “spectral collisions”. In such cases, it is not
clear yet, at least by the analysis in this work, if time-periodic solutions
do exist.

######  Remark 1.2.

Although solutions of (QS2L) are not stable in general by switching the
positions of the vorticities—unless if we consider the case $\delta=1$ as is
shown in Corollary 1.2—our elements of proof of Theorem 1.3 remain unchanged
in the case $0<b_{1}\leq b_{2}$ and the same bifurcation result holds true, as
well. In addition, the restriction on $\delta\geq 1$ can in fact be relaxed,
see Remark 4.2, below.

#### Structure of the paper

The remaining sections of this paper are organized as follows: The proof of
existence and uniqueness of global weak solutions (Theorem 1.1) is the subject
of Section 3. This is where the crucial coupling in the set of equations
(QS2L) will be discussed in details, altogether with the approximate scheme of
(QS2L) that we utilize to build global weak and Lagrangian solutions.
Afterwards, a stability result of the flow-maps associated with the weak
solutions will be established which eventually yields the uniqueness of weak
solutions.

At last, in Section 4, we build time-periodic solutions by employing Crandall-
Rabinowitz’s Theorem 2.6. This requires a precise analysis of spectral
properties of the linearized operator associated with the boundary equation
around steady states (discs) which is discussed with details in the same that
section.

The achievement of these results also hinges upon a careful understanding of
several properties of Laplace and shallow-water kernels, as well as a fine
analysis of Bessel functions. In Sections 2, below, we provide the reader with
a tool box of various abstract results on functional analysis and operator
theory that will be employed in the proof of our theorems, and which may also
serve in other different contexts.

##  2\. Functional tools and building blocks

Before we step over the elements of proof of our main theorems, we devote this
section to a short self-contained introduction of the basis analysis of
functional and operational settings as well as properties of the relevant
kernels and all notions that will play a role in the construction of weak and
time-periodic solutions in the upcoming sections.

In the sequel, we are going to use classical notations for functional spaces,
such as Lebesgue spaces, space for log-Lipschitz functions …. Moreover, for
the sake of simplicity, we will often utilize the symbol $\lesssim$ (resp.
$\lesssim_{\delta}$) instead of $\leq C$ (resp. $\leq C_{\delta}$) when the
dependance on the constant $C$ (resp. the constant $C_{\delta}$) is harmless
(resp. degenerates when the parameter $\delta$ approaches its endpoint
values).

###  2.1. Bessel functions and asymptotic expansions

Here, we recall a few important identities that we will later use to describe
the decay properties of specific parameterized integrals. These identities,
among more relevant other results, can be found in the books by Abramowitz and
Stegun [3] and by Watson [50].

We first recall that the Bessel function of the first kind and order
$\nu\in\mathbb{N}$ given by the expansion

(2.1) $J_{\nu}(z)\overset{\rm
def}{=}\sum_{m=0}^{\infty}\frac{(-1)^{m}\left(\frac{z}{2}\right)^{\nu+2m}}{m!\Gamma(\nu+m+1)},\quad\text{for
all}\quad|\mbox{arg}(z)|<\pi.$

In particular, when $\nu=n\in\mathbb{N}$, the preceding Bessel function admits
the following integral representation, which can be found in [3, Identity
9.1.21],

(2.2)
$J_{n}(z)=\frac{1}{\pi}\int_{0}^{\pi}\cos\big{(}n\theta-z\sin(\theta)\big{)}d\theta,\quad\text{for
all}\quad z\in\mathbb{C}.$

On the other hand, the Bessel functions of imaginary argument are given by

(2.3) $I_{\nu}(z)\overset{\rm
def}{=}\sum_{m=0}^{\infty}\frac{\left(\frac{z}{2}\right)^{\nu+2m}}{m!\Gamma(\nu+m+1)},$

for all $z\in\mathbb{C}$ with $|\mbox{arg}(z)|<\pi$. Thus, for all
$\nu\in\mathbb{C}\backslash\mathbb{Z}$, we define the $K$-Bessel function by

$K_{\nu}(z)\overset{\rm
def}{=}\frac{\pi}{2}\frac{I_{-\nu}(z)-I_{\nu}(z)}{\sin(\nu\pi)}$

and, when $n\in\mathbb{Z},$ we set

$K_{n}(z)\overset{\rm def}{=}\displaystyle\lim_{\nu\rightarrow n}K_{\nu}(z)$

for all $z\in\mathbb{C}$ such that $|\mbox{arg}(z)|<\pi$. Moreover, we
emphasize that

$I_{-n}\equiv I_{n}\qquad\text{and}\qquad K_{-n}\equiv K_{n},$

for all $n\in\mathbb{N}$ and that $I_{n}$ and $K_{n}$ are smooth functions
away from the origin and satisfy

(2.4) $I^{\prime}_{0}(z)=I_{1}(z)\qquad\text{and}\qquad
K^{\prime}_{0}(z)=-K_{1}(z),$

for all $z\in\mathbb{C}$ such that $|\mbox{arg}(z)|<\pi$. The preceding
identities can be found in [3, pages 357–356].

The Bessel functions $I_{n}$ and $K_{n}$ enjoy several properties and are
featured by different equivalent representation formulas. Here, we recall a
useful expansion for $K_{n}$ that will be utilized in our proof below, which
can be found in [3, Identity 9.6.11]

$\displaystyle K_{n}(z)$
$\displaystyle=\frac{1}{2}\left(\frac{z}{2}\right)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\frac{-z^{2}}{4}\right)^{k}+(-1)^{n+1}\log\left(\frac{z}{2}\right)I_{n}(z)$
$\displaystyle\quad+\frac{1}{2}\left(\frac{-z}{2}\right)^{n}\sum_{k=0}^{\infty}\frac{\Phi(k+1)+\Phi(n+k+1)}{k!(n+k)!}\left(\frac{z}{2}\right)^{2k},$

for any $n\in\mathbb{N}$ with $z\in\mathbb{C}$ such that
$|\mbox{arg}(z)|<\pi$, where $\psi(1)=-\boldsymbol{\gamma}$ stands for Euler’s
constant, and we set, for any $m\in\mathbb{N}^{*},$ that

$\Phi(m+1)=\displaystyle\sum_{k=1}^{m}\frac{1}{k}-\boldsymbol{\gamma}.$

In particular, one has the useful identity

(2.5)
$K_{0}(z)=-\log\left(\frac{z}{2}\right)I_{0}(z)+\sum_{m=0}^{\infty}\frac{\left(\frac{z}{2}\right)^{2m}}{(m!)^{2}}\Phi(m+1),$

for any $z\in\mathbb{C}$ with $|\mbox{arg}(z)|<\pi$.

The following lemma summarizes several key properties of the Bessel functions.
Note that most of the results in the next lemma are established in previous
works, whence, we only outline the idea to justify the ones that we were not
able to find in the literature.

######  Lemma 2.1.

Let $x,y\in(0,\infty)$ be fixed such that $x\leq y$. Then, the following
holds:

  1. (1)

The sequence $\big{(}(I_{n}K_{n})(x)\big{)}_{n\geq 1}$ is strictly decreasing
and converges to $0$. More generally, the mapping

$(n,x)\mapsto(I_{n}K_{n})(x)$

is strictly decreasing on $\mathbb{N}\times\mathbb{R}^{+}$.

  2. (2)

The sequence
$\left(\tfrac{(\frac{x}{y})^{n}}{2n}-I_{n}(x)K_{n}(y)\right)_{n\geq 1}$ is
positive, decreasing and converges to zero. Moreover, it holds that

(2.6)
$0<\frac{1}{2n}\left(\frac{x}{y}\right)^{n}-I_{n}(x)K_{n}(y)\leq\frac{1}{2n},$

for all $n\in\mathbb{N}^{*}$.

  3. (3)

The function

$x\mapsto\frac{I_{1}(x)}{x}$

is strictly increasing on $(0,\infty)$.

###### Proof.

The proof of the first claim of the lemma is the subject of [6] and [48]. Now,
in order to justify the second claim, we begin with writing that

$I_{n}(x)K_{n}(y)=\frac{1}{2}\int_{\log\frac{y}{x}}^{\infty}J_{0}\left(\mu\sqrt{2xy\cosh(t)-x^{2}-y^{2})}\right)e^{-nt}dt,$

where $J_{0}$ is given by (2.1). The latter identity can be found in [42, page
140]. Therefore, one deduces, for any $n\in\mathbb{N}^{*}$, that

(2.7)
$\frac{(\frac{x}{y})^{n}}{2n}-I_{n}(x)K_{n}(y)=\frac{1}{2}\int_{\log\frac{y}{x}}^{\infty}\Big{(}1-J_{0}\left(\sqrt{2xy\cosh(t)-x^{2}-y^{2})}\right)\Big{)}e^{-nt}dt.$

On the other hand, using the integral representation (2.2), we observe that

$\displaystyle 1-J_{0}\left(\sqrt{2xy\cosh(t)-x^{2}-y^{2})}\right)$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\Big{[}1-\cos\left(\sqrt{2xy\cosh(t)-x^{2}-y^{2})}\sin\theta\right)\Big{]}d\theta\geq
0.$

Therefore, this implies that the sequence
$\left(\tfrac{(\frac{x}{y})^{n}}{2n}-I_{n}(x)K_{n}(y)\right)_{n\geq 1}$ is
positive and decreasing.

Next, the asymptotic decay (2.6), directly follows from the fact that

$I_{n}(x)K_{n}(y)>0,$

for all $n\in\mathbb{N}$, and the assumption that $x\leq y$. We now turn to
justify that the function $x\mapsto\frac{I_{1}(x)}{x}$ is strictly increasing
on $(0,\infty)$. To that end, we simply need to notice, in view of the
expansion formula (2.3), that

$\frac{I_{1}(x)}{x}=\frac{1}{2}\sum_{m=0}^{\infty}\frac{\left(\frac{x}{2}\right)^{2m}}{m!(m+2)!},\quad\text{for
all}\quad x\in(0,\infty).$

The function on the right-hand side above is clearly increasing on
$(0,\infty)$, which concludes the justification of the last claim of the lemma
and completes its proof. ∎

###  2.2. Kernel estimates

In this paragraph, we intend to show the continuity and boundedness of
convolution-type operators that naturally appear through a specific
combination between velocities associated with (QS2L). But before that, let’s
introduce the kernels of our interest by first recalling that distributional
solutions of

$-\Delta\textbf{G}=\delta_{0},\quad\text{in}\quad\mathcal{S}^{\prime}(\mathbb{R}^{2})$

and, for any $\varepsilon>0$,

$(-\Delta+\varepsilon^{2})\textbf{G}_{\varepsilon}=\delta_{0},\quad\text{in}\quad\mathcal{S}^{\prime}(\mathbb{R}^{2})$

are respectively given by

${\bf{G}}(x)\overset{\rm
def}{=}-\frac{1}{2\pi}\log|x|\qquad\text{and}\qquad{\bf{G}}_{\varepsilon}(x)\overset{\rm
def}{=}\frac{1}{2\pi}K_{0}(\varepsilon|x|),$

where $K_{0}$ is the modified Bessel function of zero order introduced in the
previous section. Indeed, the first fundamental solution is a classical fact,
see for instance the book by Evans [21, Section 2.2]. As for the justification
for the fundamental solution of the second problem above, we refer to [20,
Section 4.2]. We now consider the system of equations

$\displaystyle\omega_{1}$
$\displaystyle=\Delta\psi_{1}+\lambda^{2}(\psi_{2}-\psi_{1}),$
$\displaystyle\omega_{2}$
$\displaystyle=\Delta\psi_{2}+\delta\lambda^{2}(\psi_{1}-\psi_{2}),$

for given real parameters $\delta,\lambda\in(0,\infty)$ and two functions
$\omega_{1}$ and $\omega_{2}$ with appropriate decay at infinity111In the case
of the present paper, $\omega_{1}$ and $\omega_{2}$ are assumed to belong to
$L^{1}\cap L^{\infty}(\mathbb{R}^{2})$, which is enough to give a sense to all
the computations in this section.. Therefore, it is readily seen that

$\delta\omega_{1}+\omega_{2}=\Delta(\delta\psi_{1}+\psi_{2})$

and that

$\omega_{1}-\omega_{2}=\Big{(}-\Delta+(\delta+1)\lambda^{2}\Big{)}(\psi_{2}-\psi_{1}).$

Hence, by employing the fundamental representation of Laplace and shalow–water
operators, we arrive at the identities

$\delta\psi_{1}+\psi_{2}=-\int_{\mathbb{R}^{2}}{\bf{G}}(\cdot-\xi)\big{(}\delta{\omega}_{1}+\omega_{2}\big{)}(\xi)d\xi$

and

$\psi_{2}-\psi_{1}=\int_{\mathbb{R}^{2}}{\bf{G}}_{\mu}(\cdot-\xi)\big{(}{\omega}_{1}-{\omega}_{2}\big{)}(\xi)d\xi,$

where we denote

$\mu\overset{\rm def}{=}\lambda\sqrt{1+\delta}.$

It is then readily seen that the previous two identities lead to the following
representation

(2.8) $\displaystyle\psi_{k}(z)$
$\displaystyle=\sum_{j=1}^{2}\int_{\mathbb{R}^{2}}G_{k,j}(z-\xi){\omega}_{j}(t,\xi)dA(\xi),$

where we set

(2.9) $\displaystyle G_{k,j}(x)$ $\displaystyle\overset{\rm
def}{=}\frac{\delta^{2-j}}{2\pi(\delta+1)}\log|x|+(-1)^{k+j-1}\frac{\delta^{k-1}}{2\pi(\delta+1)}K_{0}(\mu|x|),$

for any $x\in\mathbb{R}^{2}\setminus\\{0\\}$ and $k\in\\{1,2\\}$. This
representation will come in handy in the construction of time-periodic
solutions, later on.

The short introduction above motivates the study of the kernels ${\bf{G}}$ and
${\bf{G}_{\varepsilon}}$. For a later use in the proof of existence of weak
solutions, we will also need to prescribe some fine properties of the kernels
associated with the operators

$\nabla^{\perp}(-\Delta)^{-1}\qquad\text{and}\qquad\nabla^{\perp}\big{(}\Delta-\mu^{2}\big{)}^{-1},$

which we respectively denote, from now on, by $k_{+}$ and $k_{-}$ (with
$\mu=1$ for simplicity). More precisely, we now set, for all
$x\in\mathbb{R}^{2}\setminus\\{0\\}$, that

(2.10) $k_{+}(x)\overset{\rm
def}{=}-\frac{1}{2\pi}\frac{x^{\perp}}{|x|^{2}}\qquad\text{and}\qquad
k_{-}(x)=-\frac{x^{\perp}}{|x|}K_{1}(|x|),$

where, $K_{1}=K_{0}^{\prime}$ denotes the Bessel function of order one,
previously introduced in Section 2.1. Accordingly, we define the convolution
operators

$K_{\pm}f\overset{\rm def}{=}k_{\pm}\star f,$

for any suitable function $f$ for which the preceding convolutions have a
sense.

The next lemma establishes quantitative estimates on the behavior of the
kernels introduced above. This will serve later on to deduce essential
properties of operator defined through a convolution with the preceding
kernels which will be useful in the proof of uniqueness of solutions to (QS2L)
in Yudovich’s class.

######  Lemma 2.2.

Let $k_{\pm}$ be given by (2.10). Then, it holds that

(2.11) $|k_{\pm}(x)|\leq\frac{C_{1}}{|x|},$

and

(2.12) $|k_{\pm}(x)-k_{\pm}(y)|\leq C_{2}\frac{|x-y|}{|x||y|}.$

for all $x,y\in\mathbb{R}^{2}\setminus\\{0\\}$, for some universal constants
$C_{1},C_{2}>0$.

###### Proof.

Because the bounds on $k_{+}$ above are classical, we will only focus on the
proof of these bounds on $k_{-}$ which we now split into two cases:

#### Case $|x|\leq 1$

Here, we employ the observation that $k_{-}(x)$ can be seen as a perturbation
of $|x|^{-1},$ for finite values of $x\neq 0$. More precisely, we write, in
view of (2.4) and (2.5), for any $r>0$, that

$-K_{1}(r)=K_{0}^{\prime}(r)=-\frac{1}{r}+\frac{r}{2}\underbrace{\sum_{m=1}^{\infty}m\frac{\left(\frac{r}{2}\right)^{2(m-1)}}{(m!)^{2}}\left(\sum_{i=1}^{m-1}\frac{1}{i}-\gamma-\log\left(\frac{r}{2}\right)\right)}_{\overset{\rm
def}{=}H(r)},$

where, $\gamma$ denotes Euler’s constant. Therefore, noticing, for any
$r\in(0,1]$, that

$|rK_{0}^{\prime}(r)|\lesssim 1,$

we deduce that (2.11) holds for all $x\in\mathbb{R}^{2}\setminus\\{0\\}$ with
$|x|\leq 1$.

#### Case $|x|>1$

Now, we utilize the integral representation of $K_{1}$ (see [2, Identity
9.6.23]) to write, for any $r>0$, that

(2.13)
$K_{1}(r)=\frac{\pi^{\frac{1}{2}}}{2\Gamma\left(\frac{3}{2}\right)}r\int_{1}^{\infty}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt.$

In view of that, we claim that $K_{1}$ has an exponential decay as
$r\to\infty$. More generally, for a later use, we will now show that the
sequence of functions

(2.14) $r\mapsto R_{n}(r)\overset{\rm
def}{=}\int_{1}^{\infty}t^{n}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt,$

has an exponential decay at infinity, for any $n\in\mathbb{N}$. That is, we
now claim that

(2.15) $|R_{n}(r)|\lesssim_{n}e^{-r},$

for any $r\geq 1$ . The justification of the preceding bound is achieved by
recurrence and simple integration by parts. To see that, we first write, for
any $n\in\mathbb{N}$ and all $r\geq 1$, that

$\displaystyle R_{n}(r)$
$\displaystyle=\int_{1}^{2}t^{n}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt+\int_{2}^{\infty}t^{n}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt$
$\displaystyle\leq
2^{n+1}\int_{1}^{2}e^{-rt}dt+\int_{2}^{\infty}t^{n}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt$
$\displaystyle\leq
2^{n+1}e^{-r}+\int_{2}^{\infty}t^{n}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt.$

Thus, by integration by parts and using the elementary inequality

$t\leq\sqrt{2}\sqrt{t^{2}-1},$

which is valid for all $t\geq 2$, we find that

$\displaystyle R_{n}(r)$ $\displaystyle\leq
2^{n+2}e^{-r}+\frac{n+2}{r}\int_{2}^{\infty}t^{n-1}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt$
$\displaystyle\leq 2^{n+2}e^{-r}+(n+2)R_{n-1}(r).$

Therefore, it is readily seen that (2.15) follows by induction as soon as we
show, for all $r\geq 1$, that

$\int_{2}^{\infty}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt\lesssim e^{-r}.$

To that end, we perform one more integration by parts to obtain, for all
$r\geq 1$, that

$\displaystyle\int_{2}^{\infty}e^{-rt}(t^{2}-1)^{\frac{1}{2}}dt$
$\displaystyle=\frac{\sqrt{3}}{r}e^{-r}+\frac{1}{r}\int_{2}^{\infty}e^{-rt}(t^{2}-1)^{-\frac{1}{2}}dt$
$\displaystyle\leq\frac{\sqrt{3}}{r}e^{-r}+\frac{1}{\sqrt{3}r}\int_{2}^{\infty}e^{-rt}dt,$

which clearly leads to the desired bound. At last, we deduce that

$|k_{-}(x)|\lesssim\frac{1}{|x|},$

for any $x\in\mathbb{R}^{2}$ with $|x|\geq 1$, thereby concluding the proof of
(2.11).

