# Leapfrogging vortex rings for the 3-dimensional incompressible Euler
equations

Juan Dávila  Department of Mathematical Sciences, University of Bath, Bath,
Ba2 7AY, UK<EMAIL_ADDRESS>,  Manuel del Pino  Department of Mathematical
Sciences, University of Bath, Bath, Ba2 7AY, UK<EMAIL_ADDRESS>,  Monica
Musso  Department of Mathematical Sciences, University of Bath, Bath, Ba2 7AY,
UK<EMAIL_ADDRESS>and  Juncheng Wei  Department of Mathematics,
University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2.
<EMAIL_ADDRESS>
###### Abstract.

A classical problem in fluid dynamics concerns the interaction of multiple
vortex rings sharing a common axis of symmetry in an incompressible, inviscid
$3$-dimensional fluid. Helmholtz (1858) observed that a pair of similar thin,
coaxial vortex rings may pass through each other repeatedly due to the induced
flow of the rings acting on each other. This celebrated configuration, known
as leapfrogging, has not yet been rigorously established. We provide a
mathematical justification for this phenomenon by constructing a smooth
solution of the 3d Euler equations exhibiting this motion pattern.

##  1\. Introduction

We consider the 3-dimensional Euler equation for an ideal incompressible fluid
given by

$\left\\{\begin{aligned} &\ {\bf u}_{t}+({\bf u}\cdot{\nabla}){\bf
u}=-{\nabla}p,\\\ &\ {\rm div}\,{\bf u}=0,\quad x\in\mathbb{R}^{3},\ t\geq
0.\\\ \end{aligned}\right.$ (1.1)

For a solution ${\bf u}(x,t)$ of (1.1), its vorticity is defined as
$\vec{\omega}=\mathop{\rm curl}{\bf u}$. Then $\vec{\omega}$ solves the Euler
system (1.1) in its vorticity form,

$\left\\{\begin{aligned} &\ {\vec{\omega}}_{t}+({\bf
u}\cdot{\nabla}){\vec{\omega}}=(\vec{\omega}\cdot{\nabla}){\bf u},\quad{\bf
u}=\mathop{\rm curl}\vec{\psi},\\\
&\vec{\psi}(x,t)=\left(-\Delta\right)^{-1}\vec{\omega}\,=\,{1\over
4\pi}\int_{\mathbb{R}^{3}}{1\over|x-y|}\vec{\omega}(y,t)\,dy,\quad
x\in\mathbb{R}^{3},\ t\geq 0.\\\ \end{aligned}\right.$ (1.2)

A vortex ring is an axially symmetric solution of (1.2) which does not change
shape in time, whose vorticity is mostly concentrated inside a solid torus
which moves with constant speed along the symmetry axis. The vortex lines form
large circles that fill the torus, whereas fluid particles spin around the
vortex core within perpendicular cross sections characterized with a thin
torus‐shaped region in which the vorticity of the fluid is concentrated. These
objects were first described by Helmholtz in his celebrated work [28, 29]. He
considered with great attention the situation where the vorticity field is
concentrated in a circular vortex-filament of very small section, a thin
vortex ring. Helmholtz also analyzed the interaction between two or more
similar coaxial vortex rings with thin sections and similar translation
speeds. As pointed out by Jerrard and Smets [34], his description reads:

We can now see generally how two ring-formed vortex-filaments having the same
axis would mutually affect each other, since each, in addition to its proper
motion, has that of its elements of fluid as produced by the other. If they
have the same direction of rotation, they travel in the same direction; the
foremost widens and travels more slowly, the pursuer shrinks and travels
faster till finally, if their velocities are not too different, it overtakes
the first and penetrates it. Then the same game goes on in the opposite order,
so that the rings pass through each other alternately.

The motion pattern described by Helmholtz is often termed leapfrogging in
fluid mechanics. Leapfrogging vortex rings are solutions of the Euler
equations where several interacting vortex rings sharing a common axis of
symmetry move in the same direction along the symmetry axis and pass through
each other repeatedly due to the induced flow of the rings acting on each
other as depicted in Figure 1.

Figure 1. Leapfrogging vortex rings.

Even though this phenomenon has been widely studied since Helmholtz, as far as
we know it has not yet been mathematically justified. In this paper we present
what seems to be the first rigorous construction of a solution to the Euler
equations with a leapfrogging motion pattern.

The solution we construct belongs to the axisymmetric, no-swirl class, namely
with a velocity field ${\bf u}(x,t)$ which in standard cylindrical coordinates
$(r,\theta,z)$ takes the form

${\bf u}(x,t)=u^{r}(r,z,t)\,{\bf e}_{r}+u^{z}(r,z,t)\,{\bf e}_{z},\quad
x=(r\cos\theta,r\sin\theta,z)\in\mathbb{R}^{3},$

where ${\bf e}_{r}=(\cos\theta,\sin\theta,0),\quad{\bf
e}_{\theta}=(-\sin\theta,\cos\theta,0),\quad{\bf e}_{z}=(0,0,1),$ and

$(r,z)\,\in\,\Sigma=\\{(r,z)\,/\,r>0,z\in\mathbb{R}\\}.$ (1.3)

The corresponding vorticity $\vec{\omega}=\mathop{\rm curl}{\bf u}$ takes the
form

$\vec{\omega}(x,t)=\omega^{\theta}(r,z,t)\,{\bf e}_{\theta},\quad{\mbox{where
}}\omega^{\theta}={\partial}_{z}u^{r}-{\partial}_{r}u^{z}.$

The divergence free condition $\mathop{\rm div}{\bf u}=0$ becomes

${\partial}_{r}(ru^{r})+{\partial}_{z}(ru^{z})=0{\quad\hbox{in }}\Sigma.$

This implies the existence of a scalar function $\psi^{\theta}(r,z,t)$ such
that

$u^{r}=-{\partial}_{z}\psi^{\theta},\quad u^{z}={1\over
r}{\partial}_{r}(r\psi^{\theta}){\quad\hbox{in }}\Sigma.$

The relation $\vec{\omega}=\mathop{\rm curl}{\bf u}$ yields

$-\left({\partial}_{r}^{2}+{1\over r}{\partial}_{r}-{1\over
r^{2}}\right)\psi^{\theta}=\omega^{\theta}.$

We notice that the vector-valued function
$\vec{\psi}(x,t)=\psi^{\theta}(r,z,t)\,{\bf e}_{\theta}$ satisfies
$-\Delta\vec{\psi}=\vec{\omega}$. The Euler equations (1.2) thus become

$\begin{array}[]{l}{\partial}_{t}\omega^{\theta}+u^{r}{\partial}_{r}\omega^{\theta}+u^{z}{\partial}_{z}\omega^{\theta}=\frac{1}{r}u^{r}\omega^{\theta},\quad-[{\partial}_{r}^{2}+{1\over
r}{\partial}_{r}-\frac{1}{r^{2}}]\psi^{\theta}=\omega^{\theta}{\quad\hbox{in
}}\Sigma.\end{array}$ (1.4)

The axisymmetric Euler equation without swirl (1.4) has a formal singularity
at $r=0$, which sometimes is inconvenient to work with. To remove the
artificial singularity the following change of variable is usually used

$\omega=\frac{\omega^{\theta}}{r},\quad\psi=\frac{\psi^{\theta}}{r}.$

The functions $\omega$, $\psi$ are respectively known as the relative
vorticity and stream functions. In terms of these new variables equation (1.4)
becomes

$\begin{array}[]{l}r\,{\partial}_{t}\omega+\nabla^{\perp}(r^{2}\psi)\cdot\nabla\omega=0,\quad-\Delta_{5}\psi=\omega{\quad\hbox{in
}}\Sigma,\end{array}$ (1.5)

where $\nabla^{\perp}=(-{\partial}_{z},{\partial}_{r})$ and

$\Delta_{5}\psi=\partial_{rr}\Psi+\frac{3}{r}\partial_{r}\Psi+\partial_{zz}\psi,\quad
x=(r,z),$ (1.6)

supplemented with the conditions

$\psi_{r}(0,z,t)=0,\quad\lim_{|(r,z)|\to\infty}\psi(r,z,t)=0.$

It is well-known that the initial value problem for (1.5) is globally well-
posed in $L^{1}(rdrdz)\cap L^{\infty}(\Sigma)$, see [47, 40, 9]. A bounded
solution $\omega$ to (1.5) with sufficient space decay actually satisfies

$t\mapsto\int_{\mathbb{R}^{2}}r\omega(r,z,t)\,drdz=constant,\quad
t\mapsto\|w(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{2})}=constant.$ (1.7)

The same is true if the integral is taken in the time section of connected
components of the support of $\omega$. We will use these facts in the
formulation of a suitable first approximation for a solution exhibiting the
leapfrogging dynamics.

