# Existence of non-trivial non-concentrated compactly supported stationary
solutions of the 2D Euler equation with finite energy

Javier Gómez-Serrano, Jaemin Park and Jia Shi

###### Abstract

In this paper, we prove the existence of locally non-radial solutions to the
stationary 2D Euler equations with compact support but non-concentrated around
one or several points. Our solutions are of patch type, have analytic
boundary, finite energy and sign-changing vorticity and are new to the best of
our knowledge. The proof relies on a new observation that finite energy,
stationary solutions with simply-connected vorticity have compactly supported
velocity, and an application of the Nash-Moser iteration procedure.

###### Contents

  1. 1 Introduction
    1. 1.1 2D Euler rigidity and construction of stationary solutions
    2. 1.2 Structure of the proofs
    3. 1.3 Organization of the paper
  2. 2 Functional equations and functional setting for stationary vortex patches
  3. 3 Warm-up: Existence of non-radial stationary vortex patches with infinite energy
    1. 3.1 Main results for infinite energy
    2. 3.2 Proof of Theorem 3.1
      1. 3.2.1 Steps 1,2 and 3: Regularity
      2. 3.2.2 Step 4: Analysis of the linear part.
      3. 3.2.3 Step 5: Transversality
  4. 4 Existence of non-radial stationary vortex patches with finite energy
    1. 4.1 Main results for finite energy
    2. 4.2 Compactly supported velocity
    3. 4.3 Nash-Moser theorem
      1. 4.3.1 Analysis of ${T}_{s}$
      2. 4.3.2 Nash-Moser iteration
    4. 4.4 Estimates on the velocity
    5. 4.5 Checking the hypotheses in Theorem 4.7
      1. 4.5.1 Regularity of the functional $G$
      2. 4.5.2 The Dirichlet-Neumann operator
      3. 4.5.3 Estimates on the linearized operator
      4. 4.5.4 Spectral Study
    6. 4.6 Analytic regularity of the boundary
  5. A Appendix
    1. A.1 Crandall–Rabinowitz Theorem
    2. A.2 Useful integrals
    3. A.3 Integral estimates
    4. A.4 Stability of linear operators

##  1 Introduction

Throughout this paper we will work with the two-dimensional incompressible
Euler equations in vorticity form. The evolution of the vorticity $\omega$ is
given by

$\begin{cases}\partial_{t}\omega+u\cdot\nabla\omega=0\ &\text{ in
}\mathbb{R}^{2}\times\mathbb{R}_{+},\\\
u(\cdot,t)=-\nabla^{\perp}(-\Delta)^{-1}\omega(\cdot,t)&\text{ in
}\mathbb{R}^{2},\\\ \omega(\cdot,0)=\omega_{0}&\text{ in
}\mathbb{R}^{2},\end{cases}$ (1.1)

where $\nabla^{\perp}:=(-\partial_{x_{2}},\partial_{x_{1}})$. Note that we can
express $u$ as $u(\cdot,t)=\nabla^{\perp}(\omega(\cdot,t)*\mathcal{N}),$ where
$\mathcal{N}(x):=\frac{1}{2\pi}\ln|x|$ is the Newtonian potential in two
dimensions.

In this paper we will be focusing on constructing non-radial stationary
solutions to equations (1.1). We will work in the _patch_ setting, where
$\omega(\cdot,t)=\sum\Theta_{i}1_{D_{i}(t)}$ is an sum of indicator functions
of bounded sets that move with the fluid, although some of our results
translate into the smooth setting as well (where $\omega(\cdot,t)$ is smooth
and compactly-supported in $x$).  See Remark 1.5. For well-posedness results
for patch solutions, see the global well-posedness results [8, 21].

Stationary solutions of the Euler equations are an important building block
since they might play a role in many different directions: for example
understanding turbulence both in 2D [11] and in 3D realizing turbulent flows
as a superposition of Beltrami flows (which are particular stationary
solutions of 3D Euler whose curl is proportional to themselves) [31, 38, 76].
In the context of numerical simulation, steady solutions of the 2D Euler
equations were used by Chorin [27] to perform numerical simulations of the 2D
Navier-Stokes equations with small viscosity, approximating the NS solutions
as a superposition of steady eddies of constant vorticity that solve the 2D
Euler equations. In the context of convex integration [37, 9], Beltrami flows
and Mikado flows are classes of stationary solutions of the Euler equations
that have been used within an iteration scheme to generate fast oscillating
perturbations in order to construct weak solutions.

In [51] we showed that any stationary solution (in both the patch and smooth
settings) for which $\omega\geq 0$ and was compactly supported had to be
radial. In this paper we want to address the necessity of the hypothesis
$\omega\geq 0$ by answering (on the positive) the following question:

###### Question 1.1

Do there exist non-trivial stationary solutions for which $\omega$ changes
sign?

Our first main result immediately gives the answer to the above question.

######  Theorem 1.2 (Corollary 3.2)

There exist non-radial, sign-changing vortex patch solutions with analytic
boundary to the 2D Euler equation (1.1) whose kinetic energy is infinite, that
is, $\int_{\mathbb{R}^{2}}|\nabla^{\perp}\omega*\mathcal{N}|^{2}dx=\infty$.

###### Remark 1.3

In [51, Theorem A], it was shown that any non-negative stationary vortex patch
must be radially symmetric up to a translation. Theorem 3.1 implies that by
allowing an arbitrarily small portion of negative vorticity, one can find a
non-radial stationary vortex patch. More precisely, for any $\epsilon>0$, one
can find a non-radial stationary vortex patch $\omega$ such that
$\int_{\mathbb{R}^{2}}\omega^{-}(x)dx<\epsilon$ while
$\int_{\mathbb{R}^{2}}\omega^{+}(x)dx$ is uniformly bounded from below, where
$\omega^{-}(x):=-\omega(x)1_{\left\\{x\in\mathbb{R}^{2}:\omega(x)<0\right\\}}$
and
$\omega^{+}(x):=\omega(x)1_{\left\\{x\in\mathbb{R}^{2}:\omega(x)>0\right\\}}$.

Our second main theorem concerns the solutions with finite kinetic energy. For
the radial vorticity $\omega$ with zero-average,
$\int_{\mathbb{R}}\omega(x)dx=0$, its velocity vanishes outside the support of
$\omega$. With this observation, one can easily produce “globally non-radial”
solutions by placing multiple copies of such vorticity so that their supports
do not overlap. See Figure 1. However the flow on each connected component of
the support of the velocity is still circular (around different points). From
now on, we say that such a solution is locally radial. More precisely a
solution to the 2D Euler equation (1.1) is locally radial if each connected
component of $\omega$ is radial up to a translation and
$\int_{\mathcal{C}}\omega dx=0$ for each connected component $\mathcal{C}$ of
$\text{supp}(\omega)$. The next theorem states that there exist more non-
trivial stationary solutions.

Figure 1: One can choose $\Theta$ so that the radial vorticity has zero
average (left). Since the support of such velocity and vorticity coincide, one
can place such solutions to produce non-radial solutions (right).

######  Theorem 1.4 (Corollary 4.5, Theorem 4.27)

There exist vortex patch solutions to the 2D Euler equation (1.1) that are not
locally radial, with finite kinetic energy and analytic boundary. Furthermore,
the solutions have compactly supported velocity.

###### Remark 1.5

In this paper, we construct patch-type solutions with compactly supported
velocity. This is a consequence of the existence of stationary solutions with
finite kinetic energy and our key Lemma 4.3. We note that a smooth stationary
solution with finite kinetic energy also has compactly supported velocity (see
Remark 4.6).

###  1.1 2D Euler rigidity and construction of stationary solutions

In this subsection we will summarize some of the history of stationary
solutions, mostly focusing on the rigidity (only trivial solutions exist) vs
flexibility (non-trivial solutions exist) dichotomy. The first result goes
back to Fraenkel [40, Chapter 4], who proved that if $D$ is a stationary,
simply connected patch, then $D$ must be a disk. The main idea uses the fact
that in this setting, the stream function $\psi=1_{D}*\mathcal{N}$ solves a
semilinear elliptic equation $\Delta\psi=g(\psi)$ in $\mathbb{R}^{2}$ with
$g(\psi)=1_{\\{\psi<C\\}}$, and one can apply the moving plane method
developed in [79, 47] using the monotonicity of $g$ to obtain the symmetry of
$\psi$. However, this result does not cover the non simply-connected case due
to the fact that $\psi$ may take different values on the different parts of
the boundary and thus one can not apply moving plane techniques. This was
solved by the authors and Yao in [51] using a variational approach that does
not require this condition, and generalized to the smooth case as long as the
vorticity is non-negative. Fraenkel’s result (and method) was generalized to
other classes of active scalar equations such as the generalized SQG (where
the velocity in (1.1) is given by the perpendicular gradient of the
convolution with $\frac{1}{|x|^{\alpha}}$ as opposed to the Newtonian
potential) by Reichel [77, Theorem 2], Lu–Zhu [69] and Han–Lu–Zhu [55] in the
case of $\alpha\in[0,1)$ and Choksi–Neumayer–Topaloglu [26] in the case
$\alpha\in[0,\frac{5}{3})$. In [51] we closed the problem for the full range
$\alpha\in[0,2)$.

In the last few years, there has been an emergence of results on rigidity
conditions, namely under which hypotheses we can guarantee that the solution
has some rigid features in order to ultimately characterize stationary
solutions by other geometric properties (such as being a shear or being
radial). These are usually referred as ‘Liouville” type of results. In the
case of 2D Euler, Hamel–Nadirashvili in [54, 52] proved that any stationary
solution without a stagnation point must be a shear flow whenever the domain
is a strip and also in [53] proved the corresponding rigidity (radial
symmetry) result whenever the domain is a two-dimensional bounded annulus, an
exterior circular domain, a punctured disk or a punctured plane.
Constantin–Drivas–Ginsberg [29, 28] obtained rigidity and flexibility results
for Euler and other equations (such as MHD) in both 2D and 3D. Coti-
Zelati–Elgindi–Widmayer [32] constructed stationary solutions close to the
Kolmogorov and Poiseuille flows in $\mathbb{T}^{2}$. In the case of 2D
Navier–Stokes, Koch–Nadirashvili–Seregin–Šverák also proved a Liouville
theorem in [66]. See also [49, 50] where together with Yao we proved rigidity
and flexibility results for the vortex sheet problem.

We now review some additional results related to the characterization or
construction of nontrivial stationary solutions to 2D Euler (flexibility).
Nadirashvili [74], following Arnold [2, 3, 4] studied the geometry and the
stability of stationary solutions. When the problem is posed on a surface,
Izosimov–Khesin [63] characterized stationary solutions of 2D Euler.
Choffrut–Šverák [22] showed that locally near each stationary smooth solution
there exists a manifold of stationary smooth solutions transversal to the
foliation, and Choffrut–Székelyhidi [23] showed that there is an abundant set
of stationary weak ($L^{\infty}$) solutions near a smooth stationary one.
Shvydkoy–Luo [70, 71] looked at stationary smooth solutions of the form
$v=\nabla^{\perp}(r^{\gamma}f(\omega))$, where $(r,\omega)$ are polar
coordinates and were able to obtain a classification of them. In a different
direction, Turkington [82] used variational methods to construct stationary
vortex patches of a prescribed area in a bounded domain, imposing that the
patch is a characteristic function of the set $\\{\Psi>0\\}$, and also studied
the asymptotic limit of the patches tending to point vortices. He also studied
the case of unbounded domains. We emphasize that those solutions do not have
finite energy, unless the domain is bounded. Long–Wang–Zeng [68] studied the
regularity in the smooth setting (see also [14]) as well as their stability.
For other variational constructions close to point vortices, we would like to
mention the work done by Cao–Liu–Wei [12], Cao–Peng–Yan [13] and Smets–van
Schaftingen [81]. Musso–Pacard–Wei [73] constructed nonradial smooth
stationary solutions with finite energy but without compact support in
$\omega$. Our solutions are different from all of these constructions since
they are not close to point vortices.

The (nonlinear $L^{1}$) stability of circular patches was proved by
Wan–Pulvirenti [83] and later Sideris–Vega gave a shorter proof [80]. See also
Beichman–Denisov [6] for similar results on the strip. Recently, Choi–Lim [25]
generalized the stability results for radial patches to radially symmetric
monotone vorticity. Lately, Gavrilov [45, 46] managed to construct nontrivial
stationary solutions of 3D Euler with compactly supported velocity, which was
further simplified and extended to other equations by Constantin–La–Vicol
[30]. In [39], Domínguez-Vázquez–Enciso–Peralta-Salas construct a different
family of non-localizable, stationary solutions of 3D Euler that are
axisymmetric with swirl. Regarding the stability results in 3D, we refer to
the work of Choi [24] for solutions near Hill’s spherical vortex.

###  1.2 Structure of the proofs

The skeleton of our proof employs bifurcation theory, trying to find a
perturbation of trivial solutions (radial vorticity). A very natural procedure
is to frame it using a Crandall-Rabinowitz Theorem approach (see [18, 19, 16,
17, 20, 41, 42, 44, 43, 48, 56, 58, 36, 59, 57] for applications in the
context of fluid mechanics). In a suitable functional setting, the problem
boils down to finding non-trivial zeros of a nonlinear functional of the form
(see (2.3)) :

$\mathcal{F}(\Theta,R)=0,\quad F:\mathbb{R}\times H^{k+1}(\mathbb{T})\mapsto
H^{k}(\mathbb{T}),\quad k\geq 3,\text{ given that $\mathcal{F}(\Theta,0)=0$
for any $\Theta\in\mathbb{R}$}.$

Note that the loss of derivatives attributes to the fact that each component
of the functional in (2.3) takes account of the tangent vector of the
boundaries. An easy observation is that the non-degeneracy of the velocity at
the trivial solution (more precisely, the angular velocity $\neq 0$)
guarantees that the linearized functional is Fredholm. For a radial vorticity
with infinite energy, this can be more explicitly revealed in the linearized
operators obtained in Proposition 3.6, where the the angular velocity
contributes to the diagonal elements of the matrix $M_{n}$, which can also be
seen in (4.5). However, for a radial vorticity whose average vanishes, the
corresponding velocity field, which can be explicitly computed, vanishes
outside the support of the vorticity. At such trivial solutions, the
linearized functional fails to be Fredholm and one cannot directly apply the
Crandall-Rabinowitz Theorem. Indeed, the degeneracy of the velocity on the
outmost boundary yields a mismatch of the image spaces of the functional at
the nonlinear level and the linear level in terms of the regularity. One
possible attempt to overcome this issue is to find two bifurcation curves
emanating from negative-average vorticity and positive-average vorticity that
are close to each other and show that the two curves merge together forming a
loop, using the strategy in [60] (see Figure 2). It is not difficult to find
(in the three-layer setting) values for $\Theta^{+}$ and $\Theta^{-}$ and
doing numerical continuation one can observe those loops.

Figure 2: One possible strategy: Find two bifurcation curves emanating from
negative average and positive average and show that those two curves are
connected. If it is possible, then there must be a solution with zero-average
$(\Theta^{*},R^{*})$ by continuity.

Our key Lemma 4.3 shows that the degeneracy of the velocity is not a special
case at the trivial solutions, but a generic phenomenon. More precisely, if
the vorticity $\omega$ is stationary (not necessarily radial) with zero-
average, then $u=0$ outside the support of $\omega$ as long as
$\text{supp}(\omega)$ is simply connected. This gives two crucial
implications:

  * •

A non-trivial stationary solution near a trivial one cannot be obtained by
using the implicit function theorem. In Figure 2,
$D_{R}\mathcal{F}(\Theta^{*},R^{*})$, the linearized operator at
$(\Theta^{*},R^{*})$ (if it exists) cannot be an isomorphism.

  * •

The mismatch in regularity between the image spaces described above may be
treated in the sprit of Nash-Moser scheme with the use of an approximate
inverse.

The first implication shows that one cannot use the Lyapunov-Schmidt reduction
which is a crucial tool in the strategy in [60] to find a loop of bifurcation
curves and even more importantly in the original proof of the Crandall-
Rabinowitz theorem. Regarding the second implication, the Nash-Moser scheme
has been well-adapted in the context of steady-state solutions [22, 61, 62],
dynamical solutions [67, 78] and more recently, embedded into a KAM scheme in
the context of quasiperiodic solutions for the water waves problem (see [5]
and the references therein). In Section 4, we will find a bifurcation curve
without the use of such a reduction technique, and combine it with the Nash-
Moser scheme in the following way:

  * 1)

In Subsection 4.1, we slightly modify our functional setting given in Section
2 so that $(\Theta,R)$ represents a zero-average vorticity. We are led to find
non-trivial zeros of the modified functional (see (4.5) for the definition of
$G$)

$G:\mathbb{R}\times H^{k+1}\mapsto H^{k},\quad\text{given that
${G}(\Theta,0)=0$ for any $\Theta\in\mathbb{R}$ }.$

See also (4.161), where it is shown that each component of $G$ is the
tangential derivative of the stream function on each boundary component.

  * 2)

We analyze the linearized operator $D_{R}G$ to find $\Theta^{*}\in\mathbb{R}$
such that $D_{R}G(\Theta^{*},0)$ satisfies (Subsection 4.5.4)

    * $\bullet$

Ker$(D_{R}G(\Theta^{*},0))$ is one-dimensional, that is,
Ker$(D_{R}G(\Theta^{*},0))=\text{span}\left\\{v\right\\}$ for some $v\in
C^{\infty}$.

    * $\bullet$

Im$(D_{R}G(\Theta^{*},0))^{\perp}$ is one-dimensional.

    * $\bullet$

$(D_{R}G(\Theta^{*},0))$ satisfies the transversality condition:
$\partial_{\Theta}D_{R}(G(\Theta^{*},0))[v]\notin\text{Im}D_{R}G(\Theta^{*},0)$.

This indicates that a possible non-trivial solution can be found as a zero of
the following functional for sufficiently small $s>0$:

$\displaystyle\tilde{G}_{s}:H^{k+1}\mapsto
H^{k},\quad\tilde{G}_{s}(R):=G(\Theta^{*}+v\cdot R,sv+(I-P)[R]),$ (1.2)

where $P:H^{k+1}\mapsto C^{\infty}$ is the projection to
$\text{Ker}(D_{R}G(\Theta^{*},0))$, and $I$ is the identity operator (see
(4.19)).

  * 3)

We perform Newton’s method for the functional $\tilde{G}_{s}$. The two
important ingredients for Newton’s method to work are 1) a sufficiently good
initial guess and 2) the invertibility of the linearized operator
$D\tilde{G}_{s}$. For the initial guess, we already have
$\tilde{G}_{s}=O(s^{2})$, since $v\in\text{Ker}(D_{R}G(\Theta^{*},0))$.

  * 4)

Regarding the second ingredient, we observe that the operator
$D_{R}G(\Theta,R)$ can be decomposed

$D_{R}G=a+A\text{ as in \eqref{decomp_11},}$

where the operator $a(\Theta,R)$ vanishes if $(\Theta,R)$ corresponds to a
stationary solution. This is due to the fact that the velocity on the outmost
boundary vanishes by Lemma 4.3. This observation leads us to decompose
$D\tilde{G}_{s}$, which can be immediately computed from (1.2) as

$D\tilde{G}_{s}=\partial_{\Theta}G\circ
P+D_{R}G\circ(I-P)=\underbrace{\partial_{\Theta}G\circ
P+A\circ(I-P)}_{=:T_{s}}+a\circ(I-P),\quad\text{ see \eqref{T_sdef}}.$

Thanks to the transversality condition, the one dimensionality of
$\text{Im}(D_{R}G(\Theta^{*},0))^{\perp}(=\text{Im}A(\Theta^{*},0)^{\perp}$
since $a(\Theta^{*},0)=0$) is compensated by $\partial_{\Theta}G\circ P$.
Indeed, it is the main goal of Subsection 4.3.1 to show that $T_{s}$ is
invertible.

  * 5)

As described in 4) we do not have the invertibility of $D\tilde{G}_{s}$,
however, the invertibility of $T_{s}$ is enough for the iteration in (4.58) to
work since $a$ tends to $0$ along the iteration steps towards the solution.
More precisely, $T_{s}$ plays a role of the approximate inverse of
$D\tilde{G}_{s}$. The loss of derivatives occurring at each iteration step is
treated by the use of regularizing operator in (4.57) in the spirit of the
Nash-Moser scheme.

We note that our finite energy solutions consist of three-layered vortex
patches, unlike the infinite energy case (See Subsection 3.1 and 4.1). From
the technical point of view, the two-layered, zero-average trivial solutions
do not give a linearized operator that has finite dimensional Kernel. More
precisely, one cannot find a pair of parameters $(b,\Theta)$ such that 1) the
matrix $M_{n}$ in Proposition 3.6 has zero-determinant and 2) the
corresponding $\omega$ determined by $(b,\Theta)$ has zero-average (see Remark
3.12). An important aspect of such two-layered stationary vorticity is that
its stream function $\psi$ always satisfies

$\displaystyle\Delta\psi=g(\psi),\text{ for some
$g:\mathbb{R}\mapsto\mathbb{R}$}.$ (1.3)

We emphasize that the solutions that we construct in this paper cannot be
captured by (1.3) unlike the earlier works, for example, [81, 32, 28]. That
is, there is no $g$ such that the stream function $\psi$ solves (1.3). This
feature can been seen from the fact that the trivial solutions from which we
bifurcate to obtain finite energy solutions exhibit non-monotone stream
functions in the radial direction.

Lastly, we point out that the desingularization from point vortices is not
applicable to find a stationary solution with finite energy, since a steady
configuration of point vortices has zero vortex angular momentum, which
together with the finite energy hypothesis implies that the individual
circulations have to be all zero (see [75, Lemma 1.2.1]).

###  1.3 Organization of the paper

The paper is organized in the following way: Section 2 sets up the functional
framework and the equation that a stationary solution has to satisfy. Section
3 proves Theorem 1.2, in the easier case where the finite energy hypothesis is
dropped. Finally, section 4 proves Theorem 1.4 in the full generality setting
using the Nash-Moser scheme. The Appendix contains some technical background
results used throughout the proof, as well as basic bifurcation theory and
some integrals needed for the spectral study.