Now, in order for us to establish (2.12), we proceed again in two steps. Note
that w we can assume that $|y|\leq|x|$ without any loss of generality.

#### Case $|y|\leq|x|\leq 2|y|$

By employing the integral representation (2.13), we obtain that

$\left|k_{-}(x)-k_{-}(y)\right|\lesssim|x-y|R_{0}(|x|)+|y|\big{|}R_{0}(|x|)-R_{0}(|y|)\big{|},$

where $(R_{n})_{n\in\mathbb{N}}$ is defined in (2.14). On the one hand, due to
the decay estimate (2.15) and the assumption $|x|\geq|y|$, it follows that

$|x-y|R_{0}(|x|)\lesssim\frac{|x-y|}{|x||y|},$

as soon as $|x|\geq|y|>0$. On the other hand, writing, in view of Taylor’s
expansion, that

$|R_{0}(|x|)-R_{0}(|y|)|\lesssim\big{|}|x|-|y|\big{|}\int_{0}^{1}\int_{1}^{\infty}te^{-t\big{(}s|x|+(1-s)|y|\big{)}}\sqrt{t^{2}-1}dtds$

and exploiting the assumption $|x|\geq|y|$, again, we obtain that

$\displaystyle|R_{0}(|x|)-R_{0}(|y|)|$
$\displaystyle\lesssim\big{|}|x|-|y|\big{|}\int_{1}^{\infty}te^{-t|y|}\sqrt{t^{2}-1}dt$
$\displaystyle=\big{|}|x|-|y|\big{|}R_{1}(|y|).$

Therefore, by further employing the decay estimate (2.15) and the assumption
$|x|\leq 2|y|$, we arrive at the bound

$\displaystyle|y||R_{0}(|x|)-R_{0}(|y|)|$
$\displaystyle\lesssim\frac{|x-y|}{|y|^{2}}$
$\displaystyle\lesssim\frac{|x-y|}{|x||y|},$

thereby showing the validity of (2.12) whenever $x,y\in\mathbb{R}^{2}$ with
$|y|\leq|x|\leq 2|y|$.

#### Case $|y|<2|y|\leq|x|$

We first proceed in a similar way to the previous case by writing

$\left|k_{-}(x)-k_{-}(y)\right|\lesssim|x-y|R_{0}(|x|)+|y|\Big{(}R_{0}(|x|)+R_{0}(|y|)\Big{)}.$

Hence, by virtue of the decay estimate (2.15), we obtain, as long as
$|y|\leq|x|$, that

$\displaystyle\left|k_{-}(x)-k_{-}(y)\right|$
$\displaystyle\lesssim\frac{|x-y|}{|x|^{2}}+\frac{|y|}{|x|^{2}}+\frac{1}{|y|}$
$\displaystyle\lesssim\frac{|x-y|}{|x||y|}+\frac{1}{|y|}.$

At last, further employing the assumption $2|y|\leq|x|$ entails that

$\displaystyle\frac{1}{|y|}$ $\displaystyle\leq
2\left(\frac{1}{|y|}-\frac{1}{|x|}\right)$ $\displaystyle\leq
2\frac{|x-y|}{|x||y|},$

thereby yielding the desired control (2.12) in the case where $2|y|\leq|x|$,
as well. All in all, combining the foregoing estimates leads to a complete
justification of (2.12) and concludes the proof of the lemma. ∎

As a consequence of the preceding lemma, we now state a lemma which will be
essential in the proof of the uniqueness in Theorem 1.1. The proof of the next
lemma can be justified by a direct application of [16, Corollary 2.4] and [16,
Remark 2.5] that only require suitable quantitative bounds on the kernels
$k_{\pm}$, which we already established in Lemma 2.2, above.

Here and in what follows, the real function $\ell$ is defined as follows

$\ell:[0,\infty)\rightarrow[0,1]$

where we set

(2.16) $\ell(r)\overset{\rm def}{=}\left\\{\begin{array}[]{ll}0,&\text{for
}r=0,\\\ r\log\left(\frac{e}{r}\right),&\text{if }r\in(0,1]\\\
1,&\text{elsewhere}.\end{array}\right.$

######  Lemma 2.3.

For any $f\in L^{1}\cap L^{\infty}(\mathbb{R}^{2}),$ it holds that

$\left\|K_{\pm}f\right\|_{L^{\infty}}\lesssim\left\|f\right\|_{L^{1}\cap
L^{\infty}},$

where $K_{\pm}$ are the convolution operator associated with the kernels
$k_{\pm}$ defined in (2.10). Moreover, we have that

$\int_{\mathbb{R}^{2}}\left|k_{\pm}(x-z)-k_{\pm}(y-z)\right||f(z)|dz\lesssim\left\|f\right\|_{L^{1}\cap
L^{\infty}}\ell(|x-y|),$

for any $(x,y)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ with $x\neq y$.

###  2.3. Hölder-continuity of some singular operators

For convenience, we state here several results on continuity properties of a
specific type of singular operators. In particular, the ensuing bounds below
will be employed later on in the regularity analysis of contour dynamics
equation associated with time-periodic solutions of (QS2L).

######  Lemma 2.4 ([40, Lemma 2.6]).

Let $\alpha\in(0,1)$ and consider a measurable function $\mathcal{K}$ defined
on
$\mathbb{T}\times\mathbb{T}\setminus\\{(\theta,\theta),\theta\in\mathbb{T}\\}$
with values in $\mathbb{C}$. Assume further, for all
$\theta\neq\eta\in\mathbb{T}$, that

$|\mathcal{K}(\theta,\eta)|\leq\frac{C_{0}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{\alpha}},$

and

$|\mathcal{K}(\theta,\eta)|\leq\frac{C_{0}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{1+\alpha}},$

for some $C_{0}>0$. Then, the integral operator defined, for any $f\in
L^{\infty}(\mathbb{T})$, by

$\theta\mapsto\mathcal{T}f(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}\mathcal{K}(\theta,\eta)f(\eta)d\eta$

maps $L^{\infty}(\mathbb{T})$ into $C^{\alpha}(\mathbb{T})$. More precisely,
it holds that

$\left\|\mathcal{T}f\right\|_{C^{\alpha}}\lesssim
C_{0}\left\|f\right\|_{L^{\infty}}.$

Later on, we are going to deal with operators having a singularity in their
diagonal that corresponds to the endpoint case $\alpha=1$ in the preceding
lemma. The following lemma will then be useful in these situations.

######  Lemma 2.5 ([25, Proposition A.2]).

Let $\alpha\in(0,1)$ and $g$ be a $C^{\alpha}(\mathbb{T})$ function. Further
consider a measurable function $\mathcal{K}$ defined on
$\mathbb{T}\times\mathbb{T}\setminus\\{(\theta,\theta),\theta\in\mathbb{T}\\}$
with values in $\mathbb{C}$ and assume, for all $\theta\neq\eta\in\mathbb{T}$,
that

$|\mathcal{K}(\theta,\eta)|\leq\frac{C_{0}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|},$

and

$|\mathcal{K}(\theta,\eta)|\leq\frac{C_{0}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{2}},$

for some $C_{0}>0$. Then, the integral operator defined, for any $f\in
L^{\infty}(\mathbb{T})$, by

$\theta\mapsto\mathcal{T}_{g}f(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}\mathcal{K}(\theta,\eta)\big{(}g(\theta)-g(\eta)\big{)}f(\eta)d\eta$

maps $L^{\infty}(\mathbb{T})$ into $C^{\alpha}(\mathbb{T})$. More precisely,
it holds that

$\left\|\mathcal{T}_{g}f\right\|_{C^{\alpha}}\lesssim
C_{0}\left\|g\right\|_{C^{\alpha}}\left\|f\right\|_{L^{\infty}}.$

###  2.4. Crandall-Rabinowitz’s theorem

The proof of Theorem 4.1, as we will show in Section 4, relies on a careful
application of the generalized version of implicit functions theorem—known as
Crandall-Rabinowitz’s Theorem. The precise statement of that theorem reads as
follows

######  Theorem 2.6 (Crandall-Rabinowitz).

Let $X$ and $Y$ be two Banach spaces. Further consider $V\subset X$ to be a
neighborhood of $0$ and

$F:\begin{array}[t]{rcl}\mathbb{R}\times V\rightarrow Y\end{array}$

to be a function of class $C^{1}$ with the following properties

  1. (1)

(Trivial solution) For all $\Omega\in\mathbb{R}$, we assume that

$F(\Omega,0)=0.$

  2. (2)

(Regularity) Moreover, $F$ is assumed to be regular in the sense that
$\partial_{\Omega}F$, $\partial_{x}F$ and $\partial_{\Omega x}^{2}F$ exist and
are continuous.

  3. (3)

(Fredholm property) Furthermore, we assume that
$\ker\left(\partial_{x}F(0,0)\right)$ is not trivial of dimension one, i.e.,
there is $x_{0}\in V$ such that

$\ker\left(\partial_{x}F(0,0)\right)=\langle x_{0}\rangle,$

that $R\left(\partial_{x}F(0,0)\right)$ is closed in $Y$ and that $Y\setminus
R\left(\partial_{x}F(0,0)\right)$ is one dimensional.

  4. (4)

(Transversality assumption) At last, we assume that $\partial_{\Omega
x}^{2}F(0,0)x_{0}\not\in R\left(\partial_{x}F(0,0)\right).$

Under the foregoing assumptions, for $\chi$ being the complement of
$\ker\left(\partial_{x}F(0,0)\right)$ in $X$, there exist a neighborhood $U$
of $(0,0)$, an interval $(-a,a)$, for some $a>0$ and continuous functions

$\psi:(-a,a)\rightarrow\mathbb{R}\quad\textnormal{and}\quad\phi:(-a,a)\rightarrow\chi$

such that

$\psi(0)=\phi(0)=0$

and

$\Big{\\{}(\Omega,x)\in
U:\;F(\Omega,x)=0\Big{\\}}=\Big{\\{}\big{(}\psi(s),sx_{0}+s\phi(s)\big{)}:\;|s|<a\Big{\\}}\cup\Big{\\{}(\Omega,0)\in
U\Big{\\}}.$

##  3\. Weak and Lagrangian solutions

This section is devoted to the proof of Theorem 1.1. We proceed first by
introducing several notation that will be employed in the proof below. For a
given $\delta>0$, we introduce the matrix

(3.1) $\mathcal{A}_{\delta}\overset{\rm def}{=}\begin{pmatrix}1&\delta^{-1}\\\
1&-1\end{pmatrix}$

and, for a couple of real valued functions $f=(f_{1},f_{2})$, we define

$\begin{pmatrix}f_{+}\\\ f_{-}\end{pmatrix}\overset{\rm
def}{=}\mathcal{A}_{\delta}\cdot\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}.$

Note that

$\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}=\mathcal{A}^{-1}_{\delta}\cdot\begin{pmatrix}f_{+}\\\
f_{-}\end{pmatrix},$

where $\mathcal{A}_{\delta}^{-1}$ is the inverse of $\mathcal{A}_{\delta}$,
which is given by

$\mathcal{A}^{-1}_{\delta}=\left(1+\delta^{-1}\right)^{-1}A_{\delta}.$

Moreover, for another couple of real valued functions
$\widetilde{f}=(\widetilde{f}_{1},\widetilde{f}_{2})$, the following
elementary inequality can justified by a direct computation

$\displaystyle\frac{1}{2\max\\{1,\delta^{-1}\\}}$
$\displaystyle\left(\textbf{d}(f_{+},\widetilde{f}_{+})+\textbf{d}(f_{-},\widetilde{f}_{-})\right)$
$\displaystyle\qquad\qquad\leq\;\textbf{d}(f_{1},\widetilde{f}_{1})+\textbf{d}(f_{2},\widetilde{f}_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\leq\frac{1+\max\\{1,\delta^{-1}\\}}{1+\delta^{-1}}\left(\textbf{d}(f_{+},\widetilde{f}_{+})+\textbf{d}(f_{-},\widetilde{f}_{-})\right),$

where $\textbf{d}(A,B)$ denotes the distance between $A$ and $B$. For
simplicity, we shall use the following version of the preceding inequalities

(3.2) $\displaystyle
C_{\delta}^{-1}\left(\textbf{d}(f_{+},\widetilde{f}_{+})+\textbf{d}(f_{-},\widetilde{f}_{-})\right)\leq\;\textbf{d}(f_{1},\widetilde{f}_{1})+\textbf{d}(f_{2},\widetilde{f}_{2})\leq
C_{\delta}\left(\textbf{d}(f_{+},\widetilde{f}_{+})+\textbf{d}(f_{-},\widetilde{f}_{-})\right),$

where $C_{\delta}\geq 1$ is a constant that only depends on $\delta.$

###  3.1. Existence of global solutions

Here, we show existence of global solutions of (QS2L) as is claimed in Theorem
1.1.

####  3.1.1. Existence of weak-distributional solutions

Weak solutions can be constructed by analyzing the convergence of the scheme
below associated with a linearized transport equations. To that end, we assume
that we have constructed a velocity $u^{n}_{i}$, for $n\in\mathbb{N}$, then
build up the next iteration $(\omega^{n+1},u^{n+1})$ by solving the linear
system of equations

$\left\\{\begin{array}[]{ll}\partial_{t}\omega_{i}^{n+1}+v_{i}^{n}\cdot\nabla\omega_{i}^{n+1}=0,\\\
\vspace{1mm}\\\ \begin{pmatrix}\omega_{+}^{n+1}\\\
\omega_{-}^{n+1}\end{pmatrix}\overset{\rm
def}{=}\mathcal{A}_{\delta}\cdot\begin{pmatrix}\omega_{1}^{n+1}\\\
\omega_{2}^{n+1}\end{pmatrix},\\\ {}{}\\\ v_{+}^{n+1}\overset{\rm
def}{=}K_{+}\omega_{+}^{n+1},\qquad v_{-}^{n+1}\overset{\rm
def}{=}K_{-}\omega_{-}^{n+1},\\\ \vspace{1mm}\\\ \begin{pmatrix}v_{1}^{n+1}\\\
v_{2}^{n+1}\end{pmatrix}\overset{\rm
def}{=}\mathcal{A}_{\delta}^{-1}\cdot\begin{pmatrix}v_{+}^{n+1}\\\
v_{-}^{n+1}\end{pmatrix},\end{array}\right.$

supplemented with the initial data

$\omega_{i}^{n+1}{|_{t=0}}=\omega_{i}(0),\quad\text{for all}\quad
n\in\mathbb{N},$

and the initial iteration

$\begin{pmatrix}v_{1}^{0}\\\ ~{}~{}\\\ v_{2}^{0}\end{pmatrix}\overset{\rm
def}{=}\mathcal{A}_{\delta}^{-1}\cdot\begin{pmatrix}v_{+}^{0}\\\ ~{}~{}\\\
v_{-}^{0}\end{pmatrix}=\mathcal{A}_{\delta}^{-1}\cdot\begin{pmatrix}K_{+}\omega_{+}(0)\\\
~{}~{}\\\
K_{-}\omega_{-}(0)\end{pmatrix}=\mathcal{A}_{\delta}^{-1}\cdot\begin{pmatrix}K_{+}\left(\mathcal{A}_{\delta}\cdot\begin{pmatrix}\omega_{1}(0)\\\
\omega_{2}(0)\end{pmatrix}\right)\\\ ~{}~{}\\\
K_{-}\left(\mathcal{A}_{\delta}\cdot\begin{pmatrix}\omega_{1}(0)\\\
\omega_{2}(0)\end{pmatrix}\right)\end{pmatrix}.$

The adequate bounds can be proved by induction and by performing standard
energy estimates on transport equation, altogether with the employment of the
bounds on the operators $K_{\pm}$ given by Lemma 2.3. This process establishes
the existence of bounded sequence of solutions
$(\omega_{1}^{n},\omega_{2}^{n})_{n\in\mathbb{N}}$ that convergences, in the
sense of distributions, to a weak solution of the system of equations (QS2L).
The compactness of the approximate solution
$(\omega_{1}^{n},\omega_{2}^{n})_{n\in\mathbb{N}}$ and the convergence in
nonlinear terms can be done by using classical techniques. Finally, we
emphasize that the resulting limit as $n\to\infty$ enjoys the following bounds

$\omega_{1},\omega_{2}\in C(\mathbb{R}^{+},L^{p}(\mathbb{R}^{2}))\cap
C_{w}(\mathbb{R}^{+},L^{\infty}(\mathbb{R}^{2})),$ $v_{1},v_{2}\in
C(\mathbb{R}^{+},L^{q}(\mathbb{R}^{2}))\cap
L^{\infty}(\mathbb{R}^{+},LL(\mathbb{R}^{2})),$

for all $p\in[1,\infty)$ and $q\in(2,\infty].$

####  3.1.2. Existence of Lagrangian solutions

Now, we show that any weak solution is Lagrangian. To that end, for all
$i\in\\{1,2\\}$, we first define the flow-maps $X_{i}$ associated with the
velocity field $v_{i}$ as the unique solution of (ODE). Again, we emphasize
that the flow-map $X_{i}$ is uniquely determined due to the Log-lipschitz
regularity of $v_{i}$ (see [14, Section 5.2]). Therefore, we define the
Lagrangian solution $\Omega_{i}$ of (QS2L) by the push forward

$\Omega_{i}\overset{\rm
def}{=}\omega_{i,0}\left(X_{i}^{-1}(t,\cdot)\right),\quad\text{for all}\quad
t\geq 0.$

Observe that $\Omega_{i}$ satisfies the same transport equation as
$\omega_{i}$ (with velocity $v_{i}$), i.e.,

$\partial_{t}\Omega_{i}+v_{i}\cdot\nabla\Omega_{i}=0.$

Hence, by classical results on transport equations (see for instance [19]) we
deduce that $\Omega_{i}=\omega_{i}$, thereby establishing that $\omega_{i}$ is
a Lagrangian solution.