A vortex ring moving with constant speed $c$ along the $z$-axis is a
travelling wave solution of (1.5) with the form

$\omega(r,z,t)=W_{0}(r,z-c\,t),\quad\psi(r,z,t)=\Psi_{0}(r,z-c\,t),$

where $W_{0}$ and $\Psi_{0}$ solve

$\nabla^{\perp}\left(r^{2}(\Psi_{0}-{c\over 2})\right)\cdot\nabla
W_{0}=0,\quad-[\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}]\Psi_{0}=W_{0}\quad{\quad\hbox{in
}}\Sigma.$ (1.8)

It is worth mentioning that if $\Psi_{0}(r,z)$ satisfies a semilinear equation
of the form

$-[\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}]\Psi_{0}=f\left(r^{2}(\Psi_{0}-{c\over
2})\right){\quad\hbox{in }}\Sigma,$

for an arbitrary nonlinearity $f$, then $\Psi_{0},\
W_{0}=f\left(r^{2}(\Psi_{0}-{c\over 2})\right)$ solves (1.8). A first example
of a solution with a compactly supported, positive vorticity was exhibited by
Hill [37]. In 1970, Fraenkel [23] rigorously found a solution supported inside
a torus with a tiny section of radius $\varepsilon>0$ with a center located at
a fixed distance $r_{0}>0$ of the $z$-axis. See also [42]. To formally derive
the correct propagation speed $c$, let us imagine that we have a family of
solutions $(W_{\varepsilon},\Psi_{\varepsilon})$ of (1.8) with vorticity
depending on a small concentration parameter $\varepsilon>0$ in the form

$r\,W_{\varepsilon}(r,z)=\frac{1}{\varepsilon^{2}}U\left(\frac{x-(r_{0},0)}{\varepsilon}\right)(1+o(1)),\quad
x=(r,z)$ (1.9)

for a positive, rapidly decaying radial profile $U(y)$ with a mass that we
normalize as $\int_{\mathbb{R}^{2}}U(y)dy=8\pi$. To fix ideas, we consider the
Kaufmann-Scully vortex

$U(y)\,=\,\frac{8}{(1+|y|^{2})^{2}},\quad y\in\mathbb{R}^{2}.$ (1.10)

Then we have $r\,W_{\varepsilon}\rightharpoonup 8\pi\delta_{(r_{0},0)}$ as
$\varepsilon\to 0$, where $\delta_{(r_{0},0)}$ designates a Dirac mass in
$\Sigma$ at the point $(r_{0},0)$. Since

$-[\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}]\Psi_{\varepsilon}\approx\frac{1}{r_{0}\varepsilon^{2}}U\left(\frac{x-(r_{0},0)}{\varepsilon}\right){\quad\hbox{in
}}\Sigma,$

we have that $r_{0}\Psi_{\varepsilon}(r,z)$ approaches Green’s function
$G(r,z)$ where

$-[\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}]G=8\pi\delta_{(r_{0},0)}\quad{\quad\hbox{in
}}\Sigma.$

It is not hard to show that near $(r_{0},0)$ and up to an additive constant
and lower order terms we have

$G(r,z)=-2\log(|x-(r_{0},0)|^{2})\,\Bigl{(}1-\frac{3}{2r_{0}}(r-r_{0})\Bigr{)}.$

Actually, at main order we also have

$r\Psi_{\varepsilon}(r,z)=-2\log(|x-(r_{0},0)|^{2}+\varepsilon^{2})(1-\frac{3}{2r_{0}}(r-r_{0})).$

Replacing these approximate expressions into the equation
$\nabla^{\perp}\left(r^{2}(\Psi_{\varepsilon}-{c\over 2})\right)\cdot\nabla
W_{\varepsilon}=0$ near $(r_{0},0)$ yields

$[2r_{0}(4|\log\varepsilon|-{c\over
2})-r_{0}\frac{3}{2}4|\log\varepsilon|]\,{\bf e_{2}}\cdot{\nabla}U\approx 0.$

Hence, at main order the propagation speed $c$ should satisfy

$c=\frac{2}{r_{0}}|\log\varepsilon|(1+o(1))\quad\mbox{as}\quad\varepsilon\to
0.$ (1.11)

This is indeed the speed rigorously derived by Fraenkel [23]. Vortex rings
have been analyzed in larger generality in [24, 44, 1, 18]. Expression (1.11)
corresponds to a special case of formal asymptotics obtained by Da Rios in
1906 [12, 13], of thin vortex tubes around curves evolving by their binormal
flow. See [45, 32, 33]. Helicoidal travelling vortex tubes have been recently
found in [16].

### Interacting vortex rings

We shall formally derive the dynamics of $k\geq 2$ thin, coaxial vortex rings
with comparable speeds given at main order by (1.11) in (1.5). It is
convenient to introduce the new unknown $W(r,z,\tau)$ where

$\omega(r,z,t)\,=\,W(r,z-2{r_{0}^{-1}}\,|\log\varepsilon|\,t,|\log\varepsilon|\,t\,),$

so that Problem (1.5) in terms of $W(r,z,\tau)$, $\tau=|\log\varepsilon|t$,
takes the form

$|\log\varepsilon|\,r\,{\partial}_{\tau}W+{\nabla}^{\perp}(r^{2}\,(\Psi-
r_{0}^{-1}|\log\varepsilon|)\,)\cdot{\nabla}W\,=\,0,\quad-\Delta_{5}\Psi=W.$
(1.12)

Let us assume the presence of a solution $W(r,z,\tau)$ of (1.12) which, in
agreement with (1.9), has the approximate form

$rW(r,z,\tau)\,=\,\sum_{j=1}^{k}\frac{1}{\varepsilon_{j}^{2}}U\left(\frac{x-Q_{j}(\tau)}{\varepsilon_{j}}\right),\quad
x=(r,z),\quad Q_{j}(\tau)=(Q_{j}^{1}(\tau),Q_{j}^{2}(\tau)),$

for small numbers $\varepsilon_{j}\to 0$. This ansatz is consistent with the
fact that $\int rW\,drdz$ should be preserved in time on components of the
support, see (1.7). On the other hand, the fact that the $L^{\infty}$-norm of
$W$ should be preserved in time on each component suggests that
$Q_{j}^{1}(\tau)\varepsilon_{j}^{2}$ should be constant in time, say
$=r_{0}\varepsilon^{2}$. Thus, we choose

$\varepsilon_{j}=\varepsilon\frac{r_{0}}{\sqrt{Q_{j}^{1}(\tau)}},$

so that at main order, near $(r_{0},0)$ we have

$\displaystyle W(r,z,\tau)$
$\displaystyle=\,\sum_{j=1}^{k}\frac{1}{r_{0}\varepsilon^{2}}U\left(\frac{x-Q_{j}(\tau)}{\varepsilon_{j}}\right),\quad
x=(r,z),$

and correspondingly, up to lower order terms

$\displaystyle\Psi_{\varepsilon}(r,z,\tau)$
$\displaystyle=\,\sum_{j=1}^{k}\frac{2}{Q_{j}^{1}}\log\Big{(}\frac{1}{|x-Q_{j}(\tau)|^{2}+\varepsilon_{j}^{2}}\Big{)}\,(1-\frac{3}{2Q_{j}^{1}}(r-Q_{j}^{1})),\quad
x=(r,z),$

for centers $Q_{j}(\tau)$ close to $(r_{0},0)$ to be determined. Substituting
this expression into equation (1.12) we obtain that at main order and for each
$j$,

$\Big{[}-r|\log\varepsilon|\frac{dQ_{j}}{d\tau}+{\nabla}^{\perp}_{x}\sum_{i\neq
j}4r\log\frac{1}{|x-Q_{i}|}\,-2{r-r_{0}\over r_{0}}|\log\varepsilon|\,{\bf
e}_{2}\Big{]}\cdot{\nabla}U\left(\frac{x-Q_{j}(\tau)}{\varepsilon_{j}}\right)\approx
0.$ (1.13)