##  2 Functional equations and functional setting for stationary vortex
patches

Let us consider vorticity $\omega$ of the form
$\omega:=\sum_{i=1}^{n}\Theta_{i}1_{D_{i}}$ for some $n\in\mathbb{N}$, where
$\Theta_{i}\in\mathbb{R}$ and $D_{i}$ is a simply-connected domain for each
$i=1,\ldots,n$. Suppose the boundary of $D_{i}$ is parametrized by a time-
dependent $2\pi$-periodic curve
$z_{i}(\cdot,t):\mathbb{T}\mapsto\mathbb{R}^{2}$. The evolution equation of
the boundary can be written as a system of equations,

$\displaystyle\partial_{t}z_{j}(x,t)\cdot\partial_{x}z_{j}(x,t)^{\perp}$
$\displaystyle=u(z_{j}(x,t),t)\cdot\partial_{x}z_{j}(x,t)^{\perp}.$ (2.1)

where
$u(\cdot,t):=\nabla^{\perp}\left(\omega(\cdot,t)*\frac{1}{2\pi}\log|\cdot|\right)$
is the velocity vector. We will look for stationary solutions to
$\eqref{evolution_patch}$. Let us assume that each $z_{i}$ can be written as

$\displaystyle z_{i}(x)=(b_{i}+R_{i}(x))(\cos(x),\sin(x)),\quad
x\in\mathbb{T},$ (2.2)

where $b_{i}$ is a positive constant for each $i$. By plugging these
parametrizations into (2.1), we are led to solve the following system for
$b:=(b_{1},\ldots,b_{n})$, $\Theta:=(\Theta_{1},\ldots,\Theta_{n})$ and
$R:=(R_{1},\ldots,R_{n})$:

$\displaystyle 0=\mathcal{F}_{j}(b,\Theta,R):=u(z_{j}(\cdot))\cdot
z_{j}^{\prime\perp}$
$\displaystyle:=\sum_{i=1}^{n}\frac{1}{4\pi}\Theta_{i}u_{i,j}(R)\cdot
z_{j}^{\prime\perp}=\sum_{i=1}^{n}\frac{1}{4\pi}\Theta_{i}\underbrace{\left(u_{i,j}^{\theta}(R)R_{j}^{\prime}-u_{i,j}^{r}(R)(b_{j}+R_{j})\right)}_{=:S_{ij}(b_{i},b_{j},R_{i},R_{j})}.$
(2.3)

where
$u_{i,j}(R)(x):=\nabla^{\perp}\left(1_{D_{i}}*\log|\cdot|^{2}\right)(z_{j}(x))$,
which can be thought of as a contribution of the $i$th patch on the $j$th
curve, and $u_{i,j}^{\theta}$ and $u_{i,j}^{r}$ are the angular and the radial
components of $u_{i,j}$. More explicitly, we have

$\displaystyle S_{ij}(b_{i},b_{j},R_{i},R_{j})$
$\displaystyle=\int_{0}^{2\pi}\cos(x-y)((b_{i}+R_{i}(y))R_{j}^{\prime}(x)-(b_{j}+R_{j}(x))R_{i}^{\prime}(y))$
$\displaystyle\times\log((b_{j}+R_{j}(x))^{2}+(b_{i}+R_{i}(y))^{2}-2(b_{j}+R_{j}(x))(b_{i}+R_{i}(y))\cos(x-y))dy$
$\displaystyle-\int_{0}^{2\pi}\sin(x-y)((b_{i}+R_{i}(y))(b_{j}+R_{j}(x))+R_{i}^{\prime}(y)R_{j}^{\prime}(x))$
$\displaystyle\times\log((b_{j}+R_{j}(x))^{2}+(b_{i}+R_{i}(y))^{2}-2(b_{j}+R_{j}(x))(b_{i}+R_{i}(y))\cos(x-y))dy.$
(2.4)

It is clear that $S_{ij}(b_{i},b_{j},0,0)=0,$ since $u_{i,j}^{r}=0$ for radial
patches. Thus we have

$\displaystyle\mathcal{F}(b,\Theta,0)=0,\text{ for any $b,\Theta$.}$ (2.5)

In what follows, we will pick one of $\Theta_{i}$ as a bifurcation parameter,
while the others are fixed.

We now proceed to discuss the functional spaces that we will use. In Section 3
we will work with the following analytic spaces. Following [17], we denote the
space of analytic functions in the strip
$\left\\{z\in\mathbb{C}:\text{Im}(z)\leq c\right\\}$ by $\mathcal{C}_{w}(c)$.
For $k\in\mathbb{Z}$, we will consider the following spaces of
$\frac{2\pi}{m}$-periodic functions:

$\displaystyle X^{k,m}_{c}:=\left\\{f(x)\in\mathcal{C}_{w}(c),\quad
f(x)=\sum_{j=1}^{\infty}a_{jm}\cos(jmx),\quad\rVert
f\rVert_{X^{k,m}_{c}}<\infty\right\\},$ $\displaystyle
Y^{k,m}_{c}:=\left\\{f(x)\in\mathcal{C}_{w}(c),\quad
f(x)=\sum_{j=1}^{\infty}a_{jm}\sin(jmx),\quad\rVert
f\rVert_{Y^{k,m}_{c}}<\infty\right\\},$

where $\rVert f\rVert_{X^{k,m}_{c}}=\rVert
f\rVert_{Y^{k,m}_{c}}:=\sum_{j=1}^{\infty}|a_{jm}|^{2}(1+jm)^{2k}(\cosh(cjm)^{2}+\sinh(cjm)^{2})$.

In Section 4 we will consider the following spaces of
$\frac{2\pi}{m}$-periodic functions:

$\displaystyle X^{k,m}:=\left\\{f(x)\in H^{k}(\mathbb{T}):\quad
f(x)=\sum_{j=1}^{\infty}a_{jm}\cos(jmx),\quad\rVert
f\rVert_{X^{k,m}}<\infty\right\\},$ $\displaystyle Y^{k,m}:=\left\\{f(x)\in
H^{k}(\mathbb{T}):\quad f(x)=\sum_{j=1}^{\infty}a_{jm}\sin(jmx),\quad\rVert
f\rVert_{Y^{k,m}}<\infty\right\\},$

where $\rVert f\rVert_{X^{k,m}}=\rVert f\rVert_{Y^{k,m}}:=\rVert
f\rVert_{H^{k}(\mathbb{T})}$.

##  3 Warm-up: Existence of non-radial stationary vortex patches with infinite
energy

As a warm-up, in this section we aim to show that there exists a non-trivial
patch solution with infinite kinetic energy,
$\frac{1}{2\pi}\int_{\mathbb{R}^{2}}|\nabla\left(\omega*\log|\cdot|\right)|^{2}dx=\infty$.
Recall that ([72, Proposition 3.3]),

$\displaystyle\int_{\mathbb{R}^{2}}|\nabla\left(\omega*\log|\cdot|\right)|^{2}dx<\infty\iff\int_{\mathbb{R}^{2}}\omega(x)dx=0.$
(3.1)

In our proof, we will find a continuous bifurcation curve, emanating from a
two-layered vortex patch whose vorticity does not has zero mean.

###  3.1 Main results for infinite energy

Let us consider two-layer vortex patches, that is, $i\in\left\\{1,2\right\\}$
in the setting in Section 2. For $b,\Theta\in(0,1)$, using the scaling
invariance of the equations, we will choose the parameters to be

$\displaystyle b_{1}=1,\quad
b_{2}=b,\quad\Theta_{1}=\Theta,\quad\Theta_{2}=-1,$ (3.2)

so that if $R_{1}=R_{2}=0$, then

$\displaystyle\omega=\begin{cases}\Theta&\text{ in }B_{1}\backslash B_{b}\\\
\Theta-1&\text{ in }B_{b},\end{cases}$ (3.3)

where $B_{r}$ denotes the unit disk centered at the origin. Note that the case
$\Theta=1$ corresponds to an annulus. Later, we will fix $b$ as well, and let
$\Theta$ play the role as a bifurcation parameter. With this setting, the
system (2.3) is equivalent to

$\displaystyle 0=\mathcal{F}(\Theta,R)=(F^{1}(\Theta,R),F^{2}(\Theta,R)),\quad
R=(R_{1},R_{2}),$ (3.4)

where we omit the dependence of $\Theta_{2}$ and $b_{1}$, $b_{2}$ for
notational simplicity.

###### Theorem 3.1

Let $k,m$ be such that $k\geq 3$, $2\leq m\in\mathbb{N}$. Let $b$ satisfy the
condition in Lemma 3.8. Then for some $c>0$ and $s_{0}=s_{0}(k,m,c,b)>0$,
there exist two bifurcation curves $[0,s_{0})\ni
s\mapsto(\Theta^{\pm}(s),R^{\pm}(s))\in\mathbb{R}\times(X^{k,m}_{c})^{2}$ such
that for each $s\in(0,s_{0})$, $(\Theta^{\pm}(s),R^{\pm}(s))$ is a solution of
the equation (3.4) and $R^{\pm}(s)\neq 0\in(X^{k,m}_{c})^{2}$. The bifurcation
curve emanates from $(\Theta^{\pm}(0),R^{\pm}(0))=(\Theta_{m}^{\pm},0)$, where
$\Theta_{m}^{\pm}$ are defined in Lemma 3.8.

Theorem 3.1 immediately implies the existence of non-radial stationary vortex
patches with infinite energy.

###### Corollary 3.2

Let $2\leq m\in\mathbb{N}$. Then there is an $m$-fold symmetric stationary
patch solution for the 2D Euler equation with analytic boundary and infinite
kinetic energy, that is

$\int_{\mathbb{R}^{2}}|\nabla\left(\omega*\frac{1}{2\pi}\log|\cdot|\right)|^{2}dx=\infty.$

###### Proof.

By Theorem 3.1, there are two continuous bifurcation curves
$\Psi^{\pm}:[0,s_{0})\mapsto\mathbb{R}\times\left(X^{k,m}_{c}\right)^{2}$ of
solutions of (3.4) such that

$\displaystyle\Psi^{\pm}(s)=\left(\Theta^{\pm}(s),R(s)\right),\quad
F(\Psi^{\pm}(s))=0,\quad\text{ and
}\quad\Psi^{\pm}(0)=\left(\Theta_{m}^{\pm},0\right),$

for some $s_{0}>0$. From (2.1) and (2.3), it is clear that for each
$s\in(0,s_{0})$ and for each choice of $\pm$,

$\displaystyle\omega^{s}(x):=\begin{cases}\Theta-1&\text{ if }x\in
D_{2}(s),\\\ \Theta&\text{ if }x\in D_{1}(s)\backslash\overline{D_{2}(s)}\\\
0&\text{ otherwise, }\end{cases}$ (3.5)

is a stationary solution to the Euler equation, where $D_{1}(s)$ and
$D_{2}(s)$ are the bounded domains determined by (2.2) with $R(s)$. Since
$R(s)\in\left(X^{k,m}_{c}\right)^{2}$, the boundaries are analytic.

Now we consider the kinetic energy of the solution. From (3.1), it suffices to
show that $\int_{\mathbb{R}^{2}}\omega^{0}(x)dx\neq 0$. By the continuity of
the bifurcation curve, this immediately implies that
$\int_{\mathbb{R}^{2}}\omega^{s}(x)dx\neq 0$ for small $s>0$. Then it follows
from Lemma 3.11 that $b^{2}<\Theta_{m}^{\pm}$, hence

$\displaystyle\int_{\mathbb{R}^{2}}\omega^{0}(x)dx=\pi\left(-b^{2}+\Theta_{m}^{\pm}\right)>0.$

This completes the proof. ∎

The rest of this section will be devoted to prove Theorem 3.1. The proof will
be divided into 5 steps. These steps correspond to check the hypotheses of the
Crandall-Rabinowitz theorem A.1 for our functional $F$ in (3.4). The
hypotheses in Theorem A.1 can be read as follows in our setting:

  1. 1.

The functional $\mathcal{F}$ satisfies

$\mathcal{F}(\Theta,R)\,:\,(0,1)\times\\{V^{\epsilon}\\}\mapsto(Y^{k-1,m}_{c})^{2},$

where $V^{\epsilon}$ is an open neighborhood of 0,

$\displaystyle
V^{\epsilon}=\left\\{(f,g)\in(X^{k,m}_{c})^{2}\,:\,||f||_{X^{k,m}_{c}}+||g||_{X^{k,m}_{c}}<\epsilon\right\\}$

for some $\epsilon>0$ and $k\geq 3$.

  2. 2.

$\mathcal{F}(\Theta,0)=0$ for every $0<\Theta<1$.

  3. 3.

The partial derivatives $\partial_{\Theta}\mathcal{F}$, $D\mathcal{F}$,
$\partial_{\Theta}D\mathcal{F}$ exist and are continuous, where $D\mathcal{F}$
is Gateaux derivative of $\mathcal{F}$ with respect to the functional variable
$R$.

  4. 4.

Ker$(D\mathcal{F}(\Theta_{m}^{\pm},0))\subset(X^{k,m}_{c})^{2}$ and
$(Y^{k-1,m}_{c})^{2}$/Im($D\mathcal{F}(\Theta_{m}^{\pm},0)$) are one-
dimensional (see Proposition 3.8 for the definition of $\Theta_{m}^{\pm}$).

  5. 5.

$\partial_{\Theta}D\mathcal{F}(\Theta_{m}^{\pm},0)[v_{0}]\not\in$
Im($D\mathcal{F}(\Theta_{m}^{\pm},0)$), where $v_{0}$ is a non-zero element in
Ker$(D\mathcal{F}(\Theta_{m}^{\pm},0))$.

###### Remark 3.3

We remark that if $i\neq j$ then the functions inside the logarithm in
$S_{i,j}$ in (2) are uniformly bounded from below in $y$ for all $x$ by a
strictly positive constant depending on the parameters. Then we can
analytically extend the integrand in $x$ to the strip $|\Im(z)|\leq c$ in such
a way that the real part of this extension stays uniformly bounded away from 0
for a small enough $c$. The case $i=j$ can be treated similarily as in [17,
Remark 2.1].

###  3.2 Proof of Theorem 3.1

####  3.2.1 Steps 1,2 and 3: Regularity

In order to check the first three steps, it suffices to check if $S_{ij}$ in
(2) satisfies the hypotheses, since $F$ is a linear combination of $S_{ij}$.
As mentioned in Remark 3.3, the case $i\neq j$ is trivial since there is no
singularity in the integrand and analytically extended into a strip in
$\mathbb{C}$, if $(R_{1},R_{2})$ is in a sufficiently small neighborhood of
$(0,0)$. For $i=j$, the first three steps with slightly different settings
were already done in the literature. For example, step 1 can be done in the
same way as in [35]. Step 2 follows immediately from (2.5). Existence and
continuity of the Gateaux derivatives for the gSQG equation was done in [48]
and the same proof can be adapted to our setting straightforwardly.

####  3.2.2 Step 4: Analysis of the linear part.

In this section, we will focus on the spectral study of the Gateaux derivative
$D\mathcal{F}(\Theta,0):=D_{R}\mathcal{F}(\Theta,0)$.

##### Calculation of $D\mathcal{F}$

We aim to express the Gateaux derivative of $\mathcal{F}$ around $(\Theta,0)$
in the direction $(H(x),h(x))$ in terms of Fourier series.

###### Lemma 3.4

Let $S_{ij}$ be defined as in (2). Then:

$\displaystyle\left.\frac{d}{dt}S_{ij}(b_{i},b_{j},th_{i},th_{j})\right|_{t=0}$
$\displaystyle=\int
b_{i}(h_{j}^{\prime}(x)\cos(x-y)-h_{i}^{\prime}(y))\log(b_{j}^{2}+b_{i}^{2}-2b_{j}b_{i}\cos(x-y))dy=:\mathcal{L}_{1}+\mathcal{L}_{2}.$

###### Proof.

Let $V^{1,ab}_{ij}$ (resp. $V^{2,ab}_{ij}$) be the contribution of the first
term (resp. second term) of (2) where the first factor contributes with a
$t^{a}$ and the second with $t^{b}$. We are looking for all combinations such
that $a+b=1$. We start looking at the first summand. We have that:

$\displaystyle V_{ij}^{1,10}$
$\displaystyle=\int\cos(x-y)(b_{i}h_{j}^{\prime}(x)-b_{j}h_{i}^{\prime}(y))\log(b_{j}^{2}+b_{i}^{2}-2b_{j}b_{i}\cos(x-y))dy.$

Similarly, for the second one,

$\displaystyle V_{ij}^{2,01}$
$\displaystyle=-2\int\sin(x-y)(b_{i}b_{j})\frac{h_{j}(x)(b_{j}-b_{i}\cos(x-y))+h_{i}(y)(b_{i}-b_{j}\cos(x-y))}{b_{j}^{2}+b_{i}^{2}-2b_{j}b_{i}\cos(x-y)}dy,$
$\displaystyle V_{ij}^{2,10}$
$\displaystyle=-\int\sin(x-y)(b_{i}h_{j}(x)+b_{j}h_{i}(y))\log(b_{j}^{2}+b_{i}^{2}-2b_{j}b_{i}\cos(x-y))dy,$

where we have used Lemma A.2. Integrating by parts in $V_{ij}^{2,01}$:

$\displaystyle V_{ij}^{2,01}$
$\displaystyle=\int((b_{i}h_{j}(x)+b_{j}h_{i}(y))\sin(x-y)-h_{i}^{\prime}(y)(b_{i}-b_{j}\cos(x-y)))\log(b_{j}^{2}+b_{i}^{2}-2b_{j}b_{i}\cos(x-y))dy,$

Finally, adding all the log terms and the non-log terms together:

$\displaystyle V_{ij}^{1,10}+V_{ij}^{2,01}+V_{ij}^{2,10}$ $\displaystyle=\int
b_{i}(h_{j}^{\prime}(x)\cos(x-y)-h_{i}^{\prime}(y))\log(b_{j}^{2}+b_{i}^{2}-2b_{j}b_{i}\cos(x-y))dy,$

as we wanted to prove. ∎

###### Lemma 3.5

Let $h_{i}=A_{i}\cos(mx)$ and
$r=\frac{\min\\{b_{i},b_{j}\\}}{\max\\{b_{i},b_{j}\\}}$. Then:

$\displaystyle\left.\frac{d}{dt}S_{ij}(b_{i},b_{j},th_{i},th_{j})\right|_{t=0}=2\pi\sin(mx)mb_{i}\left(A_{j}r-A_{i}\frac{r^{m}}{m}\right).$

###### Proof.

From Lemma 3.4 and Corollary A.4, we have that

$\displaystyle\mathcal{L}_{1}$
$\displaystyle=-mb_{i}A_{j}\sin(mx)\mathcal{A}_{0}(r,1),$
$\displaystyle\mathcal{L}_{2}$
$\displaystyle=mb_{i}A_{i}\sin(mx)\mathcal{A}_{0}(r,m),$

Adding the two contributions gives the desired result. ∎

Note that the functional $\mathcal{F}$ is a linear combination of $S_{ij}$
(see (2.3)). Using the above two lemmas, we obtain the following proposition:

###### Proposition 3.6

Let $h(x)=\sum_{n}a_{n}\cos(nx),$ $H(x)=\sum_{n}A_{n}\cos(nx)$, then we have
that:

$\displaystyle D\mathcal{F}(\Theta,0)[H,h]=\left(\begin{array}[]{c}U(x)\\\
u(x)\end{array}\right),$

where

$\displaystyle U(x)=\sum_{n}U_{n}\sin(nx),\quad u(x)=\sum_{n}u_{n}\sin(nx),$

and the coefficients satisfy, for any $n$:

$\displaystyle\left(\begin{array}[]{c}U_{n}\\\
u_{n}\end{array}\right):=(-n)M_{n}(\Theta)\left(\begin{array}[]{c}A_{n}\\\
a_{n}\end{array}\right):=(-n)\left(\begin{array}[]{cc}\frac{b^{2}}{2}-\frac{\Theta}{2}+\frac{\Theta}{2n}&-\frac{b^{n+1}}{2n}\\\
\Theta\frac{b^{n}}{2n}&-\frac{b}{2n}+\frac{b}{2}(1-\Theta)\end{array}\right)\left(\begin{array}[]{c}A_{n}\\\
a_{n}\end{array}\right).$

###### Proof.

It follows from (3.2), the definition of $\mathcal{F}$ in (3.4), (2.3) and
Lemma 3.5. ∎

##### One dimensionality of the Kernel of the linear operator.

Our goal here is to verify the one dimensionality of
Ker($D\mathcal{F}(\Theta,0)$) for some $\Theta$. More precisely, we will prove
the following proposition:

###### Proposition 3.7

Fix $2\leq m\in\mathbb{N}$ and take any $b\in(0,b_{m})$, where $b_{m}$ is as
in Lemma 3.8. Then there exist two $\Theta_{m}^{\pm}\in(0,1)$, such that
Ker$(D\mathcal{F}(\Theta_{m}^{\pm},0))$ is one-dimensional. Furthermore,

$\displaystyle\text{Ker}(D\mathcal{F}(\Theta_{m}^{\pm},0,0))=\text{span}\left\\{v_{0}(\Theta_{m}^{\pm})\cos(mx):=\begin{pmatrix}\frac{1}{2m}b-\frac{b}{2}(1-\Theta_{m}^{\pm})\\\
\frac{\Theta_{m}^{\pm}}{2m}b^{m}\end{pmatrix}\cos(mx)\right\\}\subset
X^{k,m}_{c}\times X^{k,m}_{c}.$

The proof of the above proposition relies on the analysis of the matrix
$M_{n}$ in Lemma 3.8 and 3.9, which we will prove below.

###### Lemma 3.8

Let $\Delta_{m}(\Theta)$ be

$\displaystyle\Delta_{m}(\Theta):=\frac{4m^{2}}{b}\text{det}(M_{m}(\Theta))=\left(\Theta
b^{2m}+b^{2}m(m(1-\Theta)-1)+\Theta(1-m)(m(1-\Theta)-1)\right)$ (3.6)

Then, for any $m\geq 2$, there exists $0<b_{m}<1$ such that for any
$0<b<b_{m}<1$, there exists $0<\Theta^{-}_{m}<\Theta^{+}_{m}<1$ such that
$\Delta_{m}(\Theta_{m}^{\pm})=0$. We also have that
rk$(M_{m}(\Theta_{m}^{\pm}))=1$ for those values of $\Theta^{+}_{m}$,
$\Theta^{-}_{m}$, where $rk(A)$ is the rank of a matrix $A$.

