††thanks: The author was supported in part by Sapienza ”Giovani Ricercatori”
Grant DR n.1607

# Global Well-Posedness for
Eddy-Mean Vorticity Equations on $\mathbb{T}^{2}$

Yuri Cacchio  SBAI Sapienza
Via Antonio Scarpa, 14
00161 Roma
Italy<EMAIL_ADDRESS>
###### Abstract.

We consider the two-dimensional, $\beta$-plane, eddy-mean vorticity equations
for an incompressible flow, where the zonally averaged flow varies on scales
much larger than the perturbation. We prove global existence and uniqueness of
the solution to the equations on periodic settings.

###### Key words and phrases:

Geophysical fluid dynamics, Well-Posedness, Vorticity equation

###### 1991 Mathematics Subject Classification:

35Q86

##  1\. Introduction

Atmospheric and oceanic flows on a rotating sphere are often modeled using
equations for a single fluid layer of constant density as a consequence of
their large horizontal extent compared with their depth. However, on large
scale these flows are strongly influenced not only by rotation but also by
stratification [2, 8, 19, 20, 24].

In three-dimensional stratified configurations turbulence naturally arises as
a result of dynamical instabilities such as baroclinic instability, however,
there is no such underlying instability in the two-dimensional case [8].
Consequently, it is common to artificially force two-dimensional turbulence
through an exogenous, statistically homogeneous, forcing function. A state of
equilibrium can then be achieved by the inclusion of dissipation terms [17,
28].

In this article we take into account only the rotation by neglecting
stratification in a single layer model where we incorporate the effects of
planetary rotation by adopting a beta-plane approximation, which is a simple
device used to represent the latitudinal variation in the vertical component
of the planetary rotation [12, 23, 33].

Keeping this in mind, let us start by considering the two-dimensional beta-
plane vorticity equation

$\partial_{t}\zeta+J(\psi,\zeta+\beta y)=\nu\nabla^{2}\zeta,$ (1.1)

where $J(A,B)=A_{x}B_{y}-A_{y}B_{x}$, which is completely determined by a
single dynamical variable, the stream function $\psi$, since the vorticity
$\zeta=\nabla^{2}\psi$.

Equation (1.1) has been used in a wide range of studies investigating large
scale planetary flows in double periodic geometry [7, 9, 18, 32] and on the
sphere [13, 22, 27].

Introducing the eddy-mean decomposition, sometimes called Reynolds
decomposition, denoted by a prime and an overbar respectively,

$f(t,x,y)=\overline{f}(t,y)+f^{\prime}(t,x,y),$ (1.2)

we can split equation (1.1) into mean and fluctuating components obtaining the
following initial value problem

$\left\\{\begin{array}[]{ll}&\partial_{t}\zeta^{\prime}+C_{1}\partial_{x}\nabla^{-2}\zeta^{\prime}-\nu\nabla^{2}\zeta^{\prime}=F(\overline{u},\zeta^{\prime})\\\
&\partial_{t}\overline{u}-\nu\nabla^{2}\overline{u}=G(\zeta^{\prime})\\\
&\zeta^{\prime}(0,x,y)=\zeta^{\prime}_{0}(x,y)\\\
&\overline{u}(0,y)=\overline{u}_{0}(y)\end{array}\right.$ (1.3)

with

$\displaystyle F(\overline{u},\zeta^{\prime})$
$\displaystyle:=(\partial_{y}\nabla^{-2}\zeta^{\prime})\zeta^{\prime}_{x}-(\partial_{x}\nabla^{-2}\zeta^{\prime})\zeta^{\prime}_{y}-\overline{u}\zeta^{\prime}_{x}+\partial_{y}\overline{[(\partial_{x}\nabla^{-2}\zeta^{\prime})\zeta^{\prime}]},$
$\displaystyle G(\zeta^{\prime})$
$\displaystyle:=\partial_{y}\overline{[\partial_{x}\nabla^{-2}\zeta^{\prime}\partial_{y}\nabla^{-2}\zeta^{\prime}]}.$

We denote by $\overline{u}=\overline{u}(t,y)$ the zonal mean velocity, also
called jet velocity profile, and by $\zeta^{\prime}=\zeta^{\prime}(t,x,y)$ the
eddy-vorticity; the parameter $\nu>0$ is called kinematic viscosity. Here, the
system of equations (1.3) is posed on a periodic box (periodic boundary
condition) $\mathbb{T}^{2}_{l}=[0,l)^{2}$ of size $l>0$.

In the next section we see a more detailed description of the model (1.1) and
how to get system (1.3) from equation (1.1).

The goal of the paper is to prove existence and uniqueness of the solution of
problem (1.3) in the periodic setting, which allows us to determine the
Reynolds stress as shown in Section 2. The latter quantity turns out to be
crucial in fluid dynamics since it is the component of the total stress tensor
in a fluid to account for turbulent fluctuations in fluid momentum. Moreover,
as written in [33, p. 414], the ”closure problem” of turbulence may be thought
of as finding a representation of such Reynolds stress terms in terms of mean
flow quantities, which seems to be critical without introducing physical
assumptions not directly deducible from the equations of motion themselves.

In light of that we prove the following global well-posedness result:

######  Theorem 1.1.

If $\overline{u}_{0}\in H^{s}(\mathbb{T}_{l})$ and $\zeta^{\prime}_{0}\in
H^{s}(\mathbb{T}^{2}_{l})$, $s\geq 0$, then there exists a unique global
solution $(\overline{u},\zeta^{\prime})\in L^{\infty}\left((0,+\infty);\
H^{s}(\mathbb{T}_{l})\crossproduct H^{s}(\mathbb{T}^{2}_{l})\right)$ of the
initial value problem $\eqref{generalProblem}$.

The proof of the above theorem is first obtained locally in time via a
contraction method, and then extended using an iteration method based on the
conservation of the $L^{2}$-norm. The estimates used in the argument are
reminiscent of those used to study periodic dispersive initial value problems
and inspired by the pioneering works of Bourgain [3, 4, 6] and Kenig, Ponce
and Vega [14, 15]. Relevant results in periodic settings can also be found in
[5, 30]. We also refer the interested reader to Tao’s notes [31] for an
introduction to this method, where he shows local well-posedness of the
Navier-Stokes equations.

Therefore, we proceed as follows:

  * 1)

Duhamel’s principle. Since in Theorem 1.1 we assume very low regularity it is
clear that the initial value problem (1.3) needs to be interpreted in an
appropriate manner. To this end the first step is to use the Duhamel’s
principle so that the solution to problem (1.3) can be interpreted as the
solution of the integral system

$\displaystyle(\overline{u},\zeta^{\prime})=\bigg{(}$ $\displaystyle e^{\nu
t\partial_{yy}}\overline{u}_{0}+\int_{0}^{t}e^{(\nu\partial_{yy})(t-t^{\prime})}G(\zeta^{\prime})\
dt^{\prime},$ (1.4) $\displaystyle
e^{(\nu\nabla^{2}-\partial_{x}\nabla^{-2})t}\left.\zeta^{\prime}_{0}+\int_{0}^{t}e^{(\nu\nabla^{2}-\partial_{x}\nabla^{-2})(t-t^{\prime})}F(\overline{u},\zeta^{\prime})\
dt^{\prime}\right)$

  * 2)

Upper bounds. The second step is to define the functional

$\displaystyle\Phi(\overline{u},\zeta^{\prime}):=\bigg{(}$ $\displaystyle
e^{\nu
t\partial_{yy}}\overline{u}_{0}+\int_{0}^{t}e^{(\nu\partial_{yy})(t-t^{\prime})}G(\zeta^{\prime})\
dt^{\prime},$ $\displaystyle
e^{(\nu\nabla^{2}-\partial_{x}\nabla^{-2})t}\left.\zeta^{\prime}_{0}+\int_{0}^{t}e^{(\nu\nabla^{2}-\partial_{x}\nabla^{-2})(t-t^{\prime})}F(\overline{u},\zeta^{\prime})\
dt^{\prime}\right)$

and prove that

$\norm{\Phi(\overline{u},\zeta^{\prime})}_{L^{\infty}_{[0,\delta]}(H^{s}(\mathbb{T}_{l})\crossproduct
H^{s}(\mathbb{T}_{l}^{2}))}\leq
C(\norm{\overline{u}_{0}}_{H^{s}(\mathbb{T}_{l})}+\norm{\zeta^{\prime}_{0}}_{H^{s}(\mathbb{T}_{l}^{2})}),$
(1.5)

$\forall\ s\geq 0$, on a small enough time interval $[0,\delta]$. Similarly
one also proves that $\Phi$ is a contraction. In this way we have that

$\Phi:B(0,R)\to B(0,R)$

on a suitable ball of radius $R>0$ and contraction.

  * 3)

Local Well-Posedness. The next step is to use the fixed point theorem on the
ball defined in step 2) in order to prove existence and uniqueness of the
solution on $B(0,R)$. After that, due to the estimate (1.5), we extend this
solution from the ball over the whole space
$L^{\infty}_{[0,\delta]}(H^{s}(\mathbb{T}_{l})\crossproduct
H^{s}(\mathbb{T}_{l}^{2}))$, with $s\geq 0$.

  * 4)

Global Well-Posedness. Finally, since the small time $\delta$ in Step 2
depends only on the $L^{2}$-norms of $\overline{u}$ and $\zeta^{\prime}$, and
these can be proved to be bounded by $L^{2}$-norms of initial data, i.e.,

$\norm{\overline{u}}_{L^{2}_{y}(\mathbb{T}_{l})}\leq
C\norm{\overline{u}_{0}}_{L^{2}_{y}(\mathbb{T}_{l})}\text{ and
}\norm{\zeta^{\prime}}_{L^{2}_{xy}(\mathbb{T}_{l}^{2})}\leq
C\norm{\zeta^{\prime}_{0}}_{L^{2}_{xy}(\mathbb{T}_{l}^{2})},$ (1.6)

for some universal constant $C$, and we can extend the solution by covering
the whole time interval $[0,\infty)$ by iteration.

The organization of the paper is as follows. In Section 2, we state
assumptions, certain terminology and we derive from the vorticity equation
(1.1) the system of equations (1.3). In Section 3, as mentioned in Steps 2 and
3 above, we prove inequality (1.5), we properly define the functional $\Phi$,
and after proving that it is a contraction we obtain local well-posedness on a
small enough time interval. Finally, we extend the solution by iteration to
get the global well-posedness result of (1.3) in the last section.

##  2\. The Eddy-Mean Decomposition and Assumptions

To establish the notation used in this paper, we start with the incompressible
2D Navier-Stokes equations [2, 16, 26, 29],

$\left\\{\begin{array}[]{rl}\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}+f\vec{\bm{z}}\crossproduct\bm{u}&=\nu\nabla^{2}\bm{u}-\nabla
p\\\ \divergence{\bm{u}}&=0\end{array}\right.$ (2.1)

where we include the effect of the planetary rotation through the Coriolis
force $f$ as a tool to understand various geophysical flows [1, 10, 11, 25].
Here $\bm{u}=(u,v)$ and $p$ are unknown velocity field and pressure, $\nu>0$
is the kinematic viscosity and $(\bm{u}\cdot\nabla)$ stands for the
differential operator $u\partial_{x}+v\partial_{y}$.

For the purpose of this discussion, we do an important approximation, so-
called $\beta$-$plane$, which captures the most important dynamical effects of
sphericity, without the complicating geometric effects, which are not
essential to describe many phenomena [33]. Since the magnitude of the vertical
component of rotation varies with latitude, we can approximate this effect by
allowing the effective rotation vector to vary. Thus, Taylor-expanding the
Coriolis parameter around a latitude $\varTheta_{0}$, for small variation in
latitude, we have [12, 23, 26]

$f=2\Omega\sin\varTheta\approx
2\Omega\sin{\varTheta_{0}}+2\Omega(\varTheta-\varTheta_{0})\cos{\varTheta_{0}},$
(2.2)

where $\varTheta$ is the latitude and $\Omega$ is the angular velocity of the
sphere. Then, on the tangent plane we may mimic this by allowing the Coriolis
parameter to vary as

$f=f_{0}+\beta y$ (2.3)

where $f_{0}=2\Omega\sin{\varTheta_{0}}$, $\beta=\partial f/\partial
y=(2\Omega\cos{\varTheta_{0}})/a$ and $a$ is the radius of the planet.

Moreover, since in 2D flows the velocity field has two component which depend
on two physical space coordinates and time, $\bm{u}=(u(t,x,y),v(t,x,y),0)$,
then the vorticity field $\bm{\zeta}$ has only one non-zero component,

$\bm{\zeta}:=\nabla\crossproduct\bm{u}=(0,0,\zeta(t,x,y))=(0,0,v_{x}-u_{y}).$
(2.4)

This component satisfies the $\beta$-plane vorticity equation,

$\zeta_{t}+u\zeta_{x}+v\zeta_{y}+\beta v=\nu\nabla^{2}\zeta,$ (2.5)

which is obtained by taking the curl from (2.1).

For 2D incompressible flows, we can introduce a representation of the velocity
field in terms of the stream function $\psi(t,x,y)$ as in [21],

$\displaystyle(u,v)$ $\displaystyle=(-\psi_{y},\psi_{x}),$ (2.6)
$\displaystyle\zeta=\nabla^{2}\psi$ $\displaystyle=\psi_{xx}+\psi_{yy}.$ (2.7)

In addition, since we work in a periodic frame, we can use the eddy-mean
decomposition (1.2) in (2.5), where

$\overline{f}=\frac{1}{l}\int_{0}^{l}f\ dx$

and get the zonal mean momentum equation

$\partial_{t}\overline{u}+\partial_{y}(\overline{u^{\prime}v^{\prime}})=\nu\nabla^{2}\overline{u},$
(2.8)

and the eddy vorticity equation

$\partial_{t}\zeta^{\prime}+\overline{u}\zeta^{\prime}_{x}+(\beta-\overline{u}_{yy})v^{\prime}=\nu\nabla^{2}\zeta^{\prime}+\partial_{y}(\overline{v^{\prime}\zeta^{\prime}})-u^{\prime}\zeta^{\prime}_{x}-v^{\prime}\zeta_{y}^{\prime}.$
(2.9)

The last hypothesis we make is to assume constant $\overline{u}_{yy}$, that is
we assume that the zonally averaged flow varies on scales much larger than the
perturbation [7, 29].