It is convenient to use the ansatz

$Q_{j}(\tau)=(r_{0},0)+\frac{1}{\sqrt{|\log\varepsilon|}}q_{j}(\tau),\quad
q_{j}(\tau)=(q_{j}^{1}(\tau),q_{j}^{2}(\tau)).$

Imposing vanishing of the left factor in (1.13) at $x=Q_{j}$, neglecting lower
order terms in a fixed interval $\tau\in[0,T]$, we arrive at the limiting
system

$\frac{dq_{j}}{d\tau}=-4\sum_{\ell\not=j}\,\,\frac{(q_{j}-q_{\ell})^{\perp}}{|q_{j}-q_{\ell}|^{2}}+2\,{q_{j}^{1}\over
r_{0}^{2}}\,\,\left(\begin{matrix}0\\\
1\end{matrix}\right),\quad\tau\in[0,T].$ (1.14)

This is a Hamiltonian system for the energy

$H_{k}(q_{1},\ldots,q_{k})=2\sum_{i\neq
j}\log|q_{i}-q_{j}|-\frac{1}{r_{0}^{2}}\sum_{j=1}^{k}|q_{j}^{1}|^{2}.$

For instance, for $k=2$ and restricting ourselves to $q_{1}=-q_{2}=q$ we
arrive at the system

$\frac{dq}{d\tau}\,=\,-2\frac{q^{\perp}}{|q|^{2}}+2\,{q^{1}\over
r_{0}^{2}}\,\,\left(\begin{matrix}0\\\
1\end{matrix}\right),\quad\tau\in[0,T].$

The level curves for its Hamiltonian $q\mapsto H_{2}(q,-q)$ are depicted in
Figure 2. Any solution of this system with sufficiently small initial data is
periodic and lies on a closed level curve around the origin. These periodic
solutions clearly represent the predicted leapfrogging dynamics. As the rings
become initially more distant, the oscillation is lost.

In short, the vorticity of $k$ interacting similar thin vortex rings looks
like the superposition of vorticities of $k$ individual rings with small cross
section $\varepsilon$, whose centers evolve in time approximately following
the leapfrogging dynamics (1.14) being located at mutual distances of order
${1\over\sqrt{|\log\varepsilon|}}$.

Figure 2.  Level curves for $q\to H_{2}(q,-q)$.

Several authors investigated the reduced dynamics given by system (1.14), see
for instance [19, 20, 11, 4, 30, 36, 37, 41]. Numerical simulations of the
leapfrogging, such as in [46, 43, 38, 6], provided theoretical evidence that
this phenomenon should actually occur in the Euler equations. The leapfrogging
phenomenon was experimentally confirmed in 1978 by Yamada and Matsui [48].
They used vortex rings made of air and used smoke for visualization and
successfully created a leapfrogging pair of rings.

Our main result states the existence of a true smooth solution $\omega(r,z,t)$
of Problem (1.5) asymptotically obeying the dynamic law described above, for
each given collisionless solution of System (1.14).

It is more convenient to express, for a small $\varepsilon>0$, the problem in
terms of the equivalent formulation (1.12)

$|\log\varepsilon|\,r\,\frac{{\partial}W}{{\partial}\tau}+{\nabla}^{\perp}(r^{2}\,(\Psi-
r_{0}^{-1}|\log\varepsilon|)\,)\cdot{\nabla}W\,=\,0,\quad-\Delta_{5}\Psi=W,{\quad\hbox{in
}}\Sigma\times[0,T]$ (1.15)

where we recall that, if $W(r,z,\tau)$ solves this problem, then

$\omega(r,z,t)\,=\,W(r,z-2{r_{0}^{-1}}\,|\log\varepsilon|\,t,\,|\log\varepsilon|\,t\,),$

solves (1.5).

######  Theorem 1.

Let $q(\tau)=(q_{1}(\tau),\ldots q_{k}(\tau))$ be a solution of System
$(\ref{car1})$ without collisions in $[0,T]$ in the sense that
$\inf\\{|q_{i}(\tau)-q_{j}(\tau)|\ /\tau\in[0,T],\ i\neq j\\}\,>\,0.$ Then
there exists a smooth solution $W_{\varepsilon}(r,z,\tau)$ of Problem
$(\ref{leap01})$ such that for certain points $Q_{j}^{\varepsilon}(\tau)$ with
the form

$Q_{j}^{\varepsilon}(\tau)=(r_{0},0)+\frac{1}{\sqrt{|\log\varepsilon|}}q_{j}(\tau)+O\Big{(}\frac{\log(|\log\varepsilon|)}{|\log\varepsilon|}\Big{)},\quad$

we have

$\displaystyle W_{\varepsilon}(r,z,\tau)$
$\displaystyle=\,\frac{1}{\varepsilon^{2}r_{0}}\,\sum_{j=1}^{k}U\left(\frac{x-Q_{j}^{\varepsilon}(\tau)}{\varepsilon_{j}}\right)+\varphi(r,z,\tau),\quad
x=(r,z)$ (1.16)

where $U(y)=\frac{8}{(1+|y|^{2})^{2}}$,
$\varepsilon_{j}^{2}=\varepsilon^{2}\frac{r_{0}}{Q_{j}^{1}(\tau)}$ and for
some number $C>0$ we can estimate

$|\varphi(r,z,\tau)|\,\leq\,C\sum_{j=1}^{k}\frac{\varepsilon^{2}|\log\varepsilon|^{60}}{\varepsilon^{2}+\left|x-Q_{j}^{\varepsilon}(\tau)\right|^{2}}\quad\mbox{for
all}\quad x\in\Sigma,\ \tau\in[0,T].$

Theorem 1 corresponds to a first mathematically rigorous justification of the
leapfrogging dynamics, which in the simplest case $k=2$, $q_{1}=-q_{2}$
corresponds to one of the periodic orbits depicted in Figure 2. Motivated by
the classical question for the incompressible Euler equations, Jerrard and
Smets [34, 35] built up a solution with a vortex-ring leapfrogging pattern in
a suitable sense for the Gross-Pitaevskii equation

$iu_{t}=\Delta u+\frac{1}{\varepsilon^{2}}(1-|u|^{2})u{\quad\hbox{in
}}\mathbb{R}^{3}$

on the basis of limiting Ginzburg-Landau energy configurations. The method in
[34, 35] does not seem applicable to the Euler setting. The Kaufman-Scully
vortex (1.10) is a convenient local choice for the building blocks of the
rings. The function $\Gamma_{0}(y)=\log U(y)$ satisfies the Liouville equation

$-\Delta\Gamma_{0}=e^{\Gamma_{0}}=U(y)=\frac{8}{(1+|y|^{2})^{2}}{\quad\hbox{in
}}\mathbb{R}^{2}$ (1.17)

and fine information about the linearized equation is available. Similar
statements can be drawn for generic choices of rapidly decaying, radially
decreasing positive building blocks $U(|y|)$. Vortices (1.10) have been used
to construct solutions with concentrated vorticities in $\mathbb{R}^{2}$ in
[14] and helicoidal travelling waves in [16]. Linearization-stability in
$2d$-Euler and with a third dimension added as a vortex filament is a delicate
matter, see for instance [3, 22, 25, 26, 27, 31].

In Theorem 1, all individual rings in the vorticity expression (1.16) have
similar positive circulations, which is what leads to the leapfrogging
phenomenon. Another interesting dynamics of vortex rings is the case of
opposite sign circulations, namely dipole dynamics. This is closely related to
various scenarios for potential singularities in axi-symmetric Euler flows,
see [7, 8, 10, 21, 39]. Our approach may also be applicable to the analysis of
dipole dynamics.

### Organization of the paper.

In Section §2 we derive an explicit approximate travelling vortex ring with
highly $\varepsilon$-concentrated vorticity near a point $P_{0}$. This is an
explicit function which almost solve the travelling vortex ring equations
(1.8) in a neighbourhood of a given point $P_{0}$.