###### Proof.

For fixed $m$, we study the polynomial $\Delta_{m}(\Theta)$. We need to solve

$\displaystyle\Delta_{m}(\Theta)=m(m-1)\Theta^{2}-((m-1)^{2}+b^{2}m^{2}-b^{2m})\Theta+b^{2}m(m-1)=0.$
(3.7)

Since $\Delta_{m}(\Theta)$ is a quadratic function, we only need to show that
the discriminant is positive. We have

$\displaystyle D_{m}:=((m-1)^{2}+b^{2}m^{2}-b^{2m})^{2}-4m^{2}(m-1)^{2}b^{2}$
$\displaystyle=((m-1)^{2}+b^{2}m^{2}-b^{2m}-2m(m-1)b)((m-1)^{2}+b^{2}m^{2}-b^{2m}+2m(m-1)b)$
$\displaystyle=(m-1-bm-b^{m})(m-1-bm+b^{m})((m-1)^{2}+b^{2}m^{2}-b^{2m}+2m(m-1)b)$
$\displaystyle=:D_{m,1}\cdot D_{m,2}\cdot D_{m,3}.$

Since $m\geq 2$, $0<b<1$, we have $D_{m,3}>1-1=0$. We also have

$D_{m,2}=m(1-b)+b^{m}-1=(1-b)[m-\frac{1-b^{m}}{1-b}]=(1-b)(m-(1+b+b^{2}+...+b^{m-1}))>0.$

$D_{m,1}$ is decreasing in $b$ when $b\geq 0$ and $D_{m,1}(0)=m-1>0$,
$D_{m,1}(1)=-2$. Let $b_{m}$ be the only zero point of $D_{m,1}$ in (0,1). If
we take $0<b<b_{m}$, we have

$\displaystyle D_{1,m}>0,D_{m}>0.$ (3.8)

Hence $\Delta_{m}(\Theta)$ has two different solutions
$\Theta_{m}^{-}<\Theta_{m}^{+}$. Moreover, the matrix does not vanish when
$b\neq 0$, implying rk$(M_{m}(\Theta_{m}^{\pm}))=1$. Now we are left to show
$0<\Theta_{m}^{\pm}<1$. We have

$\Theta_{m}^{+}\Theta_{m}^{+}=b^{2}<1,$

and

$\Theta_{m}^{+}+\Theta_{m}^{-}=\frac{((m-1)^{2}+b^{2}m^{2}-b^{2m})}{m(m-1)}\geq\frac{1-1}{m(m-1)}>0.$

Hence, $0<\Theta_{m}^{-}<1$ and $0<\Theta_{m}^{+}$. If $\Theta_{m}^{+}\geq 1$,
then $\Delta_{m}(1)\leq 0$. However,

$\displaystyle\Delta_{m}(1)=b^{2m}-mb^{2}-(1-m)$ (3.9)
$\displaystyle=(1-b^{2})(m-\frac{1-b^{2m}}{1-b^{2}})$
$\displaystyle=(1-b^{2})(m-1-b^{2}-...-b^{2m-2})>0.$

Therefore $0<\Theta_{m}^{\pm}<1$. ∎

We now show that $\Delta_{jm}(\Theta_{m}^{\pm})\neq 0$ for any $j\neq 1$.

###### Lemma 3.9

Let $j>1$ and let $\Theta_{m}^{\pm}$ and $\Theta_{jm}^{\pm}$ be defined as in
the previous Lemma. Then
$\Theta_{jm}^{+}>\Theta_{m}^{+}>\Theta_{m}^{-}>\Theta_{jm}^{-}$. Hence,
$M_{jm}(\Theta_{m}^{\pm})$ is non-singular for all $j>1$.

###### Proof.

Since $\Theta_{m}^{\pm}$ solves the equation

$\Theta^{2}-\frac{((m-1)^{2}+b^{2}m^{2}-b^{2m})}{m(m-1)}\Theta+b^{2}=0.$

We only need to show $F(b,m):=\frac{((m-1)^{2}+b^{2}m^{2}-b^{2m})}{m(m-1)}$ is
strictly increasing with respect to $m$. We have

$\displaystyle F(b,m+1)-F(b,m)$
$\displaystyle=\frac{(m^{2}+b^{2}(m+1)^{2}-b^{2(m+1)})}{(m+1)m}-\frac{((m-1)^{2}+b^{2}m^{2}-b^{2m})}{m(m-1)}$
$\displaystyle=\frac{(m^{2}+b^{2}(m+1)^{2}-b^{2(m+1)})(m-1)-((m-1)^{2}+b^{2}m^{2}-b^{2m})(m+1)}{(m+1)m(m-1)}$
$\displaystyle=\frac{b^{2}((m+1)^{2}(m-1)-m^{2}(m+1))+m^{2}(m-1)-(m+1)(m-1)^{2}-b^{2m+2}(m-1)+(m+1)b^{2m}}{(m+1)m(m-1)}$
$\displaystyle=\frac{-b^{2}(m+1)+m-1-b^{2m+2}(m-1)+b^{2m}(m+1)}{(m+1)m(m-1)}.$

Thus

$\displaystyle
F(b,m+1)-F(b,m)>0\Leftrightarrow-b^{2}(m+1)+m-1-b^{2m+2}(m-1)+b^{2m}(m+1)>0$
$\displaystyle\Leftrightarrow(1+b^{2})(-1+b^{2m})-(-1+b^{2})(1+b^{2m})m>0\Leftrightarrow-(1+b^{2})(b^{2m-2}+b^{2m-4}...+b^{2}+1)+(1+b^{2m})m>0$
$\displaystyle\Leftrightarrow-\sum_{k=1}^{m}b^{2m-2k}-\sum_{k=1}^{m}b^{2k}+(1+b^{2m})m>0\Leftrightarrow\sum_{k=1}^{m}(1-b^{2k})(1-b^{2m-k})>0.$

It is easy to show the last inequality since $0<b<1$. ∎

  * Proof of Proposition 3.7: Let $m\geq 2$ and let $b$, $\Theta_{m}^{\pm}$ be as defined in Lemma 3.8. Assume that $H(x)=\sum_{j}A_{jm}\cos(jmx)$ and $h(x)=\sum_{j}a_{jm}\cos(jmx)$ satisfy

$DF(\Theta_{m}^{\pm},0)[H,h]=(0,0).$

Then it follows from Proposition 3.6 that

$\displaystyle M_{jm}(\Theta_{m}^{\pm})\begin{pmatrix}A_{jm}\\\
a_{jm}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$

For all $j>1$, it follows from Lemma 3.9 that $M_{jm}(\Theta_{m}^{\pm})$ is
invertible, thus $A_{jm}=a_{jm}=0$. For $j=1$, Lemma 3.8 tells us that
$(A_{m},a_{m})\in\text{Ker}(M_{m}(\Theta_{m}^{\pm}))=\text{span}\left\\{v_{0}(\Theta_{m}^{\pm}):=\begin{pmatrix}\frac{1}{2m}b-\frac{b}{2}(1-\Theta_{m}^{\pm})\\\
\frac{\Theta_{m}^{\pm}}{2m}b^{m}\end{pmatrix}\right\\}$. This finishes the
proof. $\Box$

##### Codimension of the image of the linear operator.

We now characterize the image of $D\mathcal{F}(\Theta_{m}^{\pm},0)$. We have
the following proposition:

###### Proposition 3.10

Let

$\displaystyle Z=\left\\{(Q,q)\in Y^{k-1,m}_{c}\times
Y^{k-1,m}_{c},Q(x)=\sum_{j=1}^{\infty}Q_{jm}\sin(jmx),q(x)=\sum_{j=1}^{\infty}q_{jm}\sin(jmx),\right.$
$\displaystyle\left.\exists\lambda_{Q,q}\in\mathbb{R}\text{
s.t.}\left(\begin{array}[]{c}Q_{m}\\\
q_{m}\end{array}\right)=\lambda_{Q,q}\left(\begin{array}[]{c}-\frac{1}{2m}b^{m+1}\\\
-\frac{1}{2m}b+\frac{b}{2}(1-\Theta_{m}^{\pm})\end{array}\right)\right\\}.$

Then $Z=\text{Im}\left(D\mathcal{F}(\Theta_{m}^{\pm},0,0)\right)$.

###### Proof.

In view of Proposition 3.6,
$\text{Im}\left(D\mathcal{F}(\Theta_{m}^{\pm},0)\right)\subset Z$ is trivial,
since $M_{jm}(\Theta_{m}^{\pm})$ is non-singular for $j>1$, and
$\text{Im}(M_{m}(\Theta_{m}^{\pm}))=\text{span}\left\\{\begin{pmatrix}-\frac{1}{2m}b^{m+1}\\\
-\frac{1}{2m}b+\frac{b}{2}(1-\Theta)\end{pmatrix}\right\\}.$ In order to prove
$\text{Im}\left(D\mathcal{F}(\Theta_{m}^{\pm},0)\right)\supset Z$, we need to
check whether the possible preimage satisfies the desired regularity. To do
so, we have the following asymptotic lemma:

###### Lemma 3.11

For fixed $m\geq 2$ and $b$ defined as in Proposition 3.7, we have
$b^{2}-\Theta_{m}^{\pm}<0$ and

$\displaystyle\frac{b}{4j^{2}m^{2}}\Delta_{jm}(\Theta_{m}^{\pm})=\frac{b}{4}\left(b^{2}-\Theta_{m}^{\pm}\right)(1-\Theta_{m}^{\pm})+O\left(\frac{1}{jm}\right),\quad\text{
as }j\to\infty,$ (3.10)

Consequently, we have

$\displaystyle\text{det}{(M_{jm}(\Theta_{m}^{\pm}))^{-1}}\lesssim_{m,\Theta}1,\quad\text{
for sufficiently large }j.$ (3.11)

###### Remark 3.12

As shown in the above lemma, there is no bifurcation curve from the two-
layered vortex patch with zero-average. This is due to the fact that the
radial vorticity $\omega$ determined by $b$ and $\Theta^{\pm}$ as in (3.3)
satisfies $\int_{\mathbb{R}^{2}}\omega
dx=\pi\left(-b^{2}+\Theta_{m}^{\pm}\right)>0$. Note that if we require
$\Theta=b^{2}$ to ensure $\int\omega dx=0$, it follows from (3.6) that
$\Delta_{m}(b^{2})=mb^{2}(1-b^{2})+b^{2}(b^{2m}-1)$, which does not vanish for
any $m\geq 2$ unless $b=0$ or $b=1$. Therefore, for any $0<b<1$, the
linearized operator is an isomorphism and the implicit function theorem shows
that there cannot be a bifurcation.

  * Proof of Lemma 3.11: First we show that $b^{2}<\Theta_{m}^{\pm}$. By Lemma 3.8, $\Theta_{m}^{+}\Theta_{m}^{-}=b^{2}$ and $0<\Theta_{m}^{\pm}<1$. Thus $\Theta_{m}^{\pm}>b^{2}$. Therefore the first assertion is proved. The second assertion follows directly from (3.6) since

$\displaystyle\frac{b}{4(jm)^{2}}\Delta_{jm}(\Theta_{m}^{\pm})=\frac{b}{4(jm)^{2}}\left((b^{2}-\Theta_{m}^{\pm})(1-\Theta_{m}^{\pm})(jm)^{2}+O\left(m\right)\right)=\frac{b}{4}\left(b^{2}-\Theta_{m}^{\pm}\right)(1-\Theta_{m}^{\pm})+O\left(\frac{1}{jm}\right).$

Lastly, by choosing $j$ large so that
$|\frac{b}{4(jm)^{2}}\Delta_{jm}(\Theta_{m}^{\pm})|>\frac{b}{8}(\Theta_{m}^{\pm}-b^{2})(1-\Theta_{m}^{\pm})>0$,
we have

$\displaystyle|\text{det}(M_{jm}(\Theta_{m}^{\pm})^{-1})|=\frac{1}{|\frac{b}{4(jm)^{2}}\Delta_{jm}(\Theta_{m}^{\pm})|}\lesssim
1,$

which proves (3.11). $\Box$

Now for an element $(Q,q)\in Z$, let $(H,h)$ be such that

$\displaystyle H(x)=\sum_{j=1}^{\infty}A_{jm}\cos(jmx),\quad
h(x)=\sum_{j=1}^{\infty}a_{jm}\cos(jmx),$

with

$\displaystyle\begin{pmatrix}A_{m}\\\
a_{m}\end{pmatrix}=-\frac{1}{m}\begin{pmatrix}0\\\
\lambda_{Q,q}\end{pmatrix},\quad\begin{pmatrix}A_{jm}\\\
a_{jm}\end{pmatrix}=(-jm)^{-1}M_{jm}(\Theta_{m}^{\pm})^{-1}\begin{pmatrix}Q_{jm}\\\
q_{jm}\end{pmatrix}\quad\text{ for $j>1$}.$

It is clear from (3.6) that $D\mathcal{F}(\Theta_{m}^{\pm},0,0)(H,h)=(Q,q).$
We will prove that $(H,h)\in X^{k,m}_{c}\times X^{k,m}_{c}$. From Lemma 3.11
and the fact that $M_{jm}(\Theta_{m}^{\pm})$ is nonsingular for $j>1$, it
follows that

$\displaystyle|A_{jm}|^{2}+|a_{jm}|^{2}\lesssim(jm)^{-2}\left(|Q_{jm}|^{2}+|q_{jm}|^{2}\right)\quad\text{
for all }j>1.$ (3.12)

Thus, we obtain

$\displaystyle\rVert H\rVert_{X^{k,m}_{c}}^{2}+\rVert
h\rVert_{X^{k,m}_{c}}^{2}$
$\displaystyle=\sum_{j=1}^{\infty}\left(|A_{jm}|^{2}+|a_{jm}|^{2}\right)(1+jm)^{2k}(\cosh(cjm)^{2}+\sinh(cjm)^{2})$
$\displaystyle\lesssim\frac{1}{m^{2}}\lambda_{Q,q}^{2}(1+m)^{2k}(\cosh(cm)^{2}+\sinh(cm)^{2})$
$\displaystyle+\sum_{j=2}^{\infty}(jm)^{-2}\left(|Q_{jm}|^{2}+|q_{jm}|^{2}\right)(1+jm)^{2k}(\cosh(cjm)^{2}+\sinh(cjm)^{2})$
$\displaystyle\lesssim 1+(\rVert Q\rVert_{Y^{k-1,m}_{c}}^{2}+\rVert
q\rVert_{Y^{k-1,m}_{c}}^{2})$ $\displaystyle<\infty.$

This proves that $(H,h)\in X^{k,m}_{c}\times X^{k,m}_{c}$, and therefore
$Z\subset\text{Im}\left(D\mathcal{F}(\Theta_{m}^{\pm},0)\right)$. ∎

####  3.2.3 Step 5: Transversality

###### Proposition 3.13

We have that

$\displaystyle\partial_{\Theta}D\mathcal{F}(\Theta_{m}^{\pm},0)[v_{0}]\not\in\text{Im}(D\mathcal{F}(\Theta_{m}^{\pm},0)),$
(3.13)

where $v_{0}=v_{0}(\Theta_{m}^{\pm})$ is as given in Proposition 3.7.

###### Proof.

For $h(x)=\sum_{j}a_{jm}\cos(jmx),$ $H(x)=\sum_{j}A_{jm}\cos(jmx)$, we have
that (see Proposition 3.6):

$\displaystyle\partial_{\Theta}D\mathcal{F}(\Theta_{m}^{\pm},0)[H,h]=\left(\begin{array}[]{c}U(x)\\\
u(x)\end{array}\right),$

where

$\displaystyle U(x)=\sum_{j}U_{jm}\sin(jmx),\quad
u(x)=\sum_{j}u_{jm}\sin(jmx),$

and the coefficients satisfy, for any $j$:

$\displaystyle\left(\begin{array}[]{c}U_{jm}\\\
u_{jm}\end{array}\right):=(-jm)\partial_{\Theta}M_{jm}(\Theta_{m}^{\pm})\left(\begin{array}[]{c}A_{jm}\\\
a_{jm}\end{array}\right):=(-jm)\left(\begin{array}[]{cc}-\frac{1}{2}+\frac{1}{2m}&0\\\
\frac{b^{m}}{2m}&-\frac{b}{2}\end{array}\right)\left(\begin{array}[]{c}A_{jm}\\\
a_{jm}\end{array}\right).$

Letting

$\displaystyle
v_{0}(\Theta_{m}^{\pm})=\left(\begin{array}[]{c}\frac{b}{2m}-\frac{b}{2}(1-\Theta_{m}^{\pm})\\\
\frac{\Theta_{m}^{\pm}}{2m}b^{m}\end{array}\right),\quad
w(\Theta_{m}^{\pm})=\left(\begin{array}[]{c}-\frac{1}{2m}b^{m+1}\\\
-\frac{1}{2m}b+\frac{b}{2}(1-\Theta_{m}^{\pm})\end{array}\right),$

be the generators of Ker$(M_{m}(\Theta_{m}^{\pm}))$ and
Im$(M_{m}(\Theta_{m}^{\pm}))$ respectively, the transversality condition is
equivalent to prove that $w_{1}(\Theta_{m}^{\pm})$ and $w(\Theta_{m}^{\pm})$
are not parallel, where

$\displaystyle w_{1}(\Theta_{m}^{\pm})$
$\displaystyle=\partial_{\Theta}M_{m}(\Theta_{m}^{\pm})v_{0}(\Theta_{m}^{\pm})=\left(\begin{array}[]{c}\frac{b}{4}\left(\frac{1}{m}-1\right)\left(\frac{1}{m}-(1-\Theta_{m}^{\pm})\right)\\\
\frac{b^{m+1}}{4m}\left(\frac{1}{m}-1\right)\end{array}\right)$

This is equivalent to prove that:

$\displaystyle
0\neq-\frac{b^{2}}{8}\left(\frac{1}{m}-1\right)\left(\frac{1}{m}-(1-\Theta_{m}^{\pm})\right)^{2}+\frac{b^{2m+2}}{8m^{2}}\left(\frac{1}{m}-1\right)\Leftrightarrow\left(\Theta_{m}^{\pm}-(1-\frac{1+b^{m}}{m})\right)\left(\Theta_{m}^{\pm}-(1-\frac{1-b^{m}}{m})\right)\neq
0.$

We prove it by contradiction. If $\Theta_{m}^{\pm}=1-\frac{1+b^{m}}{m}$, we
have $b^{m}=m(1-\Theta_{m}^{\pm})-1$. Moreover, by (3.6),we have

$\Theta_{m}^{\pm}b^{2m}+b^{m+2}m+\Theta_{m}^{\pm}(1-m)b^{m}=0.$

Hence,

$\displaystyle\Theta_{m}^{\pm}b^{m}+b^{2}m+\Theta_{m}^{\pm}(1-m)=0$
$\displaystyle\Rightarrow\Theta_{m}^{\pm}(m(1-\Theta_{m}^{\pm})-1)+b^{2}m+\Theta_{m}^{\pm}(1-m)=0$
$\displaystyle\Rightarrow-m(\Theta_{m}^{\pm})^{2}+b^{2}m=0.$

Since $\Theta_{m}^{\pm}>0,b>0$, we have

$\Theta_{m}^{\pm}=b,$

implying a contradiction since $\Theta_{m}^{+}\Theta_{m}^{-}=b^{2}$ and
$\Theta_{m}^{+}\neq\Theta_{m}^{-}$. If $\Theta_{m}^{\pm}=1-\frac{1-b^{m}}{m}$,
we can follow the same way to get $\Theta_{m}^{\pm}=b$ and get a
contradiction. ∎

  * Proof of Theorem 3.1: All the hypotheses of the Crandall-Rabinowitz theorem were checked in Propositions 3.7, 3.10 and 3.13. Therefore the desired result follows immediately. $\Box$

##  4 Existence of non-radial stationary vortex patches with finite energy

In this section, we aim to prove that there exist non-trivial patch solutions
with finite kinetic energy,
$\frac{1}{2\pi}\int_{\mathbb{R}^{2}}|\nabla\left(\omega*\log|\cdot|\right)|^{2}dx<\infty$.
As mentioned in (3.1), this property is equivalent to $\int_{\mathbb{R}}\omega
dx=0$. By Remark 3.12, we can not use two-layer patches and instead we will
consider three-layer patches.

###  4.1 Main results for finite energy

We consider vortex patches with three layers, that is,
$i\in\left\\{1,2,3\right\\}$ in the setting in Section 2. The total vorticity
that we consider is of the form $\omega=\sum_{i=1}^{3}\Theta_{i}1_{D_{i}}$,
where $D_{i}$ is determined by $\partial D_{i}=\left\\{(b_{i}+R_{i}(x))(\cos
x,\sin x):x\in\mathbb{T}\right\\}$. We will look for a bifurcation curve from
the radial one, $\sum_{i=1}^{3}\Theta_{i}1_{B_{b_{i}}}$, where $B_{r}$ denotes
the disk with radius $r$ centered at the origin. We have the following
parameters and functional variables:

  * •

$b_{i}\in\mathbb{R}$: the radii of the different layers of the annuli. We will
have $1=:b_{1}>b_{2}>b_{3}$.

  * •

$\Theta_{i}\in\mathbb{R}$: the vorticity at the different layers. We will
choose $\Theta_{1}:=1$.

  * •

$R:=(R_{1},R_{2},R_{3})\in(X^{k,m})^{3}$, for some $3\leq k\in\mathbb{N}$ :
the functional variables that determine the boundaries.

In the rest of this section, we will fix $m$, $b_{2}$ and $\Theta_{2}$ so that
for $2\leq m\in\mathbb{N}$,

$\displaystyle 0<b_{2}<\left(\frac{1}{2}\right)^{\frac{1}{2m}}\quad\text{ and
}\quad\frac{m({b_{2}}^{2}-1)}{(1-{b_{2}}^{2m})b_{2}^{2}}<\Theta_{2}<\min\bigg{\\{}2\frac{{b_{2}}^{2m-2}(b_{2}^{2}-1)m}{1-b_{2}^{2m}},\
\frac{-1}{b_{2}^{2}}\bigg{\\}}.$ (4.1)

Given $R$, $\Theta_{i}$, $b_{1}$ and $b_{2}$, we choose $b_{3}$ so that

$\displaystyle\int_{\mathbb{R}^{2}}\omega(x)dx=\sum_{i=1}^{3}\int_{0}^{2\pi}\Theta_{i}(b_{i}+R_{i}(x))^{2}dx=0.$
(4.2)

Since $b_{1}$, $b_{2}$ and $\Theta_{1}$, $\Theta_{2}$ are fixed constants,
(4.2) implies that $b_{3}$ is a function of $\Theta_{3}$ and $R$, more
precisely,

$\displaystyle b_{3}$ $\displaystyle=b_{3}(\Theta_{3},R)$
$\displaystyle=\sqrt{-\frac{1}{2\pi\Theta_{3}}\left(\Theta_{1}\int_{0}^{2\pi}(b_{1}+R_{1}(x))^{2}dx+\Theta_{2}\int_{0}^{2\pi}(b_{2}+R_{2}(x))^{2}dx+\Theta_{3}\int_{0}^{2\pi}R_{3}(x)^{2}dx\right)}$
(4.3)

If $\Theta_{3},b_{3}(\Theta_{3},R)\neq 0$, then its derivative with respect to
$R$ is given by

$\displaystyle Db_{3}(\Theta_{3},R)[h]$
$\displaystyle:=\frac{d}{dt}b_{3}(\Theta_{3},R+th)\bigg{|}_{t=0}$
$\displaystyle=-\frac{1}{2\pi\Theta_{3}b_{3}(\Theta_{3},R)}\left(\Theta_{1}\int_{\mathbb{T}}R_{1}(x)h_{1}(x)dx+\Theta_{2}\int_{\mathbb{T}}R_{2}(x)h_{2}(x)dx+\Theta_{3}\int_{\mathbb{T}}R_{3}(x)h_{3}(x)dx\right),$
(4.4)

where we used $\int_{\mathbb{T}}h_{i}(x)dx=0$ for $h_{i}\in X^{k,m}$.