###  3.2. Stability of the flow-maps and uniqueness

The proof of the uniqueness follows the idea from [16] and will be acheived by
establishing a stability result for the flow-maps associated with two
different solutions.

In the sequel, $C_{0}>0$ denotes a constant that depends only on the
$L^{1}\cap L^{\infty}(\mathbb{R}^{2})$ norm of the initial data and is allowed
to differ from one line to the next one.

Let $(v_{i},\omega_{i})$ and $(\widetilde{v}_{i},\widetilde{\omega}_{i})$ be
two Lagrangian solutions to (QS2L) with the same initial data $\omega_{i,0}$
belonging to $L^{1}\cap L^{\infty}(\mathbb{R}^{2})$. We denote by $X_{i}$ and
$\widetilde{X}_{i}$ the unique flows associated to each solution, for
$i\in\\{1,2\\}$. Further introduce

$\begin{pmatrix}X_{+}\\\ ~{}~{}\\\ X_{-}\end{pmatrix}\overset{\rm
def}{=}\mathcal{A}_{\delta}\cdot\begin{pmatrix}X_{1}\\\ ~{}~{}\\\
X_{2}\end{pmatrix},\qquad\begin{pmatrix}\widetilde{X}_{+}\\\ ~{}~{}\\\
\widetilde{X}_{-}\end{pmatrix}\overset{\rm
def}{=}\mathcal{A}_{\delta}\cdot\begin{pmatrix}\widetilde{X}_{1}\\\ ~{}~{}\\\
\widetilde{X}_{2}\end{pmatrix},$

where $A_{\delta}$ is the matrix defined in (3.1). Then, in view of the
equation (ODE) satisfied by each flow-map, we compute that

$\displaystyle X_{+}(t)-\widetilde{X}_{+}(t)$
$\displaystyle=\int_{0}^{t}\left(v_{1}(X_{1}(s))+\delta^{-1}v_{2}(X_{2}(s))-\widetilde{v}_{1}(\widetilde{X}_{1}(s))-\delta^{-1}\widetilde{v}_{2}(\widetilde{X}_{2}(s))\right)ds$
$\displaystyle=\int_{0}^{t}\left(v_{1}(X_{1}(s))-v_{1}(\widetilde{X}_{1}(s))+(v_{1}-\widetilde{v}_{1})(\widetilde{X}_{1}(s))\right)ds$
$\displaystyle\quad+\delta^{-1}\int_{0}^{t}\left(v_{2}(X_{2}(s))-v_{2}(\widetilde{X}_{2}(s))+(v_{2}-\widetilde{v}_{2})(\widetilde{X}_{2}(s))\right)ds,$

for all $t\geq 0$. Therefore, by further introducing the fluctuation

$X\overset{\rm def}{=}|X_{1}-\widetilde{X}_{1}|+|X_{2}-\widetilde{X}_{2}|,$

and owing to the fact that $v_{1}$ and $v_{2}$ are Log-Lipschitz, altogether
with (3.2), we infer that

$\displaystyle\big{|}X_{+}(t)-\widetilde{X}_{+}(t)\big{|}$
$\displaystyle\lesssim_{\delta}C_{0}\int_{0}^{t}\ell(|X_{1}-\widetilde{X}_{1}(s)|)+\ell(|X_{2}-\widetilde{X}_{2}(s)|)ds$
$\displaystyle\qquad+\int_{0}^{t}\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{1}(s))\big{|}+\int_{0}^{t}\big{|}(v_{-}-\widetilde{v}_{-})(\widetilde{X}_{1}(s))\big{|}ds$
$\displaystyle\qquad+\int_{0}^{t}\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{2}(s))\big{|}+\int_{0}^{t}\big{|}(v_{-}-\widetilde{v}_{-})(\widetilde{X}_{2}(s))\big{|}ds$
$\displaystyle\lesssim_{\delta}C_{0}\int_{0}^{t}\ell(X(s))ds$
$\displaystyle\quad+\sum_{i=1}^{2}\left(\int_{0}^{t}\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{i}(s))\big{|}ds+\int_{0}^{t}\big{|}(v_{-}-\widetilde{v}_{-})(\widetilde{X}_{i}(s))\big{|}ds\right),$

where the function $\ell$ is defined in (2.16). Moreover, in the same way, we
obtain that

$\displaystyle\big{|}X_{-}(t)-\widetilde{X}_{-}(t)\big{|}$
$\displaystyle\lesssim_{\delta}C_{0}\int_{0}^{t}\ell(X(s))ds$
$\displaystyle\quad+\sum_{i=1}^{2}\left(\int_{0}^{t}\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{i}(s))\big{|}ds+\int_{0}^{t}\big{|}(v_{-}-\widetilde{v}_{-})(\widetilde{X}_{i}(s))\big{|}ds\right).$

Hence, utilizing (3.2), again, entails that

$\displaystyle X(t)$
$\displaystyle\lesssim_{\delta}C_{0}\int_{0}^{t}\ell(X(s))ds+\sum_{i=1}^{2}\left(\int_{0}^{t}\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{i}(s))\big{|}ds+\int_{0}^{t}\big{|}(v_{-}-\widetilde{v}_{-})(\widetilde{X}_{i}(s))\big{|}ds\right).$

On the other hand, observing, for all $i\in\\{1,2\\}$, that

$\displaystyle\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{i})\big{|}$
$\displaystyle=\big{|}(K_{+}\omega_{+}(\widetilde{X}_{i})-K_{+}\widetilde{\omega}_{+}(\widetilde{X}_{i})\big{|}$
$\displaystyle\lesssim_{\delta}\sum_{j=1}^{2}\big{|}(K_{+}\omega_{j}(\widetilde{X}_{i})-K_{+}\widetilde{\omega}_{j}(\widetilde{X}_{i})\big{|},$

and exploiting the definition of $K_{+}$ allows us to write, for all
$i,j\in\\{1,2\\}$, that

$\displaystyle\big{|}(K_{+}\omega_{j}(\widetilde{X}_{i})-K_{+}\widetilde{\omega}_{j}(\widetilde{X}_{i})\big{|}$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{2}}k_{+}(\widetilde{X}_{i}(s,x),y)\omega_{j}(y)dy-
k_{+}(\widetilde{X}_{i}(s,x),y)\widetilde{\omega}_{j}(y)dy\Big{|}$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{2}}k_{+}(\widetilde{X}_{i}(s,x),X_{j}(s,y))\omega_{j,0}(y)$
$\displaystyle\qquad\qquad-
k_{+}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\omega_{0,j}(y)dy\Big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\Big{|}k_{+}(\widetilde{X}_{i}(s,x),X_{j}(s,y))$
$\displaystyle\qquad\qquad-
k_{+}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\Big{|}\omega_{0,j}(y)\Big{|}dy.$

Consequently, we find that

$\displaystyle\big{|}(v_{+}-\widetilde{v}_{+})(\widetilde{X}_{i})\big{|}$
$\displaystyle\lesssim_{\delta}\sum_{j=1}^{2}\int_{\mathbb{R}^{2}}\Big{|}k_{+}(\widetilde{X}_{i}(s,x),X_{j}(s,y))-k_{+}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\Big{|}\omega_{0,j}(y)\Big{|}dy.$

Likewise, we obtain for the minus parts

$\displaystyle\big{|}(v_{-}-\widetilde{v}_{-})(\widetilde{X}_{i})\big{|}$
$\displaystyle=\big{|}(K_{-}\omega_{-}(\widetilde{X}_{i})-K_{-}\widetilde{\omega}_{-}(\widetilde{X}_{i})\big{|}$
$\displaystyle\lesssim_{\delta}\sum_{j=1}^{2}\big{|}(K_{-}\omega_{j}(\widetilde{X}_{i})-K_{-}\widetilde{\omega}_{j}(\widetilde{X}_{i})\big{|}$
$\displaystyle\lesssim_{\delta}\sum_{j=1}^{2}\int_{\mathbb{R}^{2}}\Big{|}k_{-}(\widetilde{X}_{i}(s,x),X_{j}(s,y))-k_{-}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\Big{|}\omega_{0,j}(y)\Big{|}dy.$

All in all, gathering the foregoing estimates yields

(3.3) $\displaystyle X(t)$
$\displaystyle\lesssim_{\delta}C_{0}\int_{0}^{t}\ell(X(s))ds$
$\displaystyle\quad+\sum_{\overset{i,j=1}{m=\pm}}^{2}\int_{0}^{t}\int_{\mathbb{R}^{2}}\Big{|}k_{m}(\widetilde{X}_{i}(s,x),X_{j}(s,y))-k_{m}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\Big{|}\omega_{0,j}(y)\Big{|}dyds.$

Onwards, we omit the subscript $\delta$ and we keep in mind that the constants
in the estimates below depend on $\delta,$ as well. Let $\alpha>0$ be a
function in $L^{1}\cap L^{\infty}(\mathbb{R}^{2})$. Accordingly, we define the
density

$\eta\overset{\rm def}{=}\sum_{i=1}^{2}|\omega_{i,0}|+\alpha$

and its push-forward by the mapping $\widetilde{X}_{j}(s,\cdot)$

$\eta_{j}(s,\cdot)\overset{\rm
def}{=}\eta({\widetilde{X}_{j}}^{-1}(s,\cdot)).$

Then, we proceed by integrating (3.3) against to the measure $\eta(x)dx$ and
utilizing Fubini’s theorem to obtain that

$\displaystyle\int_{\mathbb{R}^{2}}X(t,x)\eta(x)dx\lesssim
C_{0}\int_{0}^{t}\int_{\mathbb{R}^{2}}\ell\big{(}X(s,x)\big{)}\eta(x)dxds$
$\displaystyle\quad\quad+\sum_{\overset{i,j=1}{m=\pm}}^{2}\int_{0}^{t}\int_{\mathbb{R}^{4}}\Big{|}k_{m}(\widetilde{X}_{i}(s,x),X_{j}(s,y))-k_{m}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\Big{|}\omega_{0,j}(y)\Big{|}dy\eta(x)dxds$
$\displaystyle\quad=C_{0}\int_{0}^{t}\int_{\mathbb{R}^{2}}\ell\big{(}X(s,x)\big{)}\eta(x)dxds$
$\displaystyle\quad\quad+\sum_{\overset{i,j=1}{m=\pm}}^{2}\int_{0}^{t}\int_{\mathbb{R}^{2}}\Big{|}\omega_{0,j}(y)\Big{|}\int_{\mathbb{R}^{2}}\Big{|}k_{m}(\widetilde{X}_{i}(s,x),X_{j}(s,y))-k_{m}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\eta(x)dxdyds.$

Now, owing to the fact that $\widetilde{X}_{i}(s,\cdot)$ preserves volumes,
for any $i\in\\{1,2\\}$ and $s\geq 0$, and appealing to Lemma 2.3 entails, for
any $i,j\in\\{1,2\\}$ and $m=\pm$, that

$\displaystyle\int_{\mathbb{R}^{2}}\Big{|}k_{\pm}(\widetilde{X}_{i}(s,x),X_{j}(s,y))$
$\displaystyle-
k_{\pm}(\widetilde{X}_{i}(s,x),\widetilde{X}_{j}(s,y))\Big{|}\eta(x)dx$
$\displaystyle=\int_{\mathbb{R}^{2}}\Big{|}k_{\pm}(x,X_{j}(s,y))-k_{\pm}(x,\widetilde{X}_{j}(s,y))\Big{|}\eta_{j}(s,x)dx$
$\displaystyle\lesssim\left\|\eta_{j}(s,\cdot)\right\|_{L^{1}\cap
L^{\infty}}\ell\big{(}|X_{j}(s,y)-\widetilde{X}_{j}(s,y)|\big{)}$
$\displaystyle\lesssim C_{0}\ell\big{(}X(s,y)\big{)}.$

Therefore, it follows that

$\displaystyle\int_{\mathbb{R}^{2}}X(t,x)\eta(x)dx$ $\displaystyle\lesssim
C_{0}\int_{0}^{t}\int_{\mathbb{R}^{2}}\ell(X(s,x))\eta(x)dxds$
$\displaystyle\qquad+C_{0}\sum_{j=1}^{2}\int_{0}^{t}\int_{\mathbb{R}^{2}}\Big{|}\omega_{0,j}(y)\Big{|}\ell(X(s,y))dyds$
$\displaystyle\lesssim
C_{0}\int_{0}^{t}\int_{\mathbb{R}^{2}}\ell(X(s,x))\eta(x)dxds.$

At last, since the function $\ell$ is concave and $\eta\in
L^{1}(\mathbb{R}^{2})$, we end up with

$\int_{\mathbb{R}^{2}}X(t,x)\eta(x)dx\lesssim
C_{0}\int_{0}^{t}\ell\left(\int_{\mathbb{R}^{2}}X(s,x)\eta(x)dx\right)ds.$

The uniqueness of the flow-maps follows due to Osgood’s Lemma, see [5, Lemma
3.4], for instance. Consequently, we deduce that
$\omega_{i}\equiv\widetilde{\omega}_{i}$, thereby showing the uniqueness of
the solution and completing the proof of Theorem 1.1. ∎

##  4\. Uniformly rotating solutions

This section is devoted to the proof of existence of rotating $m$-fold vortex
patches (relative) equilibriums for the Surface Quasi-Geostrophic model with
two boundaries given by (QS2L). The statement of that is given by Theorem 1.3
which we recast below in a more precise format. To that end, allow us first to
set up few notations that will be constantly used throughout this section. For
a given integer $m\in\mathbb{N}^{*}$, and real numbers $0<b_{2}\leq
b_{1}<\infty$, we introduce the set

${S}_{m,b_{1}}\overset{\rm def}{=}\left\\{s\in(0,b_{1}]:\exists
n\in\mathbb{N}^{*},\ \Omega_{m}^{-}(s,b_{1})=\Omega_{n}^{+}(s,b_{1})\right\\}$

where

$\Omega_{n}^{\pm}(b_{1},b_{2})\overset{\rm
def}{=}\frac{1}{2(\delta+1)}\left(-(A_{b_{1},b_{2},n}+B_{b_{1},b_{2},n})\pm\sqrt{(A_{b_{1},b_{2},n}-B_{b_{1},b_{2},n})^{2}+{4\delta}\gamma_{b_{1},b_{2},n}^{2}}\right),$

where, we set

$\displaystyle A_{b_{1},b_{2},n}$ $\displaystyle\overset{\rm
def}{=}(\delta+1)V_{b_{1},b_{2}}+\frac{\delta}{2n}+I_{n}(b_{1}\mu)K_{n}(b_{1}\mu)$
$\displaystyle B_{b_{1},b_{2},n}$ $\displaystyle\overset{\rm
def}{=}(\delta+1)W_{b_{1},b_{2}}+\frac{1}{2n}+{\delta}I_{n}(b_{2}\mu)K_{n}(b_{2}\mu),$
$\displaystyle\gamma_{b_{1},b_{2},n}$ $\displaystyle\overset{\rm def}{=}\
\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu),$

with $b\overset{\rm def}{=}\frac{b_{2}}{b_{1}}$ and

(4.1) $\displaystyle V_{b_{1},b_{2}}$ $\displaystyle\overset{\rm
def}{=}-\frac{\delta+b^{2}}{2(1+\delta)}-\frac{1}{1+\delta}\Big{(}I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)-bI_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\Big{)},$
$\displaystyle W_{b_{1},b_{2}}$ $\displaystyle\overset{\rm
def}{=}-\frac{1}{2}-\frac{\delta}{1+\delta}\Big{(}I_{1}(b_{2}\mu)K_{1}(b_{2}\mu)-\frac{1}{b}I_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\Big{)}.$

Above and in what follows, we denote

$\mu\overset{\rm def}{=}\lambda\sqrt{1+\delta},$

where $\Lambda,\delta>0$ are the constants appearing in the system of
equations (QS2L), whereas we recall that $I_{n}$ and $K_{n}$, for
$n\in\mathbb{N}^{*}$, stand for the modified Bessel functions previously
introduced in Section 2.1.

For a mere simplification of notation, we drop the dependence of
$\Omega_{n}^{\pm}(b_{1},b_{2})$, as well as $A_{b_{1},b_{2},n}$ and
$B_{b_{1},b_{2},n}$, on $b_{1}$ and $b_{2}$ even though this is crucial and
will be utilized in our proofs, later on.

Theorem 1.3 is a consequence of the following slightly more precise theorem.

######  Theorem 4.1 (Periodic solutions bifurcating from simple eigenvalues).

Let $m\in\mathbb{N}^{*}$ and $\lambda>0$. Further fix two radii $0<b_{2}\leq
b_{1}$ and assume that $\delta\geq\frac{b_{2}}{b_{1}}$. Then, the set
$S_{m,b_{1}}$ defined above contains at most a finite number of elements.
Moreover, the following holds:

  * •

Assuming that $b_{2}$ does not belong to $S_{m,b_{1}}$, there exist two curves
of $m$-fold symmetric pairs of simply connected V-states, for any
$m\in\mathbb{N}^{*}$, bifurcating from the steady states

$\omega_{1}=\mathds{1}_{\\{x\in\mathbb{R}^{2}:|x|<b_{1}\\}}\quad\text{and}\quad\omega_{2}=\mathds{1}_{\\{x\in\mathbb{R}^{2}:|x|<b_{2}\\}}$

at each of the angular velocities $\Omega=\Omega_{m}^{\pm}$ defined above.

  * •

Assuming $b_{1}\neq b_{2}$, there is $m_{0}\in\mathbb{N}^{*}$ such that
$S_{m,b_{1}}$ is empty for any $m\geq m_{0}$ and for which there exist two
curves of $m$-fold symmetric pairs of simply connected V-states satisfying the
same properties as in the preceding case.