In Section §3 we consider the first approximate solution $(W^{0},\Psi^{0})$ to
the leapfrogging equations (1.15). It is given by the sum of $k$ approximate
travelling vortex rings, centered at different points $P_{j}$, $j=1,\ldots,k$,
at relative distance $|\log\varepsilon|^{-{1\over 2}}$ one from each other,
all of them collapsing to $P_{0}$ as $\varepsilon\to 0$. In Subsection 3.4 we
rigorously derive the leapfrogging dynamics (1.14) for these centers. This
first approximation $(W^{0},\Psi^{0})$ turns out not to be precise enough and
it needs to be improved. We do this in Sections §4 and §5. In §4 we describe
how we find the improvement $(W^{*},\Psi^{*})$ and we describe the error of
approximation. We make this statement precise in Proposition 4.1. The proof of
Proposition 4.1 is contained in §5. The construction of the approximation
requires 10 subsequent refinements in regions close to the centers, which we
call inner improvements, and one global adjustment in the region far from the
centers, the outer improvement. At the beginning of §5 we describe the general
strategy for the improvement in §5.1 and then we proceed with the detailed
description of the 11 subsequent improvements in subsections §5.2 to §5.12.
For the sixth inner improvement and for the outer improvement we solve two
linear transport equations. We study them respectively in §6 and §7. Then we
look for a solution of equations (1.15) of the form

$\displaystyle W$ $\displaystyle=W^{*}+\varphi$ $\displaystyle\Psi$
$\displaystyle=\Psi^{*}+\psi$

where $\varphi$ and $\psi$ are small corrections of the previously found
approximations. Each remainder is decomposed as the sum of quantities
concentrated near the centers, measured in the slow space variables
${x-Q_{j}^{\varepsilon}\over\varepsilon_{j}}$, the inner contributions, plus a
more regular term taking into account the full picture away from the centers.
The full problem for the remainders is then set up as a coupled system between
the outer and inner contributions which will be solvable if the parameter
functions involved have specific values that make the system essentially
uncoupled. We find a solution of the inner-outer gluing system by means of a
degree theoretical argument involving a-priori estimates for a homotopic
deformation of the problem into a linear, uncoupled one. These arguments are
performed in §9 and §10.

The inner-outer gluing scheme has been a very powerful tool in singularity
formation problems for nonlinear elliptic and parabolic equations, see for
instance [17, 5, 15]. In those applications, the use of maximum principle is
essential. In this paper we make the first attempt to extend this scheme to
three-dimensional Euler flow and prove the Leapfrogging dynamics. We expect
that this scheme will be a useful tool in studying finite time singularities
or the Vortex Filament Conjecture for the incompressible Euler flow.

##  2\. Approximate travelling vortex ring

In this section we define the basic building block for the construction of the
leapfrogging. This object is an approximate travelling vortex ring with highly
$\varepsilon$-concentrated vorticity near a point $P_{0}=(r_{0},0)$,
$r_{0}>0$. It is achieved finding a constant $\alpha$ and a stream function
$\Psi_{\varepsilon}$ ”almost” solving in a neighbourhood of $P_{0}$ the
equations for the travelling vortex ring

$\displaystyle S_{0}(W_{\varepsilon},\Psi_{\varepsilon})$
$\displaystyle:={\nabla}^{\perp}(r^{2}(\Psi_{\varepsilon}-\alpha|\log\varepsilon|))\cdot{\nabla}W_{\varepsilon}\sim
0,\quad{\mbox{where}}\quad W_{\varepsilon}=-\Delta_{5}\Psi_{\varepsilon}$
(2.1)

with the expectation that the vorticity satisfies

$r\,W_{\varepsilon}\rightharpoonup
8\pi\delta_{P_{0}},\quad\quad\mbox{as}\quad\varepsilon\to 0.$ (2.2)

In accordance with (1.8), (1.11) and the discussion in the introduction, the
constant $\alpha$ is expected to satisfy $\alpha\sim{1\over r_{0}}$ as
$\varepsilon\to 0$ and the vorticity $r\,W_{\varepsilon}$ to have the form
(1.9). The associated stream function $\Psi_{\varepsilon}$ will correspond to
an $\varepsilon$-regularization of the following Green’s function

$-\Delta_{5}G(x;P_{0})=8\pi\delta_{P_{0}},\quad{\partial\over\partial
r}G(x;P_{0})=0,\quad{\mbox{on}}\quad\partial\Sigma,\quad G(x;P_{0})\to
0\quad\mbox{as}\quad|x|\to\infty,$

where $\Delta_{5}$ is the operator defined in (1.6). We recall it here

$\Delta_{5}\Psi=\partial_{rr}\Psi+\frac{3}{r}\partial_{r}\Psi+\partial_{zz}\Psi,\quad
x=(r,z).$

We write $G$ as

$G(x;P_{0})=\log\frac{1}{|x-P_{0}|^{4}}\big{(}1-\frac{3}{2r_{0}}(r-r_{0})+H(x;P_{0})\big{)}+K(x;P_{0}),$

where $H(\cdot;P_{0})$ and $K(\cdot;P_{0})$ satisfy respectively

$\Delta_{5}\Big{(}\log\frac{1}{|x-P_{0}|^{4}}H(x;P_{0})\Big{)}=-30\frac{(r-r_{0})^{2}}{rr_{0}|x-P_{0}|^{2}}+\frac{9}{2rr_{0}}\log\frac{1}{|x-P_{0}|^{4}},$
(2.3)

and

$\Delta_{5}K(x;P_{0})=0.$ (2.4)

We make this construction precise in the next Proposition.

######  Proposition 2.1.

Let $P_{0}=(r_{0},0)$ with $r_{0}>0$. There exist a smooth radially symmetric
function $\Gamma:\mathbb{R}^{2}\to\mathbb{R}$ with
$\Gamma(y)=\Gamma(|y|)=O({\log|y|\over|y|^{2}})$ as $|y|\to\infty$, and
$\bar{\varepsilon}>0$ such that, for all
$\varepsilon\in(0,\bar{\varepsilon})$, there exist a constant $\alpha$
satisfying

$\alpha-
r_{0}^{-1}=O(|\log\varepsilon|^{-1}),\quad\quad\mbox{as}\quad\varepsilon\to
0,$

such that, defining $\Psi_{\varepsilon}$ by

$\displaystyle r_{0}\Psi_{\varepsilon}(x;P_{0})$
$\displaystyle=\log\frac{1}{(\varepsilon^{2}+|x-P_{0}|^{2})^{2}}\big{(}1-\frac{3}{2r_{0}}(r-r_{0})+H(x;P_{0})\big{)}$
(2.5) $\displaystyle\quad+K(x;P_{0})+{r-r_{0}\over
2r_{0}}\Gamma({x-P_{0}\over\varepsilon})$

and

$W_{\varepsilon}^{0}(x;P_{0})=-\Delta_{5}\Psi_{\varepsilon}(x;P_{0})$ (2.6)

the following holds true: for any fixed $\delta>0$ small and any $x$ with
$|x-P_{0}|<\delta$, setting

$y={x-P_{0}\over\varepsilon}=\rho
e^{i\theta},\quad|y|<{\delta\over\varepsilon}$

$S_{0}(W_{\varepsilon}^{0},\Psi_{\varepsilon})$ as defined in (2.1) has the
following expansion

$\displaystyle\varepsilon^{4}$ $\displaystyle
S_{0}(W_{\varepsilon}^{0},\Psi_{\varepsilon})(\varepsilon
y+P_{0})={\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}(\rho,\theta,\varepsilon,P_{0})$ (2.7)
$\displaystyle+{\varepsilon^{3}|\log\varepsilon|\over
1+|y|^{3}}|\log(\varepsilon(1+|y|))|[E_{1}(\rho,\theta,\varepsilon,P_{0})+E_{3}(\rho,\theta,\varepsilon,P_{0})]$
$\displaystyle+O\left({\varepsilon^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}\,\right),\quad\quad\mbox{as}\quad\varepsilon\to 0.$

Here we have written $E_{k}(\rho,\theta,\varepsilon,P_{0})$ for a function of
the form

$\displaystyle
E_{k}(\rho,\theta,\varepsilon,P_{0})=E_{k,1}(\rho,\varepsilon,P_{0})\cos(k\theta)+E_{k,2}(\rho,\varepsilon,P_{0})\sin(k\theta)$
(2.8)

where

$\displaystyle\sum_{j=0}^{2}\Bigl{|}(1+\rho)^{j}\frac{\partial^{j}E_{k,i}}{\partial\rho^{j}}\Bigr{|}+\Bigl{|}\nabla_{P_{0}}E_{k,i}\Bigl{|}\lesssim
1,\quad\quad\mbox{as}\quad\varepsilon\to 0.$

The proof of this result is based on showing that $\alpha$ and
$\Psi_{\varepsilon}$ ”almost” solve the following semi-linear elliptic
equation

$\displaystyle E[\Psi](x)$
$\displaystyle:=\Delta_{5}\Psi+F(r^{2}(\Psi-\alpha|\log\varepsilon|))\equiv 0$
(2.9)

for some function $F$. Then $\Psi$ and
$W:=F(r^{2}(\Psi-\alpha|\log\varepsilon|))$ would solve (2.1).