Note that for sufficiently small $\|R_{i}\|_{L^{\infty}}$ and
$|\Theta_{3}-\Theta^{*}_{3,m}|$, where $\Theta^{*}_{3,m}$ is as defined in
Lemma 4.24, we can choose $b_{3}=b_{3}(\Theta_{3},R_{i})$ so that (4.2) is
compatible with $b_{2}>b_{3}>0$ (see Lemma 4.24). Therefore, a 4-tuple
$(\Theta_{3},R_{1},R_{2},R_{3})=:(\Theta_{3},R)$ uniquely determines
$\omega=\sum_{i=1}^{3}\Theta_{i}1_{D_{i}}$ such that the boundary of the $i$th
patch surrounds the $j$th patch if $i<j$ and $\int_{\mathbb{R}}\omega dx=0$.
In the proof, $\Theta_{3}$ will play the role of the bifurcation parameter and
we will look for a bifurcation from
$(\Theta^{*}_{3,m},0)\in\mathbb{R}\times(X^{k,m})^{3}$.

With this setting, the system (2.3) is equivalent to

$\displaystyle
0=G(\Theta_{3},R):=(G_{1}(\Theta_{3},R),G_{2}(\Theta_{3},R),G_{3}(\Theta_{3},R)),$
(4.5)

where

$\displaystyle
G_{j}(\Theta_{3},R):=\mathcal{F}_{j}(b(\Theta_{3},R),\Theta_{3},R)),\text{ and
}b(\Theta_{3},R):=(b_{1},b_{2},b_{3}(\Theta_{3},R)),\quad j=1,2,3.$ (4.6)

Now, we are ready to state the main theorem of this section:

###### Theorem 4.1

Let $k\geq 3$ and $2\leq m\in\mathbb{N}$, $\Theta_{1}=b_{1}=1$ and $b_{2}$ and
$\Theta_{2}$ as in (4.1). Then for some $s_{0}=s_{0}(m,k,b_{2},\Theta_{2})>0$,
there exists a bifurcation curve $[0,s_{0})\ni
s\mapsto(\Theta_{3}(s),R(s))\in\mathbb{R}\times(X^{k,m})^{3}$ such that for
each $s\in(0,s_{0})$, $(\Theta_{3}(s),R(s))$ is a solution of the equation
(4.5) and $(R(s))\neq 0\in(X^{k,m})^{3}$. The bifurcation curve emanates from
$(\Theta_{3}(0),R(0))=(\Theta^{*}_{3,m},0)$, where $\Theta^{*}_{3,m}$ is
defined in Lemma 4.24.

Theorem 4.1 immediately implies the existence of non-radial stationary vortex
patches with finite kinetic energy.

###### Corollary 4.2

Let $2\leq m\in\mathbb{N}$ and $k\geq 3$. Then there is an $m$-fold symmetric
stationary patch solution of the 2D Euler equation with $H^{k}$-regular
boundary and finite kinetic energy, that is

$\int_{\mathbb{R}^{2}}\left|\nabla\left(\omega*\frac{1}{2\pi}\log|\cdot|\right)\right|^{2}dx<\infty.$

###### Proof.

By the definition of $b_{3}$ in (4.2), each
$\omega=\sum_{i=1}^{3}\Theta_{i}1_{D_{i}}$ which is determined by
$(\Theta_{3}(s),R(s))$ for $s\in(0,s_{0})$, satisfies $\int_{\mathbb{R}}\omega
dx=0$. This is equivalent to
$\int_{\mathbb{R}^{2}}|\nabla\left(\omega*\frac{1}{2\pi}\log|\cdot|\right)|^{2}dx<\infty$,
(see (3.1)). ∎

The existence of the bifurcation curves will be proved by means of a Nash-
Moser iteration scheme (Theorem 4.7). The proof of Theorem 4.1 will be
accomplished in Subsection 4.5 by checking the hypotheses of Theorem 4.7.

###  4.2 Compactly supported velocity

In this subsection, we digress briefly to observe an interesting consequence
of Theorem 4.1. Thanks to a simple maximum principle lemma, it can be shown
that each stationary solution on the bifurcation curves has compactly
supported velocity.

###### Lemma 4.3

(the key Lemma) Assume that $\omega\in L^{1}\cap L^{\infty}(\mathbb{R}^{n})$
for $n\geq 2$ is compactly supported and let $\Omega$ be the unbounded
connected component of $\text{supp}(\omega)^{c}$. We additionally assume that
$\int_{\mathbb{R}^{n}}\omega dx=0$. Then for $f:=\omega*\mathcal{N}$, where

$\displaystyle\mathcal{N}(x)=\begin{cases}\frac{1}{2\pi}\log|x|&\text{ if
}n=2\\\ \frac{1}{n(2-n)V_{n}}|x|^{2-n},\quad
V_{n}:=\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}&\text{ if
}n>2,\end{cases}$

it holds that

$\displaystyle\sup_{x\in\Omega}f(x)=\max_{x\in\partial\Omega}f(x)\quad\text{and}\quad\inf_{x\in\Omega}f(x)=\min_{x\in\partial\Omega}f(x).$

Consequently, if $f$ is constant on $\partial\Omega$, then $f$ is constant in
$\Omega$.

###### Remark 4.4

The above lemma does not hold without the assumption
$\int_{\mathbb{R}^{n}}\omega dx=0$. For example, $f:=1_{B_{1}}*\mathcal{N}$ is
harmonic in $\Omega:=B_{1}^{c}$, while $\max_{\Omega}f$ is unbounded in
$\Omega^{c}$.

###### Proof.

It suffices to prove the maximum part since we can apply the argument to $-f$
for the minimum part. The proof is classical but we present a proof for the
sake of completeness.

Let $M:=\sup_{x\in\Omega}f(x)$. Then $A:=\left\\{x\in\Omega:f(x)=M\right\\}$
is relatively closed in $\Omega$, since $f$ is continuous. Furthermore, since
$f$ is harmonic in $\Omega$, the mean value property yields that $A$ is open.
Therefore $A$ must be either $\emptyset$ or $\Omega$ since $\Omega$ is
connected. If $A=\Omega$, then the result follows trivially. Now, let us
suppose that $A=\emptyset$. Towards a contradiction, assume that
$M>\max_{x\in\partial\Omega}f(x)$. Since $f$ is bounded in $\mathbb{R}^{n}$
(this property still holds when $n=2$, thanks to $\int_{\mathbb{R}^{2}}\omega
dx=0$), therefore the only possible case is that

$\displaystyle\lim_{|x|\to\infty}f(x)=M.$ (4.7)

Let us consider $\phi(r):=\frac{1}{|\partial
B_{r}|}\int_{|x|=r}f(x)d\sigma(x)$. For any sufficiently large $r$ such that
$\Omega^{c}\subset B_{r}$, where $B_{r}$ is the ball centered at the origin
with radius r, it follows that

$\displaystyle\frac{d}{dr}\phi(r)$
$\displaystyle=\frac{d}{dr}\frac{1}{|S_{n-1}|}\int_{|x|=1}f(rx)d\sigma(x)=\frac{1}{|\partial
B_{r}|}\int_{|x|<r}\omega(x)dx=0,$

Furthermore, (4.7) yields that

$\displaystyle\lim_{r\to\infty}\phi(r)=M,$

hence, we have $\phi(r)=M$ for all sufficiently large $r$. For such $r>0$, we
have

$\displaystyle M=\frac{1}{|\partial
B_{r}|}\int_{|x|=r}f(x)d\sigma\leq\max_{x\in\partial B_{r}}f(x)\leq M.$

This implies $A$ cannot be empty, which is a contradiction. Hence
$M=\max_{x\in\partial\Omega}f(x)$. ∎

###### Corollary 4.5

Let $2\leq m\in\mathbb{N}$ and $k\geq 3$. There exists an m-fold symmetric
stationary patch solution of the 2D Euler equation with $H^{k}$-regular
boundary and compactly supported velocity.

###### Proof.

Let $\omega=\sum_{i=1}^{3}\Theta_{i}1_{D_{i}}$ be the vorticity determined by
$(\Theta_{3}(s),R(s))$ for $s\in(0,s_{0})$ in Theorem 4.1. From (2.1) and
(2.3), it follows that its stream function
$f:=\frac{1}{2\pi}\omega*\log|\cdot|$ is constant on the outmost boundary
$\partial D_{1}$. However, Lemma 4.3 implies that
$\sup_{D_{1}^{c}}f=\inf_{D_{1}^{c}}f$, therefore, $f$ is constant in
$D_{1}^{c}$. This proves that $\text{supp}(\nabla^{\perp}f)\subset D_{1}$. ∎

###### Remark 4.6

In this paper we focus on patch type solutions
$\omega=\sum_{i=1}^{n}\Theta_{i}1_{D_{i}}$. Lemma 4.3 is still applicable for
smooth $\omega$ as well as long as the boundary of
$\Omega:=\text{supp}(\omega)$ can be approximated by regular level curves of
$\omega$ (See Figure 3). This is due to the fact that the stream function of
the smooth stationary $\omega$ must be constant on each regular level set of
$\omega$ (see [51, Section 1]), hence the stream function has to be constant
on each connected component of $\partial\Omega$. If $\partial\Omega$ has only
one connected component, then the velocity vanishes in the unbounded component
of $\Omega^{c}$.

Figure 3: Illustration for smooth stationary $\omega$ where
$\Omega:=\text{supp}(\omega)$ is colored in blue, whose boundary is not
necessarily connected. The dashed lines are regular level sets of $\omega$,
which converges to the outermost boundary. In such case, the stream function
is constant on the outermost boundary $\partial\Omega^{out}$, thus Lemma 4.3
implies that the velocity vanishes in the unbounded part of $\Omega^{c}$.

###  4.3 Nash-Moser theorem

We first prove a bifurcation theorem using the Nash-Moser scheme under some
assumptions, which will turn out to be satisfied by our nonlinear functional.
We follow the ideas from Berti [7].

Let $2\leq m\in\mathbb{N}$ fixed. We denote

$X^{k}:=X^{k,m},\quad Y^{k}:=Y^{k,m},\quad
C^{\infty}:=C^{\infty}(\mathbb{T}),\quad
R:=(R_{1},R_{2},R_{3})\in\left(X^{k+1}\right)^{3}\cap(C^{\infty})^{3},\quad\partial_{\Theta}:=\partial_{\Theta_{3}},$

for simplicity. Furthermore, for a Banach space $X$ and an element $R\in X$,
we denote the norm of $R$ by $|R|_{X}:=\rVert R\rVert_{X}$. In addition, we
use the notation $A\lesssim_{a,b}B$ if there exists a constant $C=C(a,b)>0$
depending on some variables $a,b$ such that $A\leq CB$. We also use
$c_{0},c_{1},....$ to denote universal constants that may vary from line to
line.

###### Theorem 4.7

Assume that there exists $\Theta_{3}^{*}\in\mathbb{R}$ and an open
neighborhood $I\times V^{3}$ of
$(\Theta^{*}_{3},0)\in\mathbb{R}\times\left(X^{3}\right)^{3}$ such that for
each $2\leq k\in\mathbb{N}$,
$G:I\times\left(X^{k+1}\right)^{3}\mapsto\left(Y^{k}\right)^{3}$ satisfies the
following: For $(\Theta_{3},R)\in
I\times\left(V^{3}\cap(C^{\infty})^{3}\right)$

  * (a)

(Existence of a curve of trivial solutions) $G(\Theta_{3},0)=0$ for all
$\Theta_{3}\in I$.

  * (b)

(Regularity) It holds that

$\displaystyle|G(\Theta_{3},R)|_{(Y^{k})^{3}}\lesssim_{k}1+|R|_{(X^{k+1})^{3}},$
(4.8)
$\displaystyle\left|\partial_{\Theta}G(\Theta_{3},R)\right|_{(Y^{k})^{3}}\lesssim_{k}1+\left|R\right|_{(X^{k+1})^{3}}$
(4.9)
$\displaystyle|D^{2}G(\Theta_{3},R)[h,h]|_{(Y^{k})^{3}}\lesssim_{k}(1+|R|_{(X^{k+3})^{3}})|h|_{(X^{k+1})^{3}}^{2}$
(4.10)
$\displaystyle|\partial_{\Theta}DG(\Theta_{3},R)[h]|_{(Y^{k})^{3}}\lesssim_{k}(1+|R|_{(X^{k+3})^{3}})|h|_{(X^{k+1})^{3}}$
(4.11)
$\displaystyle|\partial_{\Theta}D^{2}G(\Theta^{*}_{3},R)[h,h]|_{(Y^{k})^{3}}\lesssim_{k}(1+|R|_{(X^{k+3})^{3}})|h|_{(X^{k+1})^{3}}^{2}$
(4.12)
$\displaystyle|\partial_{\Theta\Theta}DG(\Theta_{3},R)[h]|_{(Y^{k})^{3}}\lesssim_{k}(1+|R|_{(X^{k+3})^{3}})|h|_{(X^{k+1})^{3}}$
(4.13)

  * (c)

(Decomposition of $DG$)
$DG(\Theta_{3},R)\in\mathcal{L}((X^{k+1})^{3};(Y^{k})^{3})$ has the following
decomposition:

$\displaystyle DG(\Theta_{3},R)[h]=a(\Theta_{3},R)[h]+A(\Theta_{3},R)[h],$

such that $a(\Theta_{3},R)\in\mathcal{L}((X^{k+1})^{3},(Y^{k})^{3})$,
$A(\Theta_{3},R)\in\mathcal{L}((X^{k+1})^{3},Y^{k+1}\times(Y^{k})^{2})$. Also,
there exists $0<\eta<1$ such that if $\rVert R\rVert_{(H^{k+2})^{3}}\leq\eta$,
then

$\displaystyle{|a(\Theta_{3},R)[h]|_{(Y^{k})^{3}}\lesssim_{k}|G(\Theta_{3},R)|_{(Y^{k})^{3}}|h|_{(X^{k+1})^{3}}},$
(4.14)
$\displaystyle|A(\Theta_{3},R)[h]|_{Y^{k+1}\times(Y^{k})^{2}}\lesssim_{k}(1+|R|_{(X^{k+3})^{3}})|h|_{(X^{k+1})^{3}},$
(4.15)

  * ($\tilde{c}$-1)

$A:\mathbb{R}\times(X^{k+3})^{3}\mapsto\mathcal{L}((X^{k+1})^{3},Y^{k+1}\times(Y^{k})^{2})$
is Lipschitz continuous. That is, if

$(\Theta_{3}^{1},R^{1}),\ (\Theta_{3}^{2},R^{2})\in
I\times(V^{3}\cap(C^{\infty})^{3}),$

and $\rVert R^{1}\rVert_{(H^{k+3}(\mathbb{T}))^{2}},\rVert
R^{2}\rVert_{(H^{k+3}(\mathbb{T}))^{2}}\leq 1$, it holds that

$\displaystyle|A(\Theta_{3}^{1},R^{1})[h]-A(\Theta_{3}^{2},R^{2})|_{(Y^{k})^{3}}\lesssim_{k}\left(|\Theta_{3}^{1}-\Theta_{3}^{2}|+|R^{1}-R^{2}|_{(X^{k+3})^{3}}\right)|h|_{(X^{k+1})^{3}}.$
(4.16)

  * ($\tilde{c}$-2)

(Tame estimates) There exists $0<\eta<1$ such that if $\rVert
R\rVert_{(H^{k+3}(\mathbb{T}))^{3}}\leq\eta$, and $A(\Theta_{3},R)[h]=z$ for
some $z\in(C^{\infty})^{3}$ and $h\in\text{Ker}(A(\Theta^{*}_{3},0))^{\perp}$,
then $h\in(C^{\infty})^{3}$. Furthermore, for any even
$\sigma\in\mathbb{N}\cup\left\\{0\right\\}$, it holds that

$\displaystyle\left|h\right|_{(X^{k+1+\sigma})^{3}}\lesssim_{k,\sigma}(1+\left|R\right|_{(X^{k+4+\sigma})^{3}})\left|z\right|_{Y^{k+1}\times(Y^{k})^{2}}+\left|z\right|_{Y^{k+1+\sigma}\times(Y^{k+\sigma})^{2}}.$
(4.17)

  * (d)

(Fredholm index zero) There exist non-zero vectors $v$ and $w$ such that $v$
and $w$ are supported on the $m$-th Fourier mode and

$\displaystyle\text{Ker}(A(\Theta_{3}^{*},0))=\text{span}\left\\{v\right\\},\quad\text{Im}(A(\Theta_{3}^{*},0))^{\perp}=\text{span}\left\\{w\right\\},$

  * (e)

(Transversality)
$\partial_{\Theta}A(\Theta_{3}^{*},0)[v]\notin\text{Im}(A(\Theta_{3}^{*},0))$.

Then, for any $k_{0}\geq 2$, there exist a constant $s_{0}=s_{0}(k_{0})>0$ and
a curve $[0,s_{0})\ni s\mapsto(\Theta_{3}(s),R(s))\in
I\times(X^{k_{0}+1})^{3}$ such that $G(\Theta_{3},R)=0$ and $R(s)\neq 0$ for
$s>0$. The curve emanates from $(\Theta_{3}(0),R(0))=(\Theta^{*}_{3},0)$.

###### Remark 4.8

Note that the evenness of $\sigma$ for the tame estimate in $(\tilde{c}-2)$ is
simply because $X^{k}$ is a space of even functions and any odd order
derivatives of $h\in X^{k}$ are even.

The rest of this section is devoted to prove Theorem 4.7.

Towards the proof, let $k_{0}\geq 2$ be fixed. We define the projections
$P:(X^{k_{0}+1})^{3}\mapsto\text{Ker}(A(\Theta_{3}^{*},0))$ and
$Q:(Y^{k_{0}})^{3}\mapsto\text{Im}(A(\Theta_{3}^{*},0))^{\perp}$ by

$\displaystyle PR:=\left(v\cdot R\right)v\quad\text{ and }\quad QR:=(w\cdot
R)w,$ (4.18)

where $(f\cdot g)$ denotes the usual $L^{2}$ inner product. Note that from the
assumptions $(d)$ and $(e)$, we make an ansatz that for sufficiently small
$s>0$, the bifurcation curve $s\mapsto(\Theta_{3}(s),R(s))$ can be written as

$(\Theta_{3}(s),R(s))=\left(\Theta^{*}_{3}+v\cdot\tilde{R}(s),sv+(I-P)\tilde{R}(s)\right),$

for some $\tilde{R}(s)\in(X^{k_{0}+1})^{3}$ such that
$|(I-P)\tilde{R}|_{(X^{k_{0}+1})^{3}}=o(s)$. From this ansatz, we define a
family of functionals $\tilde{G_{s}}:(X^{k_{0}+1})^{3}\mapsto(Y^{k_{0}})^{3}$
by

$\displaystyle\tilde{G}_{s}(R):=G(\Theta^{*}_{3}+v\cdot R,sv+(I-P)R),$ (4.19)

and look for $R\in(X^{k_{0}+1})^{3}$ such that $\tilde{G}_{s}(R)=0$ and
$|(I-P)R|_{(X^{k_{0}+1})^{3}}=o(s)$ for sufficiently small $s>0$. This will be
achieved by Newton’s method, where the first approximate solution is $R=0$. To
perform Newton’s method, we need to study the linearized operator
$D\tilde{G}_{s}$ at each approximate solution $R\neq 0$: For
$h\in(X^{k_{0}+1})^{3}$, which can be directly computed from (4.19),

$\displaystyle D\tilde{G}_{s}(R)[h]$ $\displaystyle:=D_{R}\tilde{G}_{s}(R)[h]$
$\displaystyle=\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)(v\cdot
h)+DG(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)[(I-P)h]$
$\displaystyle=:T_{s}(R)[h]+a(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)[(I-P)h],$
(4.20)

where

$\displaystyle T_{s}(R)[h]:=\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot
R,sv+(I-P)R)(v\cdot h)+A(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)[(I-P)h].$ (4.21)

The above decomposition of $D\tilde{G}_{s}$ into $T_{s}+a$ follows from the
assumption $(c)$. Recall that Newton’s method relies on the “invertibility” of
$T_{s}(R)$ for small $s>0$. However, as we will see in the next lemma, $T_{s}$
is not fully invertible between $(X^{k_{0}+1})^{3}$ and $(Y^{k_{0}})^{3}$,
because of its loss of derivatives. This is a motivation of adapting a Nash-
Moser scheme in our proof. However, (4.14) suggests that the inverse of
$T_{s}$ is a good approximate right inverse of $D\tilde{G}_{s}$. In the next
subsection, we will focus on the properties of $T_{s}(R)$.

####  4.3.1 Analysis of ${T}_{s}$

We will look for a solution $R$ to $\tilde{G}_{s}(R)=0$ in $(X^{k_{0}+1})^{3}$
for small $s>0$. In each step of Newton’s method (Nash-Moser iteration), we
will regularize the approximate solution $R_{n}$ (see (4.58)). The theorem
will be achieved by proving that $R_{n}$ converges in $(X^{k_{0}+1})^{3}$.
However, we will also obtain boundedness of the sequence $R_{n}$ in the higher
norms, which is necessary because of the extra regularity conditions as in
(c), $(\tilde{c}-1)$ and $(\tilde{c}-2)$. For this reason, we will establish
several lemmas assuming that an approximate solution $R$ is more regular then
$(X^{k_{0}+1})^{3}$, which will turn out to be true at the end of the proof.

###### Lemma 4.9

Let $0<\epsilon<1$ and $2\leq k_{0}\in\mathbb{N}$ be fixed. There exist
positive constants $s_{0}(\epsilon,k_{0}),\ c_{0}(\epsilon,k_{0})$ and
$0<\delta(\epsilon,k_{0})<1$ such that for each $0<s<s_{0}$, the following
holds: If

$\displaystyle|PR|_{(X^{k_{0}+2})^{3}}\leq s^{\epsilon},$ (4.22)
$\displaystyle|(I-P)R|_{(X^{k_{0}+2})^{3}}\leq s^{1+\epsilon},$ (4.23)
$\displaystyle|R|_{(H^{k_{0}+4})^{3}}\leq\delta,$ (4.24)

then

  * (A)

For all $t\in[0,1]$,

$\displaystyle(\Theta^{*}_{3}+v\cdot R,sv+(I-P)R)\in I\times V^{3},$ (4.25)

where $V^{3}$ is as in Theorem 4.7. Therefore,

$\displaystyle T_{s}(R)[h]:=\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot
R,sv+(I-P)R)(v\cdot h)+A(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)[(I-P)h],$

is well-defined.

  * (B)

$T_{s}(R):(X^{k_{0}+1})^{3}\mapsto Y^{k_{0}+1}\times(Y^{k_{0}})^{2}$ is an
isomorphism and

$\displaystyle|T_{s}(R)[h]|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\leq
c_{0}|h|_{(X^{k_{0}+1})^{3}},$ (4.26)
$\displaystyle|T_{s}(R)^{-1}[z]|_{(X^{k_{0}+1})^{3}}\leq\frac{c_{0}}{s}|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}},$
(4.27) $\displaystyle\left|\left(D\tilde{G}_{s}(R)\circ
T_{s}(R)^{-1}-I\right)[z]\right|_{(Y^{k_{0}})^{3}}\leq
c_{0}|\tilde{G}_{s}(R)|_{(Y^{k_{0}})^{3}}|T_{s}(R)^{-1}[z]|_{(X^{k_{0}+1})^{3}},$
(4.28)

for some $c_{0}=c_{0}(\epsilon,k_{0})>0$.