######  Remark 4.1.

In the case of identical initial discs, we will show in the proof of
Proposition 4.8 below that there are values $b_{1}=b_{2}\in(0,\infty)$ for
which spectral collisions can happen, i.e. situations where the angular
velocities satisfy $\Omega_{m}^{-}=\Omega_{n}^{+}$, for some
$n,m\in\mathbb{N}^{*}$. In that particular case, we are not able to directly
apply Crandall-Rabinowitz’s Theorem 2.6 to prove existence of time-periodic
solutions bifurcating from the same initial disc. This also shows that the set
$S_{m,b_{1}}$ is not empty in general.

######  Remark 4.2.

Note that Theorem 4.1 covers Theorem 1.3. Moreover, Theorem 4.1 emphasizes
that $\delta$ is allowed to take values in $[b^{2},1]$, as well. This
condition can probably be further relaxed so that $\delta$ would take values
in the whole half-line $(0,\infty)$, though a justification of that is not
available by in proof below. Indeed, it is to be emphasized later on that the
sequences of angular velocities $(\Omega_{n}^{\pm})_{n\in\mathbb{N}^{*}}$ are
non-decreasing for any value of $\delta$ in $(0,\infty)$, which means that
spectral collisions cannot occur from the same sequence
$(\Omega_{n}^{+})_{n\in\mathbb{N}^{*}}$ or
$(\Omega_{n}^{-})_{n\in\mathbb{N}^{*}}$. Furthermore, under the assumption
that $\delta\geq\frac{b_{1}}{b_{2}}$ with $b_{1}\neq b_{2}$, it is shown in
the proof of Proposition 4.7 below that these sequences have different limits
at infinity. This allows for avoiding spectral collusions between the
sequences $(\Omega_{n}^{+})_{n\in\mathbb{N}^{*}}$ and
$(\Omega_{n}^{-})_{n\in\mathbb{N}^{*}}$ for large symmetries. However, because
of the implicit expression of the Bessel functions, we were not able to
rigorously justify that fact in the range of parameter
$\delta\in(0,\frac{b_{1}}{b_{2}})$, even though several numerical simulations,
that we do not include in this paper, suggest that Theorem 4.1 holds for any
value of $\delta>0$.

The proof of Theorem 4.1 is carried out in several steps and is detailed in
the forthcoming sections. In particular, our strategy consists in establishing
all the prerequisites in Crandall-Rabinowitz’s Theorem 2.6 which,
subsequently, will grant us the results of Theorem 4.1.

###  4.1. Contour dynamics equation

Here, we set up the contour dynamics governing the motion of vortex patches.
Onward, we identify $\mathbb{C}$ with $\mathbb{R}^{2},$ where $\mathbb{C}$ is
naturally endowed with the Euclidean scalar product which writes, for
$z_{1}=x_{1}+{\rm i}y_{1}$ and $z_{2}=x_{2}+{\rm i}y_{2}$, as

$z_{1}\cdot z_{2}\overset{\rm
def}{=}\mbox{Re}(\overline{z_{1}}z_{2})=\tfrac{1}{2}\left(\overline{z_{1}}z_{2}+z_{1}\overline{z_{2}}\right)=x_{1}x_{2}+y_{1}y_{2}.$

Let $D_{1}$ and $D_{2}$ be two simply-connected domains, close to the discs,
of radii $b_{1}$ and $b_{2}$, respectively. Further consider polar
parametrization of the associated boundaries, for $k\in\\{1,2\\}$

(4.2) $z_{k}:\begin{array}[t]{rcl}\mathbb{T}&\mapsto&\partial D_{k}\\\
\theta&\mapsto&R_{k}(\theta)e^{{\rm i}\theta}\overset{\rm
def}{=}\sqrt{b_{k}^{2}+2r_{k}(\theta)}e^{{\rm i}\theta},\end{array}$

for some function $r(\theta).$ Accordingly, we introduce the initial
vorticities

$(\omega_{1},\omega_{2})|_{t=0}=\big{(}\mathbf{1}_{D_{1}},\mathbf{1}_{D_{2}}\big{)}$

and assume that $\omega_{1}$ and $\omega_{2}$ give rise to two rotating patch
solutions of (QS2L) about the origin, with an angular velocity
$\Omega\in\mathbb{R}$.

The vortex patch equation (1.1) provides a system of coupled nonlinear and
nonlocal PDE satisfied by the radial deformations $r_{1}$ and $r_{2}$. The
precise statement is the content of the next lemma.

######  Lemma 4.2.

The radial deformations $r\overset{\rm def}{=}(r_{1},r_{2})$ defined through
(4.2) satisfy the nonlinear coupled system

(4.3) $\mathcal{F}(\Omega,r)\overset{\rm
def}{=}\big{(}\mathcal{F}_{1}(\Omega,r),\mathcal{F}_{2}(\Omega,r)\big{)}=0,$

where we set, for all $k,j\in\\{1,2\\}$ and $\theta\in\mathbb{T}$, that

(4.4) $\displaystyle\mathcal{F}_{k}(\Omega,r)(\theta)$
$\displaystyle\overset{\rm
def}{=}\Omega\partial_{\theta}r_{k}(\theta)+\mathcal{F}_{k,1}(r)(\theta)+\mathcal{F}_{k,2}(r)(\theta),$

and

(4.5) $\mathcal{F}_{k,j}(r)(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial^{2}_{\theta\eta}\Big{(}R_{k}(\theta)R_{j}(\eta)\sin(\eta-\theta)\Big{)}d\eta,$

where the kernels $G_{k,j}$ are introduced in (2.9).

Moreover, the Rankine vorticities, corresponding to $r\equiv 0$, are
stationary solutions, for any radii $b_{1},b_{2}>0$. More precisely, one has
that

$\mathcal{F}(\Omega,0)=0,$

for all $\Omega\in\mathbb{R}$.

###### Proof.

Note first that the normal inward vector to the boundary $\partial D_{k}$ of
the patch at the point $z_{k}(\theta)$ is given by

$n_{k}(z_{k}(\theta))={\rm i}\partial_{\theta}z_{k}(\theta),$

for any $k\in\\{1,2\\}$. According to (1.1), the initial datum
$\omega_{k}|_{t=0}=\mathbf{1}_{D_{k}}$ generates a rotating patch about the
origin with uniform angular velocity $\Omega\in\mathbb{R}$ if and only if it
holds, for all $\theta\in\mathbb{T}$, that

(4.6)
$\displaystyle\Omega\,\textnormal{Re}\Big{\\{}\overline{{z}_{k}(\theta)}\partial_{\theta}{z}_{k}(\theta)\Big{\\}}=2\textnormal{Re}\Big{\\{}\partial_{\overline{z}}\psi_{k}({z}_{k}(\theta))\partial_{\theta}\overline{{z}_{k}(\theta)}\Big{\\}}.$

Therefore, observing from (4.2) that

$\displaystyle\Omega\partial_{\theta}r_{k}(\theta)=\Omega\,\textnormal{Re}\Big{\\{}\overline{{z}_{k}(\theta)}\partial_{\theta}{z}_{k}(\theta)\Big{\\}},$

entails, for all $k\in\\{1,2\\}$ and $\theta\in\mathbb{T}$, that

(4.7)
$\displaystyle\Omega\partial_{\theta}r_{k}(\theta)=2\textnormal{Re}\Big{\\{}\partial_{\overline{z}}\psi_{k}({z}_{k}(\theta))\partial_{\theta}\overline{{z}_{k}(\theta)}\Big{\\}}.$

Hence, writing, by the chain rule, that

(4.8) $\displaystyle
2\textnormal{Re}\Big{\\{}\partial_{\overline{z}}\psi_{k}({z}_{k}(\theta))\partial_{\theta}\overline{{z}_{k}(\theta)}\Big{\\}}=\partial_{\theta}\big{(}\psi_{k}(t,{z}_{k}(\theta))\big{)},$

leads, for all $k\in\\{1,2\\}$ and $\theta\in\mathbb{T}$, to the equivalence
reformulation

$\Omega\partial_{\theta}r_{k}(\theta)=\partial_{\theta}\big{(}\psi(z_{k}(\theta))\big{)}.$

Now, we recall from (2.8) that

$\displaystyle\psi_{k}(z)$
$\displaystyle=\sum_{j=1}^{2}\int_{D_{j}}G_{k,j}(z-\xi)d\xi,$

for all $z\in\mathbb{C}$, where $G_{k,j}$ is given by (2.9). Observe that the
preceding representation can be recast in polar coordinates (4.2) as

(4.9) $\displaystyle\psi_{k}(z)$
$\displaystyle=\sum_{j=1}^{2}\int_{0}^{2\pi}\int_{0}^{R_{j}(\eta)}G_{k,j}(z-\rho
e^{{\rm i}\eta})\rho d\rho d\eta.$

Therefore, employing the identity

$\partial_{\overline{z}}G_{k,j}(z,\xi)=-\partial_{\overline{\xi}}G_{k,j}(z-\xi),$

yields, for any $z\in\mathbb{C}$, that

$\displaystyle\partial_{\overline{z}}\psi_{k}\big{(}z\big{)}$
$\displaystyle=-\sum_{j=1}^{2}\int_{0}^{2\pi}\int_{0}^{R_{j}(\eta)}\partial_{\overline{\xi}}G_{k,j}\big{(}z-\rho
e^{{\rm i}\eta}\big{)}\rho d\rho d\eta.$

Thus, by further employing Gauss–Green theorem, we arrive at the
representation

(4.10) $\displaystyle 2\partial_{\overline{z}}\psi_{k}\big{(}z\big{)}$
$\displaystyle={\rm
i}\sum_{j=1}^{2}\int_{0}^{2\pi}G_{k,1}\big{(}z-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\big{(}R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\big{)}d\eta,$

whereby we deduce, for any $k\in\\{1,2\\}$ and $\theta\in\mathbb{T}$, that

$\displaystyle
2\textnormal{Re}\Big{\\{}\partial_{\overline{z}}\psi_{k}\big{(}{z}_{k}(\theta)\big{)}\partial_{\theta}\overline{{z}_{k}(\theta)}\Big{\\}}$
$\displaystyle\quad\qquad=-\sum_{j=1}^{2}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\textnormal{Im}\Big{\\{}\partial_{\theta}\big{(}R_{k}(\theta)e^{-{\rm
i}\theta}\big{)}\partial_{\eta}\big{(}R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\Big{\\}}d\eta.$

At last, inserting the latter identity above in (4.7) and employing the fact
that

$\textnormal{Im}\Big{\\{}\partial_{\theta}\big{(}f(\theta)e^{-{\rm
i}\theta}\big{)}\partial_{\eta}\big{(}g(\eta)e^{{\rm
i}\eta}\big{)}\Big{\\}}=\partial^{2}_{\theta\eta}\Big{(}f(\theta)g(\eta)\sin(\eta-\theta)\Big{)},$

for any functions $f,g$, yields the contour dynamics equation.

To conclude, by a direct computation, one sees that

$\displaystyle\mathcal{F}_{k}(\Omega,0)(\theta)$
$\displaystyle=b_{k}\sum_{j=1}^{2}b_{j}\int_{0}^{2\pi}G_{k,j}\big{(}b_{k}e^{{\rm
i}\theta}-b_{j}e^{{\rm i}\eta}\big{)}\sin(\eta-\theta)d\eta$
$\displaystyle=b_{k}\sum_{j=1}^{2}b_{j}\int_{0}^{2\pi}G_{k,j}\big{(}b_{k}-b_{j}e^{{\rm
i}(\eta-\theta)}\big{)}\sin(\eta-\theta)d\eta$
$\displaystyle=b_{k}\sum_{j=1}^{2}b_{j}\int_{0}^{2\pi}G_{k,j}\big{(}b_{k}-b_{j}e^{{\rm
i}\eta}\big{)}\sin(\eta)d\eta,$

for any $\Omega\in\mathbb{R}$ and $\theta\in\mathbb{T}$. Finally, by virtue of
symmetry properties of the kernels $G_{k,j}$,it is then readily seen that
preceding integral is identically zero, thereby concluding that $(\Omega,0)$
is a zero for the contour dynamics, for any $\Omega\in\mathbb{R}$. This
completes the proof of the lemma. ∎

######  Remark 4.3.

Let
$(\omega_{1},\omega_{2})|_{t=0}=\big{(}\mathbf{1}_{D_{1}},\mathbf{1}_{D_{2}}\big{)}$
be a rotating patch to (QS2L) with constant angular velocity $\Omega$. Further
consider a real number $a>0$ and denote by $D_{k}^{a}=aD_{k}$, for
$k\in\\{1,2\\}$. Then
$(\omega_{1},\omega_{2})|_{t=0}=\big{(}\mathbf{1}_{D_{1}^{a}},\mathbf{1}_{D_{2}^{a}}\big{)}$
is also a rotating patch with angular velocity $\Omega$ to the system (QS2L)
with $\lambda$ replaced by $a\lambda$. Indeed, according to (4.6), (4.8) and
(2.8), one has that

$\Omega\,\textnormal{Re}\Big{\\{}\overline{{z}_{k}(\theta)}\partial_{\theta}{z}_{k}(\theta)\Big{\\}}=\partial_{\theta}\bigg{(}\sum_{j=1}^{2}\int_{D_{j}}G_{k,j}\big{(}z_{k}(\theta)-\xi\big{)}dA(\xi)\bigg{)}.$

Thus, denoting $w_{k}=az_{k}$, then multiplying the preceding equation by
$a^{2}$ and implementing the change of variables $\xi^{\prime}=a\xi$ yields
that

$\Omega\,\textnormal{Re}\Big{\\{}\overline{{w}_{k}(\theta)}\partial_{\theta}{w}_{k}(\theta)\Big{\\}}=\partial_{\theta}\bigg{(}\sum_{j=1}^{2}\int_{D_{j}^{a}}G_{k,j}\big{(}\tfrac{1}{a}w_{k}(\theta)-\tfrac{1}{a}\xi^{\prime}\big{)}dA(\xi^{\prime})\bigg{)}.$

Therefore, in view of (2.9), we find that

$\displaystyle
G_{k,j}\big{(}\tfrac{1}{a}w_{k}(\theta)-\tfrac{1}{a}\xi^{\prime}\big{)}$
$\displaystyle=-\frac{\delta^{2-j}}{2\pi(\delta+1)}\log(a)+\frac{\delta^{2-j}}{2\pi(\delta+1)}\log(|w_{k}(\theta)-\xi^{\prime}|)$
$\displaystyle+(-1)^{k+j-1}\frac{\delta^{k-1}}{2\pi(\delta+1)}K_{0}\left(\tfrac{\lambda}{a}\sqrt{1+\delta}|w_{k}(\theta)-\xi^{\prime}|\right).$

Hence, we deduce that

$\Omega\,\textnormal{Re}\Big{\\{}\overline{{w}_{k}(\theta)}\partial_{\theta}{w}_{k}(\theta)\Big{\\}}=\partial_{\theta}\bigg{(}\sum_{j=1}^{2}\int_{D_{j}^{a}}\widetilde{G}_{k,j}\big{(}w_{k}(\theta)-\xi^{\prime}\big{)}dA(\xi^{\prime})\bigg{)},$

where

$\widetilde{G}_{k,j}\big{(}w-\xi^{\prime}\big{)}\overset{\rm
def}{=}\frac{\delta^{2-j}}{2\pi(\delta+1)}\log(|w-\xi^{\prime}|)+(-1)^{k+j-1}\frac{\delta^{k-1}}{2\pi(\delta+1)}K_{0}\left(\frac{\lambda}{a}\sqrt{1+\delta}|w-\xi^{\prime}|\right).$

In the next step, we establish some specific symmetry properties of the
functional $\mathcal{F}$, introduced in Lemma 4.2, above. In particular, this
is crucial and will allow us to simplify the spectral analysis of the
linearized operator associated with $\mathcal{F}$.

More importantly, it is to be emphasized that the property of $m$-fold
symmetry from the next lemma will be useful to eliminate the instabilities and
prevent their appearing at the linear level when we study spectral properties
of the linearized operator associated with $\mathcal{F}$, later on.

######  Lemma 4.3.

Let $\mathcal{F}=(\mathcal{F}_{1},\mathcal{F}_{2})$ be given by (4.4). Then,
the following holds

  1. (1)

The refection symmetry property: if $r=(r_{1},r_{2})$ are even, i.e, if

(4.11) $r(-\theta)=r(\theta),$

for any $\theta\in\mathbb{T}$, then $\mathcal{F}(\Omega,r)$ is odd, i.e.

$\mathcal{F}(\Omega,r)(-\theta)=-\mathcal{F}(\Omega,r)(\theta).$

  2. (2)

$m$-fold symmetry: if $r$ satisfy, for some $m\in\mathbb{Z}^{*}$ and all
$\theta\in\mathbb{T}$, that

$r\left(\theta+\tfrac{2\pi}{m}\right)=r(\theta),$

then, it holds that

$\mathcal{F}(\Omega,r)\left(\theta+\tfrac{2\pi}{m}\right)=\mathcal{F}(\Omega,r)(\theta),$

for all $\theta\in\mathbb{T}$.

###### Proof.

The justification of reflection symmetry property of $\mathcal{F}$ relies upon
the structure of the kernel $G_{k,j}$ defining $\mathcal{F}_{k,j}$ in (4.5).
Accordingly, by performing the change of variables $\eta\mapsto-\eta$ in
(4.5), it is readily seen, for any $k,j\in\\{1,2\\}$ and
$\theta\in\mathbb{T}$, that

$\mathcal{F}_{k,j}(\Omega,r)(-\theta)=-\mathcal{F}_{k,j}(\Omega,r)(\theta),$

whereby establishing the first claim in the lemma. Likewise, the $m-$fold
symmetry follows by making the change of variables
$\eta\mapsto\eta+\tfrac{2\pi}{m}$ in (4.5), which also follows due to the
symmetry property of the kernels $G_{k,j}$

$G_{k,j}\big{(}e^{i\alpha}(x-y)\big{)}=G_{k,j}(x-y),$

for any $x\neq y\in\mathbb{C}$ and all $\alpha\in\mathbb{T}$. The proof of the
lemma is now completed. ∎

###  4.2. Linearization around discs

Here, we explore the structure of the linearized operator associated with the
functional $\mathcal{F}$ defined in (4.3). We emphasize that the analysis we
perform in this section will be rigorously justified by a detailed regularity
study in Section 4.3, later on.