Finally, from (1.2), (2.6) and (2.7) we obtain

$\displaystyle u^{\prime}$
$\displaystyle=-\partial_{y}\nabla^{-2}\zeta^{\prime};$ $\displaystyle
v^{\prime}$ $\displaystyle=\partial_{x}\nabla^{-2}\zeta^{\prime},$ (2.10)

and we derive the initial value problem (1.3) by replacing relations (2) in
(2.8) and (2.9).

As mentioned in the introduction, proving well posedness for (1.3) will allow
us to derive the Reynolds stress using relations (2), since it is defined as
$\overline{u^{\prime}v^{\prime}}$ [20].

Before we address the computations of well posedness, with the purpose of
fixing the functions space on which we work, we show that the average of
$\overline{u}$ and $\zeta^{\prime}$ are both conserved. In fact, for
$\overline{u}$ we have

$\displaystyle\partial_{t}\int_{0}^{l}\overline{u}\ dy$
$\displaystyle=\int_{0}^{l}\partial_{t}\overline{u}\ dy$
$\displaystyle=\int_{0}^{l}\nu\partial^{2}_{y}\overline{u}+\partial_{y}\overline{[\partial_{x}\nabla^{-2}\zeta^{\prime}\partial_{y}\nabla^{-2}\zeta^{\prime}]}\
dy$
$\displaystyle=\nu\left[\partial_{y}\overline{u}\right]^{l}_{0}+\left[\overline{[\partial_{x}\nabla^{-2}\zeta^{\prime}\partial_{y}\nabla^{-2}\zeta^{\prime}]}\right]^{l}_{0}=0$

which vanishes by using periodic boundary conditions. On the other hand, for
$\zeta^{\prime}$, recalling that $\zeta=\zeta^{\prime}+\overline{\zeta}$ with
$\overline{\zeta}=\frac{1}{l}\int_{0}^{l}\zeta\ dx$, we have

$\int_{0}^{l}\int_{0}^{l}\zeta^{\prime}\ dxdy=\int_{0}^{l}\int_{0}^{l}\zeta\
dxdy-\int_{0}^{l}\int_{0}^{l}\overline{\zeta}\
dxdy=l\int_{0}^{l}\overline{\zeta}\ dy-l\int_{0}^{l}\overline{\zeta}\ dy=0.$

Then,

$\widehat{\zeta^{\prime}}(0)=\int_{0}^{l}\int_{0}^{l}\zeta^{\prime}(x,y)e^{i0\cdot(x,y)}\
dxdy=\int_{0}^{l}\int_{0}^{l}\zeta^{\prime}(x,y)\ dxdy=0,$ (2.11)

and if we define

$c_{0}=\int_{0}^{l}\overline{u}\ dy,$

we have,

$c_{0}=\int_{0}^{l}\overline{u}\ dy=\int_{0}^{l}\overline{u}(y)e^{i0\cdot y}\
dx=\widehat{\overline{u}}(0).$ (2.12)

We set

$\mu:=\overline{u}-c_{0}$

so that

$\widehat{\mu}(0)=0.$ (2.13)

Moreover, using relation (2.8),

$\displaystyle\partial_{t}\mu=\partial_{t}\overline{u}$
$\displaystyle=\nu\nabla^{2}\overline{u}+\partial_{y}\overline{[\partial_{x}\nabla^{-2}\zeta^{\prime}\partial_{y}\nabla^{-2}\zeta^{\prime}]}$
$\displaystyle=\nu\nabla^{2}\mu+\partial_{y}\overline{[\partial_{x}\nabla^{-2}\zeta^{\prime}\partial_{y}\nabla^{-2}\zeta^{\prime}]}.$
(2.14)

Then $\mu$ satisfies equation (2.8) with the additional condition (2.13). Due
to the above remarks, we study the initial value problem (1.3) equivalently in
the following functions spaces.

######  Definition 2.1.

$X^{s}:=\left\\{(f,g)\in L^{\infty}\left((0,\infty);\
H^{s}(\mathbb{T}_{l})\crossproduct
H^{s}(\mathbb{T}^{2}_{l})\right);\widehat{f}(0)=0,\
\widehat{g}(0)=0\right\\},$

and in a similar way

$X^{s,\delta}:=\left\\{(f,g)\in L^{\infty}\left([0,\delta];\
H^{s}(\mathbb{T}_{l})\crossproduct
H^{s}(\mathbb{T}^{2}_{l})\right);\widehat{f}(0)=0,\
\widehat{g}(0)=0\right\\},$

equipped with norm

$\displaystyle\norm{(f,g)}_{X^{s}}$
$\displaystyle=\norm{f}_{L^{\infty}_{[0,\infty)}H^{s}(\mathbb{T}_{l})}+\norm{g}_{L^{\infty}_{[0,\infty)}H^{s}(\mathbb{T}_{l}^{2})},$
$\displaystyle\norm{(f,g)}_{X^{s,\delta}}$
$\displaystyle=\norm{f}_{L^{\infty}_{[0,\delta]}H^{s}(\mathbb{T}_{l})}+\norm{g}_{L^{\infty}_{[0,\delta]}H^{s}(\mathbb{T}_{l}^{2})},$

respectively.

Via Plancherel’s theorem we express the $\norm{\cdot}_{H^{s}}$-norm in the
Fourier modes, namely

$\norm{f}^{2}_{H^{s}}=\sum_{k}\langle k\rangle^{2s}|\widehat{f}|^{2},$

where $\langle k\rangle:=(1+|k|^{2})^{\frac{1}{2}}$ is the Japanese bracket.

Notation 1: Throughout the paper we use $A\lesssim B$ to denote an estimate of
the form $A\leq CB$ for some absolute constant $C$. If $A\lesssim B$ and
$B\lesssim A$ we write $A\sim B$.

Notation 2: From here on we use the following notations,

$\displaystyle\norm{\cdot}_{L^{\infty}_{[0,\infty)}}$
$\displaystyle=\norm{\cdot}_{L^{\infty}_{t}}$
$\displaystyle\norm{\cdot}_{L^{\infty}_{[0,\delta]}}$
$\displaystyle=\norm{\cdot}_{L^{\infty}_{\delta}}$
$\displaystyle\norm{\cdot}_{H^{s}(\mathbb{T}^{2}_{l})}$
$\displaystyle=\norm{\cdot}_{H^{s}_{xy}}.$

##  3\. Local Well-Posedness

In this section we prove well posedness of (1.3) on the space $X^{s,\delta}$.
In order to do this, we first set the initial data of the problem. We fix

$\left(\zeta^{\prime}_{0}(x,y),\overline{u}_{0}(y)\right)\in
H^{s}(\mathbb{T}^{2}_{l})\crossproduct H^{s}(\mathbb{T}_{l}).$ (3.1)

Due to (2), we equivalently rewrite (1.3) in the following form

$\left\\{\begin{array}[]{rl}\partial_{t}\mu-\nu\partial_{yy}\mu&=G(\gamma),\\\
\partial_{t}\gamma+C_{1}\partial_{x}\nabla^{-2}\gamma-\nu\nabla^{2}\gamma&=F(\mu,\gamma),\\\
\mu(0,y)&=\mu_{0}(y)\in H^{s}(\mathbb{T}_{l}),\\\
\gamma(0,x,y)&=\gamma_{0}(x,y)\in
H^{s}(\mathbb{T}^{2}_{l}),\end{array}\right.$ (3.2)

where we define,

$\displaystyle\mu(t,y)$ $\displaystyle:=\overline{u}(t,y)-c_{0},$
$\displaystyle\gamma(t,x,y)$ $\displaystyle:=\zeta^{\prime}(t,x,y),$ (3.3)
$\displaystyle G(\gamma)$
$\displaystyle:=\partial_{y}\overline{[\partial_{x}\nabla^{-2}\gamma\partial_{y}\nabla^{-2}\gamma]},$
$\displaystyle F(\mu,\gamma)$
$\displaystyle:=(\partial_{y}\nabla^{-2}\gamma)\gamma_{x}-(\partial_{x}\nabla^{-2}\gamma)\gamma_{y}-\mu\gamma_{x}+c_{0}\gamma_{x}+\partial_{y}\overline{[(\partial_{x}\nabla^{-2}\gamma)\gamma]}.$

We now use the Duhamel’s principle so that the solution to problem (3.2) can
be interpreted as the solution of the integral system,

$\displaystyle(\mu,\gamma)=\bigg{(}$ $\displaystyle e^{\nu
t\partial_{yy}}\mu_{0}+\int_{0}^{t}e^{(\nu\partial_{yy})(t-t^{\prime})}G(\gamma)\
dt^{\prime},$ (3.4) $\displaystyle
e^{\tilde{D}^{2}t}\left.\gamma_{0}+\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)\
dt^{\prime}\right)$

with $\tilde{D}^{2}=\nu\nabla^{2}-C_{1}\partial_{x}\nabla^{-2}$. It is not
restrictive to assume $C_{1}=1$ as we will see below.

###  3.1.  Bounds and estimates

We proceed by deriving estimates of the norm of the various terms of the
Duhamel expressions above.

####  3.1.1. Zonal-Mean Momentum Equation Estimate

Let us start with the equation for $\mu$.

######  Proposition 3.1.

Let $t\in[0,\delta]$, $\delta\in\mathbb{R}_{+}$, $s\geq 0$,
$\alpha\in(\frac{1}{2},1)$. Then,

$\norm{\int_{0}^{t}e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)\
dt^{\prime}}_{L^{\infty}_{\delta}H_{y}^{s}}\lesssim\delta^{1-\alpha}\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}.$

###### Proof.

We recall that,

$\displaystyle\norm{\int_{0}^{t}e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)dt^{\prime}}_{L^{\infty}_{\delta}H_{y}^{s}}$
$\displaystyle=\sup_{t\in[0,\delta]}$
$\displaystyle\left(\sum_{k_{2}}\left|\int_{0}^{t}e^{-(t-t^{\prime})\nu
k^{2}_{2}}\widehat{G(\gamma)}(k_{2})dt^{\prime}\right|^{2}\langle
k_{2}\rangle^{2s}\right)^{\frac{1}{2}}$

where,

$\displaystyle\widehat{G(\gamma)}(k_{2})$
$\displaystyle=\mathfrak{F}_{k}\left(\partial_{y}\overline{[\partial_{x}\nabla^{-2}\gamma\partial_{y}\nabla^{-2}\gamma]}\right)$
$\displaystyle=ik_{2}\left.\left(-i\frac{k_{1}}{|k|^{2}}\widehat{\gamma}*(-i\frac{k_{2}}{|k|^{2}}\widehat{\gamma})\right)\right|_{[0,k_{2}]}$
$\displaystyle=\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\\\ m\neq 0\\\ h\neq
0\end{subarray}}i\frac{h_{1}m_{2}k_{2}}{|h|^{2}|m|^{2}}\widehat{\gamma}(h)\widehat{\gamma}(m).$
(3.5)

In fact, for a general function $f(x,y)\in H^{s}_{xy}(\mathbb{T}^{2}_{l})$,

$\overline{f(x,y)}:=\frac{1}{l}\int_{0}^{l}f(x,y)\ dx=\overline{f}(y)$

and

$\displaystyle\widehat{\overline{f(x,y)}}(k_{1},k_{2})$ $\displaystyle=\int
e^{iyk_{2}}\overline{f}(y)\ dy$ $\displaystyle=\int
e^{iyk_{2}}\left(\frac{1}{l}\int_{0}^{l}f(x,y)\ dx\right)dy$
$\displaystyle=\frac{1}{l}\int_{0}^{l}\left(\int e^{iyk_{2}}f(x,y)\
dy\right)dx$ $\displaystyle=\frac{1}{l}\int_{0}^{l}\widehat{f}(0,k_{2})\ dx$
$\displaystyle=\widehat{f}(0,k_{2})\frac{1}{l}\int_{0}^{l}dx=\widehat{f}(0,k_{2}).$

######  Remark 3.2.

Due to (2.11) and (2.13) we can assume $m\neq 0$ and $h\neq 0$ in (3.5).
Henceforth we omit this subscript.