If $R\in C^{\infty}$, then for any even
$\sigma\in\mathbb{N}\cup\left\\{0\right\\}$ and $z\in
X^{k_{0}+1+\sigma}\times(X^{k_{0}+\sigma})^{2}$, we have
$T_{s}(R)^{-1}[z]\in\left(X^{k_{0}+1+\sigma}\right)^{3}$. Also, we have that

$\displaystyle\left|T_{s}(R)^{-1}[z]\right|_{\left(X^{k_{0}+1+\sigma}\right)^{3}}\leq\frac{c_{0}}{s}\left((1+|R|_{(X^{k_{0}+4+\sigma})^{3}})|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}+|z|_{Y^{k_{0}+1+\sigma}\times(Y^{k_{0}+\sigma})^{2}}\right),$
(4.29)

where $c_{0}$ may depend on not only $\epsilon,k_{0}$ but also $\sigma$.

  * (C)

Furthermore, we can choose $c_{0}$ so that when $R=0$, the following hold:

$\displaystyle|P(T_{s}(0)^{-1}[z])|_{(X^{k_{0}+1})^{3}}\leq\frac{c_{0}}{s}|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}},$
(4.30) $\displaystyle|(I-P)(T_{s}(0)^{-1}[z])|_{(X^{k_{0}+1})^{3}}\leq
c_{0}|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}.$ (4.31)

We will frequently use

$\displaystyle|T_{s}(R)^{-1}z|_{\left(X^{k_{0}+1}\right)^{3}}\leq\frac{c_{0}}{s}|z|_{(Y^{k_{0}+1})^{3}},$
(4.32)

which is more crude than (4.27).

###### Proof.

Let us fix $\epsilon>0$ and $k_{0}\geq 2$. We will show that if (4.22)-(4.24)
hold for sufficiently small $s>0$ and for some small $\delta>0$ depending on
$\epsilon$, then (A),(B) and (C) hold for some $c_{0}>0$.

Proof of (A). To see (A), notice that (4.25) is trivial. From the assumptions
(b) and (c) in Theorem 4.7, the linear operator $T_{s}(R)$ is well-defined.

Proof of (B). The estimate (4.26), follows from (4.21), (4.15), (4.9) and

$|sv+(I-P)R|_{\left(H^{k_{0}+3}\right)^{3}}\lesssim 1+\delta\lesssim 1,$

where the last inequality follows from (4.24). Before proving (4.27), we first
prove that

$\displaystyle|Q\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot
R,sv+(I-P)R)|_{X^{k_{0}+1}\times(X^{k_{0}})^{2}}\geq cs,$ (4.33)

for some $c=c(\epsilon)>0$. The above inequality is a consequence of the
transversality condition $(e)$ in Theorem 4.7. In what follows,
$c,c_{1},c_{2},\ldots$ denote positive constants that may change from line to
line and depend only on $\epsilon$ but not on $s$.

By the regularity assumptions of $G$ in (b) in Theorem 4.7, we estimate the
quantity in (4.33) using a Taylor expansion up to linear order: For a fixed
$R$, we set $f(s,y):=Q\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot R,sv+(I-P)y)$,
so that

$\displaystyle f(s,R)=Q\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R).$
(4.34)

Using the fundamental theorem of calculus, we can write

$\displaystyle f(s,R)$
$\displaystyle:=\int_{0}^{1}\frac{d}{dt}\left(f(s,tR)\right)dt+f(s,0)$
$\displaystyle=\int_{0}^{1}\frac{d}{dt}\left(f(s,tR)\right)dt+f(0,0)+\int_{0}^{s}\frac{d}{dt}f(t,0)dt.$

In terms of $G$, (4.34) and the above equality give us that

$\displaystyle|Q\partial_{\Theta}G(\Theta_{3}^{*}$ $\displaystyle+v\cdot
R,sv+(I-P)R)|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}=|f(s,R)|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}$
$\displaystyle\geq-{\sup_{0\leq t\leq
1}\|Q\partial_{\Theta}DG(\Theta_{3}^{*}+v\cdot
R,sv+t((I-P)R))\|_{\mathcal{L}((X^{k_{0}+2})^{3},(Y^{k_{0}+1})^{3})}|(I-P)R|_{\left(X^{k_{0}+2}\right)^{3}}}$
$\displaystyle\ -{|Q\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot
R,0)|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}$ $\displaystyle\
+{\bigg{|}\int_{0}^{s}Q\partial_{\Theta}DG(\Theta_{3}^{*}+v\cdot
R,tv)[v]dt\bigg{|}_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}$
$\displaystyle=:-L_{1}-L_{2}+L_{3},$ (4.35)

where the first term after the inequality follows from b) in Theorem 4.7 and

$\displaystyle\int_{0}^{1}\left|\frac{d}{dt}\left(f(s,tR)\right)\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}dt$
$\displaystyle\leq\int_{0}^{1}\left|\frac{d}{dt}\left(f(s,tR)\right)\right|_{(Y^{k_{0}+1})^{3}}dt$
$\displaystyle\leq\sup_{0\leq t\leq
1}\left|D_{R}f(s,tR)[R]\right|_{(Y^{k_{0}+1})^{3}},$

and the rest follows simply from the triangular inequality. Then the
regularity assumption (b) in Theorem 4.7 yields that (recall $\rVert
R\rVert_{(X^{k_{0}+4})^{3}}<\delta<1$,)

$\displaystyle L_{1}\leq c_{1}|(I-P)R|_{\left(X^{k_{0}+2}\right)^{3}}\leq
c_{1}s^{1+\epsilon},$ (4.36)

where the last inequality follows from (4.23). For $L_{2}$, we use (a) in
Theorem 4.7 to obtain

$\displaystyle L_{2}=0.$ (4.37)

To estimate $L_{3}$, we compute

$\displaystyle L_{3}$
$\displaystyle\geq\bigg{|}\int_{0}^{s}Q\partial_{\Theta}DG(\Theta_{3}^{*},tv)[v]dt\bigg{|}_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}-\int_{0}^{s}\sup_{u\in(0,|v\cdot
R|)}\bigg{|}Q\partial_{\Theta\Theta}DG(\Theta_{3}^{*}+u,tv)[v]\bigg{|}_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}|v\cdot
R|dt.$ (4.38)

For the first integral, we can write

$\displaystyle Q\partial_{\Theta}DG(\Theta^{*}_{3},tv)[v]$
$\displaystyle=Q\partial_{\Theta}DG(\Theta^{*}_{3},0)[v]+\int_{0}^{t}Q\partial_{\Theta}D^{2}G(\Theta^{*}_{3},uv)[v,v]du$
$\displaystyle=QA(\Theta^{*}_{3},0)[v]+\int_{0}^{t}Q\partial_{\Theta}D^{2}G(\Theta^{*}_{3},uv)[v,v]du,$

where the last equality follows from (a) and (c) in Theorem 4.7, which shows
$a(\Theta^{*}_{3},0)=0$. For the integral term, using (4.12), we have

$\left|Q\partial_{\Theta}D^{2}G(\Theta^{*}_{3},uv)[v,v]\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\lesssim\left|Q\partial_{\Theta}D^{2}G(\Theta^{*}_{3},uv)[v,v]\right|_{(Y^{k_{0}+1})^{3}}\lesssim(1+\rVert
R\rVert_{(X^{k_{0}+4})^{3}})\rVert v\rVert_{(X^{k_{0}+2})^{3}}\leq c,$

where the last inequality follows from (4.24). Therefore

$\displaystyle\bigg{|}\int_{0}^{s}Q\partial_{\Theta}DG(\Theta_{3}^{*},tv)[v]dt\bigg{|}_{X^{k_{0}+1}\times(X^{k_{0}})^{2}}$
$\displaystyle\geq
s\left|QA(\Theta^{*}_{3},0)[v]\right|_{X^{k_{0}+1}\times(X^{k_{0}})^{2}}-\int_{0}^{s}\int_{0}^{t}cdudt$
$\displaystyle\geq
s\left|Q\partial_{\Theta}DG(\Theta^{*}_{3},0)[v]\right|_{X^{k_{0}+1}\times(X^{k_{0}})^{2}}-c_{2}s^{2}$
$\displaystyle\geq c_{3}s-c_{2}s^{2},$

where the last inequality follows from the transversality (e). For the second
integral in (4.38), we have that for sufficiently small $s>0$,

$\sup_{u\in(0,|v\cdot
R|),t\in(0,s)}\left|\partial_{\Theta\Theta}DG(\Theta^{*}_{3}+u,tv)[v]\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\lesssim\sup_{t\in(0,s)}(1+|tv|_{(X^{k_{0}+4})^{3}})|v|_{(X^{k_{0}+2})^{3}}\leq
c_{1},$

for sufficiently small $s$, which follows from (4.13). Hence using (4.22), we
obtain from (4.38) that

$\displaystyle L_{3}\geq c_{3}s-c_{2}s^{2}-c_{1}s^{1+\epsilon}$ (4.39)

Thus the claim (4.33) follows from (4.3.1) (4.36), (4.37) and (4.39) for small
$s$, depending on $\epsilon$.

Towards the proof of (4.27), we will consider how to invert
$A(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)[(I-P)]$. Since the dimension of
$\text{Im}(DG(\Theta_{3}^{*},0)^{\perp})$ is one and $w$ is the basis of
$\text{Im}(DG(\Theta_{3}^{*},0))^{\perp}$, the above claim (4.33) proves that

$\displaystyle\left|w\cdot\partial_{\Theta}G(\Theta_{3}^{*}+v\cdot
R,sv+(I-P)R)\right|>cs.$ (4.40)

To simplify the notations, we denote

$\displaystyle\tilde{A}[h]:=A(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)[(I-P)h],$
(4.41) $\displaystyle\partial_{\Theta}G:=\partial_{\Theta}G(\Theta^{*}+v\cdot
R,sv+(I-P)R).$

We denote by $Q_{1}$ the projection from
$Y^{k_{0}+1}\times\left(Y^{k_{0}}\right)^{2}$ into
$\text{Im}(\tilde{A})^{\perp}$. Since the norm of $R$ is small, we expect that
the functional structure of $\tilde{A}$ should be similar to the structure of
$A(\Theta^{*}_{3},0)$. Indeed, by continuity hypotheses (4.16), (4.24) and
Lemma A.15, where we can think of $\tilde{A}$ as a linear map from
$\text{Ker}(A(\Theta^{*}_{3},0))$, we can choose $\delta$ small enough so that
if (4.24) holds, then there exists $0\neq w_{1}\in
Y^{k_{0}+1}\times(Y^{k_{0}})^{2}$ such that

$\displaystyle\text{Im}(\tilde{A})^{\perp}=\text{span}\left\\{w_{1}\right\\}$
(4.42) $\displaystyle|w-w_{1}|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\leq
c\delta,$ (4.43)
$\displaystyle|\tilde{A}|_{\mathcal{L}(\text{Ker}\left(A(\Theta_{3}^{*},0)\right)^{\perp},\text{Im}(\tilde{A}))}\leq
c,$ (4.44)
$\displaystyle|\tilde{A}^{-1}|_{\mathcal{L}(\text{Im}(\tilde{A}),\text{Ker}\left(A(\Theta_{3}^{*},0)\right)^{\perp})}\leq
c.$ (4.45)

Now, we claim that we can further restrict $\delta$ if necessary so that

$\displaystyle\left|w_{1}\cdot\partial_{\Theta}G\right|>cs.$ (4.46)

In fact, note that (4.22)-(4.23) imply that $R\in(X^{k_{0}+2})^{3}$, thus by
(4.9), we have $\partial_{\Theta}G\in(Y^{k_{0}+1})^{3}\subset
Y^{k_{0}+1}\times(Y^{k_{0}})^{2}$. Also we have

$\displaystyle\partial_{\Theta}G$
$\displaystyle=\int_{0}^{1}\partial_{t}\left(\partial_{\Theta}G(\Theta^{*}_{3}+v\cdot
R,t(sv+(I-P)R))\right)dt$
$\displaystyle=\int_{0}^{1}\partial_{\Theta}DG(\Theta^{*}_{3}+v\cdot
R,t(sv+(I-P)R))[sv+(I-P)R]dt,$

thus

$\displaystyle\left|\partial_{\Theta}G\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\leq\left|\partial_{\Theta}G\right|_{(Y^{k_{0}+1})^{3}}\leq
c|sv+(I-P)R|_{(X^{k_{0}+2})^{2}}\leq cs,$ (4.47)

where the second inequality follows from (4.9) and the last inequality follows
from (4.23). Then it follows that (recall that $f\cdot g$ is the dot product
in $L^{2}$ space),

$\displaystyle|\partial_{\Theta}G\cdot w_{1}|$
$\displaystyle\geq|\partial_{\Theta}G\cdot
w|-|\partial_{\Theta}G\cdot(w_{1}-w)|$ $\displaystyle\geq
c_{1}s-|\partial_{\Theta}G\cdot(w_{1}-w)|$ $\displaystyle\geq
c_{1}s-cs|w-w_{1}|_{X^{k_{0}+1}\times(X^{k_{0}})^{2}}$ $\displaystyle\geq
c_{1}s-c\delta s,$

where we used (4.40), (4.47) and (4.43) for the second, third and the fourth
inequality, respectively. Hence, assuming $\delta$ is small, we have (4.46).

To prove (4.27), pick an arbitrary $z\in Y^{k_{0}+1}\times(Y^{k_{0}})^{2}$.
There exists a unique $\eta\in\text{Ker}\left(A(\Theta_{3}^{*},0)\right)$ and
a unique $h\in\text{Ker}\left(A(\Theta_{3}^{*},0)\right)^{\perp}$ such that

$\displaystyle(v\cdot\eta)Q_{1}\partial_{\Theta}G=Q_{1}z$ (4.48)
$\displaystyle(v\cdot\eta)(I-Q_{1})\partial_{\Theta}G+\tilde{A}[h]=(I-Q_{1})z.$
(4.49)

In fact, there exists a unique $\eta$ in (4.48) thanks to (4.46) and that
$Q_{1}$ is the projection to a one-dimensional space spanned by $w_{1}$. Once
$\eta$ is fixed, the existence and the uniqueness of $h$ in (4.49) follows
from (4.45). Once $\eta$ and $h$ are determined, it is clear that

$\displaystyle
T_{s}(R)[h+\eta]=(v\cdot(h+\eta))\partial_{\Theta}G+\tilde{A}[\eta+h]=(v\cdot\eta)\partial_{\Theta}G+\tilde{A}[h]=z,$
(4.50)

where we used $v\cdot h=0$ and $\tilde{A}[\eta]=\tilde{A}[(I-P)\eta]=0$, which
follows from the definition of $\tilde{A}$ in (4.41). Therefore
$T(R)^{-1}z=h+\eta$. Furthermore we have from (4.48) and (4.49) that

$\displaystyle|\eta|_{(X^{k_{0}+1})^{3}}\leq c\frac{|z\cdot
w_{1}|}{|\partial_{\Theta}G\cdot w_{1}|}$ (4.51)
$\displaystyle|h|_{(X^{k_{0}+1})^{3}}\leq|\tilde{A}^{-1}(I-Q_{1})\partial_{\Theta}G(v\cdot\eta)|_{{(X^{k_{0}+1})^{3}}}+|\tilde{A}^{-1}(I-Q_{1})z|_{(X^{k_{0}+1})^{3}}.$
(4.52)

Using (4.46) and (4.45), we obtain

$\displaystyle|\eta|_{(X^{k_{0}+1})^{3}}\leq
c\frac{|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}{s},$ (4.53)
$\displaystyle|h|_{(X^{k_{0}+1})^{3}}\leq
c|\partial_{\Theta}G|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\frac{|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}{s}+c|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\leq
c\frac{|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}{s},$ (4.54)

where we used (4.47). This proves (4.27).

To show (4.28), we compute

$\displaystyle\left(D\tilde{G}_{s}(R)\circ
T_{s}(R)^{-1}-I\right)[z]=a(\Theta_{3}^{*}+v\cdot R,sv+(I-P)R)\circ
T(R)^{-1}[z],$

which follows from (4.20). This implies

$\displaystyle|\left(D\tilde{G}_{s}(R)\circ
T_{s}(R)^{-1}-I\right)[z]|_{\left(X^{k_{0}}\right)^{3}}$ $\displaystyle\leq
c|a(\Theta_{3}^{*}-v\cdot R,sv+(I-P)R)\circ
T(R)^{-1}[z]|_{\left(X^{k_{0}}\right)^{3}}$ $\displaystyle\leq
c|G(\Theta_{3}^{*}+v\cdot
R,sv+(I-P)R)|_{\left(Y^{k_{0}}\right)^{3}}|T_{s}(R)^{-1}[z]|_{\left(X^{k_{0}+1}\right)^{3}}$
$\displaystyle\leq
c|\tilde{G}_{s}(R)|_{\left(Y^{k_{0}}\right)^{3}}|T_{s}(R)^{-1}[z]|_{\left(X^{k_{0}+1}\right)^{3}},$

where we used (4.14) to get the second inequality and used the definition of
$\tilde{G}_{s}(R)$ to obtain the last inequality.

In order to prove (4.29), we improve on the estimates in (4.51) and (4.52).
For (4.51), recall that $\eta$ is in a one-dimensional space,
span$\left\\{v\right\\}$ and $v$ is supported on the $m$-th Fourier mode.
Thus,

$\displaystyle|\eta|_{(X^{k_{0}+1+\sigma})^{3}}\leq
c|\eta|_{(X^{k_{0}+1})^{3}}\leq
c\frac{|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}{s},$ (4.55)

where $c$ may depend on $\sigma$. Furthermore, $(\tilde{c}$-2) in Theorem 4.7
and (4.49) give us that

$\displaystyle|h|_{(X^{k_{0}+1+\sigma})^{3}}$ $\displaystyle\leq
c(1+\left|R\right|_{(X^{k_{0}+4+\sigma})^{3}})\left(\left|(v\cdot\eta)\partial_{\Theta}G\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}+\left|z\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\right)$
$\displaystyle\
+c\left(\left|(v\cdot\eta)\partial_{\Theta}G\right|_{Y^{k_{0}+1+\sigma}\times(Y^{k_{0}+\sigma})^{2}}+\left|z\right|_{Y^{k_{0}+1+\sigma}\times(Y^{k_{0}+\sigma})^{2}}\right).$
(4.56)

Using (4.47) and (4.53), we have

$\displaystyle\left|(v\cdot\eta)\partial_{\Theta}G\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\leq\left|\partial_{\Theta}G\right|_{{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}|v\cdot\eta|\leq
c\left|z\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}.$

For the higher norm of $\partial_{\Theta}G$, we use (4.9) and and (4.53) and
obtain

$\displaystyle\left|(v\cdot\eta)\partial_{\Theta}G\right|_{Y^{k_{0}+1+\sigma}\times(Y^{k_{0}+\sigma})^{2}}$
$\displaystyle\leq
c\left|\partial_{\Theta}G\right|_{(Y^{k_{0}+1+\sigma})^{3}}|v\cdot\eta|$
$\displaystyle\leq\frac{c}{s}\left(1+\left|R\right|_{(X^{k_{0}+2+\sigma})^{3}}\right)\left|z\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}.$

Therefore, (4.56) gives us

$\displaystyle|h|_{(X^{k_{0}+1+\sigma})^{3}}\leq\frac{c}{s}\left((1+\left|R\right|_{(X^{k_{0}+4+\sigma})^{3}})\left|z\right|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}+\left|z\right|_{Y^{k_{0}+1+\sigma}\times(Y^{k_{0}+\sigma})^{2}}\right).$

Combining with (4.55), we obtain (4.29).

Proof of (C). Finally, if $R=0$, then we have
$|\partial_{\Theta}G|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}=|\partial_{\Theta}G(\Theta_{3},sv)|_{{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}}\leq
cs$. Therefore we can improve (4.54) and obtain

$\displaystyle|h|_{(X^{k_{0}+1})^{3}}\leq
c|z|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}},$

which implies (4.31). (4.27) follows trivially from (4.30). This completes the
proof. ∎

###### Lemma 4.10

Let $\tilde{G}_{s}$ be as in (4.19). Then there exists an open set $V^{3}$
near $0\in(X^{3})^{3}$ such that for all $R,\tilde{R}\in V^{3}$ and $k\geq 2$,

  * (a’)

(Initial value) $|\tilde{G}_{s}(0)|_{(X^{k})^{3}}\lesssim_{k}s^{2}.$

  * (b’)

(Taylor estimate) If $\rVert
R\rVert_{(X^{k+3})^{3}},\rVert\tilde{R}\rVert_{(X^{k+3})^{3}}\leq\eta$ for
some $0<\eta=\eta(k)<1$, then

$\displaystyle\begin{cases}|D\tilde{G}_{s}(R)[h]|_{(Y^{k})^{3}}\lesssim_{k}|h|_{(X^{k+1})^{3}}\\\
|\tilde{G}_{s}(\tilde{R})-\tilde{G}_{s}(R)-D\tilde{G}_{s}(R)[\tilde{R}-R]|_{(Y^{k})^{3}}\lesssim_{k}|\tilde{R}-R|_{(X^{k+1})^{3}}^{2}\\\
|\tilde{G}_{s}(\tilde{R})-\tilde{G}_{s}(R)-D\tilde{G}_{s}(R)[\tilde{R}-R]-\frac{1}{2}D^{2}\tilde{G}_{s}[\tilde{R}-R,\tilde{R}-R]|_{(Y^{k})^{3}}\lesssim_{k}|\tilde{R}-R|_{(X^{k+1})^{3}}^{3}.\end{cases}$

###### Proof.

(a’) follows from the fact that $v\in\text{Ker}(DG(\Theta_{3}^{*},0))$ and
$G(\Theta_{3}^{*},0)=0$. (b’) is due to (b) in Theorem 4.7. ∎

####  4.3.2 Nash-Moser iteration

For $\beta,N>0$ and $2\leq k\in\mathbb{N}$, we consider a regularizing
operator $S(N):(X^{k})^{3}\mapsto(C^{\infty})^{3}$ such that

$\displaystyle\begin{cases}|S(N)R|_{(X^{k+\beta})^{3}}\lesssim_{k,\beta}N^{\beta}|R|_{(X^{k})^{3}}&\forall\
R\in(H^{k})^{3}\\\
|(I-S(N))R|_{(H^{k})^{3}}\lesssim_{k,\beta}N^{-\beta}|R|_{(X^{k+\beta})^{3}}&\forall\
R\in(H^{k+\beta})^{3}.\end{cases}$ (4.57)

Note that we can choose $S$ so that $PS(N)=S(N)P$, since $v$ is supported on
the $m$-th Fourier mode (see (d) in Theorem 4.7).