######  Lemma 4.4.

The Gâteaux derivative of $\mathcal{F}$ at $r=(r_{1},r_{2})$ in the direction
$h=(h_{1},h_{2})$ is given by

(4.12) $d_{r}\mathcal{F}(\Omega,r)[h]=\Omega\begin{pmatrix}h^{\prime}_{1}&0\\\
0&h^{\prime}_{2}\end{pmatrix}+\partial_{\theta}\begin{pmatrix}V_{1}(r)h_{1}-{L}_{1,1}(r)[h_{1}]&-{L}_{1,2}(r)[h_{2}]\vspace{5mm}\\\
-{L}_{2,1}(r)[h_{1}]&V_{2}(r)h_{2}-{L}_{2,2}(r)[h_{2}]\end{pmatrix},$

for any $\Omega\in\mathbb{R}$, where we set

(4.13) $\displaystyle V_{k}(r)(\theta)$ $\displaystyle\overset{\rm
def}{=}\frac{1}{R_{k}(\theta)}\sum_{j=1}^{2}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\Big{(}R_{j}(\eta)\sin(\eta-\theta)\Big{)}d\eta,$
(4.14) $\displaystyle L_{k,n}(r)[h_{n}](\theta)$ $\displaystyle\overset{\rm
def}{=}\int_{0}^{2\pi}G_{k,n}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{n}(\eta)e^{{\rm i}\eta})h_{n}(\eta)d\eta,$

for any $\theta\in\mathbb{T}$.

###### Proof.

Let $\theta\in\mathbb{T}$ and $k\in\\{1,2\\}$ be fixed. Now, in view of Lemma
4.2 above, notice that it is sufficient to differentiate the stream function.
To that end, we first write, by virtue of (4.9), that

$d_{r_{k}}\psi_{k}(z)[h_{k}](\theta)=\int_{0}^{2\pi}G_{k,k}(z-R_{k}(\eta)e^{{\rm
i}\eta})h_{k}(\eta)d\eta$

and

$d_{r_{3-k}}\psi_{k}(z)[h_{3-k}](\theta)=\int_{0}^{2\pi}G_{k,3-k}(z-R_{3-k}(\eta)e^{{\rm
i}\eta})h_{3-k}(\eta)d\eta,$

for any $z\in\mathbb{C}$. On the other hand, by differentiating (4.2) we
infers that

$d_{r_{k}}\overline{z}_{k}(\theta)[h_{k}](\theta)=\frac{h_{k}(\theta)}{R_{k}(\theta)}e^{-{\rm
i}\theta}.$

Therefore, due to (4.10), it follows that

$\displaystyle
2\textnormal{Re}\Big{\\{}\partial_{\overline{z}}\psi_{k}\big{(}{z}_{k}(\theta)\big{)}$
$\displaystyle d_{r_{k}}\overline{{z}_{k}[h_{k}](\theta)}\Big{\\}}$
$\displaystyle\qquad=-\frac{h_{k}(\theta)}{R_{k}(\theta)}\sum_{j=1}^{2}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\,\textnormal{Im}\Big{\\{}R_{j}(\eta)e^{{\rm
i}(\eta-\theta)}\Big{\\}}d\eta.$

At last, combining the foregoing identities yields that

$d_{r_{k}}\big{(}\psi_{k}(z_{k})\big{)}[h_{k}]=-V_{k}(r)h_{k}+L_{k,k}(r)[h_{k}]$

and

$d_{r_{3-k}}\big{(}\psi_{k}(z_{k})\big{)}[h_{3-k}]=L_{k,3-k}(r)[h_{3-k}],$

thereby completing the proof of the lemma. ∎

Before we move on to a more refined analysis of linearized operator
$d_{r}\mathcal{F}$, allow us first to recall that $\delta$ and $\lambda$ are
considered here as fixed non-negative parameters defined by the system of
equations (QS2L), above. Accordingly, we recall the notations

$\mu\overset{\rm def}{=}\lambda\sqrt{1+\delta},\qquad\text{and}\qquad
b\overset{\rm def}{=}\frac{b_{2}}{b_{1}}\in(0,1),$

where $b_{1}$ and $b_{2}$ refer to the radii of the initial patches. At last,
we recall that the modified Bessel functions $I_{n}$ and $K_{n}$, for
$n\in\mathbb{N}$, are introduced in Section 2.1 as well as their useful
properties which will come in handy later on in this section.

In the next lemma, we compute the differential of $\mathcal{F}$ at the
equilibrium $(\Omega,0)$ and show that it acts as a Fourier multiplier.

######  Lemma 4.5.

Given $(h_{1},h_{2})$ with the Fourier series expansions

$h_{k}(\theta)=\displaystyle\sum_{n\in\mathbb{N}^{*}}c_{n,k}\cos(n\theta),$

for some sequence $(c_{n,k})_{n\in\mathbb{N}^{*}}\subset\mathbb{R}$, all
$k\in\\{1,2\\}$ and $\theta\in\mathbb{T}$, it then holds that the linearized
operator of $\mathcal{F}$ at $r=0$ writes, for any $\theta\in\mathbb{T}$, as

$d_{r}\mathcal{F}(\Omega,0)\begin{pmatrix}{h}_{1}\vspace{0.1cm}\\\
{h}_{2}\end{pmatrix}(\theta)=-\sum_{n\in\mathbb{N}^{*}}nM_{n}(\Omega)\begin{pmatrix}c_{n,1}\vspace{0.1cm}\\\
c_{n,2}\end{pmatrix}\sin(n\theta),$

where we set

$M_{n}(\Omega)\overset{\rm
def}{=}\begin{pmatrix}\Omega+V_{b_{1},b_{2}}+\tfrac{\tfrac{\delta}{2n}+I_{n}(b_{1}\mu)K_{n}(b_{1}\mu)}{\delta+1}&\tfrac{\tfrac{b^{n}}{2n}-I_{n}(\mu
b_{2})K_{n}(b_{1}\mu)}{\delta+1}\vspace{5mm}\\\
\tfrac{\delta}{\delta+1}\Big{(}\tfrac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\Big{)}&\Omega+W_{b_{1},b_{2}}+\tfrac{\tfrac{1}{2n}+{\delta}I_{n}(b_{2}\mu)K_{n}(b_{2}\mu)}{\delta+1}\end{pmatrix},$

for all $n\in\mathbb{N}^{*}$, and where $V_{b_{1},b_{2}}$ and
$W_{b_{1},b_{2}}$ are defined in (4.1).

###### Proof.

Let $\theta\in\mathbb{T}$ and $k\in\\{1,2\\}$ be fixed in the proof and we
proceed first by recasting the following identities: for any
$n\in\mathbb{N}^{*}$ and $x,y,\lambda\in(0,\infty)$ with $x\leq y$, that

(4.15) $\frac{1}{2\pi}\int_{0}^{2\pi}\log\left(\big{|}1-xe^{{\rm
i}\theta}\big{|}\right)\cos(n\theta)d\theta=-\frac{x^{n}}{2n}$

and

(4.16)
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}K_{0}\left(\lambda|x-ye^{{\rm
i}\theta}|\right)\cos(n\theta)d\theta=I_{n}(\lambda x)K_{n}(\lambda y).$

The proof of the first identity can be found in [13, Lemma A.3], whereas the
justification of the second one follows from the Beltrami’s summation formula
[50, page 361]

$K_{0}\left(|x-ye^{{\rm
i}\theta}|\right)=\sum_{m\in\mathbb{Z}}I_{m}(x)K_{m}(y)\cos(m\theta).$

Now, observe that substituting the value $r=0$ in (4.13) gives

$\displaystyle
V_{k}(0)(\theta)=\sum_{j=1}^{2}\frac{b_{j}}{b_{k}}\int_{0}^{2\pi}G_{k,j}\big{(}b_{k}e^{{\rm
i}\theta}-b_{j}e^{{\rm i}\eta}\big{)}\cos(\eta-\theta)d\eta.$

On the one hand, in view of (2.9), we write that

$\displaystyle\int_{0}^{2\pi}G_{k,k}\big{(}b_{k}e^{{\rm i}\theta}-b_{k}e^{{\rm
i}\eta}\big{)}\cos(\eta-\theta)d\eta$
$\displaystyle=\frac{\delta^{2-k}}{2\pi(\delta+1)}\int_{0}^{2\pi}\log\big{(}b_{k}\big{|}1-e^{{\rm
i}(\eta-\theta)}\big{|}\big{)}\cos(\eta-\theta)d\eta$
$\displaystyle\quad-\frac{\delta^{k-1}}{2\pi(\delta+1)}\int_{0}^{2\pi}K_{0}\big{(}\mu
b_{k}\big{|}1-e^{{\rm i}(\eta-\theta)}\big{|}\big{)}\cos(\eta-\theta)d\eta.$

Therefore, changing the variable $\eta\mapsto\eta+\theta$ and employing the
identities (4.16) and (4.15) entails that

$\displaystyle\int_{0}^{2\pi}G_{k,k}\big{(}b_{k}e^{{\rm i}\theta}-b_{k}e^{{\rm
i}\eta}\big{)}\cos(\eta-\theta)d\eta$
$\displaystyle=-\frac{\delta^{2-k}}{2(\delta+1)}-\frac{\delta^{k-1}}{\delta+1}I_{1}(b_{k}\mu)K_{1}(b_{k}\mu).$

On the other hand, by virtue of (2.9), writing

$\displaystyle\int_{0}^{2\pi}G_{k,3-k}\big{(}b_{k}e^{{\rm
i}\theta}-b_{3-k}e^{{\rm i}\eta}\big{)}$ $\displaystyle\cos(\eta-\theta)d\eta$
$\displaystyle=\frac{\delta^{k-1}}{2\pi(\delta+1)}\int_{0}^{2\pi}\log\big{(}b_{1}\big{|}1-be^{{\rm
i}(\eta-\theta)}\big{|}\big{)}\cos(\eta-\theta)d\eta$
$\displaystyle\quad+\frac{\delta^{k-1}}{2\pi(\delta+1)}\int_{0}^{2\pi}K_{0}\big{(}\mu\big{|}b_{k}-b_{3-k}e^{{\rm
i}(\eta-\theta)}\big{|}\big{)}\cos(\eta-\theta)d\eta,$

it then follows, by the same arguments as before, that

$\displaystyle\int_{0}^{2\pi}G_{k,3-k}\big{(}b_{k}e^{{\rm
i}\theta}-b_{3-k}e^{{\rm i}\eta}\big{)}\cos(\eta-\theta)d\eta$
$\displaystyle=-\frac{\delta^{k-1}b}{2(\delta+1)}+\frac{\delta^{k-1}}{\delta+1}I_{1}(b_{2}\mu)K_{1}(b_{1}\mu).$

Next, according to (4.14) and (2.9), one has that

$\displaystyle L_{k,j}(0)[h_{j}](\theta)$ $\displaystyle\overset{\rm
def}{=}\sum_{n\geq 1}c_{n,j}\int_{0}^{2\pi}G_{k,j}(b_{k}e^{{\rm
i}\theta}-b_{j}e^{{\rm i}\eta})\cos(n\eta)d\eta$ $\displaystyle=\sum_{n\geq
1}\tfrac{c_{n,j}}{2\pi(\delta+1)}\bigg{[}\delta^{2-j}\int_{0}^{2\pi}\log\big{(}\big{|}b_{k}e^{{\rm
i}\theta}-b_{j}e^{{\rm i}\eta}\big{|}\big{)}\cos(n\eta)d\eta$
$\displaystyle\qquad-(-1)^{k+j}\delta^{k-1}\int_{0}^{2\pi}K_{0}\big{(}\mu\big{|}b_{k}e^{{\rm
i}\theta}-b_{j}e^{{\rm i}\eta}\big{|}\big{)}\cos(n\eta)d\eta\bigg{]},$

for any $j\in\\{1,2\\}$. Thus, by a change of variables, we find that

$\displaystyle L_{k,j}(0)[h_{j}](\theta)$ $\displaystyle=\sum_{n\geq
1}\frac{c_{n,j}}{2\pi(\delta+1)}\bigg{(}\delta^{2-j}\int_{0}^{2\pi}\log\big{(}\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\cos(n\eta+n\theta)d\eta$
$\displaystyle\qquad-(-1)^{k+j}\delta^{k-1}\int_{0}^{2\pi}K_{0}\big{(}\mu\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\cos(n\eta+n\theta)d\eta\bigg{)}$
$\displaystyle=\sum_{n\geq
1}\frac{c_{n,j}}{2\pi(\delta+1)}\bigg{(}\delta^{2-j}\cos(n\theta)\int_{0}^{2\pi}\log\big{(}\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\cos(n\eta)d\eta$
$\displaystyle\qquad-\delta^{2-j}\sin(n\theta)\int_{0}^{2\pi}\log\big{(}\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\sin(n\eta)d\eta$
$\displaystyle\qquad-(-1)^{k+j}\delta^{k-1}\cos(n\theta)\int_{0}^{2\pi}K_{0}\big{(}\mu\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\cos(n\eta)d\eta$
$\displaystyle\qquad-(-1)^{k+j}\delta^{k-1}\sin(n\theta)\int_{0}^{2\pi}K_{0}\big{(}\mu\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\sin(n\eta)d\eta\bigg{)}.$

Therefore, noting, by symmetry, that

$\displaystyle\int_{0}^{2\pi}\log\big{(}\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\sin(n\eta)d\eta$ $\displaystyle=0$

and

$\displaystyle\int_{0}^{2\pi}K_{0}\big{(}\mu\big{|}b_{k}-b_{j}e^{{\rm
i}\eta}\big{|}\big{)}\sin(n\eta)d\eta$ $\displaystyle=0,$

we conclude, by utilizing (4.16) and (4.15), again, that

$\displaystyle L_{k,k}(0)[h_{k}](\theta)$ $\displaystyle=-\sum_{n\geq
1}\frac{c_{n,k}}{\delta+1}\left(\frac{\delta^{2-k}}{2n}+\delta^{k-1}I_{n}(b_{k}\mu)K_{n}(b_{k}\mu)\right)\cos(n\theta),$
$\displaystyle L_{k,3-k}(r)[h_{3-k}](\theta)$ $\displaystyle=\sum_{n\geq
1}\frac{c_{n,3-k}\delta^{k-1}}{\delta+1}\left(-\frac{b^{n}}{2n}+I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\right)\cos(n\theta).$

At last, gathering the foregoing identities and plugging them in (4.12) at the
equilibrium $r=0$ completes the proof of the lemma. ∎

###  4.3. Regularity properties

Here, we justify the regularity properties of the nonlinear functional
$\mathcal{F}$ introduced in (4.3). In particular, by virtue of the results
laid out in this section, all the formal computations in the Section 4.2 above
will be fully justified.

Let us first set a few notations to be used afterwards. For $\alpha\in(0,1)$,
and $m\in\mathbb{N}^{+}$, consider the $m$-fold Banach spaces

$X_{m}^{\alpha}\overset{\rm def}{=}\Big{\\{}h\in
C^{1+\alpha}(\mathbb{T}):\,h(\theta)=\sum_{n\geq
1}c_{n}\cos(nm\theta),\,c_{n}\in\mathbb{R},\,\theta\in\mathbb{T}\Big{\\}}$

and

$Y_{m}^{\alpha}\overset{\rm def}{=}\Big{\\{}h\in
C^{\alpha}(\mathbb{T}):\,h(\theta)=\sum_{n\geq
1}c_{n}\sin(nm\theta),\,c_{n}\in\mathbb{R},\,\theta\in\mathbb{T}\Big{\\}}$

equipped with their usual norms. Accordingly, we define the product spaces

$\mathcal{X}_{m}^{\alpha}\overset{\rm def}{=}X_{m}^{\alpha}\times
X_{m}^{\alpha},\qquad\mathcal{Y}_{m}^{\alpha}\overset{\rm
def}{=}Y_{m}^{\alpha}\times Y_{m}^{\alpha},$

and, for all $\epsilon\in(0,1)$, we denote by
$\mathcal{B}_{m,\epsilon}^{\alpha}$ the open ball in
$\mathcal{X}_{m}^{\alpha}$ centered at the origin and with radius $\epsilon$,
i.e.,

$\displaystyle\mathcal{B}_{m,\epsilon}^{\alpha}\overset{\rm
def}{=}\big{\\{}r\in\mathcal{X}_{m}:\lVert
r\rVert_{\mathcal{X}_{m}}<\epsilon\big{\\}}.$

All regularity properties of the functional $\mathcal{F}$ that we need are now
established in the next proposition.

######  Proposition 4.6.

Let $\lambda>0,$ $b\in(0,1)$, $\alpha\in(0,1)$ and $m\in\mathbb{N}^{*}.$ Then,
there exists $\epsilon>0$ such that the functional $\mathcal{F}$ introduced in
(4.3) is well defined as a mapping

$\mathcal{F}:\mathbb{R}\times\mathcal{B}_{m,\epsilon}^{\alpha}\rightarrow\mathcal{Y}_{m}^{\alpha},$

and is of class $C^{1}.$ Moreover, the partial derivative

$\partial_{\Omega,r}^{2}\mathcal{F}:\mathbb{R}\times
B_{m,\epsilon}^{\alpha}\rightarrow\mathcal{L}(\mathcal{X}_{m}^{\alpha},\mathcal{Y}_{m}^{\alpha})$

exists and is continuous.