For now, we only consider

$\displaystyle\norm{\int_{0}^{t}e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)dt^{\prime}}^{2}_{H_{y}^{s}}$
$\displaystyle\leq\int_{0}^{t}\norm{e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)}^{2}_{H_{y}^{s}}dt^{\prime}$
$\displaystyle=\int_{0}^{t}\sum_{k_{2}}\left|e^{-(t-t^{\prime})\nu
k^{2}_{2}}\widehat{G(\gamma)}(k_{2})\right|^{2}\langle
k_{2}\rangle^{2s}dt^{\prime}.$

Using (3.5),

$\displaystyle=\int_{0}^{t}\sum_{k_{2}}\Bigg{|}e^{-(t-t^{\prime})\nu
k^{2}_{2}}\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}i\frac{h_{1}m_{2}k_{2}}{|h|^{2}|m|^{2}}\widehat{\gamma}(h)\widehat{\gamma}(m)\Bigg{|}^{2}\langle
k_{2}\rangle^{2s}dt^{\prime}$
$\displaystyle\leq\int_{0}^{t}\sum_{k_{2}}\Bigg{(}e^{-(t-t^{\prime})\nu
k^{2}_{2}}\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|h||m||k_{2}|\langle
k_{2}\rangle^{s}}{|h|^{2}|m|^{2}}\widehat{\gamma}(h)\widehat{\gamma}(m)\Bigg{)}^{2}dt^{\prime}$
$\displaystyle=\int_{0}^{t}\sum_{k_{2}}\Bigg{(}\frac{e^{-(t-t^{\prime})\nu
k^{2}_{2}}((t-t^{\prime})\nu k_{2}^{2})^{\alpha}}{((t-t^{\prime})\nu
k_{2}^{2})^{\alpha}}\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k_{2}|\langle
k_{2}\rangle^{s}}{|h||m|}\widehat{\gamma}(h)\widehat{\gamma}(m)\Bigg{)}^{2}dt^{\prime}.$

Since

$e^{-(t-t^{\prime})\nu k^{2}_{2}}((t-t^{\prime})\nu k^{2}_{2})^{\alpha}\leq
C,\ \ \alpha\in[0,1],$ (3.6)

is uniformly bounded, we can focus on

$\sum_{k_{2}}\Bigg{(}\frac{1}{(\nu
k_{2}^{2})^{\alpha}}\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k_{2}|\langle
k_{2}\rangle^{s}}{|h||m|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|\Bigg{)}^{2},$

which is equivalent to

$\sum_{k_{2}}\Bigg{(}\frac{1}{(\nu
k_{2}^{2})^{\alpha}}\sum_{\begin{subarray}{c}h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k_{2}|\langle
k_{2}\rangle^{s}}{|h||\tilde{m}|}|\widehat{\gamma}(h)||\widehat{\gamma}(\tilde{m})|\Bigg{)}^{2}$

with

$\tilde{m}=(-h_{1},m_{2}).$

By duality, we want to show

$\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k_{2}}\sum_{\begin{subarray}{c}h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{V(k_{2},h,\tilde{m})}{\langle
h\rangle^{s}\langle\tilde{m}\rangle^{s}}f_{1}(h)f_{2}(\tilde{m})g(k_{2})\lesssim\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}},$

where

$\displaystyle V(k_{2},h,\tilde{m})$ $\displaystyle:=\frac{\langle
k_{2}\rangle^{s}}{|k_{2}|^{2\alpha-1}|h||\tilde{m}|};$ $\displaystyle
f_{1}(h)$ $\displaystyle:=\langle h\rangle^{s}|\widehat{\gamma}(h)|;$
$\displaystyle f_{2}(\tilde{m})$
$\displaystyle:=\langle\tilde{m}\rangle^{s}|\widehat{\gamma}(\tilde{m})|.$

We remark that if $\gamma\in H^{s}$ then $f_{i}\in l^{2}$ for $i=1,2$.
Moreover, we have

$\langle k\rangle^{s}=\left((1+|k|^{2})^{\frac{1}{2}}\right)^{s}\sim|k|^{s}.$

Hence,

$\frac{V(k_{2},h,\tilde{m})}{\langle
h\rangle^{s}\langle\tilde{m}\rangle^{s}}\sim\frac{|k_{2}|^{s-2\alpha+1}}{|h|^{s+1}|\tilde{m}|^{s+1}}.$

We are ready to study all possible cases as $h,m$ and $k$ varies:

  * 1)

$|h_{2}|\gg|m_{2}|\Rightarrow|h_{2}|\sim|k_{2}|$.

$\displaystyle\frac{|k_{2}|^{s-2\alpha+1}}{|h|^{s+1}|\tilde{m}|^{s+1}}\sim\frac{|h_{2}|^{s-2\alpha+1}}{(|h_{1}|^{2}+|h_{2}|^{2})^{\frac{s+1}{2}}(|h_{1}|^{2}+|m_{2}|^{2})^{\frac{s+1}{2}}}.$

Since $|\tilde{m}|^{s+1}=(|h_{1}|^{2}+|m_{2}|^{2})^{\frac{s+1}{2}}\geq 1$ and
$|h_{2}|\neq 0$ because $|h_{2}|\gg|m_{2}|$,

$\displaystyle\leq\frac{|h_{2}|^{s-2\alpha+1}}{|h_{2}|^{s+1}}\leq\frac{1}{|h_{2}|^{2\alpha}}.$

Using three times Cauchy-Schwartz inequality we have

$\displaystyle\sum_{k_{2}}\sum_{\begin{subarray}{c}h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{V(k_{2},h,\tilde{m})}{\langle
h\rangle^{s}\langle\tilde{m}\rangle^{s}}f_{1}(h)f_{2}(\tilde{m})g(k_{2})$
$\displaystyle\leq\sum_{k_{2}=h_{2}+m_{2}}\frac{1}{|h_{2}|^{2\alpha}}g(k_{2})\sum_{h_{1}}f_{1}(h_{1},h_{2})f_{2}(-h_{1},m_{2})$
$\displaystyle\leq\sum_{k_{2}=h_{2}+m_{2}}\frac{g(k_{2})}{|h_{2}|^{2\alpha}}\left(\sum_{h_{1}}|f_{1}(h_{1},h_{2})|^{2}\right)^{\frac{1}{2}}\left(\sum_{h_{1}}|f_{2}(-h_{1},m_{2})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{h_{2}}\sum_{k_{2}}\frac{g(k_{2})}{|h_{2}|^{2\alpha}}\left(\sum_{h_{1}}|f_{1}(h_{1},h_{2})|^{2}\right)^{\frac{1}{2}}\left(\sum_{h_{1}}|f_{2}(-h_{1},k_{2}-h_{2})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\norm{g}_{l_{2}}\norm{f_{2}}_{l_{2}}\sum_{h_{2}}\frac{1}{|h_{2}|^{2\alpha}}\left(\sum_{h_{1}}|f_{1}(h_{1},h_{2})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\norm{g}_{l_{2}}\norm{f_{1}}_{l_{2}}\norm{f_{2}}_{l_{2}}\left(\sum_{h_{2}}\frac{1}{|h_{2}|^{4\alpha}}\right)^{\frac{1}{2}}.$

The last term is summable if $4\alpha>1$, i.e.

$\alpha>\frac{1}{4}.$ (3.7)

  * 2)

$|h_{2}|\ll|m_{2}|\Rightarrow|m_{2}|\sim|k_{2}|$.

$\displaystyle\frac{|k_{2}|^{s-2\alpha+1}}{|h|^{s+1}|\tilde{m}|^{s+1}}$
$\displaystyle\sim\frac{|m_{2}|^{s-2\alpha+1}}{(|h_{1}|^{2}+|h_{2}|^{2})^{\frac{s+1}{2}}(|h_{1}|^{2}+|m_{2}|^{2})^{\frac{s+1}{2}}}$
$\displaystyle\leq\frac{|m_{2}|^{s-2\alpha+1}}{|m_{2}|^{s+1}}\leq\frac{1}{|m_{2}|^{2\alpha}}$

because $|h|^{s+1}=(|h_{1}|^{2}+|h_{2}|^{2})^{\frac{s+1}{2}}\geq 1$ and
$|m_{2}|\neq 0$ since $|m_{2}|\gg|h_{2}|$. Then,

$\displaystyle\sum_{k_{2}}\sum_{\begin{subarray}{c}h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{V(k_{2},h,\tilde{m})}{\langle
h\rangle^{s}\langle\tilde{m}\rangle^{s}}f_{1}(h)f_{2}(\tilde{m})g(k_{2})$
$\displaystyle\leq\sum_{k_{2}=h_{2}+m_{2}}\frac{1}{|m_{2}|^{2\alpha}}g(k_{2})\sum_{h_{1}}f_{1}(h_{1},h_{2})f_{2}(-h_{1},m_{2})$
$\displaystyle\leq\sum_{k_{2}=h_{2}+m_{2}}\frac{g(k_{2})}{|m_{2}|^{2\alpha}}\left(\sum_{h_{1}}|f_{1}(h_{1},h_{2})|^{2}\right)^{\frac{1}{2}}\left(\sum_{h_{1}}|f_{2}(-h_{1},m_{2})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{m_{2}}\sum_{k_{2}}\frac{g(k_{2})}{|m_{2}|^{2\alpha}}\left(\sum_{h_{1}}|f_{1}(h_{1},k_{2}-m_{2})|^{2}\right)^{\frac{1}{2}}\left(\sum_{h_{1}}|f_{2}(-h_{1},m_{2})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\norm{g}_{l_{2}}\norm{f_{1}}_{l_{2}}\sum_{m_{2}}\frac{1}{|m_{2}|^{2\alpha}}\left(\sum_{h_{1}}|f_{2}(-h_{1},m_{2})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\norm{g}_{l_{2}}\norm{f_{1}}_{l_{2}}\norm{f_{2}}_{l_{2}}\left(\sum_{m_{2}}\frac{1}{|m_{2}|^{4\alpha}}\right)^{\frac{1}{2}}.$

As above, the last term is summable if

$\alpha>\frac{1}{4}.$

  * 3)

$|h_{2}|\sim|m_{2}|\Rightarrow|k_{2}|\sim|h_{2}|\sim|m_{2}|$.

$\displaystyle\frac{|k_{2}|^{s-2\alpha+1}}{|h|^{s+1}|\tilde{m}|^{s+1}}$
$\displaystyle\sim\frac{|h_{2}|^{s-2\alpha+1}}{(|h_{1}|^{2}+|h_{2}|^{2})^{\frac{s+1}{2}}(|h_{1}|^{2}+|m_{2}|^{2})^{\frac{s+1}{2}}}$

Since $|k_{2}|\neq 0$ at least one between $|h_{2}|$ and $|m_{2}|$ is not
zero. Moreover $|k_{2}|\sim|h_{2}|\sim|m_{2}|$, then

$\displaystyle\frac{|k_{2}|^{s-2\alpha+1}}{|h|^{s+1}|\tilde{m}|^{s+1}}\lesssim\frac{1}{|h_{2}|^{2\alpha}}$

and we conclude as in the previous case.

By $1),2),3)$ we have that

$\displaystyle s$ $\displaystyle\geq 0;$ $\displaystyle\alpha$
$\displaystyle>\frac{1}{4}.$

Then, by (3.6) and definitions of $f_{1}$ and $f_{2}$,

$\displaystyle\norm{\int_{0}^{t}e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)dt^{\prime}}_{H_{y}^{s}}$
$\displaystyle\lesssim\norm{\gamma}^{2}_{H^{s}_{xy}}\int_{0}^{t}\frac{1}{(t-t^{\prime})^{\alpha}}dt^{\prime}$
$\displaystyle\sim t^{1-\alpha}\norm{\gamma}^{2}_{H^{s}_{xy}},$

for $\frac{1}{4}<\alpha<1$ and $s\geq 0$. Finally

$\displaystyle\norm{\int_{0}^{t}e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)dt^{\prime}}_{L^{\infty}_{\delta}H_{y}^{s}}\lesssim\delta^{1-\alpha}\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}$

for $\frac{1}{4}<\alpha<1$ and $s\geq 0$. ∎

We now define the following functional,

$\Phi_{1}(\mu,\gamma)=e^{\nu
t\partial_{yy}}\mu_{0}+\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}G(\gamma)dt^{\prime},$
(3.8)

and as a consequence of the previous proposition we have,

######  Corollary 3.3.

Let $t\in[0,\delta]$, $\delta\in\mathbb{R}_{+}$, $s\geq 0$,
$\alpha\in(\frac{1}{2},1)$. Then,

$\norm{\Phi_{1}(\mu,\gamma)}_{L^{\infty}_{\delta}H^{s}_{y}}\lesssim\norm{\mu_{0}}_{H^{s}_{y}}+\delta^{1-\alpha}\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}.$

###### Proof.

$\norm{\Phi_{1}(\mu,\gamma)}_{L^{\infty}_{\delta}H^{s}_{y}}\leq\underbrace{\norm{e^{\nu
t\partial_{yy}}\mu_{0}}_{L^{\infty}_{\delta}H^{s}_{y}}}_{A}+\underbrace{\norm{\int_{0}^{t}e^{(t-t^{\prime})\nu\partial_{yy}}G(\gamma)dt^{\prime}}^{2}_{L^{\infty}_{t}H_{y}^{s}}}_{B}.$

By Proposition 3.1 we have

$B\lesssim\delta^{1-\alpha}\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}.$

On the other hand,

$\displaystyle A=\norm{e^{\nu
t\partial_{yy}}\mu_{0}}_{L^{\infty}_{\delta}H^{s}_{y}}$
$\displaystyle=\sup_{t\in[0,\delta]}\left(\sum_{k_{2}}\left|e^{-t\nu
k^{2}_{2}}\widehat{\mu}_{0}(k_{2})\right|^{2}\langle
k_{2}\rangle^{2s}\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\left(\sum_{k_{2}}\left|\widehat{\mu}_{0}(k_{2})\right|^{2}\langle
k_{2}\rangle^{2s}\right)^{\frac{1}{2}}=\norm{\mu_{0}}_{H^{s}_{y}}.$

∎

####  3.1.2. Eddy-Vorticity Equation Estimate

Similarly, we study the equation for $\gamma$.