We expect the vorticity to locally satisfy around $P_{0}$ the asymptotic
behaviour (2.2). This can be achieved choosing

$F(r^{2}(\Psi-\alpha|\log\varepsilon|))={8\over
r_{0}}e^{-K(P_{0};P_{0})}\varepsilon^{2-\alpha
r_{0}}f\Big{(}\frac{r^{2}}{r_{0}}(\Psi-\alpha|\log\varepsilon|)\Big{)},\quad{\mbox{with}}\quad
f(s)=e^{s}.$ (2.10)

We will show that in this case the vorticity has the following expansion in a
neighbourhood of $P_{0}$: for $|x-P_{0}|<\delta$, setting
$y={x-P_{0}\over\varepsilon}$,

$\displaystyle W_{\varepsilon}^{0}(x;P_{0})$
$\displaystyle={1\over\varepsilon^{2}r_{0}}w[\varepsilon,P_{0}](y)$ (2.11)
$\displaystyle w[\varepsilon,P_{0}](y)$
$\displaystyle=U(y)\left(1-{\varepsilon y_{1}\over
2r_{0}}(\Gamma_{0}+\bar{A}+\Gamma)+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\right)$

where $U(y)$ and $\Gamma_{0}(y)$ are defined in (1.17), $\bar{A}$ is

$\bar{A}=-6-{\int_{\mathbb{R}^{2}}U(y)y_{1}\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\Gamma_{0}\,dy\over\int_{\mathbb{R}^{2}}U(y)y_{1}\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\,dy},$
(2.12)

and $\Gamma=\Gamma(y)$ is the function introduced in (2.5). Moreover,

$\displaystyle\theta_{1}[P_{0}](y)$
$\displaystyle=\Biggl{[}a(P_{0})+b_{1}(P_{0})\cos 2\theta+b_{2}(P_{0})\sin
2\theta$ (2.13)
$\displaystyle+|\log(\varepsilon(1+|y|))|(a(P_{0})+b_{1}(P_{0})\cos
2\theta+b_{2}(P_{0})\sin 2\theta)\Biggl{]}\,O(|y|^{2})$

and

$\theta_{2}[P_{0}](y)=a(P_{0})O(|y|^{3}|\log(\varepsilon(1+|y|)|))$ (2.14)

for $a$, $b_{1}$ and $b_{2}$ smooth functions of $P_{0}$, uniformly bounded as
$\varepsilon\to 0$.

From (2.11) we see that $rW_{\varepsilon}^{0}(x)$ approaches, locally around
$P_{0}$, a Dirac delta, for $\varepsilon\to 0$.

We will make use of these estimates in the next Section. We next present the
proof of Proposition 2.1.

###### Proof of Proposition 2.1.

Let us fix $\delta>0$ and consider the region of points $x$ with
$|x-P_{0}|<\delta$. We use the expanded variable

$y={x-P_{0}\over\varepsilon}$

which we also identify with polar coordinates $y=\rho e^{i\theta}$,
$\rho=|y|$.

The strategy of the proof consists in constructing $\Psi_{\varepsilon}$ and
$\alpha$ so that the following expansion for $E[\Psi_{\varepsilon}]$ holds
true, where $E$ is given by (2.9)-(2.10): for all
$|y|<{\delta\over\varepsilon}$ we have

$\displaystyle\varepsilon^{2}\,r_{0}\,E[\Psi_{\varepsilon}](x)$
$\displaystyle={\varepsilon^{2}\over
1+|y|^{2}}\Biggl{[}a(P_{0})+b_{1}(P_{0})\cos 2\theta+b_{2}(P_{0})\sin 2\theta$
(2.15) $\displaystyle+|\log(\varepsilon(1+|y|))|(a(P_{0})+b_{1}(P_{0})\cos
2\theta+b_{2}(P_{0})\sin 2\theta)\Biggl{]}$
$\displaystyle+O({\varepsilon^{3}\over
1+|y|})|\log(\varepsilon(|y|+1))|,\,\quad$

uniformly as $\varepsilon\to 0$. Here $a$, $b_{1}$ and $b_{2}$ denote smooth
functions of $P_{0}$, uniformly bounded together with their derivative as
$\varepsilon\to 0$, whose definition may change from line to line.

Assume expansion (2.15) holds true. By construction we have that, in the same
region $|y|<{\delta\over\varepsilon}$

$\displaystyle\frac{r^{2}}{r_{0}}\Big{(}\Psi_{\varepsilon}(x;P_{0})-\alpha|\log\varepsilon|\Big{)}=(1+{\varepsilon\over
r_{0}}y_{1})^{2}\left(r_{0}\Psi_{\varepsilon}(\varepsilon
y+P_{0};P_{0})-\alpha r_{0}|\log\varepsilon|\right)$ (2.16)
$\displaystyle=\Gamma_{0}(y)-(4-\alpha r_{0})\log\varepsilon-\log
8+K(P_{0};P_{0})$
$\displaystyle\quad+\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)+\bar{A}+\Gamma(y)\Big{)}\,\varepsilon
y_{1}+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y),$

where $\theta_{1}$ and $\theta_{2}$ are functions also described by (2.13) and
(2.14) respectively. Here $\Gamma_{0}(y)=\log U(y)$ as in (1.17).

Formula (2.7) readily follows from (2.16) and (2.15).

The rest of the proof is devoted to show the validity of (2.15). We start with
the observation that, since the operator in (2.9) is invariant under
translations in the $z$-direction, it is not restrictive to work in the class
of functions that are even in the variable $z$. Let the
$\varepsilon$-regularization of the Green’s function $G$ be given by

$G_{\varepsilon}(x;P_{0}):=\log\frac{1}{(\varepsilon^{2}+|x-P_{0}|^{2})^{2}}\big{(}1-\frac{3}{2r_{0}}(r-r_{0})+H(x;P_{0})\big{)}+K(x;P_{0}).$

To simplify notation, we drop the dependence on $P_{0}$ in the functions
$G_{\varepsilon}$, $H$ and $K$.

Let $y=\frac{x-P_{0}}{\varepsilon}$ and $\delta>0$ small, independent of
$\varepsilon$. Then for $|x-P_{0}|<\delta$ we have

$G_{\varepsilon}(x)=(\Gamma_{0}(y)-4\log(\varepsilon)-\log
8)\Big{(}1-\frac{3}{2r_{0}}\varepsilon y_{1}+H(P_{0}+\varepsilon
y)\Big{)}+K(P_{0}+\varepsilon y),$

where we mean $H(P_{0}+\varepsilon y)=H(P_{0}+\varepsilon y;P_{0})$, etc.
Observe that ${\partial}_{z}K(P_{0})=0$ by symmetry. From (2.3) we get that
$H(x;P_{0})$ has the following expansion as $x\to P_{0}$

$\displaystyle H(x;P_{0})$
$\displaystyle=a_{1}(P_{0})(r-r_{0})^{2}+a_{2}(P_{0})z^{2}+b(P_{0}){|x-P_{0}|^{2}\over\log|x-P_{0}|^{2}}+O(|x-P_{0}|^{3})$
(2.17)

where $a_{1}$, $a_{2}$, $b$ are constants whose value depends on $P_{0}$.
Using (2.17), we expand

$\displaystyle G_{\varepsilon}(x)$
$\displaystyle=\Gamma_{0}(y)-4\log\varepsilon-\log
8+K(P_{0})-\frac{3}{2r_{0}}\left[\Gamma_{0}(y)-4\log\varepsilon-\log
8-{2r_{0}\over 3}\partial_{r}K(P_{0})\right]\varepsilon y_{1}$
$\displaystyle\quad+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)$

where $\theta_{1}$, $\theta_{2}$ as in (2.13) and (2.14).