###### Proof.

Throughout the proof, $\alpha\in(0,1)$ and $r\in B_{m,\varepsilon}^{\alpha}$
are fixed parameters. In view of (4.3), notice that it suffices to establish
the regularity properties for $\mathcal{F}_{k,j}$, for all $j,k\in\\{1,2\\}$,
where $\mathcal{F}_{k,j}$ are introduced in (4.5) and can be recast here as

$\mathcal{F}_{k,j}(r(\theta))={\rm
i}\partial_{\theta}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}\big{)}\cdot\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\big{(}R_{j}(\eta)e^{{\rm i}\eta}\big{)}d\eta,$

where we recall that

$R_{k}(\theta)=\sqrt{b_{k}^{2}+2r_{k}(\theta)},$

for all $k\in\\{1,2\\}$. Thus, it is readily seen that

$\theta\mapsto\partial_{\theta}\big{(}R_{k}(\theta)e^{{\rm i}\theta}\big{)}\in
C^{\alpha}(\mathbb{T}),$

as soon as the deformation $\theta\mapsto r(\theta)$ belongs to
$C^{1+\alpha}(\mathbb{T})$, which holds by assumption. Next, observe, for
$r\in\mathcal{X}_{m}^{\alpha}$, that

(4.17) $b_{k}\leq|R_{k}(\theta)|\leq\sqrt{b_{k}^{2}+2\epsilon},\quad\text{for
all}\quad\theta\in\mathbb{T}.$

Therefore, in view of the identities (2.9) and (2.5), altogether with the fact
that

$\big{|}\log|x|\big{|}\lesssim|x|^{-\alpha},$

for all $x\in\mathbb{R}^{2}\setminus\\{0\\}$ and any $\alpha\in(0,1),$ it is
then readily seen that

$\bigg{|}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm i}\theta}-R_{n}(\eta)e^{{\rm
i}\eta}\big{)}\bigg{|}\lesssim\big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{n}(\eta)e^{{\rm i}\eta}\big{|}^{-\alpha},$

for any $j,k\in\\{1,2\\}$. Accordingly, one deduces for $\epsilon$ small
enough (see for [40, inequality (61)]) that

(4.18) $\Big{|}G_{k,n}\big{(}R_{k}(\theta)e^{{\rm i}\theta}-R_{n}(\eta)e^{{\rm
i}\eta}\big{)}\Big{|}\lesssim\Big{|}\sin\left(\tfrac{\theta-\eta}{2}\right)\Big{|}^{-\alpha},$

for all $\theta\neq\eta\in\mathbb{T}$. Moreover, noticing that

$\Big{|}\nabla\left(G_{k,j}(|x|)\right)\Big{|}\lesssim|k_{+}(x)|+|k_{-}(x)|,$

where $k_{\pm}$ are introduced in (2.10), and employing (2.11) yields that

$\Big{|}\nabla\left(G_{k,j}(|x|)\right)\Big{|}\lesssim|x|^{-1}.$

Thus, in view of (4.17), we deduce that

(4.19) $\Big{|}\partial_{\theta}\left(G_{k,j}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta})\right)\Big{|}\lesssim\Big{|}\sin\left(\tfrac{\theta-\eta}{2}\right)\Big{|}^{-(1+\alpha)},$

for all $\theta\neq\eta\in\mathbb{T}$. All in all, with (4.18) and (4.19) in
hand, Lemma 2.4 allows us to deduce that

$\theta\mapsto\mathcal{F}_{k,j}(r(\theta))\in C^{\alpha}(\mathbb{T}),$

for any $j,k\in\\{1,2\\}$. In addition to that, the reflection symmetry of
$\mathcal{F}_{k}$ is already established in (4.11), whereby we arrive at the
conclusion that the mapping

$\mathcal{F}:\mathbb{R}\times
B_{m,\varepsilon}^{\alpha}\rightarrow\mathcal{Y}_{m}^{\alpha}$

is well defined. Now, we show that this mapping is $C^{1}$ with respect to the
variable $r$, as its $C^{1}$ regularity wit respect to variable $\Omega$
clearly holds true as it follows from the observation that

$\partial_{\Omega}\mathcal{F}(\Omega,r)=r^{\prime}(\theta).$

To that end, allow us first to recall the expression (4.12) here for
convenience

$d_{r}\mathcal{F}(\Omega,r)[h]=\Omega\begin{pmatrix}h^{\prime}_{1}&0\\\
0&h^{\prime}_{2}\end{pmatrix}+\partial_{\theta}\begin{pmatrix}V_{1}(r)h_{1}-{L}_{1,1}(r)[h_{1}]&-{L}_{1,2}(r)[h_{2}]\vspace{5mm}\\\
-{L}_{2,1}(r)[h_{1}]&V_{2}(r)h_{2}-{L}_{2,2}(r)[h_{2}]\end{pmatrix},$

which holds for any $h=(h_{1},h_{2})\in\mathcal{X}_{m}^{\alpha}$, where
$V_{k}$ and $L_{k,n}$ are respectively given by (4.13) and (4.14) which can
also be recast as

$V_{k}(r)(\theta)=\frac{e^{i\theta}}{R_{k}(\theta)}\cdot\sum_{j=1}^{2}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\Big{(}R_{j}(\eta)e^{i\eta}\Big{)}d\eta$

and

$L_{k,n}(r)[h_{n}](\theta)=\int_{0}^{2\pi}G_{k,n}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{n}(\eta)e^{{\rm i}\eta})h_{n}(\eta)d\eta.$

Now, we shall prove that $d_{r}\mathcal{F}(\Omega,r)[\cdot]$ is a well defined
linear mapping from $\mathcal{X}_{m}^{\alpha}$ into $\mathcal{Y}_{m}^{\alpha}$
and we emphasize that we can restrict our focus on the regularity property,
for the symmetry follows by similar arguments to the proof of Lemma 4.3.
Moreover, observe that the $C^{\alpha}$ regularity of
$d_{r}\mathcal{F}(\Omega,r)[h]$, for any $h\in C^{\alpha+1}\times
C^{\alpha+1}$, directly follows from the $C^{\alpha+1}$ regularity of
$V_{k}(r)[h]$ and $L_{k,n}(r)[h]$, which we will now prove in details. Thus,
we now claim that

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\Big{(}R_{j}(\eta)e^{i\eta}\Big{)}d\eta$

and

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,j}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm i}\eta})h_{j}(\eta)d\eta$

belong to $C^{\alpha}(\mathbb{T})$ and we split the proof of that into two
parts:

#### Regularity of the anti-diagonal

This is the case where the kernels $G_{j,k}$ are regular due to a crucial
cancellation that only appears in the elements of the anti-diagonal of the
gradient-matrix. This cancellation holds to be a consequence of the specific
combination of the kernels associated with $\Delta^{-1}$ and
$(\Delta-\lambda^{2}(1+\delta)\operatorname{Id})^{-1}$ through the coupling in
(QS2L).

In order to observe this phenomenon, we first emphasize that the expression of
$G_{j,k}$ in (2.9), for $j\neq k,$ can be recast as

$G_{1,2}(z)=\frac{1}{2\pi(1+\delta)}\Big{(}\log(|z|)+K_{0}(\mu|z|)\Big{)}$

and

$G_{2,1}(z)=\frac{\delta}{2\pi(1+\delta)}\Big{(}\log(|z|)+K_{0}(\mu|z|)\Big{)}.$

Therefore, we expand $K_{0}$ by utilizing (2.3) to find, after performing
minor simplifications, that

(4.20)
$K_{0}(\mu|z|)=-\log(|z|)-\left(\underbrace{\log\left(\frac{\mu}{2}\right)I_{0}(\mu|z|)+\log\left(\frac{|z|}{2}\right)\sum_{m=1}^{\infty}\frac{\left(\mu\frac{|z|}{2}\right)^{2m}}{m!\Gamma(m+1)}}_{\overset{\rm
def}{=}Q(|z|)}\right).$

Thus, we deduce that

$G_{1,2}(z)=\frac{1}{2\pi(1+\delta)}Q(|z|)$
$G_{2,1}(z)=\frac{\delta}{2\pi(1+\delta)}Q(|z|).$

The crucial observation here is that $Q$ is more regular than the previous
kernels, $\log$ and $K_{0}$, themselves. More precisely, we emphasize that one
can show by a direct computations that

$Q(|\cdot|)\in C^{1+\alpha}_{\mathrm{loc}}(\mathbb{R}^{+}).$

Therefore, it is readily seen that

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,j}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\Big{(}R_{j}(\eta)e^{i\eta}\Big{)}d\eta$

and

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,j}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm i}\eta})h_{j}(\eta)d\eta$

belong to $C^{\alpha}(\mathbb{T}),$ for any $i\neq j\in\\{1,2\\}$, as soon as
$r\in C^{1+\alpha}(\mathbb{T})$, which holds by assumption.

#### Regularity of the diagonal

Owing again to (2.9) and (4.20), we write that

$G_{1,1}(z)=\frac{1}{2\pi(1+\delta)}\Big{(}(\delta+1)\log(|z|)-Q(|z|)\Big{)}$

and

$G_{2,2}(z)=\frac{1}{2\pi(1+\delta)}\Big{(}(\delta+1)\log(|z|)-\delta
Q(|z|)\Big{)},$

where the function $Q$ is defined in (4.20) and belongs to
$C^{1+\alpha}(\mathbb{T})$ as it is emphasized above. Accordingly, in order to
show the regularity of the diagonal elements, i.e. the fact that

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,k}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}d\eta$

and

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,k}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta})h_{k}(\eta)d\eta$

belong to $C^{\alpha}(\mathbb{T}),$ for any $k\in\\{1,2\\}$, it only remains
to prove that

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}\log\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}d\eta$

and

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}\log\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta}\Big{|}h_{k}(\eta)d\eta$

belong to $C^{\alpha}(\mathbb{T})$. Note that this is equivalent to showing
that the functions

$\theta\mapsto\mathcal{U}(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\cdot\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}d\eta$

and

$\theta\mapsto\mathcal{W}(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\cdot\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}h_{k}(\eta)d\eta$

enjoy the $C^{\alpha}$ regularity. The achievement of the preceding claims
relies on the observation that

(4.21) $\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\cdot\partial_{\eta}\Big{(}R_{k}(\eta)e^{{\rm
i}\eta}\Big{)}d\eta=0,$

for any $k\in\\{1,2\\}$, which, in particular, allows us to write, for all
$\theta\in\mathbb{T}$, that

(4.22) $\displaystyle\mathcal{U}(\theta)$
$\displaystyle=\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}\cdot\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\Bigg{(}\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}-\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}\Bigg{)}d\eta$
$\displaystyle\quad+\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\cdot\Bigg{(}\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}-\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}\Bigg{)}d\eta$

and that

$\displaystyle\mathcal{W}(\theta)$
$\displaystyle=\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}\cdot\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm
i}\theta}-R_{j}(\eta)e^{{\rm i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\Big{(}h_{k}(\eta)-h_{k}(\theta)\Big{)}d\eta$
$\displaystyle\quad+h_{k}(\theta)\int_{0}^{2\pi}\frac{R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\cdot\Bigg{(}\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}-\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}\Bigg{)}d\eta.$

On the other hand, due to the $C^{1+\alpha}$ regularity of $h_{j}$ and $r$,
one has that

$\Bigg{|}\partial_{\theta}\Big{(}R_{k}(\theta)e^{{\rm
i}\theta}\Big{)}-\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}\Bigg{|}\lesssim\big{|}\theta-\eta\big{|}^{\alpha}$

and that

$\Big{|}h_{k}(\eta)-h_{k}(\theta)\Big{|}\lesssim\big{|}\theta-\eta\big{|}^{\alpha},$

for any $\theta,\eta\in\mathbb{T}$. Therefore, combining the preceding bounds
with

(4.23) $\left|\frac{R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\right|\lesssim\frac{1}{\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}}\lesssim\Big{|}\sin\left(\tfrac{\theta-\eta}{2}\right)\Big{|}^{-1}$

and

$\left|\partial_{\theta}\left(\frac{R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta}}{\Big{|}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\Big{|}^{2}}\right)\right|\lesssim\Big{|}\sin\left(\tfrac{\theta-\eta}{2}\right)\Big{|}^{-2},$

which can be proved by means of a straitforward computations, the desired
regularity of $\mathcal{U}$ and $\mathcal{W}$ follows then by a direct
application of Lemma 2.5. This completes the proof of the fact that the
operator $d_{r}\mathcal{F}[\cdot]$ is well defined from $\mathbb{R}\times
B_{m,\epsilon}^{\alpha}$ to $C^{\alpha}(\mathbb{T})$.

As for its continuity with respect to the $r$ variable, we emphasize that we
can focus on establishing that for the functions

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,k}\big{(}R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm
i}\eta}\big{)}\partial_{\eta}\Big{(}R_{k}(\eta)e^{i\eta}\Big{)}d\eta$

and

$\theta\mapsto\partial_{\theta}\int_{0}^{2\pi}G_{k,k}(R_{k}(\theta)e^{{\rm
i}\theta}-R_{k}(\eta)e^{{\rm i}\eta})h_{k}(\eta)d\eta,$

as the treatment of the remaining terms in the expression of
$d_{r}\mathcal{F}[h]$ is straitforward. Again, this reduces to proving the
continuity, with respect to $r$, of the functions $\mathcal{U}$ and
$\mathcal{W}$ defined above.

For simplicity, we only outline here the proof of that for $\mathcal{U}$, for
the same arguments apply for $\mathcal{W}$, as well. To that end, we consider
two deformations $r=(r_{1},r_{2})$ and
$\widetilde{r}=(\widetilde{r}_{1},\widetilde{r}_{2})$ and we are going to use
the notation $\widetilde{f}$, for any given functional $f$, to precise its
dependence on $\widetilde{r}$ instead of $r$. Moreover, let us introduce the
functions

$M(\theta)\overset{\rm def}{=}R_{k}(\theta)e^{{\rm
i}\theta},\qquad\mathcal{J}_{k}(\theta)\overset{\rm
def}{=}M^{\prime}(\theta),$ $\mathcal{H}_{k}(\theta,\eta)\overset{\rm
def}{=}\frac{M(\theta)-M(\eta)}{\Big{|}M(\theta)-M(\eta)\Big{|}^{2}},$

and we recall that

$R_{k}=\sqrt{b_{k}^{2}+2r_{k}}.$

In view of these notations, we recast (4.21) as

$\int_{0}^{2\pi}\mathcal{H}_{k}(\theta,\eta)\cdot\mathcal{J}_{k}(\eta)d\eta=0,$

for all $\theta\in\mathbb{T}$. Therefore, we write, in view of (4.22), that

$\displaystyle\mathcal{U}(\theta)-\mathcal{\widetilde{U}}(\theta)$
$\displaystyle=\mathcal{J}_{k}(\theta)\cdot\int_{0}^{2\pi}\mathcal{H}_{k}(\theta,\eta)\Big{(}\mathcal{J}_{k}(\eta)-\mathcal{J}_{k}(\theta)\Big{)}d\eta$
$\displaystyle\quad+\mathcal{J}_{k}(\theta)\int_{0}^{2\pi}\mathcal{H}_{k}(\theta,\eta)\cdot\Big{(}\mathcal{J}_{k}(\theta)-\mathcal{J}_{k}(\eta)\Big{)}d\eta$
$\displaystyle\quad-\mathcal{\widetilde{J}}_{k}(\theta)\cdot\int_{0}^{2\pi}\mathcal{\widetilde{H}}_{k}(\theta,\eta)\Big{(}\mathcal{\widetilde{J}}_{k}(\eta)-\mathcal{\widetilde{J}}_{k}(\theta)\Big{)}d\eta$
$\displaystyle\quad-\mathcal{\widetilde{J}}_{k}(\theta)\int_{0}^{2\pi}\mathcal{\widetilde{H}}_{k}(\theta,\eta)\cdot\Big{(}\mathcal{\widetilde{J}}_{k}(\theta)-\mathcal{\widetilde{J}}_{k}(\eta)\Big{)}d\eta$
$\displaystyle=\big{(}\mathcal{J}_{k}(\theta)-\mathcal{\widetilde{J}}_{k}(\theta)\big{)}\cdot\int_{0}^{2\pi}\mathcal{H}_{k}(\theta,\eta)\Big{(}\mathcal{J}_{k}(\eta)-\mathcal{J}_{k}(\theta)\Big{)}d\eta$
$\displaystyle\quad+\mathcal{\widetilde{J}}_{k}(\theta)\cdot\int_{0}^{2\pi}\big{(}\mathcal{H}_{k}(\theta,\eta)-\mathcal{\widetilde{H}}_{k}(\theta,\eta)\big{)}\Big{(}\mathcal{J}_{k}(\eta)-\mathcal{J}_{k}(\theta)\Big{)}d\eta$
$\displaystyle\quad+\mathcal{\widetilde{J}}_{k}(\theta)\cdot\int_{0}^{2\pi}\mathcal{\widetilde{H}}_{k}(\theta,\eta)\Big{(}\big{(}\mathcal{J}_{k}-\mathcal{\widetilde{J}}_{k}\big{)}(\eta)-\big{(}\mathcal{J}_{k}-\mathcal{\widetilde{J}}_{k}\big{)}(\theta)\Big{)}d\eta$
$\displaystyle\overset{\rm def}{=}\sum_{j=1}^{3}\mathcal{A}_{j}(\theta).$

Thus, by virtue of the same argument laid out in the regularity study of
$\mathcal{U}$ in the previous step of the proof, we observe that a direct
application of Lemma 2.5 leads to the bound

$\left\|\mathcal{A}_{1}\right\|_{C^{\alpha}}+\left\|\mathcal{A}_{3}\right\|_{C^{\alpha}}\lesssim\|\mathcal{J}_{k}-\mathcal{\widetilde{J}}_{k}\|_{C^{\alpha}}\lesssim\left\|r_{k}-\widetilde{r}_{k}\right\|_{C^{1+\alpha}}.$

As for the estimate of $A_{2}$, in order to apply Lemma 2.5, we first need to
show that

(4.24)
$\big{|}\big{(}\mathcal{H}_{k}-\mathcal{\widetilde{H}}_{k}\big{)}(\theta,\eta)\big{|}\lesssim\left\|r-\widetilde{r}\right\|_{C^{1+\alpha}}\Big{|}\sin\left(\tfrac{\theta-\eta}{2}\right)\Big{|}^{-1}$

and

(4.25)
$\big{|}\partial_{\theta}\big{(}\mathcal{H}_{k}-\mathcal{\widetilde{H}}_{k}\big{)}(\theta,\eta)\big{|}\lesssim\left\|r-\widetilde{r}\right\|_{C^{1+\alpha}}\Big{|}\sin\left(\tfrac{\theta-\eta}{2}\right)\Big{|}^{-2},$

for all $\theta\neq\eta\in\mathbb{T}$. To that end, we only need to write, by
a direct computation, that

$\displaystyle\big{|}\mathcal{H}_{k}(\theta,\eta)$
$\displaystyle-\mathcal{\widetilde{H}}_{k}(\theta,\eta)\big{|}$
$\displaystyle\lesssim\big{|}\big{(}M-\widetilde{M}\big{)}(\theta)-\big{(}M-\widetilde{M}\big{)}(\eta)\big{|}\left(\frac{1}{\Big{|}{M}(\theta)-{M}(\eta)\Big{|}}+\frac{1}{\Big{|}\widetilde{M}(\theta)-\widetilde{M}(\eta)\Big{|}}\right)^{2}.$

Therefore, by using the Lipschitz property of the functions $M$ and
$\widetilde{M}$ combined with (4.23), one deduces that

$\displaystyle\big{|}\mathcal{H}_{k}(\theta,\eta)-\mathcal{\widetilde{H}}_{k}(\theta,\eta)\big{|}$
$\displaystyle\lesssim\|M-\widetilde{M}\|_{C^{1}}\frac{|\theta-\eta|}{\Big{|}\sin\left(\frac{\theta-\eta}{2}\right)\Big{|}^{2}}$
$\displaystyle\lesssim\left\|r-\widetilde{r}\right\|_{C^{1+\alpha}}\Big{|}\sin\left(\frac{\theta-\eta}{2}\right)\Big{|}^{-1},$

whereby showing (4.24). The justification of (4.25) can be done along the same
lines, whence we skip its proof here. This completes the proof of the
proposition. ∎

###  4.4. Spectral analysis of the linearized operator

This section is devoted, first, to perform a refine analysis of the
eigenvalues associated with the matrix $M_{n}(\Omega)$, introduced in in Lemma
4.5 and defining the linearized operator $d_{r}\mathcal{F}(\Omega,0)$, and,
second, to establish the remaining prerequisite properties of that operator
before we apply Crandall-Rabinowitz’s theorem.