######  Proposition 3.4.

Let $t\in[0,\delta]$, $\delta\in\mathbb{R}_{+}$, $s\geq 0$,
$\alpha\in(\frac{3}{4},1)$. Then,

$\displaystyle\norm{\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)\
dt^{\prime}}_{L^{\infty}_{\delta}H_{xy}^{s}}$
$\displaystyle\lesssim\delta^{1-\alpha}\left(\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}\right.$
$\displaystyle\left.+\norm{\mu}_{L^{\infty}_{\delta}H^{s}_{y}}\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}\right).$

###### Proof.

By definition,

$\displaystyle\norm{\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)\
dt^{\prime}}_{L^{\infty}_{\delta}H_{xy}^{s}}$
$\displaystyle=\sup_{t\in[0,\delta]}\left(\sum_{k}\left|\int_{0}^{t}\right.\right.$
$\displaystyle\left.\left.e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})(t-t^{\prime})}\widehat{F(\mu,\gamma)}(k)dt^{\prime}\right|^{2}\langle
k\rangle^{2s}\right)^{\frac{1}{2}}$

where,

$\displaystyle\widehat{F(\mu,\gamma)}$ $\displaystyle(k)$
$\displaystyle=\mathfrak{F}\left((\partial_{y}\nabla^{-2}\zeta^{\prime})\zeta^{\prime}_{x}-(\partial_{x}\nabla^{-2}\zeta^{\prime})\zeta^{\prime}_{y}-\mu\gamma_{x}+c_{0}\gamma_{x}+\partial_{y}\overline{[(\partial_{x}\nabla^{-2}\zeta^{\prime})\zeta^{\prime}]}\right)$
$\displaystyle=-\left(i\frac{k_{2}}{|k|^{2}}\widehat{\gamma}\right)*(ik_{1}\widehat{\gamma})+\left(i\frac{k_{1}}{|k|^{2}}\widehat{\gamma}\right)*\left(ik_{2}\widehat{\gamma}\right)-\widehat{\mu}(k_{2})*ik_{1}\widehat{\gamma}(k)$
$\displaystyle\left.+c_{0}ik_{1}\widehat{\gamma}(k)-ik_{2}\left(i\frac{k_{1}}{|k|^{2}}\widehat{\gamma}*\widehat{\gamma}\right)\right|_{[0,k_{2}]}$
$\displaystyle=\sum_{k=h+m}\left(\frac{h_{2}m_{1}}{|h|^{2}}-\frac{h_{1}m_{2}}{|h|^{2}}\right)\widehat{\gamma}(h)\widehat{\gamma}(m)-\sum_{\begin{subarray}{c}k_{1}=h_{1}+0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}ih_{1}\widehat{\mu}(m_{2})\widehat{\gamma}(h)$
$\displaystyle+c_{0}ik_{1}\widehat{\gamma}(k)+\sum_{\begin{subarray}{c}0=k_{1}=h_{1}+m_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{k_{2}h_{1}}{|h|^{2}}\widehat{\gamma}(h)\widehat{\gamma}(m).$

For now, we only consider

$\displaystyle\bigg{\|}\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}$
$\displaystyle F(\mu,\gamma)\ dt^{\prime}\bigg{\|}^{2}_{H_{xy}^{s}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\norm{\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)\
dt^{\prime}}^{2}_{H_{xy}^{s}}dt^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{k}\left|e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})(t-t^{\prime})}\widehat{F(\mu,\gamma)}(k)\right|^{2}\langle
k\rangle^{2s}dt^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{k}\left|e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})(t-t^{\prime})}\bigg{(}\sum_{k=h+m}\left(\frac{h_{2}m_{1}}{|h|^{2}}-\frac{h_{1}m_{2}}{|h|^{2}}\right)\widehat{\gamma}(h)\widehat{\gamma}(m)\right.$
$\displaystyle+\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}ih_{1}\widehat{\mu}(m_{2})\widehat{\gamma}(h)+c_{0}ik_{1}\widehat{\gamma}(k)$
$\displaystyle+\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\left.\frac{k_{2}h_{1}}{|h|^{2}}\widehat{\gamma}(h)\widehat{\gamma}(m)\bigg{)}\right|^{2}\langle
k\rangle^{2s}dt^{\prime}$
$\displaystyle\leq\int_{0}^{t}\sum_{k}\Bigg{[}e^{-\nu|k|^{2}(t-t^{\prime})}\langle
k\rangle^{s}$
$\displaystyle\bigg{(}\sum_{k=h+m}\left(\frac{|h||m|+|h||m|}{|h|^{2}}\right)|\widehat{\gamma}(h)||\widehat{\gamma}(m)|$
$\displaystyle+\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}|h||\widehat{\mu}(m_{2})||\widehat{\gamma}(h)|+c_{0}|k||\widehat{\gamma}(k)|$
$\displaystyle+\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k||h|}{|h|^{2}}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|\bigg{)}\Bigg{]}^{2}dt^{\prime}.$

Multiplying and dividing by $(\nu|k|^{2}(t-t^{\prime}))^{\alpha}$,

$\displaystyle=\int_{0}^{t}\sum_{k}\Bigg{[}\frac{e^{-\nu|k|^{2}(t-t^{\prime})}(\nu|k|^{2}(t-t^{\prime}))^{\alpha}}{(\nu|k|^{2}(t-t^{\prime}))^{\alpha}}$
$\displaystyle\langle
k\rangle^{s}\Bigg{(}\sum_{k=h+m}2\frac{|m|}{|h|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|$
$\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}|h||\widehat{\mu}(m_{2})||\widehat{\gamma}(h)|+c_{0}|k||\widehat{\gamma}(k)|$
$\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}k_{1}=h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|\Bigg{)}\Bigg{]}^{2}dt^{\prime}.$

Since

$e^{-\nu|k|^{2}(t-t^{\prime})}(\nu|k|^{2}(t-t^{\prime}))^{\alpha}\leq C,\ \ \
C\in\mathbb{R}$ (3.9)

is uniformly bounded, we study

$\displaystyle\sum_{k}\left[\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\right.\bigg{(}\sum_{k=h+m}$ $\displaystyle
2\frac{|m|}{|h|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|+\sum_{\begin{subarray}{c}k_{1}=h_{1}+0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}|h||\widehat{\mu}(m_{2})||\widehat{\gamma}(h)|$
$\displaystyle+c_{0}|k||\widehat{\gamma}(k)|+\sum_{\begin{subarray}{c}h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|\bigg{)}\bigg{]}^{2}.$

By duality we want to show

$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\Bigg{(}\sum_{k=h+m}$ $\displaystyle
2\frac{|m|}{|h|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|+\sum_{\begin{subarray}{c}k_{1}=h_{1}+0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}|h||\widehat{\mu}(m_{2})||\widehat{\gamma}(h)|$
$\displaystyle+c_{0}|k||\widehat{\gamma}(k)|+\sum_{\begin{subarray}{c}h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|}|\widehat{\gamma}(h)||\widehat{\gamma}(m)|\Bigg{)}g(k)$
$\displaystyle\lesssim\big{(}\norm{f_{1}}^{2}_{l^{2}}$
$\displaystyle\norm{g}_{l^{2}}+\norm{f_{1}}_{l^{2}}\norm{g}_{l^{2}}+\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}}\big{)}$
(3.10)

where

$\displaystyle f_{1}(k)$ $\displaystyle:=\langle
k\rangle^{s}|\widehat{\gamma}(k)|,$ $\displaystyle f_{2}(k_{2})$
$\displaystyle:=\langle k_{2}\rangle^{s}|\widehat{\mu}(k_{2})|.$

We observe that if $\gamma,\ \mu\in H^{s}$ then $f_{i}\in l^{2}$ for $i=1,2$.
Since

$\sup(a+b)\leq\sup(a)+\sup(b),$

we split (3.1.2) and study the following problems

$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{k=h+m}\frac{|m|}{|h|}\frac{1}{\langle
h\rangle^{s}\langle
m\rangle^{s}}f_{1}(h)f_{1}(m)g(k)\lesssim\norm{f_{1}}^{2}_{l^{2}}\norm{g}_{l^{2}}$
(3.11) $\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{\begin{subarray}{c}h_{1}+m_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|}\frac{1}{\langle
h\rangle^{s}\langle
m\rangle^{s}}f_{1}(h)f_{1}(m)g(k)\lesssim\norm{f_{1}}_{l^{2}}^{2}\norm{g}_{l^{2}}$
(3.12) $\displaystyle\sup_{\norm{g}_{l^{2}}\leq
1}\sum_{k}c_{0}\frac{|k|}{|k|^{2\alpha}}f_{1}(k)g(k)\lesssim\norm{f_{1}}_{l^{2}}\norm{g}_{l^{2}}$
(3.13) $\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|h|}{\langle h\rangle^{s}\langle
m_{2}\rangle^{s}}f_{1}(h)f_{2}(m_{2})g(k)\lesssim\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}}.$
(3.14)

We start with (3.11) and since

$\langle k\rangle^{s}\sim|k|^{s},$ (3.15)

we have

$\frac{\langle k\rangle^{s}}{|k|^{2\alpha}}\frac{|m|}{|h|}\frac{1}{\langle
h\rangle^{s}\langle
m\rangle^{s}}\sim\frac{|k|^{s-2\alpha}}{|h|^{s+1}|m|^{s-1}}.$

  * 1)

$|h|\gg|m|\Rightarrow|k|\sim|h|$.

$\frac{|k|^{s-2\alpha}}{|h|^{s+1}|m|^{s-1}}\sim\frac{1}{|h|^{2\alpha+1}|m|^{s-1}}\leq\frac{1}{|m|^{2\alpha+s}}.$

Proceeding as in Proposition 3.1, we get

$\displaystyle\sum_{k}\sum_{k=h+m}$
$\displaystyle\frac{1}{|m|^{2\alpha+s}}f_{1}(h)f_{1}(m)g(k)$ $\displaystyle=$
$\displaystyle\sum_{m}\frac{1}{|m|^{2\alpha+s}}f_{1}(m)\sum_{k}f_{1}(k-m)g(k)$
$\displaystyle\leq$
$\displaystyle\sum_{m}\frac{1}{|m|^{2\alpha+s}}f_{1}(m)\left(\sum_{k}f_{1}^{2}(k-m)\right)^{\frac{1}{2}}\left(\sum_{k}g^{2}(k)\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{m}\frac{1}{|m|^{4\alpha+2s}}\right)^{\frac{1}{2}}\left(\sum_{m}f_{1}^{2}(m)\right)^{\frac{1}{2}}\norm{f_{1}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{m}\frac{1}{|m|^{4\alpha+2s}}\right)^{\frac{1}{2}}\norm{f_{1}}^{2}_{l^{2}}\norm{g}_{l^{2}}.$

The first term is summable if

$4\alpha+2s>2.$

The worst case is when $s=0$, but in this situation it is sufficient to choose

$\alpha>\frac{1}{2}.$

  * 2)

$|h|\ll|m|\Rightarrow|k|\sim|m|$.

$\frac{|k|^{s-2\alpha}}{|h|^{s+1}|m|^{s-1}}\sim\frac{1}{|h|^{s+1}|m|^{2\alpha-1}}\overbrace{\leq}^{\alpha>\frac{1}{2}}\frac{1}{|h|^{2\alpha+s}}\overbrace{\leq}^{s\geq
0}\frac{1}{|h|^{2\alpha}}.$

Then,

$\displaystyle\sum_{k}\sum_{k=h+m}$
$\displaystyle\frac{1}{|h|^{2\alpha}}f_{1}(h)f_{1}(m)g(k)$
$\displaystyle=\sum_{h}\frac{1}{|h|^{2\alpha}}f_{1}(h)\sum_{m}f_{1}(m)g(h+m)$
$\displaystyle\leq\left(\sum_{h}\frac{1}{|h|^{4\alpha}}\right)^{\frac{1}{2}}\norm{f_{1}}^{2}_{l^{2}}\norm{g}_{l^{2}}$

which is summable if $\alpha>\frac{1}{2}$.

  * 3)

$|h|\sim|m|\Rightarrow|k|\sim|h|\sim|m|$.

$\frac{|k|^{s-2\alpha}}{|h|^{s+1}|m|^{s-1}}\sim\frac{1}{|h|^{2\alpha+s}}\leq\frac{1}{|h|^{2\alpha}}.$

we conclude as in the previous case.

By $1),2),3)$ we get

$\displaystyle s$ $\displaystyle\geq 0;$ $\displaystyle\alpha$
$\displaystyle>\frac{1}{2}.$

Similarly, we study (3.12). We have

$\frac{|k|}{|h|\langle h\rangle^{s}\langle
m\rangle^{s}}\sim\frac{|k|}{|h|^{s+1}|m|^{s}},$

so that we write

$\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}g(k)\sum_{\begin{subarray}{c}k_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|^{s+1}|m|^{s}}f_{1}(h)f_{1}(m)$
(3.16)

If we define $\tilde{m}=(-h_{1},m_{2})$, equation (3.16) becomes

$\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}g(k)\sum_{k=h+\tilde{m}}\frac{|k|}{|h|^{s+1}|\tilde{m}|^{s}}f_{1}(h)f_{1}(\tilde{m})$

  * 1)

$|h|\gg|\tilde{m}|\Rightarrow|k|\sim|h|$.

$\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\frac{|k|}{|h|^{s+1}|\tilde{m}|^{s}}\sim\frac{1}{|h|^{2\alpha}|\tilde{m}|^{s}}\leq\frac{1}{|\tilde{m}|^{2\alpha}|\tilde{m}|^{s}}\leq\frac{1}{|\tilde{m}|^{2\alpha}}.$

Then,

$\displaystyle\sum_{k}\sum_{k=h+\tilde{m}}$
$\displaystyle\frac{1}{|\tilde{m}|^{2\alpha}}f_{1}(h)f_{1}(\tilde{m})g(k)$
$\displaystyle=$
$\displaystyle\sum_{\tilde{m}}\frac{1}{|\tilde{m}|^{2\alpha}}f_{1}(\tilde{m})\sum_{k}f_{1}(k-\tilde{m})g(k)$
$\displaystyle\leq$
$\displaystyle\sum_{\tilde{m}}\frac{1}{|\tilde{m}|^{2\alpha}}f_{1}(\tilde{m})\left(\sum_{k}f_{1}^{2}(k-\tilde{m})\right)^{\frac{1}{2}}\left(\sum_{k}g^{2}(k)\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{\tilde{m}}\frac{1}{|\tilde{m}|^{4\alpha}}\right)^{\frac{1}{2}}\left(\sum_{\tilde{m}}f_{1}^{2}(\tilde{m})\right)^{\frac{1}{2}}\norm{f_{1}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{\tilde{m}}\frac{1}{|\tilde{m}|^{4\alpha}}\right)^{\frac{1}{2}}\norm{f_{1}}^{2}_{l^{2}}\norm{g}_{l^{2}}.$

The first term is summable if

$\alpha>\frac{1}{2}.$

  * 2)

$|h|\ll|\tilde{m}|\Rightarrow|k|\sim|\tilde{m}|$.