We next introduce

$\Psi_{P_{0}}(x)=\frac{1}{r_{0}}G_{\varepsilon}(x),$

and readily get

$\displaystyle\frac{r^{2}}{r_{0}}\Big{(}\Psi_{P_{0}}(x)-\alpha|\log\varepsilon|\Big{)}=\Gamma_{0}(y)-(4-\alpha
r_{0})\log\varepsilon-\log 8+K(P_{0})$
$\displaystyle\quad+\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+2r_{0}\partial_{r}K(P_{0})+4K(P_{0})-\log(8)\Big{)}\,\varepsilon
y_{1}$
$\displaystyle\quad+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y),$

and

$\displaystyle{8\over r_{0}}e^{-K(P_{0})}\varepsilon^{2-\alpha
r_{0}}f\Big{(}\frac{r^{2}}{r_{0}}(\Psi_{P_{0}}(x)-\alpha|\log\varepsilon|)\Big{)}$
(2.18)
$\displaystyle=\frac{1}{\varepsilon^{2}r_{0}}U(y)\exp\Big{[}\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+2r_{0}\partial_{r}K(P_{0})+4K(P_{0})-\log
8\Big{)}\,\varepsilon y_{1}$
$\displaystyle\quad+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\Big{]},$

for functions $\theta_{1}$, $\theta_{2}$ of the form (2.13) and (2.14).

Next we compute

$\displaystyle\Delta_{5}\Psi_{P_{0}}=(1)+(2)$

where

$\displaystyle(1)$
$\displaystyle=\frac{1}{r_{0}}\Delta_{5}\Big{[}(\Gamma_{0}({x-P_{0}\over\varepsilon})-4\log\varepsilon-\log
8)\Big{(}1-\frac{3}{2r_{0}}(r-r_{0})\Big{)}\Big{]}$ $\displaystyle(2)$
$\displaystyle=\frac{1}{r_{0}}\Delta_{5}[(\Gamma_{0}({x-P_{0}\over\varepsilon})-4\log\varepsilon-\log
8)H],$

since by definition $\Delta_{5}K=0$. Setting again $\varepsilon y=x-P_{0}$, we
get

$\displaystyle(1)$
$\displaystyle=\frac{1}{r_{0}}\Big{[}\Big{(}-\frac{1}{\varepsilon^{2}}U(y)+\frac{3}{\varepsilon(r_{0}+\varepsilon
y_{1})}\Gamma_{0}^{\prime}(y)\frac{y_{1}}{\rho}\Big{)}\Big{(}1-\frac{3}{2r_{0}}\varepsilon\
y_{1}\Big{)}-\frac{3}{\varepsilon
r_{0}}\Gamma_{0}^{\prime}(\rho)\frac{y_{1}}{\rho}-\frac{9}{2rr_{0}}(\Gamma_{0}-4\log\varepsilon-\log
8)\Big{]}$
$\displaystyle=\frac{1}{r_{0}}\Big{[}-\frac{1}{\varepsilon^{2}}U(y)+\frac{3}{2\varepsilon
r_{0}}U(y)y_{1}-\frac{15}{2}\frac{\Gamma_{0}^{\prime}(\rho)}{\rho}\frac{y_{1}^{2}}{rr_{0}}-\frac{9}{2rr_{0}}(\Gamma_{0}-4\log\varepsilon-\log
8)\Big{]}$
$\displaystyle=\frac{1}{\varepsilon^{2}r_{0}}\Big{[}-U(y)+\frac{3}{2r_{0}}\varepsilon
U(y)y_{1}+30{(r-r_{0})^{2}\over
r_{0}r}{\varepsilon^{2}\over\varepsilon^{2}+|x-P_{0}|^{2}}-\frac{9}{2rr_{0}}\varepsilon^{2}\log{1\over(\varepsilon^{2}+|x-P_{0}|^{2})^{2}}\Big{]}.$

From (2.3) we get

$\displaystyle(2)$
$\displaystyle=-30\frac{(r-r_{0})^{2}}{rr_{0}^{2}|x-P_{0}|^{2}}+\frac{9}{2rr_{0}^{2}}\log\frac{1}{|x-P_{0}|^{4}}+\frac{1}{r_{0}}\Delta_{5}[(\Gamma_{0}({x-P_{0}\over\varepsilon})-4\log\varepsilon-\log
8-\log{1\over|x-P_{0}|^{4}})H]$
$\displaystyle=-30\frac{(r-r_{0})^{2}}{rr_{0}^{2}|x-P_{0}|^{2}}+\frac{9}{2rr_{0}^{2}}\log\frac{1}{|x-P_{0}|^{4}}+\frac{2}{r_{0}}\Delta_{5}[\log{|{x-P_{0}\over\varepsilon}|^{2}\over
1+|{x-P_{0}\over\varepsilon}|^{2}}H]$

Combining the above expressions we conclude that

$\displaystyle-\Delta_{5}\Psi_{P_{0}}$
$\displaystyle=\frac{1}{r_{0}}\Big{[}\frac{1}{\varepsilon^{2}}U(y)-\frac{3}{\varepsilon
r_{0}}U(y)y_{1}+O(\frac{1}{1+\rho^{2}})\Big{]}$ (2.19)
$\displaystyle=\frac{1}{r_{0}\varepsilon^{2}}U(y)\Big{[}1-\frac{3}{r_{0}}\varepsilon
y_{1}+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\Big{]},$

where $\theta_{1}$ and $\theta_{2}$ are described in (2.13) and (2.14).

Putting together (2.18) and (2.19) we find that

$\varepsilon^{2}E[\Psi_{P_{0}}]=\frac{1}{r_{0}}[\varepsilon
E_{0}+\Theta_{\varepsilon}(y)]$ (2.20)

where

$\displaystyle E_{0}(y)$
$\displaystyle=\frac{y_{1}}{2r_{0}}U(y)\Big{(}\Gamma_{0}(y)-4(1-\alpha
r_{0})\log\varepsilon+A_{0}\Big{)},$

with

$A_{0}=2r_{0}\partial_{r}K(P_{0})+4K(P_{0})-\log 8+6,$

and

$\displaystyle\Theta_{\varepsilon}(y)$
$\displaystyle=U(y)\exp\Big{[}\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+2r_{0}\partial_{r}K(P_{0})+4K(P_{0})-\log
8\Big{)}\,\varepsilon y_{1}$
$\displaystyle\quad+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\Big{]}$
$\displaystyle-U(y)\Big{[}1+\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+2r_{0}\partial_{r}K(P_{0})+4K(P_{0})-\log
8\Big{)}\,\varepsilon y_{1}\Big{]}$
$\displaystyle+U(y)\left[\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\right],$

where $\theta_{1}$, $\theta_{2}$ denote again generic functions of the form
(2.13), (2.14).

A closer look at this expression gives that $\Theta_{\varepsilon}(y)$ can be
described as follows

$\displaystyle\Theta_{\varepsilon}(y)$ $\displaystyle={\varepsilon^{2}\over
1+|y|^{2}}\Biggl{[}a(P_{0})+b_{1}(P_{0})\cos 2\theta+b_{2}(P_{0})\sin 2\theta$
(2.21) $\displaystyle+|\log(\varepsilon(1+|y|))|(a(P_{0})+b_{1}(P_{0})\cos
2\theta+b_{2}(P_{0})\sin 2\theta)\Biggl{]}$
$\displaystyle+O({\varepsilon^{3}\over
1+|y|})\log(\varepsilon(1+|y|))a(P_{0})$

uniformly as $\varepsilon\to 0$. Also here $a$, $b_{1}$, $b_{2}$ stand for
generic smooth functions of $P_{0}$, uniformly bounded as $\varepsilon\to 0$.

In order to reduce the size of the error term in (2.20) we solve

$\displaystyle\Delta\bar{\psi}+U(y)\bar{\psi}+\varepsilon
E_{0}(y)=0\quad{\mbox{in}}\quad\mathbb{R}^{2},\quad\lim_{|y|\to\infty}\bar{\psi}(y)=0.$
(2.22)

It is here when we introduce the function $\Pi$. The error term $E_{0}$ is by
definition even with respect to the variable $y_{2}$. It is known that all
bounded solutions to

$\Delta\bar{\psi}+U(y)\bar{\psi}=0\quad{\mbox{in}}\quad\mathbb{R}^{2}$

are given by linear combinations of

$\frac{\partial\Gamma_{0}}{{\partial}y_{i}}(y),\quad i=1,2,\quad
2+\nabla\Gamma_{0}\cdot y.$

This result can be found in [2]. By the standard Fredholm alternative for this
problem we need the following solvability condition satisfied

$\int_{\mathbb{R}^{2}}E_{0}(y)\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\,dy\
=\ 0.$