######  Proposition 4.7.

For a given $n\in\mathbb{N}^{*},$ the matrix $M_{n}(\Omega)$, introduced in
Lemma 4.5 above, is not invertible if and only if $\Omega=\Omega_{n}^{\pm},$
where

(4.26) $\Omega_{n}^{\pm}\overset{\rm
def}{=}\frac{1}{2(\delta+1)}\left(-(A_{n}+B_{n})\pm\sqrt{(A_{n}-B_{n})^{2}+{4\delta}\left(\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\right)^{2}}\right),$

where, we set

(4.27) $\displaystyle A_{n}$ $\displaystyle\overset{\rm
def}{=}(\delta+1)V_{b_{1},b_{2}}+\frac{\delta}{2n}+I_{n}(b_{1}\mu)K_{n}(b_{1}\mu)$
$\displaystyle B_{n}$ $\displaystyle\overset{\rm
def}{=}(\delta+1)W_{b_{1},b_{2}}+\frac{1}{2n}+{\delta}I_{n}(b_{2}\mu)K_{n}(b_{2}\mu),$

and we recall that $V_{b_{1},b_{2}}$ and $W_{b_{1},b_{2}}$ are defined in
(4.1), above. Moreover, for any $0<b_{2}\leq b_{1}$ and $\delta>0$, the
sequences $(\Omega_{n}^{\pm})_{n\in\mathbb{N}}$ are strictly increasing. In
addition to that, there is $p_{0}\in\mathbb{N}^{*}$ such that, for all
$m,n\geq p_{0}$, any $0<b_{2}<b_{1}$ and $\delta\geq\frac{b_{2}}{b_{1}}$, it
holds that

$\Omega_{n}^{+}>\Omega_{m}^{-}.$

Furthermore, regarding $\Omega_{n}^{\pm}$ as functions of $b_{2}\in(0,b_{1})$,
for fixed $b_{1}\in(0,\infty)$ and $n\in\mathbb{N}^{*}$, and introducing the
set

${S}_{m,b_{1}}\overset{\rm def}{=}\left\\{b_{2}\in(0,b_{1}):\exists
n\in\mathbb{N}^{*},\ \Omega_{m}^{-}(b_{2})=\Omega_{n}^{+}(b_{2})\right\\},$

it then holds, for any $m\in\mathbb{N}^{*}$, that the set ${S}_{m,b_{1}}$
contains at most a finite number of elements.

At last, in the particular case $b_{1}=b_{2}$, there exists
$b_{1}\in(0,\infty)$ for which one has that

$\Omega_{n}^{+}=\Omega_{m}^{-},\quad\text{for some}\quad
n,m\in\mathbb{N}^{*}.$

######  Remark 4.4.

It is readily seen that the set ${S}_{m,b_{1}}$ is empty, for any $m>p_{0}$,
where $p_{0}$ is as per given in the statement of Proposition 4.7, above.
However, the arguments in our proof below are not enough to show that this set
is empty for low $m$-fold symmetries. This remains then unclear for now and
can probably be wrong in view to the example of spectral collisions given in
the statement the same lemma above in the case $b_{1}=b_{2}$.

###### Proof.

We split the proof into four steps for a better readability.

#### Computing the eigenvalues

We first proceed by noticing that the matrix introduced in Lemma 4.5 can be
recast as

$M_{n}(\Omega)=\begin{pmatrix}\Omega+\frac{A_{n}}{\delta+1}&\frac{1}{\delta+1}\Big{(}\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\Big{)}\vspace{5mm}\\\
\frac{\delta}{\delta+1}\Big{(}\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\Big{)}&\Omega+\frac{B_{n}}{\delta+1},\end{pmatrix},$

for any $n\in\mathbb{N}^{*}$, where $A_{n}$ and $B_{n}$ are defined in the
statement of Proposition 4.7, above. Accordingly, it is readily seen that

$\displaystyle\textnormal{det}M_{n}$
$\displaystyle=\left(\Omega+\frac{A_{n}}{\delta+1}\right)\left(\Omega+\frac{B_{n}}{\delta+1}\right)-\frac{\delta}{(\delta+1)^{2}}\left(\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\right)^{2}$
$\displaystyle=\Omega^{2}+\frac{A_{n}+B_{n}}{\delta+1}\Omega+\frac{A_{n}B_{n}}{(\delta+1)^{2}}-\frac{\delta}{(\delta+1)^{2}}\left(\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\right)^{2}.$

Therefore, noting that the discriminant of the preceding second order
polynomial satisfies

$\displaystyle(\delta+1)^{2}\Delta_{n}$
$\displaystyle=\big{(}A_{n}+B_{n}\big{)}^{2}-4A_{n}B_{n}+{4\delta}\left(\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\right)^{2}$
$\displaystyle=\big{(}A_{n}-B_{n}\big{)}^{2}+{4\delta}\left(\frac{b^{n}}{2n}-I_{n}(b_{2}\mu)K_{n}(b_{1}\mu)\right)^{2}>0,$

it then follows, for all $n\geq 1$, that there exist two angular velocities
$\Omega_{n}^{\pm}$ which are as given they are introduced in the lemma, and
for which the matrix $M_{n}(\Omega_{n}^{\pm})$ is singular.

#### Monotony of the sequences $(\Omega_{n}^{\pm})_{n\in\mathbb{N}^{*}}$

Now, we provide a more precise analysis of the angular velocities
$\Omega_{n}^{\pm}$. To that end, note first, due to the representations

$A_{n}=-\delta\left(\frac{1}{2}-\frac{1}{2n}\right)-b\left(\frac{b}{2}-I_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\right)-\Big{(}I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)-I_{n}(b_{1}\mu)K_{n}(b_{1}\mu)\Big{)}$

and

$B_{n}=-\left(\frac{1}{2}-\frac{1}{2n}\right)-\frac{\delta}{b}\left(\frac{b}{2}-I_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\right)-\delta\Big{(}I_{1}(b_{2}\mu)K_{1}(b_{2}\mu)-I_{n}(b_{2}\mu)K_{n}(b_{2}\mu)\Big{)},$

together with Lemma 2.1, that the sequences $(A_{n})_{n\in\mathbb{N}^{*}}$ and
$(B_{n})_{n\in\mathbb{N}^{*}}$ are non-positive and non-increasing.

Now, we claim that $(\Omega_{n}^{\pm})_{n\in\mathbb{N}^{*}}$, defined in
(4.26), are both non-negative, non-decreasing sequences. To see that, we
regard these sequences as functions of the real variable $x\in[1,\infty)$,
i.e., we now consider the functions $x\mapsto\Omega_{x}^{\pm}$ which extend
the preceding sequences on $[1,\infty)$. Note that it is easy to check that
these functions are differentiable. Therefore, denoting

$M_{x}\overset{\rm
def}{=}\frac{A_{x}-B_{x}}{\delta+1},\qquad\Gamma_{x}\overset{\rm
def}{=}\frac{\frac{b^{x}}{2x}-I_{x}(b_{2}\mu)K_{x}(b_{1}\mu)}{\delta+1},$

we compute, for any $x\in[1,\infty)$, that

$\displaystyle 2(\delta+1)\partial_{x}\Omega_{x}^{\pm}$
$\displaystyle=\left(\pm\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}-1\right)\partial_{x}A_{x}-\left(\pm\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}+1\right)\partial_{x}B_{x}$
$\displaystyle\qquad\pm\frac{4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\partial_{x}\left(\frac{b^{x}}{2x}-I_{x}(b_{2}\mu)K_{x}(b_{1}\mu)\right).$

Then, from the expression of $A_{x}$ and $B_{x}$, given by (4.27), we get

$\displaystyle 2(\delta+1)\partial_{x}\Omega_{x}^{\pm}$
$\displaystyle=-\left(\delta+1\mp\frac{(\delta-1)M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\right)\partial_{x}\left(\frac{1}{2x}\right)$
$\displaystyle\qquad-\left(1\mp\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\right)\partial_{x}\Big{(}I_{x}(b_{1}\mu)K_{x}(b_{1}\mu)\Big{)}$
$\displaystyle\qquad-\delta\left(1\pm\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\right)\partial_{x}\left(I_{x}(b_{2}\mu)K_{x}(b_{2}\mu)\right)$
$\displaystyle\qquad\pm\frac{4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\partial_{x}\left(\frac{b^{x}}{2x}-I_{x}(b_{2}\mu)K_{x}(b_{1}\mu)\right).$

Hence, one deduces that

$\displaystyle 2(\delta+1)\partial_{x}\Omega_{x}^{-}$
$\displaystyle=-(\delta+1)\Bigg{(}\frac{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}+M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\Bigg{)}\partial_{x}\left(\frac{1}{2x}\right)$
$\displaystyle\quad-\Bigg{(}1+\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\Bigg{)}\partial_{x}\Big{(}I_{x}(b_{1}\mu)K_{x}(b_{1}\mu)\Big{)}$
$\displaystyle\quad-\delta\Bigg{(}1-\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\Bigg{)}\partial_{x}\Big{(}I_{x}(b_{2}\mu)K_{x}(b_{2}\mu)\Big{)}$
$\displaystyle\quad-\frac{4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\partial_{x}\Big{(}\frac{b^{x}}{2x}-I_{x}(b_{2}\mu)K_{x}(b_{1}\mu)\Big{)}.$

Likewise for $\Omega_{x}^{+}$, by further employing the straitforward
computation

$\displaystyle\delta+1-\frac{(\delta-1)M_{x}+4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}$
$\displaystyle=\frac{(\delta+1)\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}-(\delta-1)M_{x}-4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}$
$\displaystyle=\frac{(\delta+1)^{2}({M_{x}^{2}+4\delta\Gamma_{x}^{2}})-\big{(}(\delta-1)M_{x}+4\delta\Gamma_{x}\big{)}^{2}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}\big{(}(\delta+1)\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}+(\delta-1)M_{x}+4\delta\Gamma_{x}\big{)}}$
$\displaystyle=4\delta\left(\frac{M_{x}^{2}+(\delta-1)^{2}\Gamma_{x}^{2}-2(\delta-1)M_{x}\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}\big{(}(\delta+1)\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}+(\delta-1)M_{x}+4\delta\Gamma_{x}\big{)}}\right)$
$\displaystyle=\frac{4\delta\big{(}M_{x}-(\delta-1)\Gamma_{x}\big{)}^{2}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}\big{(}(\delta+1)\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}+(\delta-1)M_{x}+4\delta\Gamma_{x}\big{)}},$

it is then readily seen that

$\displaystyle 2(\delta+1)\partial_{x}\Omega_{x}^{+}$
$\displaystyle=-\bigg{(}\frac{4\delta\big{(}M_{x}-(\delta-1)\Gamma_{x}\big{)}^{2}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}\big{(}(\delta+1)\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}+(\delta-1)M_{x}+4\delta\Gamma_{x}\big{)}}\bigg{)}\partial_{x}\left(\frac{1}{2x}\right)$
$\displaystyle\quad-\frac{4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\partial_{x}\left(\frac{1}{2x}-\frac{b^{x}}{2x}\right)$
$\displaystyle\quad-\bigg{(}1-\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\bigg{)}\partial_{x}\Big{(}I_{x}(b_{1}\mu)K_{x}(b_{1}\mu)\Big{)}$
$\displaystyle\quad-\delta\bigg{(}1+\frac{M_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\bigg{)}\partial_{x}\Big{(}I_{x}(b_{2}\mu)K_{x}(b_{2}\mu)\Big{)}$
$\displaystyle\quad-\frac{4\delta\Gamma_{x}}{\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}}\partial_{x}\Big{(}I_{x}(b_{2}\mu)K_{x}(b_{1}\mu)\Big{)}.$

At last, the conclusion of the proof of the monotonicity if the function
$x\mapsto\Omega_{x}^{\pm}$ follows from the fact that the functions

$x\mapsto\frac{1}{x},\qquad x\mapsto I_{x}(b_{k}\mu)K_{x}(b_{j}\mu),\qquad
x\mapsto\frac{1}{2x}-\frac{b^{x}}{2x}$

are decreasing, for any $k,j\in\\{1,2\\}$ as soon as $b_{k}\leq b_{j}$, and
$b\in(0,1)$, altogether with the fact that $\Gamma_{x}\geq 0$ and that

$-\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}}\leqslant
M_{x}\leqslant\sqrt{M_{x}^{2}+4\delta\Gamma_{x}^{2}},$

for any $x\in[1,\infty)$. As a consequence of the preceding computations, we
deduce that

$\Omega_{n}^{\pm}\neq\Omega_{m}^{\pm},$

for all $m\neq n\in\mathbb{N}^{*}$. We will now precise sufficient conditions
that will ensure that

$\Omega_{n}^{+}\neq\Omega_{m}^{-}.$

To that end, we first observe that

$\Omega_{n}^{-}<\Omega_{n}^{+},$

for all $n\in\mathbb{N}^{*}$. Moreover, by denoting

$A_{\infty}\overset{\rm def}{=}\lim_{n\to\infty}A_{n}\quad\text{and}\quad
B_{\infty}\overset{\rm def}{=}\lim_{n\to\infty}B_{n},$

it then follows that that

$\Omega_{\infty}^{\pm}\overset{\rm
def}{=}-\frac{1}{2(\delta+1)}\Big{(}A_{\infty}+B_{\infty}\mp|A_{\infty}-B_{\infty}|\Big{)}.$

On the other hand, we observe that

$A_{\infty}=-\left(\frac{b^{2}+\delta}{2}+I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)-bI_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\right),$

whereas

$B_{\infty}=-\left(\frac{1+\delta}{2}+\delta
I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)-\frac{\delta}{b}I_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\right).$

Therefore, one sees that

$|A_{\infty}-B_{\infty}|=\left|\left(\delta-b^{2}\right)\left(\frac{1}{2}-\frac{1}{b}I_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\right)-\left(\delta-1\right)\left(\frac{1}{2}-I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)\right)\right|.$

Observe that the value of preceding quantity is identically zero when
$b_{1}=b_{2}$. However, we are going to show now that it is strictly positive
elsewhere. To see that, we notice, by using the strict monotonicity of the
function

$x\mapsto\frac{1}{x}I_{1}(x)$

on $(0,\infty)$, which is proved in Lemma 2.1, that

$\frac{1}{b}I_{1}(b_{2}\mu)-I_{1}(b_{1}\mu)=b_{1}\mu\left(\frac{1}{b_{2}\mu}I_{1}(b_{2}\mu)-\frac{1}{b_{1}\mu}I_{1}(b_{1}\mu)\right)<0,$

as long as $0<b_{1}<b_{2}$. In view of the preceding inequality, we now
observe, thanks to the assumption $\delta\geq b^{2}$, that

$\displaystyle\left(\delta-b^{2}\right)\left(\frac{1}{2}-\frac{1}{b}I_{1}(b_{2}\mu)K_{1}(b_{1}\mu)\right)$
$\displaystyle-\left(\delta-1\right)\left(\frac{1}{2}-I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)\right)$
$\displaystyle\qquad\qquad\qquad\qquad>(1-b^{2})\left(\frac{1}{2}-I_{1}(b_{1}\mu)K_{1}(b_{1}\mu)\right)>0.$

Thus, we deduce, for any $\delta>b^{2}$ and $b_{1}\neq b_{2}$, that
$A_{\infty}\neq B_{\infty}$ and, therefore, that

$\Omega_{\infty}^{+}=-\frac{1}{\delta+1}\min\\{A_{\infty},B_{\infty}\\}>\Omega_{\infty}^{-}=-\frac{1}{\delta+1}\max\\{A_{\infty},B_{\infty}\\}.$

Accordingly, due to the monotonicity of the sequences
$(\Omega_{n}^{\pm})_{n\in\mathbb{N}^{*}}$, it is then readily seen that one
can find $p_{0}\in\mathbb{N}^{*}$ such that

$\Omega_{m}^{+}\geq\Omega_{p_{0}}^{+}>\Omega_{\infty}^{-}>\Omega_{n}^{-},$

for all $m\geq p_{0}$ and $n\in\mathbb{N}^{*}$, thereby concluding that

$\Omega_{m}^{+}>\Omega_{n}^{-},$

for all $n,m\geq p_{0}.$

#### Employing analyticity to study the cardinal of spectral collisions

Let us now assume that $1\leq n<p_{0}$ and fix $m\in\mathbb{N}^{*}$ with
$1\leq n<m$. Also, we now regard $\Omega_{n}^{\pm}$ as analytic functions of
the real variable $b_{2}\in(0,b_{1})$. We emphasize that this is a consequence
of the analytic property of the Bessel functions defining the eigenvalues
$\Omega_{n}^{\pm}$, see [43, Appendix B.2].