$\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\frac{|k|}{|h|^{s+1}|\tilde{m}|^{s}}\sim\frac{1}{|k|^{2\alpha-1}|h|^{s+1}}\overbrace{\leq}^{\alpha>\frac{1}{2}}\frac{1}{|h|^{2\alpha-1}|h|^{s+1}}\leq\frac{1}{|h|^{2\alpha}}.$

Then,

$\displaystyle\sum_{k}\sum_{k=h+\tilde{m}}$
$\displaystyle\frac{1}{|h|^{2\alpha}}f_{1}(h)f_{1}(\tilde{m})g(k)$
$\displaystyle=\sum_{h}\frac{1}{|h|^{2\alpha}}f_{1}(h)\sum_{\tilde{m}}f_{1}(\tilde{m})g(h+\tilde{m})$
$\displaystyle\leq\left(\sum_{h}\frac{1}{|h|^{4\alpha}}\right)^{\frac{1}{2}}\norm{f_{1}}^{2}_{l^{2}}\norm{g}_{l^{2}},$

which is summable if $\alpha>\frac{1}{2}$.

  * 3)

$|h|\sim|\tilde{m}|\Rightarrow|k|\sim|h|\sim|\tilde{m}|$.

$\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\frac{|k|}{|h|^{s+1}|\tilde{m}|^{s}}\lesssim\frac{1}{|h|^{2\alpha}}.$

We conclude as in the previous case.

By $1),2),3)$ we get

$\displaystyle s$ $\displaystyle\geq 0;$ $\displaystyle\alpha$
$\displaystyle>\frac{1}{2}.$

For (3.13), if we assume $\alpha\geq\frac{1}{2}$

$\displaystyle\sum_{k}c_{0}\frac{1}{|k|^{2\alpha-1}}f_{1}(k)g(k)$
$\displaystyle\leq
c_{0}\left(\sum_{k}f_{1}^{2}(k)\right)^{\frac{1}{2}}\left(\sum_{k}g^{2}(k)\right)^{\frac{1}{2}}$
$\displaystyle\sim\norm{f_{1}}_{l^{2}}\norm{g}_{l_{2}},$

where we used $|k|\geq 1$.

Finally, we study (3.14). We have

$\displaystyle\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}g(k)\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|h|}{\langle h\rangle^{s}\langle
m_{2}\rangle^{s}}f_{1}(h)f_{2}(m_{2})$ $\displaystyle\sim$
$\displaystyle\sum_{k}\frac{|k|^{s}}{|k|^{2\alpha}}g(k)\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|h|}{|h|^{s}|m_{2}|^{s}}f_{1}(h)f_{2}(m_{2}).$

  * 1)

$|h|\gg|m|=|m_{2}|\Rightarrow|k|\sim|h|$. Then,

$\frac{|k|^{s}}{|k|^{2\alpha}}\frac{|h|}{|h|^{s}|m_{2}|^{s}}\sim\frac{1}{|h|^{2\alpha-1}|m_{2}|^{s}}.$

If $\alpha>1/2$,

$\leq\frac{1}{|m_{2}|^{s-1+2\alpha}},$

we get,

$\displaystyle\sum_{k}\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{1}{|m_{2}|^{2\alpha-1+s}}f_{1}(h_{1},h_{2})f_{2}(m_{2})g(k_{1},k_{2})$
$\displaystyle\leq\sum_{k_{2}=h_{2}+m_{2}}\frac{1}{|m_{2}|^{2\alpha-1+s}}f_{2}(m_{2})\overbrace{\left(\sum_{k_{1}}|f_{1}(k_{1},h_{2})|^{2}\right)^{\frac{1}{2}}}^{w(h_{2})}\overbrace{\left(\sum_{k_{1}}|g(k_{1},k_{2})|^{2}\right)^{\frac{1}{2}}}^{v(k_{2})}$
$\displaystyle=\sum_{k_{2}}\sum_{m_{2}}\frac{1}{|m_{2}|^{2\alpha-1+s}}f_{2}(m_{2})w(k_{2}-m_{2})v(k_{2})$
$\displaystyle\leq\sum_{m_{2}}\frac{1}{|m_{2}|^{2\alpha-1+s}}f_{2}(m_{2})\norm{f_{1}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\leq\left(\sum_{m_{2}}\frac{1}{|m_{2}|^{4\alpha-2+2s}}\right)^{\frac{1}{2}}\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}}$

which is summable if $4\alpha-2+2s>1$. If $s=0$, then $\alpha>\frac{3}{4}$.

  * 2)

$|m_{2}|\gg|h|$ $\Rightarrow$ $|k|\sim|m_{2}|$. Then,

$\frac{|k|^{s}}{|k|^{2\alpha}}\frac{|h|}{|h|^{s}|m_{2}|^{s}}\lesssim\frac{|m_{2}|}{|m_{2}|^{2\alpha}|h|^{s}}=\frac{1}{|m_{2}|^{2\alpha-1}|h|^{s}}\leq\frac{1}{|m_{2}|^{2\alpha-1}}$

recalling that $|h|\geq 1$ and $s\geq 0$. We conclude as in the previous case.

  * 3)

$|m_{2}|\sim|h|$ $\Rightarrow$ $|k|\sim|m_{2}|\sim|h|$.Then,

$\frac{|k|^{s}}{|k|^{2\alpha}}\frac{|h|}{|h|^{s}|m_{2}|^{s}}\sim\frac{1}{|m_{2}|^{2\alpha-1+s}}$

and we conclude as in the previous case.

Putting all the conditions on $\alpha$ and $s$ together, we get

$\displaystyle s$ $\displaystyle\geq 0;$ $\displaystyle 1>\alpha$
$\displaystyle>\frac{3}{4}.$

By definition of $f_{1}$ and $f_{2}$ and using (3.9), we derive

$\displaystyle\bigg{\|}\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}$
$\displaystyle F(\gamma,t^{\prime})dt^{\prime}\bigg{\|}_{H_{xy}^{s}}$
$\displaystyle\lesssim\int_{0}^{t}\frac{1}{(t-t^{\prime})^{\alpha}}dt^{\prime}\left(\norm{\gamma}^{2}_{H^{s}_{xy}}+\norm{\mu}_{H^{s}_{y}}\norm{\gamma}_{H^{s}_{xy}}+\norm{\gamma}_{H^{s}_{xy}}\right)$
$\displaystyle\sim
t^{1-\alpha}\left(\norm{\gamma}^{2}_{H^{s}_{xy}}+\norm{\mu}_{H^{s}_{y}}\norm{\gamma}_{H^{s}_{xy}}+\norm{\gamma}_{H^{s}_{xy}}\right).$

Finally,

$\displaystyle\bigg{\|}\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}$
$\displaystyle
F(\gamma,t^{\prime})dt^{\prime}\bigg{\|}_{L^{\infty}_{\delta}H_{xy}^{s}}$
$\displaystyle\lesssim\delta^{1-\alpha}\left(\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\mu}_{L^{\infty}_{\delta}H^{s}_{y}}\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}\right).$

for $s\geq 0$ and $\frac{3}{4}<\alpha<1$. ∎

Let us introduce the following functional,

$\Phi_{2}(\mu,\gamma)=e^{t\tilde{D}^{2}}\gamma_{0}+\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)dt^{\prime}.$
(3.17)

An immediate consequence of the previous proposition is,

######  Corollary 3.5.

Let $t\in[0,\delta]$, $\delta\in\mathbb{R}_{+}$, $s\geq 0$,
$\alpha\in(\frac{3}{4},1)$. Then,

$\displaystyle\|\Phi_{2}(\mu,\gamma)$
$\displaystyle\|_{L^{\infty}_{\delta}H^{s}_{xy}}$
$\displaystyle\lesssim\norm{\gamma_{0}}_{H^{s}_{xy}}+\delta^{1-\alpha}\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}\left(\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\mu}_{L^{\infty}_{\delta}H^{s}_{y}}+1\right).$

###### Proof.

$\norm{\Phi_{2}(\gamma)}_{L^{\infty}_{\delta}H^{s}_{xy}}\leq\underbrace{\norm{e^{t\tilde{D}^{2}}\gamma_{0}}_{L^{\infty}_{\delta}H^{s}_{xy}}}_{A}+\underbrace{\norm{\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\gamma,t^{\prime})dt^{\prime}}_{L^{\infty}_{\delta}H_{xy}^{s}}}_{B}$

By Proposition 3.4 we have

$B\lesssim\delta^{1-\alpha}\left(\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\mu}_{L^{\infty}_{\delta}H^{s}_{y}}\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}\right).$

On the other hand,

$\displaystyle A=\norm{\mathfrak{F}(e^{t\tilde{D}^{2}}\gamma_{0})}_{H^{s}}$
$\displaystyle=\left(\sum_{k}\left|e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})}\widehat{\gamma}_{0}(k)\right|^{2}\langle
k\rangle^{2s}\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\left(\sum_{k}|\widehat{\gamma}_{0}(k)|^{2}\langle
k\rangle^{2s}\right)^{\frac{1}{2}}$
$\displaystyle=\norm{\gamma_{0}}_{H^{s}_{xy}}.$

∎

####  3.1.3. The $\Phi$-Functional

With $\Phi_{1}$ defined in (3.8) and $\Phi_{2}$ defined in (3.17), let us
define

$\displaystyle\Phi$ $\displaystyle(\mu,\gamma)=(\Phi_{1},\Phi_{2})$
$\displaystyle=\left(e^{\nu
t\partial_{yy}}\mu_{0}+\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}G(\gamma)\
dt^{\prime},\
e^{\tilde{D}^{2}t}\gamma_{0}+\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)\
dt^{\prime}\right)$

By Corollary 3.3 and 3.5 we have

######  Corollary 3.6.

Let $t\in[0,\delta]$, $\delta\in\mathbb{R}_{+}$, $s\geq 0$,
$\alpha\in(\frac{3}{4},1)$. Then,

$\displaystyle\norm{\Phi(\mu,\gamma)}_{X^{s,\delta}}\leq
C_{1}\norm{\mu_{0}}_{H^{s}_{y}}+C_{2}\norm{\gamma_{0}}_{H^{s}_{xy}}$ (3.18)
$\displaystyle+C_{3}\delta^{1-\alpha}\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}+C_{4}\delta^{1-\alpha}\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}\left(\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\mu}_{L^{\infty}_{\delta}H^{s}_{y}}+1\right).$

###### Proof.

Since

$\norm{\Phi(\mu,\gamma)}_{X^{s,\delta}}=\norm{\Phi_{1}(\mu,\gamma)}_{L^{\infty}_{t}H^{s}_{y}}+\norm{\Phi_{2}(\mu,\gamma)}_{L^{\infty}_{\delta}H^{s}_{xy}},$

we use the computations made in Corollary 3.3 and 3.5 which give us

$\displaystyle\norm{\Phi_{1}(\mu,\gamma)}_{L^{\infty}_{t}H^{s}_{xy}}$
$\displaystyle\leq
C_{1}\norm{\mu_{0}}_{H^{s}_{y}}+C_{3}\delta^{1-\alpha}\norm{\gamma}^{2}_{L^{\infty}_{\delta}H^{s}_{xy}}$
$\displaystyle\norm{\Phi_{2}(\mu,\gamma)}_{L^{\infty}_{t}H^{s}_{xy}}$
$\displaystyle\leq C_{3}\norm{\gamma_{0}}_{H^{s}_{xy}}$
$\displaystyle+C_{4}\delta^{1-\alpha}\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}\left(\norm{\gamma}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\mu}_{L^{\infty}_{\delta}H^{s}_{y}}+1\right).$

Combining the results we get (3.18). ∎

###  3.2. Local Well-Posedness

Let us consider

$B(0,R)\subset{X^{s,\delta}}$

where
$R:=2\left(C_{1}\norm{\mu_{0}}_{H^{s}_{y}}+C_{2}\norm{\gamma_{0}}_{H^{s}_{xy}}\right)$.
If we set $C_{1},C_{2}\gg 1$ such that $R\geq 1$ and we fix $\delta$
sufficiently small such that

$\displaystyle C_{i}\delta^{1-\alpha}R<\frac{1}{8},\text{ with }i=3,4,$ (3.19)

then by (3.18) we get

$\Phi:B(0,R)\to B(0,R).$

We remark that (3.19) implies

$\delta\sim
R^{-\frac{1}{1-\alpha}}\sim(\norm{\gamma_{0}}_{H^{s}_{xy}}+\norm{\mu_{0}}_{H^{s}_{y}})^{-\frac{1}{1-\alpha}}.$
(3.20)

####  3.2.1. Contraction

We now show that $\Phi$ is a contraction on $B(0,R)$. As in the previous
section, we proceed by doing one estimate at a time and then combine the
results.