For $N_{n}>1$ and even integer $\beta>0$, which will be chosen later, we set

$\displaystyle\begin{cases}R_{n+1}=R_{n}-S(N_{n})T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]&\text{
for }n\geq 0\\\ R_{0}=0,\end{cases}$ (4.58)

and

$\displaystyle\begin{cases}a_{n}:=|R_{n+1}-R_{n}|_{(X^{k_{0}+1})^{3}}\\\
a^{\prime}_{n}:=|R_{n+1}-R_{n}|_{(X^{k_{0}+2})^{3}}\\\
a^{\prime\prime}_{n}=|R_{n+1}-R_{n}|_{(X^{k_{0}+4})^{3}}\\\
b_{n}:=|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}\\\
C_{n}:=|T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1+\beta})^{3}}.\end{cases}$
(4.59)

Note that our goal is to show that $\sum_{n=0}^{\infty}a_{n}<\infty$, which
implies that $R_{n}$ converges in $(X^{k_{0}+1})^{3}$. In order to prove the
assumptions in (4.22)-(4.24), we will need the boundedness of $a_{n}^{\prime}$
and $a_{n}^{\prime\prime}$ as well.

###### Lemma 4.11

Let $\epsilon>0$ and $k_{0}\geq 2$ fixed and let $s_{0}$ and $\delta$ be
defined as in Lemma 4.9. We also assume that $\delta$ is even smaller if
necessary, so that $\delta<\eta(k_{0})$ where $\eta(k_{0})$ is as in (b’) in
Lemma 4.10. For any $n\geq 0$ such that $R_{n}$ satisfies (4.22), (4.23) and
(4.24) for some $s\in(0,s_{0})$, then we have

$\displaystyle a_{0}\lesssim_{\epsilon,k_{0}}s,$ (4.60) $\displaystyle
a_{n}\lesssim_{\epsilon,k_{0}}\frac{1}{s}N_{n}b_{n}$ $\displaystyle\text{ for
}n\geq 1,$ (4.61) $\displaystyle b_{0}\lesssim_{\epsilon,k_{0},\beta}s^{2}$
(4.62) $\displaystyle
b_{1}\lesssim_{\epsilon,k_{0},\beta}(N_{0}^{3}+1)s^{3}+N_{0}^{-\beta}C_{0}.$
(4.63) $\displaystyle
b_{n+1}\lesssim_{\epsilon,k_{0},\beta}a_{n}^{2}+\frac{1}{s}N_{n}b_{n}^{2}+N_{n}^{-\beta}(1+b_{n})C_{n}$
$\displaystyle\text{ for }n\geq 1,$ (4.64) $\displaystyle
a^{\prime}_{n}\lesssim_{\epsilon,k_{0}}\frac{1}{s}N_{n}^{2}b_{n}$
$\displaystyle\text{ for }n\geq 0,$ (4.65) $\displaystyle
a^{\prime\prime}_{n}\lesssim_{\epsilon,k_{0}}\frac{1}{s}N_{n}^{4}b_{n}$
$\displaystyle\text{ for }n\geq 0.$ (4.66)

Furthermore, we have

$\displaystyle|PR_{1}|_{(X^{k_{0}+2})^{3}}\lesssim_{\epsilon,k_{0}}N_{0}s$
(4.67)
$\displaystyle|(I-P)R_{1}|_{(X^{k_{0}+2})^{3}}\lesssim_{\epsilon,k_{0}}N_{0}s^{2}.$
(4.68)

###### Proof.

Proof of (4.60), (4.61), (4.67) and (4.68). We first claim that

$\displaystyle|PR_{1}|_{(X^{k_{0}+1})^{3}}\lesssim s$ (4.69)
$\displaystyle|(I-P)R_{1}|_{(X^{k_{0}+1})^{3}}\lesssim s^{2},$ (4.70)

Let us prove (4.69) first. Thanks to (a’) in Lemma 4.10 and (4.30), we have

$\displaystyle|PR_{1}|_{(X^{k_{0}+1})^{3}}=|S(N_{0})PT_{s}(0)^{-1}[\tilde{G}_{s}(0)]|_{(X^{k_{0}+1})^{3}}\lesssim|PT_{s}(0)^{-1}[\tilde{G}_{s}(0)]|_{(X^{k_{0}+1})^{3}}\lesssim\frac{1}{s}|\tilde{G}_{s}(0)|_{(Y^{k_{0}}+1)^{3}}\lesssim
s,$

where we used that $S(N_{0})$ and $P$ commute to obtain the first equality. To
prove (4.70), we use (4.31) and compute

$\displaystyle|(I-P)R_{1}|_{(X^{k_{0}+1})^{3}}$
$\displaystyle\lesssim|S(N_{0})(I-P)T_{s}(0)^{-1}[\tilde{G}_{s}(0)]|_{(X^{k_{0}+1})^{3}}$
$\displaystyle\lesssim|(I-P)T_{s}(0)^{-1}[\tilde{G}_{s}(0)]|_{(X^{k_{0}+1})^{3}}$
$\displaystyle\lesssim|\tilde{G}_{s}(0)|_{(Y^{k_{0}+1})^{3}}$
$\displaystyle\lesssim s^{2},$

which proves (4.70). With the claim, (4.60) follows immediately. (4.61) (4.67)
and (4.68) can be proved in exactly the same way as above using (4.57).

Proof of (4.62), (4.63) and (4.64). Note that Lemma 4.10 immediately implies
(4.62). Now, let us prove (4.63) and (4.64). We have

$\displaystyle b_{n+1}:=|\tilde{G}_{s}(R_{n+1})|_{(Y^{k_{0}})^{3}}$
$\displaystyle\lesssim{|\tilde{G}_{s}(R_{n+1})-\tilde{G}_{s}(R_{n})-D\tilde{G}_{s}(R_{n})[R_{n+1}-R_{n}]|_{(Y^{k_{0}})^{3}}}$
$\displaystyle\
+{|\tilde{G}_{s}(R_{n})+D\tilde{G}_{s}(R_{n})[R_{n+1}-R_{n}]|_{(Y^{k_{0}})^{3}}}$
$\displaystyle=:J_{1}+J_{2},$ (4.71)

We claim that

$\displaystyle\begin{cases}J_{1}\lesssim s^{3}&\text{ if }n=0\\\ J_{1}\lesssim
a_{n}^{2}=|R_{n+1}-R_{n}|_{(X^{k_{0}+1})^{3}}^{2}&\text{ if }n\geq
1.\end{cases}$ (4.72)

If $n=0$, we have

$\displaystyle J_{1}$
$\displaystyle\lesssim\bigg{|}\tilde{G}_{s}({R_{1}})-\tilde{G}_{s}(0)-D\tilde{G}_{s}(0)[R_{1}]-\frac{1}{2}D^{2}\tilde{G}_{s}(0)\left[R_{1},R_{1}\right]\bigg{|}_{(Y^{k_{0}})^{3}}+\frac{1}{2}|D^{2}\tilde{G}_{s}(0)[R_{1},R_{1}]|_{(Y^{k_{0}})^{3}}$
$\displaystyle\lesssim|R_{1}|^{3}_{(X^{k_{0}+1})^{3}}+\frac{1}{2}|D^{2}\tilde{G}_{s}(0)[R_{1},R_{1}]|_{(Y^{k_{0}})^{3}}$
$\displaystyle\lesssim
s^{3}+\frac{1}{2}|D^{2}\tilde{G}_{s}(0)[R_{1},R_{1}]|_{(Y^{k_{0}})^{3}},$

where we used (b’) in Lemma 4.10 in the second inequality and (4.69) and
(4.70) in the last inequality. To estimate $D^{2}\tilde{G}$, we recall the
definition of $\tilde{G}_{s}$ and compute

$\displaystyle D^{2}\tilde{G}_{s}(0)[R_{1},R_{1}]$
$\displaystyle=\left(\frac{d}{dt}\right)^{2}G(\Theta_{3}^{*}+tv\cdot
R_{1},sv+(I-P)tR_{1})\bigg{|}_{t=0}$
$\displaystyle=\partial_{\Theta\Theta}G(\Theta_{3}^{*},sv)(v\cdot
R_{1})^{2}+2(v\cdot
R_{1})\partial_{\Theta}DG(\Theta_{3}^{*},sv)[(I-P)[R_{1}]]$ $\displaystyle\
+D^{2}G(\Theta_{3}^{*},sv)\left[(I-P)R_{1},(I-P)R_{1}\right].$

Since $\partial_{\Theta\Theta}G(\Theta_{3}^{*},0)=0$, therefore

$\left|\partial_{\Theta\Theta}G(\Theta^{*}_{3},sv)\right|_{(Y^{k_{0}})^{3}}=\left|\int_{0}^{s}\frac{d}{dt}\left(\partial_{\Theta\Theta}G(\Theta^{*}_{3},tv)\right)dt\right|_{(Y^{k_{0}})^{3}}\lesssim
s,$

which implies $|\partial_{\Theta\Theta}G(\Theta_{3}^{*},sv)(v\cdot
R_{1})^{2}|_{(Y^{k_{0}})^{3}}\lesssim s^{3}$, since $|v\cdot
R_{1}|\lesssim|PR_{1}|_{(X^{k_{0}+1})^{3}}\lesssim s$, which follows from
(4.69). Also it follows from (b), (4.69) and (4.70) that

$\displaystyle|2(v\cdot
R_{1})\partial_{\Theta}DG(\Theta_{3}^{*},sv)[(I-P)[R_{1}]]|_{(Y^{k_{0}})^{3}}\lesssim
s^{3},$
$\displaystyle|D^{2}G(\Theta_{3}^{*},sv)[(I-P)R_{1},(I-P)R_{1}]|_{(Y^{k_{0}})^{3}}\lesssim
s^{4},$

therefore, $|D^{2}\tilde{G}_{s}(0)[R_{1},R_{1}]|_{(Y^{k_{0}})^{3}}\lesssim
s^{3}$. Thus, we have $J_{1}\lesssim s^{3}$, which proves (4.72) for $n=0$. If
$n\geq 1$, the claim in (4.72) follows immediately from (b’) in Lemma 4.10.

In order to estimate $J_{2}$ in (4.3.2), we have that for $n\geq 0$,

$\displaystyle J_{2}$ $\displaystyle\lesssim{|(I-D\tilde{G}_{s}(R_{n})\circ
T_{s}(R_{n})^{-1})[\tilde{G}_{s}(R_{n})]|_{(Y^{k_{0}})^{3}}}$ $\displaystyle\
+{|D\tilde{G}_{s}(R_{n})\left(I-S(N_{n})\right)T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(Y^{k_{0}})^{3}}}$
$\displaystyle=:J_{21}+J_{22}.$

Then it follows from (4.57) and (4.28) that

$\displaystyle J_{21}$
$\displaystyle\lesssim|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}|T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1})^{3}}$
$\displaystyle\lesssim|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}|S(N_{n})T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1})^{3}}$
$\displaystyle\
+|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}|\left(I-S(N_{n})\right)T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1})^{3}}$
$\displaystyle\lesssim
N_{n}|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}|T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}})^{3}}$
$\displaystyle\
+N_{n}^{-\beta}|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}|T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1+\beta})^{3}}$
$\displaystyle\lesssim\frac{1}{s}N_{n}|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}^{2}+N_{n}^{-\beta}|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}})^{3}}|T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1+\beta})^{3}}$
$\displaystyle\lesssim\frac{1}{s}N_{n}b_{n}^{2}+N_{n}^{-\beta}b_{n}C_{n},$

where the fourth inequality follows from (4.32). Also we have

$\displaystyle
J_{22}\lesssim|\left(I-S(N_{n})\right)T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{(X^{k_{0}+1})^{3}}\lesssim
N_{n}^{-\beta}C_{n}.$

Hence we have for $n\geq 0$ that

$\displaystyle
J_{2}\lesssim\frac{1}{s}N_{n}b_{n}^{2}+N_{n}^{-\beta}(1+b_{n})C_{n}.$ (4.73)

Thus with (4.3.2), (4.72), (4.73) and (a’) in lemma 4.10, we have that

$\displaystyle\begin{cases}b_{1}\lesssim(1+N_{0})s^{3}+N_{0}^{-\beta}C_{0}\\\
b_{n+1}\lesssim
a_{n}^{2}+\frac{1}{s}N_{n}b_{n}^{2}+N_{n}^{-\beta}(1+b_{n})C_{n}&\text{ for
}n\geq 1.\end{cases}$ (4.74)

Proof of (4.65) and (4.66). Finally, for (4.65) and (4.66), we use (4.32) to
compute

$\displaystyle|R_{n+1}-R_{n}|_{(X^{k_{0}+2})^{3}}=|S(N_{n})T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{\left(X^{k_{0}+2}\right)^{3}}\lesssim
N_{n}^{2}\frac{1}{s}b_{n},$ (4.75)
$\displaystyle|R_{n+1}-R_{n}|_{(X^{k_{0}+4})^{3}}=|S(N_{n})T_{s}(R_{n})^{-1}[\tilde{G}_{s}(R_{n})]|_{\left(X^{k_{0}+4}\right)^{3}}\lesssim
N_{n}^{4}\frac{1}{s}b_{n}.$ (4.76)

This finishes the proof. ∎

Let us now derive a recursive formula for $C_{n}$.

###### Lemma 4.12

For $j=0,\ldots,n-1$ assume that $R_{j}$ satisfies the same assumptions as in
Lemma 4.11, that is, given $\epsilon>0$ and $k_{0}\geq 2$, $R_{j}$ satisfies
(4.22), (4.23) and (4.24) and the assumptions of Lemma 4.10 for
$s\in(0,s_{0}(\epsilon,k_{0}))$ and $\delta=\delta(\epsilon,k_{0})>0$. Then,

$\displaystyle
C_{n}\lesssim_{\epsilon,k_{0},\beta}\frac{1}{s}\left(1+n\sup_{j=0,\ldots
n-1}N_{j}^{4}C_{j}\right),\quad\text{ for $n\geq 1$ and }\quad
C_{0}\lesssim_{\epsilon,k_{0},\beta}s.$

###### Proof.

It follows from (a’) in Lemma 4.10 and (4.29) in Lemma 4.9 that

$\displaystyle
C_{0}\lesssim\frac{1}{s}|\tilde{G}_{s}(0)|_{Y^{k_{0}+1+\beta}\times(Y^{k_{0}+\beta})^{2}}\lesssim\frac{1}{s}|\tilde{G}_{s}(0)|_{(Y^{k_{0}+1+\beta})^{3}}\lesssim
s.$

Now let us assume that $n\geq 1$. It follows from the definition of $C_{n}$ in
(4.59) and (4.29) that (recall that $\beta$ is even)

$\displaystyle
C_{n}\lesssim\frac{1}{s}\left((1+|R_{n}|_{(X^{k_{0}+4+\beta})^{3}})|\tilde{G}_{s}(R_{n})|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}+|\tilde{G}_{s}(R_{n})|_{Y^{k_{0}+1+\beta}\times(Y^{k_{0}+\beta})^{2}}\right).$

Using (4.8), we have

$\displaystyle|\tilde{G}_{s}(R_{n})|_{Y^{k_{0}+1}\times(Y^{k_{0}})^{2}}\lesssim|\tilde{G}_{s}(R_{n})|_{(Y^{k_{0}+1})^{3}}\lesssim
1+|R_{n}|_{(X^{k_{0}+2})^{3}}\lesssim 1,$
$\displaystyle|\tilde{G}_{s}(R_{n})|_{Y^{k_{0}+1+\beta}\times(Y^{k_{0}+\beta})^{2}}\lesssim
1+|R_{n}|_{(X^{k_{0}+2+\beta})^{3}}.$

therefore, we have

$\displaystyle C_{n}\lesssim\frac{1}{s}(1+|R_{n}|_{(X^{k_{0}+4+\beta})^{3}}).$

For $R_{n}$, we have

$\displaystyle|R_{n}|_{(X^{k_{0}+4+\beta})^{3}}$
$\displaystyle\lesssim\sum_{j=0}^{n-1}\left|R_{j+1}-R_{j}\right|_{(X^{k_{0}+4+\beta})^{3}}$
$\displaystyle\lesssim\sum_{j=0}^{n-1}\left|S(N_{j})T_{s}(R_{j})^{-1}[\tilde{G}_{s}(R_{j})]\right|_{(X^{k_{0}+4+\beta})^{3}}$
$\displaystyle\lesssim\sum_{j=0}^{n-1}N_{j}^{3}C_{j}$ $\displaystyle\lesssim
n\sup_{j=0,\ldots,n-1}N_{j}^{3}C_{j}.$

Hence, we obtain the desired result. ∎

Now we are ready to prove the main theorem of this section.

  * Proof of Theorem 4.7: We fix $k_{0}\geq 2$ and pick $\tilde{\epsilon},\ \tilde{s}>0$ so that

$\displaystyle\sum_{k=1}^{\infty}s^{-1+2\left(\frac{67}{64}\right)^{k}}\leq
s^{1+\tilde{\epsilon}}\text{ for all $s\in(0,\tilde{s})$.}$ (4.77)

And we choose

$\displaystyle\ \beta:=816\quad\text{ and
}\quad\epsilon:=\min\left\\{\frac{1}{4},\frac{\tilde{\epsilon}}{2}\right\\}>0,$
(4.78)

and let $s_{0}=s_{0}(\epsilon,k_{0}),$ $\delta=\delta(\epsilon,k_{0})$ be as
in Lemma 4.9. As before, we can also assume $\delta$ is small enough so that
$\delta<\eta(k_{0})$ where $\eta$ is as in (b’) in Lemma 4.10. Since $k_{0}$,
$\epsilon$ and $\beta$ are fixed, by Lemma 4.11 and 4.12, we can find a
constant $K>0$ such that as long as $R_{n}$ satisfies (4.22),(4.23) and
(4.24), for some $s\in(0,s_{0})$, it holds that for any sequence of positive
numbers $N_{n}$, and

$\displaystyle a_{0}\leq Ks$ (4.79)
$\displaystyle|PR_{1}|_{(X^{k_{0}+2})^{3}}\leq KN_{0}s$ (4.80)
$\displaystyle|(I-P)R_{1}|_{(X^{k_{0}+2})^{3}}\leq KN_{0}s^{2}.$ (4.81)
$\displaystyle a_{n}\leq\frac{K}{s}N_{n}b_{n}\quad\text{for $n\geq 1$,}$
(4.82) $\displaystyle b_{0}\leq Ks^{2},$ (4.83) $\displaystyle b_{1}\leq
K\left((N_{0}^{3}+1)s^{3}+N_{0}^{-\beta}C_{0}\right)$ (4.84) $\displaystyle
b_{n+1}\leq
K(a_{n}^{2}+\frac{1}{s}N_{n}b_{n}^{2}+N_{n}^{-\beta}(1+b_{n})C_{n})\quad\text{
for $n\geq 1$},$ (4.85) $\displaystyle
a^{\prime}_{n}\leq\frac{K}{s}N_{n}^{2}b_{n}\quad\text{ for $n\geq 0$},$ (4.86)
$\displaystyle a^{\prime\prime}_{n}\leq\frac{K}{s}N_{n}^{4}b_{n}\quad\text{
for $n\geq 0$},$ (4.87) $\displaystyle
C_{n}\leq\frac{K}{s}\left(1+n\sup_{j=0,\ldots
n-1}\left|N_{j}^{4}C_{j}\right|\right)\quad\text{ for $n\geq 1$,}$ (4.88)
$\displaystyle C_{0}\leq Ks.$ (4.89)

In what follows, we assume without loss of generality that

$\displaystyle K>1,\quad\text{ and }\quad\delta<1.$ (4.90)

For such $K$, we can find $s^{*}$ such that for all $s\in(0,s^{*})$ (each of
them can be easily verified for small enough $s>0$),

$\displaystyle 2Kn<s^{-2\left(\frac{67}{64}\right)^{n-1}+1}\quad\text{ for all
$n\geq 1$},$ (4.91) $\displaystyle 4K^{3}s^{\frac{7}{16}}<\frac{1}{2},$ (4.92)
$\displaystyle Ks^{\frac{\tilde{\epsilon}}{2}}\leq\frac{1}{2},$ (4.93)
$\displaystyle s<1,$ (4.94) $\displaystyle
K^{2}s^{\frac{3}{4}}\leq\frac{1}{2}\delta,$ (4.95) $\displaystyle
K\sum_{k=1}^{\infty}s^{-1+\frac{127}{64}\left(\frac{67}{64}\right)^{k}}\leq\frac{1}{2}\delta,$
(4.96) $\displaystyle K(n+2)\leq s^{1-2\left(\frac{67}{64}\right)^{n}}\text{
for all $n\geq 0$},$ (4.97) $\displaystyle 3K^{3}\leq
s^{2+\left(-\frac{139}{32}+\frac{143}{64}\cdot\frac{67}{64}\right)\left(\frac{67}{64}\right)^{n}}\text{
for all $n\geq 1$},$ (4.98) $\displaystyle 3K\leq
s^{1+\left(-\frac{141}{32}+\frac{143}{64}\left(\frac{67}{64}\right)\right)\left(\frac{67}{64}\right)^{n}}\text{
for all $n\geq 1$},$ (4.99) $\displaystyle 6K\leq
s^{\left(-\frac{816}{16}+\frac{768}{16}+\frac{143}{64}\frac{67}{64}\right)\left(\frac{67}{64}\right)^{n}}\text{
for all $n\geq 1$}.$ (4.100)

Lastly, we fix $s\in(0,\min\left\\{s_{0},s^{*},\tilde{s}\right\\})$ and

$\displaystyle
N_{n}:=s^{-\frac{1}{16}\left(\frac{67}{64}\right)^{n}}\quad\text{ for $n\geq
0$.}$ (4.101)

We claim that for all $n\geq 0$,

    * $(P1)_{n}$

: $R_{n+1}$ satisfies (4.22),(4.23) and (4.24).

    * $(P2)_{n}$

: $C_{n}\leq N_{n}^{768}$.

    * $(P3)_{n}$

: $b_{n+1}\leq s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n+1}}$.

    * $(P4)_{n}$

: $a_{n+1}\leq Ks^{-1+\frac{139}{64}\left(\frac{67}{64}\right)^{n+1}}$.

Once we have the above claims, then $(P1)_{n}$ justifies all the recurrence
formulae above for all $n\in\mathbb{N}$, which follows from Lemma 4.11 and
4.12. Also, it is clear from (4.82) and $(P4)_{n}$ that

$\displaystyle\sum_{n=0}^{\infty}a_{n}\leq
KN_{0}s+K\sum_{n=1}^{\infty}s^{-1+\frac{139}{64}\left(\frac{67}{64}\right)^{n}}<\infty.$

Therefore $R_{n}$ is a Cauchy sequence in $(X^{k_{0}+1})^{3}$, and we can find
a limit $R_{\infty}:=\lim_{n\to\infty}R_{n}$. Then, $(P3)_{n}$ and the
continuity of the functional $R\mapsto\tilde{G}_{s}$ implies that
$\tilde{G}_{s}(R_{\infty})=0$. From the definition of $\tilde{G}_{s}$ in
(4.19), this implies the existence of a solution $R=R(s)\neq 0$ for each
$s\in(0,\min\left\\{s_{0},s^{*},\tilde{s}\right\\})$ and finishes the proof.