In fact, the remaining solvability conditions

$\int_{\mathbb{R}^{2}}E_{0}(y)\frac{\partial\Gamma_{0}}{{\partial}y_{2}}(y)\,dy\
=\ 0,\quad\int_{\mathbb{R}^{2}}E_{0}(y)\left(2+\nabla\Gamma_{0}\cdot
y\right)\,dy\ =\ 0$

are automatically satisfied by symmetry. We compute

$\displaystyle
2r_{0}\int_{\mathbb{R}^{2}}E_{0}(y)\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\,dy\
$
$\displaystyle=\left(\int_{\mathbb{R}^{2}}U(y)y_{1}\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\,dy\right)(A_{0}-4(1-\alpha
r_{0})\log\varepsilon)$
$\displaystyle+\int_{\mathbb{R}^{2}}U(y)y_{1}\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\Gamma_{0}\,dy.$

Then
$\int_{\mathbb{R}^{2}}E_{0}(y)\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\,dy\
=\ 0$ is satisfied if

$A_{0}-4(1-\alpha
r_{0})\log\varepsilon=-{\int_{\mathbb{R}^{2}}U(y)y_{1}\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\Gamma_{0}\,dy\over\int_{\mathbb{R}^{2}}U(y)y_{1}\frac{\partial\Gamma_{0}}{{\partial}y_{1}}(y)\,dy}:=A.$

Observe that $A$ is a constant, independent of $\varepsilon$ and of $P_{0}$.
This equation is satisfied if we choose $\alpha$ in terms of $r_{0}$ to be

$\alpha=\frac{1}{r_{0}}+\frac{\beta(r_{0})}{\log\varepsilon},$ (2.23)

for an explicit function $\beta=\beta(r_{0})$ which is smooth and uniformly
bounded in $r_{0}$, together with its derivative, as $\varepsilon\to 0$.

With this choice of $\alpha$ we can solve (2.22). Writing in polar coordinates
$y=\rho e^{i\theta}$ we observe that $E_{0}$ has the form
$E_{0}(y)=Q(\rho)\cos(\theta)$ with $Q(\rho)=O(\frac{\log\rho}{\rho^{3}})$ as
$\rho\to\infty$. A direct computation yields to the existence of a function
$\Gamma(y)$, radially symmetric and such that
$\Gamma(\rho)=O(\frac{\log\rho}{\rho^{2}})$ as $\rho\to\infty$, so that the
solution $\bar{\psi}(y;P_{0})=\bar{\psi}(y)$ of (2.22) has the form

$\bar{\psi}(y;P_{0})={\varepsilon\over 2r_{0}}\Gamma(\rho)y_{1}.$ (2.24)

Using now the whole expression of $\Psi_{\varepsilon}$ as in (2.5), we
decompose

$\displaystyle\varepsilon^{2}E[\Psi_{\varepsilon}]$
$\displaystyle=\varepsilon^{2}E[\Psi_{P_{0}}]+{\varepsilon^{3}\over
r_{0}}[({\partial}_{rr}+{\partial}_{zz})\bar{\Gamma}+F^{\prime}(\Psi_{P_{0}})\bar{\Gamma}]+\varepsilon^{3}{3\over
r_{0}r}{\partial}_{r}\bar{\Gamma}$
$\displaystyle+\varepsilon^{2}[F(\Psi_{P_{0}}+\varepsilon\bar{\Gamma})-F(\Psi_{P_{0}})-F^{\prime}(\Psi_{P_{0}})\varepsilon\bar{\Gamma}]$

where $\bar{\Gamma}={r-r_{0}\over 2r_{0}}\Gamma({x-P_{0}\over\varepsilon})$
and $F$ is

$F(s)={8\over r_{0}}e^{-K(P_{0};P_{0})}\varepsilon^{2-\alpha
r_{0}}f\Big{(}\frac{r^{2}}{r_{0}}(s-\alpha|\log\varepsilon|)\Big{)}.$

For $|y|<{\delta\over\varepsilon}$,

$\displaystyle F^{\prime}(\Psi_{P_{0}})$
$\displaystyle=\frac{1}{\varepsilon^{2}}U(y)\exp\Big{[}\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+2r_{0}\partial_{r}K(P_{0})+4K(P_{0})-\log
8\Big{)}\,\varepsilon y_{1}$
$\displaystyle\quad+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\Big{]}\times(1+{\varepsilon
y_{1}\over r_{0}})^{2}$
$\displaystyle={1\over\varepsilon^{2}}U(y)\Big{[}1+\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+a(P_{0})\Big{)}\varepsilon
y_{1}+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\Big{]}$

where $\theta_{1}$, $\theta_{2}$ satisfy (2.13), (2.14). Since
$\varepsilon\Gamma(x)=\bar{\psi}(y)$, for $y={x-P_{0}\over\varepsilon}$ and
$\bar{\psi}$ defined in (2.24), we have

$\displaystyle{\varepsilon^{3}\over r_{0}}$
$\displaystyle[({\partial}_{rr}+{\partial}_{zz})\bar{\Gamma}+F^{\prime}(\Psi_{P_{0}})\bar{\Gamma}]={1\over
r_{0}}[\Delta_{y}\bar{\psi}+U(y)\bar{\psi}]$
$\displaystyle+U(y)\bar{\psi}\Big{[}\frac{1}{2r_{0}}\Big{(}\Gamma_{0}(y)-(4-4\alpha
r_{0})\log\varepsilon+a(P_{0})\Big{)}\varepsilon
y_{1}+\varepsilon^{2}\theta_{1}[P_{0}](y)+\varepsilon^{3}\theta_{2}[P_{0}](y)\Big{]}.$

Recall now that $\varepsilon^{2}E[\Psi_{0}]$ can be written as in (2.20).
Since $\bar{\psi}$ solves (2.22) and has the form (2.24), we get

$\displaystyle\varepsilon^{2}E$
$\displaystyle[\Psi_{P_{0}}]+{\varepsilon^{3}\over
r_{0}}[({\partial}_{rr}+{\partial}_{zz})\bar{\Gamma}+F^{\prime}(\Psi_{P_{0}})\bar{\Gamma}]=\Theta_{\varepsilon}(y)$

where $\Theta_{\varepsilon}$ can be described as in (2.21). Besides, using the
form of the function $\Gamma$, as described in (2.5), one sees with a direct
inspection that

$\varepsilon^{3}{3\over
r_{0}r}{\partial}_{r}\bar{\Gamma}+\varepsilon^{2}[F(\Psi_{P_{0}}+\varepsilon\bar{\Gamma})-F(\Psi_{P_{0}})-F^{\prime}(\Psi_{P_{0}})\varepsilon\bar{\Gamma}]=\Theta_{\varepsilon}(y),$

with $\Theta_{\varepsilon}$ another function of the form (2.21). This
concludes the proof of (2.15).

∎

##  3\. First approximate leapfrogging

Take $r_{0}>0$ and define

$\alpha_{0}={1\over r_{0}}.$ (3.1)

The rest of the paper is devoted to produce a solution to Problem (1.12). With
a little abuse of notation, we will use the variable $t$ instead of $\tau$
from now on. More precisely, we will prove the existence of $(\Psi,W)$ solving

$\displaystyle\left\\{\begin{aligned}
&|\log\varepsilon|\,r\,{\partial}_{t}W+{\nabla}^{\perp}(r^{2}(\Psi-\alpha_{0}\,|\log\varepsilon|))\cdot{\nabla}W=0\quad{\mbox{in}}\quad\Sigma\times[0,T)\\\
&-\Delta_{5}\Psi=W,\quad\quad{\mbox{in}}\quad\Sigma\times[0,T)\\\
&{\partial\over\partial
r}\Psi(x,t)=0\quad{\mbox{on}}\quad\partial\Sigma\times[0,T),\quad|\Psi(x,t)|\to
0\quad\text{as }|x|\to\infty.\end{aligned}\right.$ (3.2)

We recall that $\Sigma=\\{x=(r,z)\,:\,r>0,\,x\in\mathbb{R}\\}$, see (1.3), and

$\Delta_{5}\Psi=\partial_{rr}\Psi+\frac{3}{r}\partial_{r}\Psi+\partial_{zz}\Psi.$

This Section is devoted to define a first approximate solution to (3.2), given
as a sum of approximate travelling vortex rings, as built in Section 2. These
traveling vortex rings are centered at different points, at relative distance
$|\log\varepsilon|^{-{1\over 2}}$ one from each other, all of them collapsing
to $(r_{0},0)$ as $\varepsilon\to 0$. Let us be more precise.