######  Proposition 3.7.

Let $(\mu_{1},\gamma_{1}),\ (\mu_{2},\gamma_{2})\in B(0,R)$ be vectors
functions such that $\mu_{1}(0,y)=\mu_{2}(0,y)=:\mu_{0}$ and let $s\geq 0$,
$\alpha\in(\frac{1}{2},1)$ Then,

$\norm{\Phi_{1}(\mu_{1},\gamma_{1})-\Phi_{1}(\mu_{2},\gamma_{2})}_{L^{\infty}_{\delta}H^{s}_{y}}\lesssim\delta^{1-\alpha}R\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\delta}}.$

###### Proof.

By definition we have

$\displaystyle\Phi_{1}(\mu_{1},\gamma_{1})$ $\displaystyle=e^{\nu
t\partial_{yy}}\mu_{0}+\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}G(\gamma_{1})dt^{\prime},$
$\displaystyle\Phi_{1}(\mu_{2},\gamma_{2})$ $\displaystyle=e^{\nu
t\partial_{yy}}\mu_{0}+\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}G(\gamma_{2})dt^{\prime}.$

Then,

$\Phi_{1}(\mu_{1},\gamma_{1})-\Phi_{1}(\mu_{2},\gamma_{2})=\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}\left(G(\gamma_{1})-G(\gamma_{2})\right)dt^{\prime}$

where,

$\displaystyle G(\gamma_{1})-G(\gamma_{2})$
$\displaystyle=\partial_{y}\left(\overline{[\partial_{x}\nabla^{-2}\gamma_{1}\partial_{y}\nabla^{-2}\gamma_{1}]-[\partial_{x}\nabla^{-2}\gamma_{2}\partial_{y}\nabla^{-2}\gamma_{2}]}\right)$
$\displaystyle=$
$\displaystyle\partial_{y}\left(\overline{[\partial_{x}\nabla^{-2}(\gamma_{1}-\gamma_{2})\partial_{y}\nabla^{-2}\gamma_{1}]+[\partial_{x}\nabla^{-2}\gamma_{2}\partial_{y}\nabla^{-2}(\gamma_{1}-\gamma_{2})]}\right)$

and

$\displaystyle\norm{\Phi_{1}(\mu_{1},\gamma_{1})-\Phi_{1}(\mu_{2},\gamma_{2})}_{L^{\infty}_{\delta}H_{y}^{s}}=\norm{\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}\left(G(\gamma_{1})-G(\gamma_{2})\right)dt^{\prime}}_{L^{\infty}_{\delta}H_{y}^{s}}.$

As seen in Proposition 3.1, by duality we can study the following two
equivalent quantities

$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}$
$\displaystyle\sum_{k_{2}}\sum_{\begin{subarray}{c}h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{V(k_{2},h,\tilde{m})}{\langle
h\rangle^{s}\langle\tilde{m}\rangle^{s}}f_{1}(h)f_{2}(\tilde{m})g(k_{2})\lesssim\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}$
$\displaystyle\sum_{k_{2}}\sum_{\begin{subarray}{c}h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{V(k_{2},h,\tilde{m})}{\langle
h\rangle^{s}\langle\tilde{m}\rangle^{s}}f_{3}(h)f_{1}(\tilde{m})g(k_{2})\lesssim\norm{f_{1}}_{l^{2}}\norm{f_{3}}_{l^{2}}\norm{g}_{l^{2}}.$

where

$\displaystyle\tilde{m}$ $\displaystyle:=(-h_{1},m_{2});$ $\displaystyle
V(k_{2},h,\tilde{m})$ $\displaystyle:=\frac{\langle
k_{2}\rangle^{s}}{|k_{2}|^{2\alpha-1}|h||\tilde{m}|};$ $\displaystyle
f_{1}(k)$ $\displaystyle:=\langle
k\rangle^{s}|(\widehat{\gamma_{1}-\gamma_{2}})(k)|;$ $\displaystyle f_{2}(k)$
$\displaystyle:=\langle k\rangle^{s}|\widehat{\gamma_{1}}(k)|;$ $\displaystyle
f_{3}(k)$ $\displaystyle:=\langle k\rangle^{s}|\widehat{\gamma_{2}}(k)|.$

Since we have the same coefficients as in the Proposition 3.1, using the same
computations we get

$\displaystyle\|\Phi_{1}(\mu_{1},\gamma_{1})-\Phi_{1}(\mu_{2},\gamma_{2})$
$\displaystyle\|_{L^{\infty}_{\delta}H_{y}^{s}}$
$\displaystyle\lesssim\delta^{1-\alpha}\norm{\gamma_{1}-\gamma_{2}}_{L^{\infty}_{\delta}H^{s}_{xy}}\left(\norm{\gamma_{1}}_{L^{\infty}_{\delta}H^{s}_{xy}}+\norm{\gamma_{2}}_{L^{\infty}_{\delta}H^{s}_{xy}}\right)$
$\displaystyle\lesssim\delta^{1-\alpha}R\norm{\gamma_{1}-\gamma_{2}}_{L^{\infty}_{\delta}H^{s}_{xy}}$
$\displaystyle\lesssim\delta^{1-\alpha}R\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\delta}}.$

for $s\geq 0$ and $\frac{1}{2}<\alpha<1$. ∎

######  Proposition 3.8.

Let $(\mu_{1},\gamma_{1}),\ (\mu_{2},\gamma_{2})\in B(0,R)$ be vectors
functions such that $\gamma_{1}(0,x,y)=\gamma_{2}(0,x,y)=:\gamma_{0}(x,y)$ and
let $s\geq 0$, $\alpha\in(\frac{3}{4},1)$. Then,

$\norm{\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}(\mu_{2},\gamma_{2})}_{L^{\infty}_{\delta}H^{s}_{xy}}\lesssim(1+3R)\delta^{1-\alpha}\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\delta}},$

with $\alpha\in\left(\frac{3}{4},1\right)$.

###### Proof.

By definition we have

$\displaystyle\Phi_{2}(\mu_{1},\gamma_{1})$
$\displaystyle=e^{t\tilde{D}^{2}}\gamma_{0}+\int_{0}^{t}e^{(t-t^{\prime})\tilde{D}^{2}}F(\mu_{1},\gamma_{1})\
dt^{\prime}$ $\displaystyle\Phi_{2}(\mu_{2},\gamma_{2})$
$\displaystyle=e^{t\tilde{D}^{2}}\gamma_{0}+\int_{0}^{t}e^{(t-t^{\prime})\tilde{D}^{2}}F(\mu_{2},\gamma_{2})\
dt^{\prime},$

where

$\displaystyle
F(\mu,\gamma)=\overbrace{\gamma_{x}\partial_{y}\nabla^{-2}\gamma}^{A}-\overbrace{\gamma_{y}\partial_{x}\nabla^{-2}\gamma}^{B}+\overbrace{\partial_{y}[\overline{(\gamma\partial_{x}\nabla^{-2}\gamma)}}^{C}]-\overbrace{\mu\gamma_{x}}^{D}+\overbrace{c_{0}\gamma_{x}}^{E}.$

Then,

$\displaystyle\norm{\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}(\mu_{2},\gamma_{2})}_{L^{\infty}_{\delta}H^{s}_{xy}}$
$\displaystyle=\norm{\int_{0}^{t}e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})(t-t^{\prime})}(F(\mu_{1},\gamma_{1})-F(\mu_{2},\gamma_{2}))\
dt^{\prime}}_{L^{\infty}_{\delta}H^{s}_{x,y}}$
$\displaystyle=\sup_{t\in[0,\delta]}\left(\sum_{k}\left|\int_{0}^{t}e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})(t-t^{\prime})}(\widehat{F(\mu_{1},\gamma_{1})-F(\mu_{2},\gamma_{2})})\
dt^{\prime}\right|^{2}\langle k\rangle^{2s}\right)^{\frac{1}{2}}.$

We observe that

$\displaystyle A_{\gamma_{1}}$
$\displaystyle={\gamma_{1}}_{x}\partial_{y}\nabla^{-2}\gamma_{1};\ \ \ \
A_{\gamma_{2}}={\gamma_{2}}_{x}\partial_{y}\nabla^{-2}\gamma_{2}.$
$\displaystyle A_{\gamma_{1}}-A_{\gamma_{2}}$
$\displaystyle={\gamma_{1}}_{x}\partial_{y}\nabla^{-2}\gamma_{1}-{\gamma_{2}}_{x}\partial_{y}\nabla^{-2}{\gamma_{2}}$
$\displaystyle=({\gamma_{1}}_{x}-{\gamma_{2}}_{x})\partial_{y}\nabla^{-2}\gamma_{1}+{\gamma_{2}}_{x}\partial_{y}\nabla^{-2}(\gamma_{1}-{\gamma_{2}}).$
$\displaystyle B_{\gamma_{1}}$
$\displaystyle={\gamma_{1}}_{y}\partial_{x}\nabla^{-2}\gamma_{1};\ \ \ \
B_{\gamma_{2}}={\gamma_{2}}_{y}\partial_{x}\nabla^{-2}{\gamma_{2}}.$
$\displaystyle B_{\gamma_{1}}-B_{\gamma_{2}}$
$\displaystyle={\gamma_{1}}_{y}\partial_{x}\nabla^{-2}\gamma_{1}-{\gamma_{2}}_{y}\partial_{x}\nabla^{-2}{\gamma_{2}}$
$\displaystyle=({\gamma_{1}}_{y}-{\gamma_{2}}_{y})\partial_{x}\nabla^{-2}\gamma_{1}+{\gamma_{2}}_{y}\partial_{x}\nabla^{-2}(\gamma_{1}-{\gamma_{2}}).$
$\displaystyle C_{\gamma_{1}}$
$\displaystyle=\partial_{y}\overline{(\gamma_{1}\partial_{x}\nabla^{-2}\gamma_{1})};\
\ \ \
C_{\gamma_{2}}=\partial_{y}\overline{({\gamma_{2}}\partial_{x}\nabla^{-2}{\gamma_{2}})}.$
$\displaystyle C_{\gamma_{1}}-C_{\gamma_{2}}$
$\displaystyle=\partial_{y}\overline{(\gamma_{1}\partial_{x}\nabla^{-2}\gamma_{1})}-\partial_{y}\overline{({\gamma_{2}}\partial_{x}\nabla^{-2}{\gamma_{2}})}$
$\displaystyle=\partial_{y}\overline{((\gamma_{1}-{\gamma_{2}})\partial_{x}\nabla^{-2}\gamma_{1})}+\partial_{y}\overline{({\gamma_{2}}\partial_{x}\nabla^{-2}(\gamma_{1}-{\gamma_{2}}))}$
$\displaystyle D_{\mu_{1},\gamma_{1}}$
$\displaystyle=\mu_{1}{\gamma_{1}}_{x};\ \ \ \
D_{\mu_{2},\gamma_{2}}=\mu_{2}{\gamma_{2}}_{x}.$ $\displaystyle
D_{\mu_{1},\gamma_{1}}-D_{\mu_{2},\gamma_{2}}$
$\displaystyle=\mu_{1}{\gamma_{1}}_{x}-\mu_{2}{\gamma_{2}}_{x}$
$\displaystyle=\mu_{1}(\gamma_{1}-\gamma_{2})_{x}+(\mu_{1}-\mu_{2}){\gamma_{2}}_{x}$
$\displaystyle E_{\gamma_{1}}$ $\displaystyle=c_{0}{\gamma_{1}}_{x};\ \ \ \
E_{\gamma_{2}}=c_{0}{\gamma_{2}}_{x}.$ $\displaystyle
E_{\gamma_{1}}-E_{\gamma_{2}}$
$\displaystyle=c_{0}\left({\gamma_{1}}_{x}-{\gamma_{2}}_{x}\right).$

Then,

$\displaystyle F(\mu_{1},\gamma_{1})-F({\mu_{2},\gamma_{2}})$
$\displaystyle=({\gamma_{1}}_{x}-{\gamma_{2}}_{x})\partial_{y}\nabla^{-2}\gamma_{1}+{\gamma_{2}}_{x}\partial_{y}\nabla^{-2}(\gamma_{1}-{\gamma_{2}})$
$\displaystyle-({\gamma_{1}}_{y}-{\gamma_{2}}_{y})\partial_{x}\nabla^{-2}\gamma_{1}-{\gamma_{2}}_{y}\partial_{x}\nabla^{-2}(\gamma_{1}-{\gamma_{2}})$
$\displaystyle+\partial_{y}\overline{((\gamma_{1}-{\gamma_{2}})\partial_{x}\nabla^{-2}\gamma_{1})}+\partial_{y}\overline{({\gamma_{2}}\partial_{x}\nabla^{-2}(\gamma_{1}-{\gamma_{2}}))}$
$\displaystyle-\mu_{1}(\gamma_{1}-\gamma_{2})_{x}-(\mu_{1}-\mu_{2}){\gamma_{2}}_{x}+c_{0}\left({\gamma_{1}}_{x}-{\gamma_{2}}_{x}\right),$

and

$\displaystyle\left(\widehat{F(\mu_{1},\gamma_{1})-F({\mu_{2},\gamma_{2}})}\right)(k)$
$\displaystyle=ik_{1}(\widehat{\gamma_{1}-{\gamma_{2}}})(k)*(-i\frac{k_{2}}{|k|^{2}}\widehat{\gamma_{1}})(k)+(ik_{1}\widehat{{\gamma_{2}}})(k)*(-i\frac{k_{2}}{|k|^{2}}(\widehat{\gamma_{1}-{\gamma_{2}}}))(k)$
$\displaystyle-
ik_{2}(\widehat{\gamma_{1}-{\gamma_{2}}})(k)*(-i\frac{k_{1}}{|k|^{2}}\widehat{\gamma_{1}})(k)-(ik_{2}\widehat{{\gamma_{2}}})(k)*(-i\frac{k_{1}}{|k|^{2}}(\widehat{\gamma_{1}-{\gamma_{2}}}))(k)$
$\displaystyle+\left.ik_{2}\left((\widehat{\gamma_{1}-{\gamma_{2}}})(k)*(-i\frac{k_{1}}{|k|^{2}}\widehat{\gamma_{1}})(k)+\widehat{{\gamma_{2}}}(k)*(-i\frac{k_{1}}{|k|^{2}})(\widehat{\gamma_{1}-{\gamma_{2}}})(k)\right)\right|_{[0,k_{2}]}$
$\displaystyle-\widehat{\mu_{1}}(k_{2})*ik_{1}(\widehat{\gamma_{1}-{\gamma_{2}}})(k)-(\widehat{\mu_{1}-\mu_{2}})(k_{2})*ik_{1}\widehat{\gamma_{2}}(k)+c_{0}ik_{1}(\widehat{\gamma-{\gamma_{2}}})(k)$
$\displaystyle=\sum_{k=h+m}\frac{m_{1}h_{2}-m_{2}h_{1}}{|h|^{2}}\big{[}(\widehat{\gamma_{1}-{\gamma_{2}}})(m)\widehat{\gamma_{1}}(h)+(\widehat{\gamma_{1}-{\gamma_{2}}})(h)\widehat{{\gamma_{2}}}(m)\big{]}$
$\displaystyle+\sum_{(0,k_{2})}\frac{k_{2}h_{1}}{|h|^{2}}\big{[}(\widehat{\gamma_{1}-{\gamma_{2}}})(m)\widehat{\gamma_{1}}(h)+(\widehat{\gamma_{1}-{\gamma_{2}}})(h)\widehat{{\gamma_{2}}}(m)\big{]}$
$\displaystyle-\sum_{\begin{subarray}{c}k_{1}=h_{1}+0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}ih_{1}\big{[}\widehat{\mu_{1}}(m_{2})(\widehat{\gamma_{1}-{\gamma_{2}}})(h)+(\widehat{\mu_{1}-\mu_{2}})(m_{2})\widehat{\gamma_{2}}(h)\big{]}$
$\displaystyle+c_{0}ik_{1}(\widehat{\gamma_{1}-{\gamma_{2}}})(k).$