Thus, by virtue of the monotony of the eigenvalues $n\mapsto\Omega_{n}^{\pm}$,
we infer, for any $1\leq n\leq p_{0}$, that there is at most one index $m>n$
for which the equality

$\Omega_{m}^{-}=\Omega_{n}^{+}$

can probably hold for some $b_{2}\in(0,b_{1})$. Accordingly, for $1\leq n\leq
p_{0}$, we introduce the (possibly empty) set

$S_{n,b_{1}}\overset{\rm def}{=}\left\\{b_{2}\in(0,b_{1}):\exists
m=m(n)>n,\quad\text{such
that}\quad\Omega_{m(n)}^{-}(b_{2})=\Omega_{n}^{+}(b_{2})\right\\}.$

Therefore, we claim that there is $q_{0}\in\mathbb{N}^{*}$ and a finite
sequence of real numbers $(c_{j})_{1\leq j\leq q_{0}}$ such that

(4.28) $\bigcup_{1\leq n\leq
p_{0}}S_{n,b_{1}}\subset\left\\{c_{j}\in(0,1):1\leq j\leq q_{0}\right\\}.$

In other words, loosely speaking, we claim that the set of values
$b_{2}\in(0,b_{1})$ for which the sequence of eigenvalues of the matrix
$M_{n}(\Omega)$ can match is, in worse cases, negligible for Lebesgue’s
measure. Again, showing that relies on the analytic property of the function

$b_{2}\mapsto\Omega_{n}^{+}(b_{2})-\Omega_{m(n)}^{-}(b_{2})$

on $(0,b_{1})$, for all $n\in\mathbb{N}^{*},$ which yields that this non
constant function has at most isolated zeros on open connected sets.
Consequently, we deduce that

$S_{n,b_{1}}\subset\left\\{c_{j,n}\in(0,b_{1}):1\leq j\leq
j_{*}\right\\},\quad\text{for some}\quad j_{*}\in\mathbb{N}^{*},$

and some real numbers $c_{j,n}\in(0,b_{1})$, whereby (4.28) follows. At last,
we deduce that

$\Omega_{n}^{+}\neq\Omega_{m}^{-},$

for any $b_{2}\in(0,b_{1})\backslash\\{c_{j}\\}_{1\leq j\leq q_{0}}$ and all
$n,m\in\mathbb{N}^{*}$.

#### Example of spectral collisions

Let us now consider the case of identical initial discs, i.e., $b_{1}=b_{2}$
and show that, in this particular situation, there exist values of
$b_{1}\in(0,\infty)$ for which the sequence of matrix
$\big{(}M_{n}(\Omega)\big{)}_{n\in\mathbb{N}^{*}}$ possess a simple sequence
of eigenvalues, i.e., that

$\Omega_{n}^{+}\neq\Omega_{n}^{-},\quad\text{for all}\quad
n\in\mathbb{N}^{*},$

but with possible “counterpart collision”, i.e,

$\Omega_{n}^{+}=\Omega_{m}^{-},\quad\text{for some}\quad
n,m\in\mathbb{N}^{*}.$

To see that, we emphasize that a direct computation of the angular velocities
$\Omega_{n}^{\pm}$, by setting $b_{1}=b_{2}$ in (4.26), yields that

(4.29)
$\Omega_{n}^{+}=\frac{1}{2}-I_{n}K_{n}(b_{1}\mu)\qquad\text{and}\qquad\Omega_{n}^{-}=\frac{1}{2}-\frac{1}{2n}.$

Therefore, in view of Lemma 2.1 above, it is easy to see that both sequences
$(\Omega_{n}^{\pm})_{n\in\mathbb{N}^{*}}$ are increasing and converge to the
same limit $\frac{1}{2}$. Also, it is readily seen, again by virtue of Lemma
2.1, that

$\Omega_{n}^{+}>\Omega_{n}^{-},\quad\text{for all}\quad n\in\mathbb{N}^{*},$

and, whence, by monotony of the sequence
$(\Omega_{n}^{-})_{n\in\mathbb{N}^{*}}$, that

$\Omega_{n}^{+}\neq\Omega_{m}^{-},\quad\text{for all}\quad n\geq
m\in\mathbb{N}^{*}.$

However, using the asymptotic properties of the function $x\mapsto
I_{n}K_{n}(x)$ to write that

$\lim_{x\to
0}I_{1}K_{1}(x)=\frac{1}{2}\qquad\text{and}\qquad\lim_{x\to\infty}I_{1}K_{1}(x)=0,$

meaning that $x\mapsto I_{1}K_{1}(x)$ is a continuous function and takes
values in $(0,\frac{1}{2})$, allows us to deduce the existence of
$x_{0}\in(0,\infty)$ and $m\in\mathbb{N}^{*}$ such that

$I_{1}K_{1}(x_{0})=\frac{1}{2m},$

thereby deducing that there is $b_{1}=b_{2}\in(0,\infty)$ such that

$\Omega_{1}^{+}=\Omega_{m}^{-},\quad\text{for some}\quad m\in\mathbb{N}^{*}.$

This shows the existence of spectral collisions in the case $b_{1}=b_{2}$ and
concludes the proof of the proposition. ∎

###  4.5. Applying Crandall-Rabinowitz’s theorem and proof of Theorem 1.3

Now, we are in position to establish the last prerequisites before we apply
Crandall-Rabinowitz’s theorem. In particular, in the following proposition, we
prove essential properties of the kernel and image of the linearized operator
$d_{r}\mathcal{F}$ along with the transversality condition.

######  Proposition 4.8.

Let $m\in\mathbb{N}^{*}$ and $\alpha\in(0,1)$ be fixed and assume that the
radii of the initial discs are such that $0<b_{2}<b_{1}$. Further assume that
$\delta\geq\frac{b_{2}}{b_{1}}$ and either $m\geq p_{0}$ or $b_{2}\notin
S_{m,b_{1}}$, where $p_{0}$ and $S_{m,b_{1}}$ are introduced in Proposition
4.7 above. Then, the following holds true:

  1. (1)

The linearized operator $d_{r}\mathcal{F}(\Omega,0)$ has a non trivial kernel
in $\mathcal{X}_{m}^{\alpha}$ if and only if $\Omega=\Omega^{\pm}_{mn}$ for
some $n\in\mathbb{N}^{*}$, where $\Omega^{\pm}_{mn}$ is given by (4.26). In
this case, the kernel of $d_{r}\mathcal{F}(\Omega_{m}^{\pm},0)$ is a one-
dimensional vector space in $\mathcal{X}_{m}^{\alpha}$ generated by

$\theta\mapsto\begin{pmatrix}\Omega_{m}^{\pm}+\frac{B_{m}}{\delta+1}\vspace{3mm}\\\
-\frac{\delta}{\delta+1}\gamma_{m}\end{pmatrix}\cos(m\theta),$

where, $B_{m}$ is defined by (4.27) and we set

(4.30) $\gamma_{m}\overset{\rm
def}{=}\frac{b^{m}}{2m}-I_{m}(b_{2}\mu)K_{m}(b_{1}\mu),\qquad
b=\frac{b_{2}}{b_{1}}.$

  2. (2)

The range of $d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)$ is closed in
$\mathcal{Y}_{m}^{\alpha}$ and is of co-dimension one.

  3. (3)

At last, the transversality condition holds, i.e., we have that

$\partial_{\Omega}d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)\begin{pmatrix}1\\\
\sigma_{m}\end{pmatrix}\cos(m\cdot)\notin
R\big{(}\partial_{r}\mathcal{F}(\Omega^{\pm}_{m},0)\big{)}.$

###### Proof.

Let $\rho=(\rho_{1},\rho_{2})\in\mathcal{X}_{m}^{\alpha}$ with Fourier series
expansions

$\rho_{k}(\theta)=\displaystyle\sum_{n\geq 1}c_{nm,k}\cos(nm\theta),$

for some $c_{n,k}\in\mathbb{R}$ and all $k\in\\{1,2\\}$. According to Lemma
4.5 above, one has that

(4.31) $d_{r}\mathcal{F}(\Omega,0)\begin{pmatrix}{\rho}_{1}\vspace{0.1cm}\\\
{\rho}_{2}\end{pmatrix}=-\sum_{n\geq
1}nmM_{nm}(\Omega)\begin{pmatrix}c_{nm,1}\vspace{0.1cm}\\\
c_{nm,2}\end{pmatrix}\sin(nm\theta).$

Hence, in view of Proposition 4.7, the determinant of the matrix
$M_{m}(\Omega)$ vanishes if and only if $\Omega=\Omega^{\pm}_{m}$. Thus, the
kernel of $d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)$ is non trivial and it is one-
dimensional if and only if for all $n\geq 2$,

(4.32) $\textnormal{det}\big{(}M_{nm}(\Omega^{\pm}_{m})\big{)}\neq 0$

which is equivalent to

$\Omega^{\pm}_{m}\neq\Omega^{\pm}_{nm}\quad\textnormal{for all}\quad n\neq 1.$

The preceding condition is ensured by Proposition 4.7 whenever one the two
following conditions are fulfilled: either $m>p_{0}$ or that $b_{2}\notin
S_{m,b_{1}}$. Now, observe that $\rho=(\rho_{1},\rho_{2})$ belongs to the
kernel of $d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)$ if and only the Fourier
coefficients in its Fourier expansion (4.31) vanish, i.e, if

$c_{nm,1}=c_{nm,2}=0,$

for all $n\neq 1$, and

$(c_{m,1},c_{m,2})\in\textnormal{ker}\big{(}M_{m}(\Omega^{\pm}_{m})\big{)}.$

Hence, by noticing that

$\begin{pmatrix}\Omega_{m}^{\pm}+\tfrac{A_{n}}{\delta+1}&\tfrac{\gamma_{m}}{\delta+1}\vspace{5mm}\\\
\tfrac{\delta}{\delta+1}\gamma_{m}&\Omega_{m}^{\pm}+\tfrac{B_{n}}{\delta+1},\end{pmatrix}\cdot\begin{pmatrix}\Omega_{m}^{\pm}+\frac{B_{m}}{\delta+1}\vspace{3mm}\\\
-\frac{\delta}{\delta+1}\gamma_{m}\end{pmatrix}=\begin{pmatrix}0\vspace{2mm}\\\
0\end{pmatrix},$

one then deduces that the generator of
$\textnormal{ker}\big{(}d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)\big{)}$ can be
chosen as the pair of functions

$\theta\mapsto\begin{pmatrix}\Omega_{m}^{\pm}+\frac{B_{m}}{\delta+1}\vspace{3mm}\\\
-\frac{\delta}{\delta+1}\gamma_{m}\end{pmatrix}\cos(m\theta).$

This takes care of the first claim in the proposition. Now, we turn our
attention to show that the range of $d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)$
coincides with the space of functions
$(g_{1},g_{2})\in\mathcal{Y}_{m}^{\alpha}$ such that

(4.33) $\begin{pmatrix}g_{1}\\\ g_{2}\end{pmatrix}=\sum_{n\geq
1}\begin{pmatrix}g_{nm,1}\\\ g_{nm,2}\end{pmatrix}\sin(nm\theta),$

where $g_{nm,1},g_{nm,2}\in\mathbb{R}$, for any $n\geq 2$, and there exists
$c_{m,1},c_{m,2}\in\mathbb{R}$ such that

(4.34) $M_{m}(\Omega_{m}^{\pm})\begin{pmatrix}c_{m,1}\vspace{0.1cm}\\\
c_{m,2}\end{pmatrix}=\begin{pmatrix}g_{m,1}\vspace{0.1cm}\\\
g_{m,2}\end{pmatrix}.$

First, observe that the range of $d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)$ is
obviously included in the space introduced above, which is clearly closed and
of co-dimension one in $\mathcal{Y}_{m}^{\alpha}$. Therefore, it only remains
to check the converse inclusion. To see that, fixing
$(g_{1},g_{2})\in\mathcal{Y}_{m}^{\alpha}$, we shall prove that the equation

(4.35)
$d_{r}\mathcal{F}(\Omega^{\pm}_{m},0)[(\rho_{1},\rho_{2})]=\begin{pmatrix}g_{1}\\\
g_{2}\end{pmatrix}$

admits a solution in the space $\mathcal{X}_{m}^{\alpha}$, as soon as
$(g_{1},g_{2})$ satisfies (4.33) and (4.34) for some
$c_{nm,1},c_{nm,2}\in\mathbb{R}$. To that end, we first notice that (4.35) is
equivalent to

$-nmM_{nm}(\Omega_{m}^{\pm})\begin{pmatrix}c_{nm,1}\vspace{0.1cm}\\\
c_{nm,2}\end{pmatrix}=\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix},\quad\text{for all}\quad n\geq 1.$

The existence of $c_{m,1}$ and $c_{m,2}$ is obviously ensured by (4.34). As
for the remaining coefficients, it is readily seen that

$\begin{pmatrix}c_{nm,1}\vspace{0.1cm}\\\
c_{nm,2}\end{pmatrix}=-\frac{1}{nm}M_{nm}^{-1}(\Omega_{m}^{\pm})\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix},\quad\text{for all}\quad n\geq 2,$

where we have used the fact that the matrix $M_{nm}(\Omega_{m}^{\pm})$ is
invertible for all $n\geq 2$, which is established at the beginning of this
proof. This defines the coefficients $c_{nm,1}$ and $c_{nm,2}$.

Next, for a given couple of functions
$(g_{1},g_{2})\in\mathcal{Y}_{m}^{\alpha}$ as in (4.33), we need to show that
the solution of (4.35) belongs to the space $\mathcal{X}_{m}^{\alpha}$. More
precisely, we claim that

$\begin{pmatrix}\rho_{1}\\\
\rho_{2}\end{pmatrix}=\begin{pmatrix}c_{m,1}\vspace{0.1cm}\\\
c_{m,2}\end{pmatrix}\cos(m\theta)-\sum_{n\geq
2}\frac{1}{nm}M_{nm}^{-1}(\Omega_{m}^{\pm})\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix}\cos(nm\theta)\in C^{1+\alpha}(\mathbb{T})\times
C^{1+\alpha}(\mathbb{T}),$

where, for $c_{m,1}$ and $c_{m,2}$ are defined in (4.34). It is clear that we
can restrict our selves to showing that

$\sum_{n\geq
2}\frac{1}{nm}M_{nm}^{-1}(\Omega_{m}^{\pm})\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix}\cos(nm\theta)\in C^{1+\alpha}(\mathbb{T})\times
C^{1+\alpha}(\mathbb{T}),$

which is equivalent to proving that

(4.36) $\begin{pmatrix}\widetilde{\rho}_{1}\\\
\widetilde{\rho}_{2}\end{pmatrix}\overset{\rm def}{=}\sum_{n\geq
2}M_{nm}^{-1}(\Omega_{m}^{\pm})\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix}\sin(nm\theta)\in C^{\alpha}(\mathbb{T})\times
C^{\alpha}(\mathbb{T}).$

Now, by a direct computation, it is easy to check that

$M_{nm}^{-1}(\Omega_{m}^{\pm})=\frac{1}{\det(M_{nm}(\Omega_{m}^{\pm}))}\begin{pmatrix}\Omega_{m}^{\pm}+\frac{B_{mn}}{\delta+1}&-\frac{\gamma_{nm}}{\delta+1}\vspace{0.1cm}\\\
-\frac{\delta\gamma_{nm}}{\delta+1}&\Omega_{m}^{\pm}+\frac{A_{mn}}{\delta+1}\end{pmatrix},\quad\text{for
all}\quad n\geq 2,$

where $\gamma_{n}$ is defined in (4.30) whereas $A_{n}$ and $B_{n}$ are
introduced in (4.27). Note that the preceding matrix is well defined thanks to
(4.32) which ensures that

$\displaystyle\det\big{(}M_{nm}(\Omega_{m}^{\pm})\big{)}\neq 0,\quad\text{for
all}\quad n\geq 2.$

We further introduce the matrix

$P_{n,m}\overset{\rm
def}{=}M_{nm}^{-1}(\Omega_{m}^{\pm})-M_{\infty}^{-1}(\Omega_{m}^{\pm}),\quad\text{for
all}\quad n\geq 2,$

where

$M_{\infty}^{-1}(\Omega_{m}^{\pm})\overset{\rm
def}{=}\lim_{n\rightarrow\infty}M_{mn}^{-1}(\Omega_{m}^{\pm}),$

and we claim that

(4.37) $|P_{n,m}|=O\left(\frac{1}{n}\right),\quad\text{for all}\quad n\geq 2.$

It is easy to see that the preceding claim follows as a consequence of the
following asymptotics:

$\left|\frac{1}{\det(M_{nm}(\Omega_{m}^{\pm}))}-\frac{1}{\det(M_{\infty}(\Omega_{m}^{\pm}))}\right|=O\left(\frac{1}{n}\right)$

and

$\left|\begin{pmatrix}\frac{B_{mn}}{\delta+1}&-\frac{\gamma_{nm}}{\delta+1}\vspace{0.1cm}\\\
-\frac{\delta\gamma_{nm}}{\delta+1}&\frac{A_{mn}}{\delta+1}\end{pmatrix}-\begin{pmatrix}\frac{B_{\infty}}{\delta+1}&0\vspace{0.1cm}\\\
0&\frac{A_{\infty}}{\delta+1}\end{pmatrix}\right|=O\left(\frac{1}{n}\right),$

as $n\to\infty$. These asymptotic identities are a direct consequence of the
decay properties of Bessel functions, see in particular (2.6). This justifies
our claim (4.37).

Now, we are ready to conclude the proof of our main claim (4.36). To that end,
we first write that

$\displaystyle\begin{pmatrix}\widetilde{\rho}_{1}\\\
\widetilde{\rho}_{2}\end{pmatrix}=\sum_{n\geq
2}P_{n,m}(\Omega_{m}^{\pm})\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix}\sin(nm\theta)-M_{\infty}^{-1}(\Omega_{m}^{\pm})\sum_{n\geq
2}\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix}\sin(nm\theta).$

On the one hand, due to the assumption that
$(g_{1},g_{2})\in\mathcal{Y}_{m}^{\alpha}$, it is readily seen that

$M_{\infty}^{-1}(\Omega_{m}^{\pm})\sum_{n\geq
2}\begin{pmatrix}g_{nm,1}\vspace{0.1cm}\\\
g_{nm,2}\end{pmatrix}\sin(nm\theta)\in C^{\alpha}(\mathbb{T})\times
C^{\alpha}(\mathbb{T}).$

On the other hand, since we have proved that

$P_{n,m}=O\left(\frac{1}{n}\right),\quad\text{for all}\quad n\geq 2,$

it then follows that