Now we prove the claim $(Pi)_{n}$ for $i=1,\ldots,4$. We will follow the usual
induction argument.

Initial step. In the initial case, we assume that $n=0$. We first prove
$(P1)_{0}$. It follows from (4.80),(4.81) and (4.101) that

$\displaystyle|PR_{1}|_{(X^{k_{0}+2})^{3}}\leq Ks^{\frac{15}{16}}\leq
s^{\frac{1}{4}}\leq s^{\epsilon}$ (4.102)
$\displaystyle|(I-P)R_{1}|_{(X^{k_{0}+2})^{3}}\leq Ks^{1+\frac{15}{16}}\leq
s^{1+\frac{1}{4}}\leq s^{1+\epsilon},$ (4.103)

where the two second inequalities follow from (4.92) ($Ks^{\frac{15}{16}}\leq
4K^{3}s^{\frac{7}{16}}s^{\frac{1}{2}}<s^{\frac{1}{4}}$) and the last
inequalities follow from (4.94) and (4.78). Furthermore, it follows from
(4.87), (4.83) and (4.101) that

$\displaystyle|R_{1}|_{(X^{k_{0}+4})^{3}}=a^{\prime\prime}_{0}\leq\frac{K}{s}\cdot\left(s^{-\frac{1}{16}}\right)^{4}\cdot\left(Ks^{2}\right)\leq
K^{2}s^{\frac{3}{4}}\leq\frac{1}{2}\delta,$ (4.104)

where the last inequality follows from (4.95). Therefore $(P1)_{0}$ holds for
$n=0$. $(P2)_{0}$ follows immediately from (4.89) and (4.101). In order to
prove $(P3)_{0}$, note that thanks to (4.84), it is enough to show that

$K\left((N_{0}^{3}+1)s^{3}+N_{0}^{-\beta}C_{0}\right)\leq
s^{\frac{143}{64}\cdot\frac{67}{64}},$

in other words,

$K\left(s^{\frac{45}{16}}+s^{3}+Ks^{\frac{816}{16}}s\right)\leq
s^{\frac{143}{64}\cdot\frac{67}{64}},$

where we used (4.78), (4.89) and (4.101). By (4.94), $s^{\frac{45}{16}}$ is
the largest value among the three in the parentheses, hence it is sufficient
to show that $3K^{2}<s^{\frac{143}{64}\cdot\frac{67}{64}-\frac{45}{16}}.$
Since $\frac{143}{64}\cdot\frac{67}{64}-\frac{45}{16}>-\frac{1}{2}$, it is
enough to show that $3K^{3}\leq s^{-\frac{1}{2}}$, which follows from (4.92).
$(P4)_{0}$ follows from (4.82), (4.101) and $(P3)_{0}$.

Induction step. In this step, we assume that $(Pi)_{k}$ is true for all $0\leq
k\leq n_{0}$ and aim to prove $(Pi)_{n_{0}+1}$. Let us prove $(P1)_{n_{0}+1}$
first. It follows from (4.86), (4.101) and $(P3)_{k}$ for $k=0,\ldots,n_{0}$,
that

$\sum_{k=1}^{n_{0}+1}a^{\prime}_{k}\leq
K\sum_{k=1}^{n_{0}+1}s^{-1}\left(s^{-\frac{1}{16}\left(\frac{67}{64}\right)^{k}}\right)^{2}\cdot
s^{\frac{143}{64}\left(\frac{67}{64}\right)^{k}}\leq
K\sum_{k=1}^{n_{0}+1}s^{-1+2\left(\frac{67}{64}\right)^{k}}\leq
Ks^{1+\tilde{\epsilon}},$

where the last inequality follows from our choice on $\tilde{\epsilon}$ and
$s\in\tilde{s}$ in (4.77). Hence, we have

$\displaystyle\sum_{k=1}^{n_{0}+1}a^{\prime}_{k}\leq\left(Ks^{\frac{\tilde{\epsilon}}{2}}\right)s^{1+\frac{\tilde{\epsilon}}{2}}\leq\frac{1}{2}s^{1+\frac{\tilde{\epsilon}}{2}}\leq\frac{1}{2}s^{1+\epsilon},$

where the second inequality follows from (4.93) and the last inequality
follows from (4.78) and (4.94). Therefore we obtain

$|PR_{n_{0}+2}|_{(X^{k_{0}+2})^{3}}\leq|PR_{1}|_{(X^{k_{0}+2})^{3}}+\sum_{k=1}^{n_{0}+1}a^{\prime}_{k}\leq
Ks^{\frac{15}{16}}+\frac{1}{2}s^{1+\epsilon}\leq\left(Ks^{\frac{7}{16}}\right)s^{\frac{1}{2}}+\left(\frac{1}{2}s\right)s^{\epsilon}\leq
s^{\epsilon},$

where the second inequality follows from (4.102) and the last inequality
follows from $\epsilon<\frac{1}{2}$ and $Ks^{\frac{7}{16}}\leq\frac{1}{2}$,
which can be deduced from (4.93) and (4.78). Using (4.103), instead of
(4.102), one can easily obtain

$|(I-P)R_{n_{0}+2}|_{(X^{k_{0}+2})^{3}}\leq s^{1+\epsilon}.$

To prove (4.24) for $R_{n_{0}+2}$, we compute

$\displaystyle|R_{n_{0}+2}|_{(H^{k_{0}+4})^{3}}$ $\displaystyle\leq
a^{\prime\prime}_{0}+\sum_{k=1}^{n_{0}+1}a^{\prime\prime}_{k}\leq\frac{1}{2}\delta+K\sum_{k=1}^{n_{0}+1}s^{-1}\left(s^{-\frac{1}{16}\left(\frac{67}{64}\right)^{k}}\right)^{4}\cdot
s^{\frac{143}{64}\left(\frac{67}{64}\right)^{k}}$
$\displaystyle\leq\frac{1}{2}\delta+K\sum_{k=1}^{\infty}s^{-1+\frac{127}{64}\left(\frac{67}{64}\right)^{k}}\leq\delta,$
(4.105)

where the second inequality follows from (4.104), (4.87) and $(P3)_{k}$, for
$0\leq k\leq n_{0}$, and the last inequality follows from (4.96). This proves
$(P1)_{n_{0}+1}$.

We turn to $(P2)_{n_{0}+1}$. Using (4.88) and $(P2)_{n_{0}}$, we have

$C_{n_{0}+1}\leq\frac{K}{s}(2+n_{0})N_{n_{0}}^{4}\sup_{j=0,\ldots,n_{0}}C_{j}\leq\frac{K}{s}(2+n_{0})N_{n_{0}}^{772},$

where the last inequality follows from $(P2)_{n_{0}}$. Hence, it suffices to
show that

$\frac{K}{s}(2+n_{0})N_{n_{0}}^{772}\leq(N_{n_{0}+1})^{768}.$

Plugging (4.101), this is equivalent to

$K(n_{0}+2)\leq
s^{1+\left(\frac{67}{64}\right)^{n_{0}}\left(\frac{772}{16}-\frac{768}{16}\cdot\frac{67}{64}\right)}=s^{1-2\left(\frac{67}{64}\right)^{n_{0}}},$

which is true thanks to our choice of $s$ in (4.97). This proves
$(P2)_{n_{0}+1}$.

For $(P3)_{n_{0}+1}$, thanks to (4.85), it is enough to show that

$K\left(a_{n_{0}+1}^{2}+\frac{1}{s}N_{n_{0}+1}b_{n_{0}+1}^{2}+N_{n_{0}+1}^{-\beta}(1+b_{n_{0}+1})C_{n_{0}+1})\right)\leq
s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n_{0}+2}}.$

Using $(P4)_{n_{0}}$, (4.101), $(P3)_{n_{0}}$, (4.78), $(P2)_{n_{0}+1}$ and
(4.94), which implies $b_{n_{0}+1}\leq 1$ , we only need to show that

$\displaystyle
K\left(\underbrace{\left(Ks^{-1+\frac{139}{64}\left(\frac{67}{64}\right)^{n_{0}+1}}\right)^{2}}_{=:A_{1}}+\underbrace{s^{-1}s^{-\frac{1}{16}\left(\frac{67}{64}\right)^{n_{0}+1}}s^{\frac{143}{32}\left(\frac{67}{64}\right)^{n_{0}+1}}}_{=:A_{2}}+\underbrace{2s^{\frac{816}{16}\left(\frac{67}{64}\right)^{n_{0}+1}}s^{-\frac{768}{16}\left(\frac{67}{64}\right)^{n_{0}+1}}}_{=:A_{3}}\right)\leq
s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n_{0}+2}}.$ (4.106)

We will show that
$KA_{i}\leq\frac{1}{3}s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n_{0}+2}}$
for $i=1,2,3$. For $A_{1}$, it suffices to show that

$K^{3}s^{-2+\frac{139}{32}\left(\frac{67}{64}\right)^{n_{0}+1}}\leq\frac{1}{3}s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n_{0}+2}},$

equivalently,

$3K^{3}\leq
s^{2+\left(-\frac{139}{32}+\frac{143}{64}\cdot\frac{67}{64}\right)\left(\frac{67}{64}\right)^{n_{0}+1}},$

and this follows from our choice of $s$ in (4.98). For $A_{2}$, we need to
show that

$Ks^{-1+\frac{141}{32}\left(\frac{67}{64}\right)^{n_{0}+1}}\leq\frac{1}{3}s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n_{0}+2}},$

eqvalently,

$3K\leq
s^{1+\left(-\frac{141}{32}+\frac{143}{64}\left(\frac{67}{64}\right)\right)\left(\frac{67}{64}\right)^{n_{0}+1}},$

and this follows from (4.99). For $A_{3}$, it is enough to show that

$2Ks^{\frac{816}{16}\left(\frac{67}{64}\right)^{n_{0}+1}}\cdot
s^{-\frac{768}{16}\left(\frac{67}{64}\right)^{n_{0}+1}}\leq\frac{1}{3}s^{\frac{143}{64}\left(\frac{67}{64}\right)^{n_{0}+2}},$

equivalently,

$6K\leq
s^{\left(-\frac{816}{16}+\frac{768}{16}+\frac{143}{64}\frac{67}{64}\right)\left(\frac{67}{64}\right)^{n_{0}+1}},$

which follows from (4.100). This proves $(P3)_{n_{0}+1}$.

Lastly, $(P4)_{n_{0}+1}$ can be proved by (4.82) and $(P3)_{n_{0}+1}$, that
is,

$a_{n_{0}+2}\leq Ks^{-1}N_{n_{0}+2}b_{n_{0}+2}\leq
Ks^{-1}s^{\left(-\frac{1}{16}+\frac{143}{64}\right)\left(\frac{67}{64}\right)^{n_{0}+2}}=Ks^{-1+\frac{139}{64}\left(\frac{67}{64}\right)^{n_{0}+2}},$

which finishes the proof. $\Box$

###  4.4 Estimates on the velocity

In Subsection 4.5, we will check whether our functional $G$ in (4.5) satisfies
the hypotheses in Theorem 4.7. In this section, we will derive some useful
estimates on the velocity vector generated by each patch. Recall that given
$b_{1},b_{2}>0$, $b_{1}\neq b_{2}$, and $R_{1},R_{2}\in
C^{\infty}(\mathbb{T})$, we denote for $i,j=1,2$,

$\displaystyle R=(R_{1},R_{2})\in(C^{\infty}(\mathbb{T}))^{2},$ (4.107)
$\displaystyle z_{i}(R)(x)=(b_{i}+R_{i}(x))(\cos x,\sin x),$ (4.108)
$\displaystyle u_{i,j}(R)(x):=\begin{pmatrix}u^{1}_{i,j}(R)\\\
u^{2}_{i,j}(R)\end{pmatrix}=\int_{\mathbb{T}}\log(|z_{j}(R)(x)-z_{i}(R)(y)|^{2})z_{i}(R)^{\prime\perp}(y)dy.$
(4.109)

###### Proposition 4.13

For $k\geq 2$, there exists $\epsilon=\epsilon(k,b_{1},b_{2})>0$ such that if
$\rVert R\rVert_{(H^{3}(\mathbb{T}))^{2}}\leq\epsilon$, then

  * (A)

$u_{i,j}:(H^{k+1}(\mathbb{T}))^{2}\mapsto(H^{k}(\mathbb{T}))^{2},\text{ and
}\rVert
u_{i,j}\rVert_{(H^{k}(\mathbb{T}))^{2}}\lesssim_{k,b_{1},b_{2}}1+\rVert
R\rVert_{(H^{k+1}(\mathbb{T}))^{2}}.$

  * (B)

The Gateaux derivative
$Du_{i,j}(R):(H^{k+1}(\mathbb{T}))^{2}\mapsto(H^{k+1}(\mathbb{T}))^{2}$ exists
and

$\rVert
Du_{i,j}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim_{k,b_{1},b_{2}}\rVert
h\rVert_{(H^{k+1}(\mathbb{T}))^{2}}+\rVert
R\rVert_{(H^{k+3}(\mathbb{T}))^{2}}\rVert
h\rVert_{(H^{1}(\mathbb{T}))^{2}},\text{ for
}h\in(C^{\infty}(\mathbb{T}))^{2}.$

  * (C)

The map $R\mapsto Du_{{i,j}}(R)$ is Lipschitz continuous, in the sense that
for $R,r,h\in(C^{\infty}(\mathbb{T}))^{2}$ such that $\rVert
R\rVert_{(H^{3}(\mathbb{T}))^{2}},\rVert
r\rVert_{(H^{3}(\mathbb{T}))^{2}}\leq\epsilon,$ and $\rVert
R\rVert_{(H^{k+3}(\mathbb{T}))^{2}},\rVert
r\rVert_{(H^{k+3}(\mathbb{T}))^{2}}\leq 1$,

$\rVert
Du_{i,j}(R)[h]-Du_{i,j}(r)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim_{k,b_{1},b_{2}}\rVert
R-r\rVert_{(H^{2}(\mathbb{T}))^{2}}\rVert
h\rVert_{(H^{k+1}(\mathbb{T}))^{2}}+\rVert
R-r\rVert_{(H^{k+3}(\mathbb{T}))^{2}}\rVert
h\rVert_{(H^{1}(\mathbb{T}))^{2}}.$

  * (D)

For $\sigma\in\mathbb{N}\cup\left\\{0\right\\}$, there exists a linear
operator $T_{i,j}^{\sigma}(R)$ such that for
$h\in\left(C^{\infty}(\mathbb{T})\right)^{2}$,

$\left(\frac{d}{dx}\right)^{\sigma}\left(Du_{i,j}(R)[h]\right)=Du_{i,j}(R)[h^{(\sigma)}]+T_{i,j}^{\sigma}(R)[h],$

where $T_{i,j}^{\sigma}(R)$ satisfies

$\rVert
T_{i,j}^{\sigma}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim_{k}(1+\rVert
R\rVert_{(H^{k+4+\sigma}(\mathbb{T}))^{2}})\rVert
h\rVert_{(L^{2}(\mathbb{T}))^{2}}+(1+\rVert
R\rVert_{(H^{4}(\mathbb{T}))^{2}})\rVert
h^{(k+\sigma)}\rVert_{(L^{2}(\mathbb{T}))^{2}}.$

###### Remark 4.14

It is well known that roughly speaking, if $\partial D$ is
$C^{k+\alpha}$-regular for some $k\geq 1,$ $\alpha>0$ then the velocity
$\nabla^{\perp}(1_{D}*\log|x|)$ is also $C^{k+\alpha}$-regular up to the
boundary. In $(A)$ in the above Proposition, we do not aim to prove the
optimal regularity since it is not necessary in the proof of the main theorem.

The proof of the proposition will be given after Lemmas 4.15 and 4.16. We will
deal with the case $i=j$ only. If $i\neq j$, then the integrand in $u_{i,j}$
has no singularity thus the result follows straightforwardly in a similar
manner. Hence, by abuse of notation, we denote for $b>0$ and $R\in
C^{\infty}(\mathbb{T})$,

$\displaystyle z(R)(x):=(b+R(x))(\cos x,\sin x),$ (4.110) $\displaystyle
u(R)(x):=\begin{pmatrix}u^{1}(R)\\\
u^{2}(R)\end{pmatrix}:=\int_{\mathbb{T}}\log\left(|z(R)(x)-z(R)(y)|^{2}\right)z(R)^{\prime\perp}(y)dy.$
(4.111)

Let us write $u(R)$ as

$\displaystyle u(R)(x)$
$\displaystyle=\int_{\mathbb{T}}\log\left(2-2\cos(x-y)\right)z(R)^{\prime\perp}(y)dy+\int_{\mathbb{T}}\log\left(\frac{|z(R)(x)-z(R)(y)|^{2}}{2-2\cos(x-y)}\right)z(R)^{\prime\perp}(y)dy$
$\displaystyle=\int_{\mathbb{T}}\log\left(2-2\cos(x-y)\right)z(R)^{\prime\perp}(y)dy+\int_{\mathbb{T}}K(R)(x,y)z(R)^{\prime}(y)^{\perp}dy$
$\displaystyle=:u_{L}(R)(x)+u_{N}(R)(x),$ (4.112)

where

$\displaystyle K(R)(x,y)=F(R(x),R(y),J(R)(x,y)),$ $\displaystyle
F(u,v,w):=\log(b^{2}+b(u+v)+uv+w^{2}),$ (4.113) $\displaystyle
J(R)(x,y)=\frac{R(x)-R(y)}{2\sin(\frac{x-y}{2})}.$

If $\rVert R\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$ for sufficiently small
$\epsilon>0$, it follows from Lemmas A.10 and A.5 that

$\displaystyle\rVert J(R)\rVert_{H^{2}(\mathbb{T}^{2})}\lesssim_{b}\rVert
R\rVert_{H^{3}(\mathbb{T})}\lesssim\epsilon.$ (4.114)

Therefore, for sufficiently small $\epsilon>0$, Lemmas A.8 and A.10 imply that
for $l\geq 2$,

$\displaystyle\rVert K(R)\rVert_{H^{l}(\mathbb{T}^{2})}\lesssim_{l,b}1+\rVert
R\rVert_{H^{l}(\mathbb{T})}+\rVert
J(R)\rVert_{H^{l}(\mathbb{T}^{2})}\lesssim_{l,b}1+\rVert
R\rVert_{H^{l+1}(\mathbb{T})},,$ (4.115)

The next lemma will be used to prove the growth condition of the velocity in
(A) in Proposition 4.13.

###### Lemma 4.15

For $k\geq 2$, there exists $\epsilon=\epsilon(k,b)>0$ such that if $R\in
C^{\infty}(\mathbb{T})$ and $\rVert R\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$,
then

$\rVert u\rVert_{(H^{k}(\mathbb{T}))^{2}}\lesssim_{k,b}1+\rVert
R\rVert_{H^{k+1}(\mathbb{T})}.$

###### Proof.

For $u_{L}$, then it follows from Lemma A.14 and the fact that $R\mapsto
u_{L}(R)$ is linear that

$\rVert u_{L}(R)\rVert_{(H^{k}(\mathbb{T}))^{2}}\lesssim\rVert
z(R)^{\prime}\rVert_{(H^{k-1}(\mathbb{T}))^{2}}\lesssim 1+\rVert
R\rVert_{H^{k}(\mathbb{T})}.$

Now, let us consider the nonlinear part. We have

$\rVert u_{N}\rVert_{(H^{k}(\mathbb{T}))^{2}}\lesssim\rVert
K(R)\rVert_{H^{k}(\mathbb{T}^{2})}\rVert
z(R)^{\prime}\rVert_{(L^{\infty}(\mathbb{T}))^{2}}\lesssim 1+\rVert
R\rVert_{H^{k+1}(\mathbb{T})},$

where the last inequality follows from (4.115). ∎

Now we turn to (B), (C) and (D) in Proposition 4.13. We will use the following
notations:

$\displaystyle DK(R)[h]:=\frac{d}{dt}K(R+th)|_{t=0}$
$\displaystyle=\underbrace{\partial_{u}F(R(x),R(y),J(R)(x,y))}_{=:\partial_{u}F(R)(x,y)}h(x)$
$\displaystyle\
+\underbrace{\partial_{v}F(R(x),R(y),J(R)(x,y))}_{=:\partial_{v}F(R)(x,y)}h(y)$
$\displaystyle\
+\underbrace{\partial_{w}F(R(x),R(y),J(R)(x,y))}_{=:\partial_{w}F(R)(x,y)}J(h)(x,y),$
(4.116)

where $J(h):=\frac{h(x)-h(y)}{2\sin\left(\frac{x-y}{2}\right)}$ and

$\displaystyle
Dz(R)^{\prime\perp}[h]:=\frac{d}{dt}z(R+th)^{\prime\perp}|_{t=0}=-h^{\prime}(x)(\sin
x,\cos x)-h(x)(\cos x,\sin x).$ (4.117)

With the above notations, we can write the derivative of the nonlinear term as

$\displaystyle Du_{N}(R)[h]$
$\displaystyle=\int_{\mathbb{T}}K(R)(x,y)Dz(R)^{\prime\perp}[h](y)dy+\int_{\mathbb{T}}DK(R)[h](x,y)z(R)^{\prime\perp}(y)dy$
$\displaystyle=:Du_{N}^{1}(R)[h]+Du_{N}^{2}(R)[h].$ (4.118)

Again, Lemma A.8 and (4.114) yield that if $\rVert
R\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$ for sufficiently small $\epsilon>0$,
then for $l\geq 2$,

$\displaystyle\rVert\partial_{u}F(R)\rVert_{(H^{l}(\mathbb{T}))^{2}},\rVert\partial_{v}F(R)\rVert_{(H^{l}(\mathbb{T}))^{2}},\rVert\partial_{w}F(R)\rVert_{(H^{l}(\mathbb{T}))^{2}}$
$\displaystyle\lesssim_{l}1+\rVert R\rVert_{H^{l}(\mathbb{T})}+\rVert
J(R)\rVert_{H^{l}(\mathbb{T}^{2})}$ $\displaystyle\lesssim_{l}1+\rVert
R\rVert_{H^{l+1}(\mathbb{T})}.$ (4.119)

Also, if $R_{1},R_{2}\in C^{\infty}({\mathbb{T}})$ and $\rVert
R_{1}\rVert_{H^{3}(\mathbb{T})},\rVert
R_{2}\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$, then from Lemma A.8, it follows
that

$\displaystyle\rVert\partial_{u}F(R_{1})-\partial_{u}F(R_{2})\rVert_{(H^{l}(\mathbb{T}))^{2}},\rVert\partial_{v}F(R_{1})-\partial_{v}F(R_{2})\rVert_{(H^{l}(\mathbb{T}))^{2}},\rVert\partial_{w}F(R_{1})-\partial_{w}F(R_{2})\rVert_{(H^{l}(\mathbb{T}))^{2}}\lesssim_{l}\rVert
R_{1}-R_{2}\rVert_{H^{l+1}(\mathbb{T})}.$ (4.120)