Therefore,

$\displaystyle\|\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}($
$\displaystyle\mu_{2},\gamma_{2})\|_{L^{\infty}_{\delta}H^{s}_{xy}}$
$\displaystyle=\sup_{t\in[0,\delta]}\left(\sum_{k}\right.$
$\displaystyle\left|\int_{0}^{t}\right.e^{(i\frac{k_{1}}{|k|^{2}}-\nu|k|^{2})(t-t^{\prime})}$
$\displaystyle\cdot\left(\sum_{k=h+m}\right.$
$\displaystyle\frac{m_{1}h_{2}-m_{2}h_{1}}{|h|^{2}}\big{[}(\widehat{\gamma_{1}-{\gamma_{2}}})(m)\widehat{\gamma_{1}}(h)+(\widehat{\gamma_{1}-{\gamma_{2}}})(h)\widehat{{\gamma_{2}}}(m)\big{]}$
$\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}k_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{k_{2}h_{1}}{|h|^{2}}\big{[}(\widehat{\gamma_{1}-{\gamma_{2}}})(m)\widehat{\gamma_{1}}(h)+(\widehat{\gamma_{1}-{\gamma_{2}}})(h)\widehat{{\gamma_{2}}}(m)\big{]}$
$\displaystyle-$ $\displaystyle\sum_{\begin{subarray}{c}k_{1}=h_{1}+0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}ih_{1}\big{[}\widehat{\mu_{1}}(m_{2})(\widehat{\gamma_{1}-{\gamma_{2}}})(h)+(\widehat{\mu_{1}-\mu_{2}})(m_{2})\widehat{\gamma_{2}}(h)\big{]}$
$\displaystyle+$ $\displaystyle
c_{0}ik_{1}(\widehat{\gamma_{1}-{\gamma_{2}}})(k)\Bigg{)}dt^{\prime}\bigg{|}^{2}\langle
k\rangle^{2s}\Bigg{)}^{\frac{1}{2}}.$

As in the proof of Proposition 3.4, by duality it is sufficient to show

$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}2\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{k=h+m}\frac{|m|}{|h|}\frac{1}{\langle
h\rangle^{s}\langle
m\rangle^{s}}f_{1}(h)f_{2}(m)g(k)\lesssim\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}2\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{k=h+m}\frac{|m|}{|h|}\frac{1}{\langle
h\rangle^{s}\langle
m\rangle^{s}}f_{2}(h)f_{3}(m)g(k)\lesssim\norm{f_{2}}_{l^{2}}\norm{f_{3}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{\begin{subarray}{c}k_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|\langle h\rangle^{s}\langle
m\rangle^{s}}f_{1}(h)f_{2}(m)g(k)\lesssim\norm{f_{1}}_{l^{2}}\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{\begin{subarray}{c}k_{1}=0\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|k|}{|h|\langle h\rangle^{s}\langle
m\rangle^{s}}f_{2}(h)f_{3}(m)g(k)\lesssim\norm{f_{2}}_{l^{2}}\norm{f_{3}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|h|}{\langle h\rangle^{s}\langle
m_{2}\rangle^{s}}f_{2}(h)f_{4}(m_{2})g(k)\lesssim\norm{f_{2}}_{l^{2}}\norm{f_{4}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq 1}\sum_{k}\frac{\langle
k\rangle^{s}}{|k|^{2\alpha}}\sum_{\begin{subarray}{c}k_{1}=h_{1}\\\
k_{2}=h_{2}+m_{2}\end{subarray}}\frac{|h|}{\langle h\rangle^{s}\langle
m_{2}\rangle^{s}}f_{3}(h)f_{5}(m_{2})g(k)\lesssim\norm{f_{3}}_{l^{2}}\norm{f_{5}}_{l^{2}}\norm{g}_{l^{2}}$
$\displaystyle\sup_{\norm{g}_{l^{2}}\leq
1}\sum_{k}c_{0}\frac{|k|}{|k|^{2\alpha}}f_{2}(k)g(k)\lesssim\norm{f_{2}}_{l^{2}}\norm{g}_{l^{2}},$

where we define

$\displaystyle f_{1}(k):=$ $\displaystyle\langle
k\rangle^{s}|\widehat{\gamma_{1}}(k)|$ $\displaystyle f_{2}(k):=$
$\displaystyle\langle k\rangle^{s}|(\widehat{\gamma_{1}-{\gamma_{2}}})(k)|$
$\displaystyle f_{3}(k):=$ $\displaystyle\langle
k\rangle^{s}|\widehat{{\gamma_{2}}}(k)|$ $\displaystyle f_{4}(k_{2}):=$
$\displaystyle\langle k_{2}\rangle^{s}|\widehat{\mu_{1}}(k_{2})|$
$\displaystyle f_{5}(k_{2}):=$ $\displaystyle\langle
k_{2}\rangle^{s}|(\widehat{\mu_{1}-\mu_{2}})(k_{2})|.$

Since the coefficients of the sum are exactly those of Proposition 3.4, we
have

$\displaystyle s$ $\displaystyle\geq 0,$ $\displaystyle\alpha$
$\displaystyle>\frac{3}{4}$

and

$\displaystyle\|\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}({\mu_{2},\gamma_{2}})$
$\displaystyle\|_{H^{s}_{xy}}\lesssim\int_{0}^{t}\frac{1}{(t-t^{\prime})^{\alpha}}\
dt^{\prime}(2\norm{\gamma_{1}-{\gamma_{2}}}_{H^{s}_{xy}}\norm{\gamma_{1}}_{H^{s}_{xy}}$
$\displaystyle+2\norm{\gamma_{1}-{\gamma_{2}}}_{H^{s}_{xy}}\norm{{\gamma_{2}}}_{H^{s}_{xy}}+\norm{\gamma_{1}-{\gamma_{2}}}_{H^{s}_{xy}}\norm{\mu_{1}}_{H^{s}_{y}}$
$\displaystyle+\norm{{\gamma_{2}}}_{H^{s}_{xy}}\norm{\mu_{1}-\mu_{2}}_{H^{s}_{y}}+\norm{\gamma_{1}-{\gamma_{2}}}_{H^{s}_{xy}})$
$\displaystyle\sim t^{1-\alpha}$
$\displaystyle\left(\norm{\gamma_{1}-{\gamma_{2}}}_{H^{s}_{xy}}(\norm{\gamma_{1}}_{H^{s}_{xy}}+\norm{{\gamma_{2}}}_{H^{s}_{xy}}+\norm{\mu_{1}}_{H^{s}_{xy}}+1)\right.$
$\displaystyle\left.+\norm{{\gamma_{2}}}_{H^{s}_{xy}}\norm{\mu_{1}-\mu_{2}}_{H^{s}_{y}}\right)$
$\displaystyle\lesssim(1+$ $\displaystyle
3R)t^{1-\alpha}\left(\norm{\gamma_{1}-{\gamma_{2}}}_{H^{s}_{xy}}+\norm{\mu_{1}-\mu_{2}}_{H^{s}_{y}}\right).$

Then,

$\norm{\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}({\mu_{2},\gamma_{2}})}_{L^{\infty}_{\delta}H^{s}_{x,y}}\lesssim(1+3R)\delta^{1-\alpha}\norm{(\mu_{1},\gamma_{1})-(\mu_{2},{\gamma_{2}})}_{X^{s,\delta}}.$

with $s\geq 0$ and $\frac{3}{4}<\alpha<1$. ∎

Finally, combining Proposition 3.7 and 3.8 we get

######  Proposition 3.9.

Let $(\mu_{1},\gamma_{1}),\ (\mu_{2},\gamma_{2})\in
B(0,R)\subset{X^{s,\delta}}$ be vector functions and let $s\geq 0$ and
$\alpha\in(\frac{3}{4},1)$. Then

$\norm{\Phi(\mu_{1},\gamma_{1})-\Phi(\mu_{2},\gamma_{2})}_{{X^{s,\delta}}}\leq
C(1+3R)\delta^{1-\alpha}\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\delta}}.$
(3.21)

###### Proof.

Since

$\displaystyle\norm{\Phi(\mu_{1},\gamma_{1})-\Phi(\mu_{2},\gamma_{2})}_{{X^{s,\delta}}}$
$\displaystyle=\norm{\Phi_{1}(\mu_{1},\gamma_{1})-\Phi_{1}(\mu_{2},\gamma_{2})}_{L^{\infty}_{\delta}H^{s}_{y}}$
$\displaystyle+\norm{\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}(\mu_{2},\gamma_{2})}_{L^{\infty}_{\delta}H^{s}_{xy}},$

and recalling that from Propositions 3.7 and 3.8 we have

$\displaystyle\norm{\Phi_{1}(\mu_{1},\gamma_{1})-\Phi_{1}(\mu_{2},\gamma_{2})}$
${}_{L^{\infty}_{\delta}H^{s}_{y}}\leq
C_{1}\delta^{1-\alpha}R\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\delta}}$
$\displaystyle\norm{\Phi_{2}(\mu_{1},\gamma_{1})-\Phi_{2}(\mu_{2},\gamma_{2})}$
${}_{L^{\infty}_{\delta}H^{s}_{xy}}\leq
C_{2}(1+3R)\delta^{1-\alpha}\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\delta}}$

we get (3.21). ∎

If we set $\delta$ sufficiently small such that

$C(1+3R)\delta^{1-\alpha}\leq\frac{1}{2},$

then $\Phi$ is a contraction on $B(0,R)$.

####  3.2.2. Well-Posedness on $X^{s,\delta}$

######  Corollary 3.10.

Under the previous assumption,

$\exists!\ (\mu,\gamma)\in B(0,R)\subset X^{s,\delta}\ |\
(\mu,\gamma)=\Phi(\mu,\gamma).$

###### Proof.

By fixed point theorem. ∎

Next step is to extend the existence and uniqueness of the solution from
$B(0,R)$ to the whole space $X^{s,\delta}$.

######  Proposition 3.11.

Suppose there exists $(\tilde{\mu},\tilde{\gamma})\in X^{s,\delta}$, $s\geq
0$, solution of (3.2) such that

$\displaystyle(\tilde{\mu},\tilde{\gamma})=$ $\displaystyle\left(e^{\nu
t\partial_{yy}}\mu_{0}+\int_{0}^{t}e^{\nu(t-t^{\prime})\partial_{yy}}G(\tilde{\gamma})\
dt^{\prime};\right.$
$\displaystyle\left.e^{\tilde{D}^{2}t}\gamma_{0}+\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\tilde{\mu},\tilde{\gamma})\
dt^{\prime}\right)\text{ on }{X^{s,\delta}}.$

Let $(\mu,\gamma)\in B(0,R)$ be the functions defined in Corollary 3.10. Then,

$(\tilde{\mu},\tilde{\gamma})=(\mu,\gamma)\text{ on }X^{s,\delta}.$

###### Proof.

Fix $0<\varepsilon<\delta$. By (3.21)

$\displaystyle\norm{(\mu,\gamma)-(\tilde{\mu},\tilde{\gamma})}_{X^{s,\varepsilon}}$
$\displaystyle=\norm{\Phi(\mu,\gamma)-\Phi(\tilde{\mu},\tilde{\gamma})}_{L^{\infty}_{\varepsilon}H^{s}_{xy}}$
$\displaystyle\leq
C(1+3R)\delta^{1-\alpha}\norm{(\mu_{1},\gamma_{1})-(\mu_{2},\gamma_{2})}_{X^{s,\varepsilon}}.$

Set $\varepsilon>0$ sufficiently small such that

$C(1+3R)\varepsilon^{1-\alpha}\leq\frac{1}{2}.$

Then,

$\norm{(\mu,\gamma)-(\tilde{\mu},\tilde{\gamma})}_{X^{s,\varepsilon}}\leq\frac{1}{2}\norm{(\mu,\gamma)-(\tilde{\mu},\tilde{\gamma})}_{X^{s,\varepsilon}}$

which is possible if and only if $(\tilde{\mu},\tilde{\gamma})=(\mu,\gamma)$
on $X^{s,\varepsilon}$.
In particular
$(\tilde{\mu}(\varepsilon),\tilde{\gamma}(\varepsilon))=(\mu(\varepsilon),\gamma(\varepsilon))$,
therefore we can repeat the same argument on interval
$[\varepsilon,2\varepsilon]$ until we cover $[0,\delta]$. ∎

##  4\. Global Well-Posedness for Eddy-Mean Vorticity System

###  4.1. Upper Bounds of $L^{2}$-Norms by Initial Data

To provide global well-posedness of (3.2), an upper bound for the $L^{2}$
norms of $\overline{u}$ and $\zeta^{\prime}$ is required, as mentioned in Step
4) of the introduction. This upper bound can be obtained from the norm of the
initial data, as follows.

Let us start by recalling the equation (1.1) that governs the dynamics of the
problem, which is given by

$\partial_{t}\zeta+J(\psi,\zeta+\beta y)=\nu\nabla^{2}\zeta,$

where $J(\psi,\zeta+\beta y)=u(\zeta+\beta y)_{x}+v(\zeta+\beta y)_{y}$,
$u=-\psi_{y},\ v=\psi_{x}$ and $\zeta=\nabla^{2}\psi$.

Energy equation: Multiplying the above equation by the stream function $\psi$
and integrating in space we get

$\left<\psi\partial_{t}\zeta\right>+\left<\psi J(\psi,\zeta+\beta
y)\right>=\left<\nu\psi\nabla^{2}\nabla^{2}\psi\right>,$

where

$\left<\cdot\right>=\frac{1}{l^{2}}\iint\cdot\ dxdy.$

######  Remark 4.1.

  * 1)

$div(f\nabla v)=\nabla f\cdot\nabla v+f\nabla^{2}v$;
$f,v:\mathbb{R}^{n}\to\mathbb{R}$.

  * 2)

$J(\psi,\psi(\zeta+\beta y))=\psi J(\psi,\zeta+\beta y)$.
In fact,

$\displaystyle J(\psi,\psi(\zeta+\beta y))$ $\displaystyle=u(\psi(\zeta+\beta
y))_{x}+v(\psi(\zeta+\beta y))_{y}$ $\displaystyle=u\psi_{x}(\zeta+\beta
y)+u\psi(\zeta+\beta y)_{x}+v\psi_{y}(\zeta+\beta y)+v\psi(\zeta+\beta y)_{y}$
$\displaystyle=J(\psi,(\zeta+\beta y))\psi.$

Then,

$\displaystyle\oint\psi\nabla\psi_{t}\cdot\widehat{n}dl-\left<\nabla\psi\cdot\nabla\psi_{t}\right>+\left<J(\right.$
$\displaystyle\left.\psi,\psi(\zeta+\beta y))\right>$ (4.1)
$\displaystyle=\nu\oint\psi\nabla\zeta\cdot\widehat{n}dl-\left<\nu\nabla\psi\cdot\nabla\nabla^{2}\psi\right>.$

Since we have periodic boundary conditions and incompressibility, equation
(4.1) becomes

$\partial_{t}\left(\frac{1}{2}\left<|\nabla\psi|^{2}\right>\right)=-\nu\left<\psi_{xx}^{2}+2\psi_{xy}^{2}+\psi_{yy}^{2}\right>.$

In particular

$\partial_{t}\left(\frac{1}{2}\left<|\nabla\psi|^{2}\right>\right)\leq 0,$

so that

$\left<|\nabla\psi|^{2}\right>\leq\left<|\nabla\psi_{0}|^{2}\right>.$

We can now separate mean and eddy contributions:

$\displaystyle\left<|\nabla\psi|^{2}\right>$
$\displaystyle=\left<u^{2}+v^{2}\right>=\left<(\overline{u}+u^{\prime})^{2}+v^{\prime
2}\right>$
$\displaystyle=\left<\overline{u}^{2}+2u^{\prime}\overline{u}+u^{\prime
2}+v^{\prime 2}\right>$
$\displaystyle=\left<\overline{u}^{2}\right>+\left<u^{\prime 2}+v^{\prime
2}\right>\leq\left<|\nabla\psi_{0}|^{2}\right>.$ (4.2)

This implies that both $\left<\overline{u}^{2}\right>$ and $\left<u^{\prime
2}+v^{\prime 2}\right>$ are bounded.

Enstrophy equation: Similarly to the previous case, we have

$\left<\zeta\zeta_{t}\right>+\left<\zeta J(\psi,\zeta+\beta
y)\right>=\nu\left<\zeta\nabla^{2}\zeta\right>$
$\Rightarrow\partial_{t}\left(\frac{1}{2}\left<\zeta^{2}\right>\right)=-\nu\left<|\nabla\zeta|^{2}\right>.$

Then,

$\left<\zeta^{2}\right>\leq\left<\zeta_{0}^{2}\right>.$

Separating mean and eddy contributions

$\displaystyle\left<\zeta^{2}\right>$
$\displaystyle=\left<(-\overline{u}_{y}+\zeta^{\prime})^{2}\right>$
$\displaystyle=\left<\overline{u}^{2}_{y}-2\overline{u}_{y}\zeta^{\prime}+\zeta^{\prime
2}\right>$
$\displaystyle=\left<\overline{u}_{y}^{2}\right>+\left<\zeta^{\prime
2}\right>\leq\left<\zeta_{0}^{2}\right>$ (4.3)

and therefore $\left<\zeta^{\prime 2}\right>$ is also bounded.

We have thus proved

######  Proposition 4.2.

Let $\left(\mu_{0}(y),\gamma_{0}(x,y)\right)\in
H^{s}(\mathbb{T}_{l})\crossproduct H^{s}(\mathbb{T}^{2}_{l})$ be the initial
data of problem (3.2). Then, using notation (3), we have

$\norm{\gamma(t)}_{L^{2}_{xy}}\leq\norm{\gamma_{0}}_{L^{2}_{xy}}\text{ and }\
\norm{\mu(t)}_{L^{2}_{y}}\lesssim\norm{\mu_{0}}_{L^{2}_{y}}\ \ \ \ \forall\
t\geq 0,$ (4.4)

with $C>0$.

###  4.2. Global Well-Posedness

In this section we complete the proof of Theorem 1.1. In Section 3, we
established a priori upper bounds for the solution over a finite time interval
and proved local well-posedness in time and space using the contraction
theorem. We also obtained spatial extension of the solution through
Proposition 3.11. The final step to be addressed is time extension.

More specifically, we proved that the problem (3.2) has a unique solution on

${X^{s,\delta}}=\left\\{(f,g)\in L^{\infty}_{\delta}(H^{s}_{y}\times
H^{s}_{xy});\widehat{f}(0)=0,\ \widehat{g}(0)=0\right\\},$

with

$s\geq 0\text{ and
}\delta\sim(\norm{\gamma_{0}}_{H^{s}_{xy}}+\norm{\mu_{0}}_{H^{s}_{y}})^{-\frac{1}{1-\alpha}}\
\ \alpha\in\left(\frac{3}{4},1\right),$

which we can explicitly write as

$\displaystyle(\mu,\gamma)=\bigg{(}e^{\nu t\partial_{yy}}$
$\displaystyle\mu_{0}+\int_{0}^{t}e^{(-\nu\partial_{yy})(t-t^{\prime})}G(\gamma)\
dt^{\prime},$ $\displaystyle e^{\tilde{D}^{2}t}$
$\displaystyle\left.\gamma_{0}+\int_{0}^{t}e^{\tilde{D}^{2}(t-t^{\prime})}F(\mu,\gamma)\
dt^{\prime}\right)$ (4.5)

where $\tilde{D}^{2}=\nu\nabla^{2}-\partial_{x}\nabla^{-2}$. We are now ready
to close the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

We start by choosing

$\delta\sim(\norm{\gamma_{0}}_{L^{2}_{xy}}+\norm{\mu_{0}}_{L^{2}_{y}})^{-\frac{1}{1-\alpha}}$
(4.6)

instead of (3.20). By previous computations we have existence and uniqueness
of the solution to (3.2) on $X^{s,\delta}$ with $\delta$ as in (4.6).
Moreover, by Proposition 4.2, we have

$\norm{\gamma(t)}_{L_{xy}^{2}}+\norm{\mu(t)}_{L_{y}^{2}}\leq\left(\norm{\gamma_{0}}_{L^{2}_{xy}}+C\norm{\mu_{0}}_{L_{y}^{2}}\right)\
\ \ \forall\ t\geq 0.$ (4.7)

Then, if we set as initial data

$\left(\mu(\delta),\gamma(\delta)\right),$

we can use the same techniques as in previous sections to establish the
existence and uniqueness of a solution on $X^{s,[\delta,\delta+\delta^{*}]}$
with a new parameter, which we call $\delta^{*}$, of the following form

$\delta^{*}\sim(\norm{\gamma(\delta)}_{L^{2}_{xy}}+\norm{\mu(\delta)}_{L^{2}_{y}})^{-\frac{1}{1-\alpha}}.$
(4.8)

Due to (4.7),

$\delta\leq\delta^{*},$

and in particular we have well-posedness of (3.2) on $X^{s,[\delta,2\delta]}$.
Iterating in this way, we obtain the solution on $X^{s}$. ∎

### Acknowledgment

The author would like to thank G. Staffilani and R. Ferrari for suggesting the
problem and for many useful conversations during the preparation of this
paper. The author would also like to thank MIT for its hospitality.

## References

  * [1] M.A.H. Al-Jaboori and D. Wirosoetisno, Navier-Stokes equations on the $\beta$-plane. Discrete and Continuous Dynamical Systems - B, 16 (3) : 687-701, (2011).
  * [2] G. K. Batchelor, An Introduction to Fluid Dynamics. Cambridge University Press, (2000).
  * [3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II. Geom. Funct. Anal., 3 : 107–156, 209–262, (1993).
  * [4] J. Bourgain, On the Cauchy problem for periodic KdV-type equations. J. Fourier Anal. Appl., (1995).
  * [5] J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data. Selecta Math. (N.S.), 3 (2):115–159, (1997).
  * [6] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Math. Research Notices, 5, 253–283, (1998).
  * [7] N.C. Constantinou, B.F. Farrell and P.J. Ioannou, Statistical state dynamics of jet–wave coexistence in barotropic beta-plane turbulence. Journal of the Atmospheric Sciences, 73 (5):2229–2253, (2016).
  * [8] L. Cope, The Dynamics of Geophysical and Astrophysical Turbulence. University of Cambridge, (2020).
  * [9] S. Danilov and V. M. Gryanik, Barotropic beta-plane turbulence in a regime with strong zonal jets revisited. Journal of the Atmospheric Sciences, 61, 2283–2295, (2004).
  * [10] I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: Equatorial waves and convergence results. Mémoires de la Société Mathématique de France, no. 107, 122 p., (2006).
  * [11] I. Gallagher and L. Saint-Raymond, On the influence of the Earth’s rotation on geophysical flows. Handbook of Mathematical Fluid Dynamics, Elsevier, (S. Friedlander and D. Serre, eds), (2006).
  * [12] J.R. Holton. An introduction to dynamical meteorology. Academic Press, (2004).
  * [13] H. P. Huang and W. A. Robinson, Two-dimensional turbulence and persistent zonal jets in a global barotropic model. Nonlinear Processes Geophys., 55, 611–632, (1998).
  * [14] C. Kenig, G. Ponce, and L. Vega, Well-Posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle. Communications on Pure and Applied Mathematics, (1993).
  * [15] C. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc., 9 : 573–603, (1996).
  * [16] S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence. Cambridge University Press (2012).
  * [17] J. Liu, P. M. Goldreich, and D. J. Stevenson, Constraints on deep-seated zonal winds inside jupiter and saturn.  Icarus, 196 (2) : 653-664, (2008).
  * [18] S.G. Llewellyn Smith, The motion of a non-isolated vortex on the beta-plane. Journal of Fluid Mechanics, Volume 346, pp. 149-179, (1997).
  * [19] A.J. Majda and A.L. Bertozzi, Vorticity and Incompressible Flow. Cambridge University Press, (2000).
  * [20] J.C. McWilliams, Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press, (2011).
  * [21] S. Nazarenko, Fluid Dynamics via Examples and Solutions.  Taylor and Francis Inc, (2014).
  * [22] T. Nozawa and S. Yoden, Formation of zonal-band structure in forced two-dimensional turbulence on a rotating sphere. Physics of Fluids, 9, 2081-2093, (1997).
  * [23] J. Pedlosky, Geophysical fluid dynamics. Springer-Verlag, (1992).
  * [24] S.B. Pope, Turbulent flows. Cambridge University Press, (2000).
  * [25] P.B. Rhines, Waves and turbulence on a beta-plane. Journal of Fluid Mechanics, 69, 417-443, (1975).
  * [26] R. Salmon, Lectures on Geophysical Fluid Dynamics. Oxford University Press, (1998).
  * [27] R. K. Scott and L. M. Polvani, Forced-Dissipative Shallow-Water Turbulence on the Sphere and the Atmospheric Circulation of the Giant Planets. Journal of the Atmospheric Sciences, 64, 3158–3176, (2007).
  * [28] R.K. Scott and L.M. Polvani, Equatorial super rotation in shallow atmospheres. Geophysical Research Letters, 35 (24) : L24202, (2008).
  * [29] K. Srinivasan and W.R. Young, Zonostrophic Instability. Journal of the Atmospheric Sciences, 69 (5) : 1633-1656, (2012).
  * [30] G. Staffilani, On Solutions for Periodic Generalized KdV Equations. IMRN International Mathematics Research Notices, No. 18, (1997).
  * [31] T. Tao, 254A, Notes 1: Local well-posedness of the Navier-Stokes equations. terrytao.wordpress.com, (2018).
  * [32] G. K. Vallis and M. E. Maltrud, Generation of mean flow and jets on a beta plane and over topography. J. Phys. Oceanogr., 23, 1346–1362, (1993).
  * [33] G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, (2017).