For ${\partial_{w}F^{\\#}(R)}(x):=\lim_{y\to
x}\partial_{w}F(R)(x,y)=\partial_{w}F(R(x),R(x),R^{\prime}(x))$, we have

$\displaystyle\rVert{\partial_{w}F^{\\#}(R)}\rVert_{H^{l}(\mathbb{T})}\lesssim_{l}1+\rVert
R\rVert_{H^{l+1}(\mathbb{T})},$ (4.121)
$\displaystyle\rVert{\partial_{w}F^{\\#}(R)}\rVert_{L^{\infty}(\mathbb{T})}\lesssim
1+\rVert R\rVert_{H^{2}(\mathbb{T})}\lesssim 1.$ (4.122)
$\displaystyle\rVert{\partial_{w}F^{\\#}(R_{1})}-{\partial_{w}F^{\\#}(R_{2})}\rVert_{H^{l}(\mathbb{T})}\lesssim_{l}\rVert
R_{1}-R_{2}\rVert_{H^{l+1}(\mathbb{T})},$ (4.123)
$\displaystyle\rVert{\partial_{w}F^{\\#}(R_{1})}-{\partial_{w}F^{\\#}(R_{2})}\rVert_{L^{\infty}(\mathbb{T})}\lesssim\rVert
R_{1}-R_{2}\rVert_{H^{2}(\mathbb{T})}.$ (4.124)

We decompose $Du_{N}^{2}(R)[h]$ into

$\displaystyle Du_{N}^{2}(R)[h]$
$\displaystyle=\int_{\mathbb{T}}\underbrace{\partial_{u}F(R)(x,y)z(R)^{\prime\perp}}_{=:K_{1}(R)(x,y)}h(x)dy+\int_{\mathbb{T}}\underbrace{\partial_{v}F(R)(x,y)z(R)^{\prime\perp}}_{=:K_{2}(R)(x,y)}h(y)dy$
$\displaystyle\
+\int_{\mathbb{T}}\underbrace{\partial_{w}F(R)(x,y)z(R)^{\prime\perp}}_{=:K_{3}(R)(x,y)}J(h)(x,y)dy$
(4.125)
$\displaystyle=:Du_{N1}^{2}(R)[h]+Du_{N2}^{2}(R)[h]+Du_{N3}^{2}(R)[h].$
(4.126)

Then it follows from Lemma A.5, (4.4), (4.121) and (4.122) that for $l\geq 2$,

$\displaystyle\rVert
K_{i}(R)\rVert_{H^{l}(\mathbb{T}^{2})}\lesssim_{l}1+\rVert
R\rVert_{H^{l+1}(\mathbb{T})},\text{ for }i=1,2,3,$ (4.127)
$\displaystyle\rVert{K_{3}}^{\\#}(R)\rVert_{H^{l}(\mathbb{T})}\lesssim
1+\rVert R\rVert_{H^{l+1}(\mathbb{T})},\text{ where
}{K_{3}}^{\\#}(R)(x):=K_{3}(R)(x,x),$ (4.128)
$\displaystyle\rVert{K_{3}}^{\\#}(R)\rVert_{L^{\infty}(\mathbb{T})}\lesssim\rVert{K_{3}}^{\\#}(R)\rVert_{H^{1}(\mathbb{T})}\lesssim\rVert{K_{3}}^{\\#}(R)\rVert_{H^{2}(\mathbb{T})}\lesssim
1+\rVert R\rVert_{H^{3}({\mathbb{T}})}\lesssim 1,$ (4.129)
$\displaystyle\rVert
K_{i}(R_{1})-K_{2}(R_{2})\rVert_{H^{l}(\mathbb{T}^{2})}\lesssim_{l}\rVert
R_{1}-R_{2}\rVert_{H^{l+1}(\mathbb{T})},\text{ for }i=1,2,3,$ (4.130)
$\displaystyle\rVert{K_{3}}^{\\#}(R_{1})-{K_{3}}^{\\#}(R_{2})\rVert_{H^{l}(\mathbb{T})}\lesssim\rVert
R_{1}-R_{2}\rVert_{H^{l+1}(\mathbb{T})},$ (4.131)
$\displaystyle\rVert{K_{3}}^{\\#}(R_{1})-{K_{3}}^{\\#}(R_{2})\rVert_{L^{\infty}(\mathbb{T})}\lesssim\rVert
R_{1}-R_{2}\rVert_{H^{2}(\mathbb{T})}.$ (4.132)

Furthermore, it follows from the definitions of $K_{3}$ and $\partial_{w}F$ in
(4.125), (4.116) and (4.113) that

$\displaystyle\rVert\nabla
K_{3}(R)(x,y)\rVert_{L^{\infty}(\mathbb{T}^{2})}\lesssim 1+\rVert
R^{\prime\prime}\rVert_{L^{\infty}(\mathbb{T})}+\rVert\nabla
J(R)\rVert_{L^{\infty}(\mathbb{T}^{2})}\lesssim 1+\rVert
R^{\prime\prime}\rVert_{L^{\infty}(\mathbb{T})}\lesssim 1.$ (4.133)

###### Lemma 4.16

For $k\geq 2$, there exists $\epsilon=\epsilon(k,b)>0$ such that if $R\in
C^{\infty}(\mathbb{T})$ and $\rVert R\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$,
the Gateaux derivative
$Du(R):H^{k+1}(\mathbb{T})\mapsto(H^{k+1}(\mathbb{T}))^{2}$ exists and

$\displaystyle\rVert
Du(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim_{k,b}\rVert
h\rVert_{(H^{k+1}(\mathbb{T}))^{2}}+\rVert R\rVert_{H^{k+3}(\mathbb{T})}\rVert
h\rVert_{H^{1}(\mathbb{T})},\text{ for }h\in C^{\infty}(\mathbb{T}).$ (4.134)

Furthermore, if $R_{1},R_{2}\in C^{\infty}(\mathbb{T})$, $\rVert
R_{1}\rVert_{H^{3}(\mathbb{T})},\rVert
R_{2}\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$, and $\rVert
R_{1}\rVert_{H^{k+3}(\mathbb{T})},\rVert R_{2}\rVert_{H^{k+3}(\mathbb{T})}\leq
1$, then

$\displaystyle\rVert
Du(R_{1})-Du(R_{2})[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim_{k,b}\rVert
R_{1}-R_{2}\rVert_{H^{2}(\mathbb{T})}\rVert
h\rVert_{H^{k+1}(\mathbb{T})}+\rVert
R_{1}-R_{2}\rVert_{H^{k+3}(\mathbb{T})}\rVert h\rVert_{H^{1}(\mathbb{T})},$
(4.135)

$\text{ for }h\in C^{\infty}(\mathbb{T}).$

Lastly, for $\sigma\in\mathbb{N}\cup\left\\{0\right\\}$, there exists a linear
operator $T_{\sigma}(R)$ such that for $h\in C^{\infty}(\mathbb{T})$,

$\displaystyle\left(\frac{d}{dx}\right)^{\sigma}\left(Du(R)[h]\right)=Du(R)[h^{(\sigma)}]+T^{\sigma}(R)[h],$
(4.136)

where $T^{\sigma}(R)$ satisfies

$\displaystyle\rVert
T^{\sigma}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim_{k}((1+\rVert
R\rVert_{H^{k+4+\sigma}(\mathbb{T})})\rVert
h\rVert_{L^{2}(\mathbb{T})}+(1+\rVert R\rVert_{H^{4}(\mathbb{T})})\rVert
h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}.$ (4.137)

###### Proof.

We omit the proof of (4.135) since it can be proved exactly same way as
(4.134) with (A.3) in Lemma A.8. In order to prove (4.134), we deal with
$Du_{L}$. Since it is linear, we have

$Du_{L}(R)[h](x):=\int_{\mathbb{T}}\log(2-2\cos(x-y))Dz(R)^{\prime\perp}[h](y)dy.$

Hence, it follows from Lemma A.14 that

$\displaystyle\rVert
Du_{L}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim\rVert
Dz(R)^{\prime\perp}[h]\rVert_{(H^{k}(\mathbb{T}))^{2}}\lesssim\rVert
h\rVert_{H^{k+1}(\mathbb{T})}.$ (4.138)

Now, we estimate $Du_{N}$. Recalling the decomposition in (4.4) we will
estimate $Du_{N}^{1}$ and $Du_{N}^{2}$ separately. For $Du_{N}^{1}$, it
follows from (4.115) and (4.117) that

$\displaystyle\rVert
Du_{N}^{1}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim\rVert
K(R)\rVert_{H^{k+1}(\mathbb{T}^{2})}\rVert
Dz(R)^{\prime\perp}\rVert_{(L^{2}(\mathbb{T}))^{2}}\lesssim(1+\rVert
R\rVert_{H^{k+2}(\mathbb{T})})\rVert h\rVert_{H^{1}(\mathbb{T})}.$ (4.139)

For $Du_{N}^{2}$, we recall the decomposition in (4.126), and use (4.127) and
Lemma A.11 to obtain

$\rVert Du_{N1}^{2}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}},\rVert
Du_{N2}^{2}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim\rVert
h\rVert_{H^{k+1}(\mathbb{T})}+\rVert R\rVert_{H^{k+2}(\mathbb{T})}\rVert
h\rVert_{L^{\infty}(\mathbb{T})}.$

For $Du_{N3}^{2}(R)[h]$, we use (4.127), (4.128), (4.129), (4.133) and Lemma
A.12 to show that

$\rVert Du_{N3}^{2}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim\rVert
R\rVert_{H^{k+3}(\mathbb{T})}\rVert h\rVert_{H^{1}(\mathbb{T})}+\rVert
h\rVert_{H^{k+1}(\mathbb{T})}.$

Therefore $\rVert
Du_{N}^{2}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim\rVert
h\rVert_{H^{k+1}(\mathbb{T})}+\rVert R\rVert_{H^{k+3}(\mathbb{T})}\rVert
h\rVert_{H^{1}(\mathbb{T})}$. With (4.138), (4.139) and (4.4), the desired
result follows.

Now, we turn to (4.136) and (4.137). Recall from (4.4), (4.4) and (4.126) that

$\displaystyle Du(R)[h]$
$\displaystyle=\int_{\mathbb{T}}\log(2-2\cos(y))h(x-y)dy+\int_{\mathbb{T}}K(R)(x,y)Dz(R)^{\prime\perp}[h](y)dy$
$\displaystyle\
+\int_{\mathbb{T}}K_{1}(R)(x,y)z(R)^{\prime\perp}(y)h(x)dy+\int_{\mathbb{T}}K_{2}(R)(x,y)z(R)^{\prime\perp}(y)h(y)dy$
$\displaystyle\
+\int_{\mathbb{T}}K_{3}(R)(x,y)z(R)^{\prime\perp}(y)J(h)(x,y)dy$
$\displaystyle=I_{1}(R)[h]+I_{2}(R)[h]+I_{3}(R)[h]+I_{4}(R)[h]+I_{5}(R)[h]$

where $K,K_{1},K_{2},K_{3}$ are of the form $H(R(x),R(y),J(R)(x,y))$ for some
smooth function $H:\mathbb{R}^{3}\mapsto\mathbb{R}$. It suffices to show that
for each $i=1,\ldots,5$ and $\sigma\in\mathbb{N}\cup\left\\{0\right\\}$,

$\left(\frac{d}{dx}\right)^{\sigma}I_{i}(R)[h]=I_{i}(R)[h^{\sigma}]+T_{i}^{\sigma}(R)[h],$

for some $T_{i}^{\sigma}(R)$ such that

$\displaystyle\rVert
T_{i}^{\sigma}(R)[h]\rVert_{(H^{k+1}(\mathbb{T}))^{2}}\lesssim(1+\rVert
R\rVert_{H^{k+4+\sigma}(\mathbb{T})})\rVert
h\rVert_{L^{2}(\mathbb{T})}+(1+\rVert R\rVert_{H^{4}(\mathbb{T})})\rVert
h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}.$ (4.140)

We only deal with $I_{5}$ since the other terms can be done in the same way.
We also assume $\sigma\geq 1$ since $\sigma=0$ follows trivially. Let
$\tilde{K}_{3}(R)(x,y):=K_{3}(R)(x,x-y)z(R)^{\prime\perp}(x-y)$. Then it
follows from (A.5), (4.127), (4.128) and (4.129) that for $l\geq 2$,

$\displaystyle\rVert\tilde{K}_{3}(R)\rVert_{H^{l}(\mathbb{T}^{2})}\lesssim_{l}1+\rVert
R\rVert_{H^{l+1}(\mathbb{T})},$ (4.141)
$\displaystyle\rVert{\tilde{K}_{3}}^{\\#}(R)\rVert_{H^{l}(\mathbb{T})}\lesssim
1+\rVert R\rVert_{H^{l+1}(\mathbb{T})},\text{ where
}{\tilde{K}_{3}}^{\\#}(R)(x):=\tilde{K}_{3}(R)(x,x),.$ (4.142)

Using the change of variables, $y\mapsto x-y$, we have

$\displaystyle\left(\frac{d}{dx}\right)^{\sigma}I_{5}(R)[h]$
$\displaystyle=\int_{\mathbb{T}}\tilde{K}_{3}(R)(x,y)J(h^{(\sigma)})(x,x-y)dy+\sum_{p+q=\sigma,q\leq\sigma-1}C_{p,q,\sigma}\int_{\mathbb{T}}(\partial_{x})^{p}\tilde{K}_{3}(R)(x,y)J(h^{(q)})(x,x-y)dy,$
$\displaystyle=:I_{5}(R)[h^{(\sigma)}](x)+\sum_{p+q=\sigma,q\leq\sigma-1}C_{p,q,\sigma}\underbrace{\int_{\mathbb{T}}(\partial_{x})^{p}\tilde{K}_{3}(R)(x,x-y)J(h^{(q)})(x,y)dy}_{=:T_{5}^{\sigma,p,q}(R)[h](x)}$
(4.143) $\displaystyle=:I_{5}(R)[h^{(\sigma)}](x)+T_{5}^{\sigma}(R)[h](x),$
(4.144)

where $C_{p,q,\sigma}$ is some constant and we used
$\partial_{x}\left(J(h)(x,x-y)\right)=J(h^{\prime})(x,x-y)$. It suffices to
show that $T_{5}^{\sigma,p,q}(R)[h]$ satisfies (4.140). Let
$\tilde{K}_{3}^{*}(R)(x,y):=(\partial_{x})^{p}\tilde{K}_{3}(R)(x,x-y)$. Then
it follows from (4.141) and (4.142) that for $l\geq 2$,

$\displaystyle\rVert\tilde{K}^{*}_{3}(R)\rVert_{(H^{l}(\mathbb{T}^{2}))^{2}}\lesssim_{l}1+\rVert
R\rVert_{H^{l+p+1}(\mathbb{T})},$ (4.145)
$\displaystyle\rVert({\tilde{K}^{*}_{3}})^{\\#}(R)\rVert_{(H^{l}(\mathbb{T}))^{2}}\lesssim
1+\rVert R\rVert_{H^{l+p+1}(\mathbb{T})},\text{ where
}({\tilde{K}^{*}_{3}})^{\\#}(R)(x):=\tilde{K}^{*}_{3}(R)(x,x),$ (4.146)
$\displaystyle\rVert({\tilde{K}^{*}_{3}})^{\\#}(R)\rVert_{(L^{\infty}(\mathbb{T}))^{2}}\lesssim\rVert\tilde{K}_{3}(R)\rVert_{(W^{p,\infty}(\mathbb{T}^{2}))^{2}}\lesssim\rVert\tilde{K}_{3}(R)\rVert_{(H^{p+2}(\mathbb{T}^{2}))^{2}}.$
(4.147)

Furthermore, it follows similarly as (4.133) that

$\displaystyle\rVert\nabla\tilde{K}^{*}_{3}(R)\rVert_{L^{\infty}(\mathbb{T}^{2})}\lesssim\rVert\nabla^{(p+1)}\tilde{K}_{3}(R)\rVert_{L^{\infty}(\mathbb{T}^{2})}\lesssim
1+\rVert R^{(p+2)}\rVert_{L^{\infty}(\mathbb{T})}\lesssim 1+\rVert
R\rVert_{H^{p+3}(\mathbb{T})}.$ (4.148)

Hence it follows from Lemma A.12 that

$\displaystyle\rVert T^{\sigma,p,q}_{5}(R)[h]\rVert_{H^{k+1}(\mathbb{T})}$
$\displaystyle\lesssim\left(\rVert\tilde{K}^{*}_{3}(R)\rVert_{(H^{k+2}(\mathbb{T}^{2}))^{2}}+\rVert({\tilde{K}^{*}_{3}})^{\\#}(R)\rVert_{(H^{k+1}(\mathbb{T}^{2}))^{2}}\right)\rVert
h^{(q)}\rVert_{H^{1}(\mathbb{T})}$
$\displaystyle+\left(\rVert({\tilde{K}^{*}_{3}})^{\\#}(R)\rVert_{(L^{\infty}(\mathbb{T}^{2}))^{2}}+\rVert\nabla\tilde{K}^{*}_{3}(R)\rVert_{(L^{\infty}(\mathbb{T}^{2}))^{2}}\right)\rVert
h^{(q)}\rVert_{H^{k+1}(\mathbb{T})}$ $\displaystyle\lesssim(1+\rVert
R\rVert_{H^{k+3+p}(\mathbb{T})})\rVert
h^{(q)}\rVert_{H^{1}(\mathbb{T})}+(1+\rVert
R\rVert_{H^{p+3}(\mathbb{T})})\rVert h^{(q)}\rVert_{H^{k+1}(\mathbb{T})}$
$\displaystyle\lesssim(1+\rVert R\rVert_{H^{k+3+p}(\mathbb{T})})\rVert
h\rVert_{H^{1+q}(\mathbb{T})}+(1+\rVert R\rVert_{H^{p+3}(\mathbb{T})})\rVert
h\rVert_{H^{k+1+q}(\mathbb{T})}$

where the second last inequality follows from (4.148). Now, recalling that
$\rVert R\rVert_{H^{3}(\mathbb{T})}\leq\epsilon$ and plugging the following
inequalities into the above estimate,

$\rVert R\rVert_{H^{l}(\mathbb{T})}\lesssim\rVert
R\rVert_{L^{2}(\mathbb{T})}+\rVert
R^{(l)}\rVert_{L^{2}(\mathbb{T})},\quad\rVert
h\rVert_{H^{l}(\mathbb{T})}\lesssim\rVert h\rVert_{L^{2}(\mathbb{T})}+\rVert
h^{(l)}\rVert_{L^{2}(\mathbb{T})},\text{ for }l\geq 0,$

we obtain

$\displaystyle\rVert T^{\sigma,p,q}_{5}(R)[h]\rVert_{H^{k+1}(\mathbb{T})}$
$\displaystyle\lesssim(1+\rVert
R^{(k+p+3)}\rVert_{L^{2}(\mathbb{T})})\left(\rVert
h\rVert_{L^{2}(\mathbb{T})}+\rVert h^{(1+q)}\rVert_{L^{2}(\mathbb{T})}\right)$
$\displaystyle\ +(1+\rVert R^{(p+3)}\rVert_{L^{2}(\mathbb{T})})(\rVert
h\rVert_{L^{2}(\mathbb{T})}+\rVert h^{(k+1+q)}\rVert_{L^{2}(\mathbb{T})})$
$\displaystyle=:L_{1}+L_{2}.$ (4.149)

For $L_{1}$, note that $k+p+3\leq k+3+\sigma$ and $q\leq\sigma-1$, hence we
have

$\displaystyle L_{1}\lesssim(1+\rVert
R^{(k+\sigma+3)}\rVert_{L^{2}(\mathbb{T})})\rVert
h\rVert_{L^{2}(\mathbb{T})}+\rVert
h^{(\sigma)}\rVert_{L^{2}(\mathbb{T})}+\rVert
R^{(k+p+3)}\rVert_{L^{2}(\mathbb{T})}\rVert
h^{(1+q)}\rVert_{L^{2}(\mathbb{T})}.$ (4.150)

Using the interpolation inequality in Lemma A.6, we have

$\displaystyle\rVert
R^{(k+p+3)}\rVert_{L^{2}(\mathbb{T})}=\rVert(R^{(4)})^{(k+p-1)}\rVert_{L^{2}(\mathbb{T})}\lesssim\rVert
R^{(4)}\rVert_{L^{2}(\mathbb{T})}^{1-\frac{k+p-1}{k+\sigma}}\rVert(R^{(4)})^{(k+\sigma)}\rVert_{L^{2}({\mathbb{T})}}^{\frac{k+p-1}{k+\sigma}}$
$\displaystyle\rVert h^{(1+q)}\rVert_{L^{2}(\mathbb{T})}\lesssim\rVert
h\rVert_{L^{2}(\mathbb{T})}^{1-\frac{1+q}{k+\sigma}}\rVert
h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}^{\frac{1+q}{k+\sigma}}.$

Thus, we have

$\displaystyle\rVert R^{(k+p+3)}\rVert_{L^{2}(\mathbb{T})}\rVert
h^{(1+q)}\rVert_{L^{2}(\mathbb{T})}$ $\displaystyle\lesssim\left(\rVert
R^{(4)}\rVert_{L^{2}(\mathbb{T})}\rVert
h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}\right)^{\frac{1+q}{k+\sigma}}\left(\rVert(R^{(4)})^{(k+\sigma)}\rVert_{L^{2}({\mathbb{T})}}\rVert
h\rVert_{L^{2}(\mathbb{T})}\right)^{\frac{k+p-1}{k+\sigma}}$
$\displaystyle\lesssim\rVert R^{(4)}\rVert_{L^{2}(\mathbb{T})}\rVert
h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}+\rVert
R^{(k+4+\sigma)}\rVert_{L^{2}({\mathbb{T})}}\rVert
h\rVert_{L^{2}(\mathbb{T})},$

where we used $p+q=\sigma$ and Young’s inequality. Plugging this inequality
into (4.150) and using $\rVert
h^{(\sigma)}\rVert_{L^{2}(\mathbb{T})}\lesssim\rVert
h\rVert_{L^{2}(\mathbb{T})}+\rVert h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}$
to estimate the second term on the right-hand side in (4.150), we obtain

$L_{1}\lesssim(1+\rVert R\rVert_{H^{k+4+\sigma}(\mathbb{T})})\rVert
h\rVert_{L^{2}(\mathbb{T})}+(1+\rVert R\rVert_{H^{4}(\mathbb{T})})\rVert
h^{(k+\sigma)}\rVert_{L^{2}(\mathbb{T})}.$

Similarly, we can obtain

$L_{2}\lesssim(1+\rVert R\rVert_{H^{k+4+\sigma}(\mathbb{T})})\rVert