###  3.1. The parameter functions

Fix an integer $k\geq 2$ and consider points $P_{j}=P_{j}(t)$, for
$j\in\\{1,\ldots,k\\}$, which evolve in time and have the form

$\displaystyle P_{j}$ $\displaystyle=P_{j}(t)=(r_{j}(t),z_{j}(t)),\quad
t\in[0,T)\quad{\mbox{with}}$ (3.3) $\displaystyle P_{j}$
$\displaystyle=P_{0}+P_{j}^{0}(t)+P^{1}_{j}(t)+{\bf a}_{j}(t),\quad
P_{0}:=(r_{0},0).$

Let us describe the different terms in the decomposition of $P_{j}$. The
points $P_{j}^{0}(t)$ are explicit and will be determined towards the end of
this section, and their size will be

$\displaystyle P_{j}^{0}$
$\displaystyle=(r^{0}_{j}(t),z^{0}_{j}(t)),\quad\|P^{0}_{j}\|_{L^{\infty}[0,T)}+\|{\partial}_{t}P^{0}_{j}\|_{L^{\infty}[0,T)}\lesssim|\log\varepsilon|^{-{1\over
2}}$ (3.4) $\displaystyle|P^{0}_{j}(t)-P_{i}^{0}(t)|\geq
C|\log\varepsilon|^{-{1\over 2}}\quad\forall i\not=j,\quad\forall t\in[0,T)$

for some $C>0$, independent of $\varepsilon$. These points will satisfy at
main order the reduced system for the leapfrogging dynamics (1.14).

The points $P_{j}^{1}(t)=(r_{j}^{1}(t),z_{j}^{1}(t))$ in (3.3) will also be
determined in the process of the construction of an approximate solution and
they will satisfy

$\|P^{1}_{j}\|_{L^{\infty}[0,T)}+\|{\partial}_{t}P^{1}_{j}\|_{L^{\infty}[0,T)}\lesssim\varepsilon^{2-\sigma},$
(3.5)

for some $\sigma>0$, small and independent of $\varepsilon$.

The points ${\bf a}_{j}(t)=(a_{j1}(t),a_{j2}(t))$ are free parameters to
adjust at the end of our proof. For the moment, we ask they are continuous
functions in $[0,T]$ for which ${\partial}_{t}{\bf a}$ exists and such that

$\|{\bf a}_{j}\|_{C^{1}[0,T)}\lesssim\varepsilon^{3+\sigma}.$ (3.6)

The following notation will be used to identify the different sets of points
in the decomposition of $P_{j}$ given in (3.3)

$\displaystyle P$ $\displaystyle=(P_{1},\ldots,P_{k}),\quad
P^{b}=(P_{1}^{b},\ldots,P_{k}^{b}),\quad b=0,1,\quad{\bf a}=({\bf
a}_{1},\ldots,{\bf a}_{k}).$

Since the relative distance between two points is of order
$|\log\varepsilon|^{-{1\over 2}}$ we have

$\\{x\,:\,|x-P_{i}|<|\log\varepsilon|^{-1}\\}\cap\\{x\,:\,|x-P_{\ell}|<|\log\varepsilon|^{-1}\\}=\emptyset$
(3.7)

for all $\varepsilon$ small and $i\not=\ell$. Besides, under assumptions
(3.3), (3.4), (3.5) and (3.6), we have that

$P_{j}\to(r_{0},0),\quad\quad\mbox{as}\quad\varepsilon\to 0,\quad\forall
j=1,\ldots,k,$

uniformly for $t\in[0,T]$.

Given the points $P_{j}$ as described in (3.3), we introduce positive
functions $\varepsilon_{j}=\varepsilon_{j}(t)>0$ such that, for all
$j=1,\ldots,k$,

$t\to r_{j}(t)\,\varepsilon_{j}^{2}(t)\quad{\mbox{is independent of}}\quad t.$

For convenience we choose

$r_{j}(t)\,\varepsilon_{j}^{2}(t)=r_{0}\,\varepsilon^{2}\quad\quad\mbox{for
all}\quad\quad t\in[0,T).$ (3.8)

From (3.3)–(3.6), we have that for all $j=1,\ldots,k$,

$\|\varepsilon_{j}-\varepsilon\|_{L^{\infty}[0,T)}+\|{\partial}_{t}\varepsilon_{j}\|_{L^{\infty}[0,T)}\lesssim\varepsilon|\log\varepsilon|^{-{1\over
2}},\quad\forall\quad j=1,\ldots,k,\quad\quad\mbox{as}\quad\varepsilon\to 0.$
(3.9)

For such $\varepsilon_{j}(t)$ and $P_{j}(t)$, let $\alpha_{j}$ be given as in
Proposition 2.1 so that

$\alpha_{j}[r_{j}](t)={1\over
r_{j}(t)}+{\beta(r_{j}(t))\over|\log\varepsilon_{j}(t)|},$

where $\beta(s)$ is the smooth function defined as in (2.23). Observe that

$\alpha_{j}[r_{0}](t)={1\over
r_{0}}+{\beta(r_{0})\over|\log\varepsilon|}=\alpha_{0}+{\beta(r_{0})\over|\log\varepsilon|}$

is constant in time. From (3.1) we also have

$\alpha_{0}-\alpha_{j}[r_{j}](t)={r_{j}-r_{0}\over
r_{0}r_{j}}-{\beta(r_{j})\over|\log\varepsilon_{j}|},$ (3.10)

and $\|\alpha_{0}-\alpha_{j}\|_{L^{\infty}[0,T)}=O(|\log\varepsilon|^{-{1\over
2}})$ as $\varepsilon\to 0$.

Having introduced these parameters, we define $k$ approximate traveling vortex
rings:

$\Psi_{j}^{0}(x;P_{j}):=\Psi_{\varepsilon_{j}(t)}(x;P_{j}(t)),\quad
j=1,\ldots,k$

where $\Psi_{\varepsilon}(x;P)$ is given in (2.5) in Proposition 2.1. As in
(2.6), we define

$W_{j}^{0}(x;P_{j}):=-\Delta_{5}\Psi_{\varepsilon_{j}(t)}(x;P_{j}(t)),\quad
j=1,\ldots,k.$ (3.11)

The starting point of our construction is to take these vorticities and add
them up together. We do the same for the stream functions, which we then
modify with a cut off function.

For $N\in\mathbb{N}$, we set

$\eta_{N}(s)=1,\quad{\mbox{for}}\quad s<N,\quad=0\quad{\mbox{for}}\quad s>2N$
(3.12)

for a smooth cut-off function.

For $P(t)=(P_{1},P_{2},\ldots,,P_{k})(t),$ define

$\bar{\Psi}^{0}(x;P)=\eta_{0}(x)\sum_{j=1}^{k}\Psi_{j}^{0}(x;P_{j}),\quad
W^{0}(x;P)=\sum_{j=1}^{k}W_{j}^{0}(x;P_{j}),$

where $\eta_{0}$ is the smooth cut-off function given by

$\eta_{0}(x)=\eta_{1}\left({4|x-(r_{0},0)|\over r_{0}}\right),$

with $\eta_{1}$as in (3.12).

###  3.2. The function $H^{0}$ and definition of the very first approximation

We shall slightly modify $\bar{\Psi}_{0}$ by a function $H^{0}$ to have

$\Delta_{5}\left(\bar{\Psi}^{0}+H^{0}\right)(x,t)+W^{0}(x,t)=0,$

for $x\in\Sigma$, $t\in[0,T]$.

For this purpose, consider the linear problem

$\Delta_{5}\psi+h=0,\quad{\mbox{in }}\Sigma,\quad{\partial\psi\over\partial
r}=0\quad{\mbox{on}}\,\,{\partial}\Sigma,\quad\psi(x)\to
0,\,\,\quad\mbox{as}\quad|x|\to\infty,$ (3.13)

for a smooth function $h$ satisfying

$|h(x)|\leq{C\over 1+|x|^{2+\nu}},$ (3.14)

where $C>0$. Recall that
$\Delta_{5}={{\partial}^{2}\over{\partial}r^{2}}+{3\over
r}{{\partial}\over{\partial}r}+{{\partial}^{2}\over{\partial}z^{2}}$, for
$x=(r,z)\in\Sigma$.

We have

######  Lemma 3.1.

Assume $h$ satisfies (3.14), with $\nu>0$. Then there exist a solution
$\psi={\mathcal{T}}(h)$ to (3.13) and a constant $C_{1}>0$ such that

$(1+|x|)|\nabla\psi(x)|+|\psi(x)|\leq{C_{1}\over 1+|x|^{\min(\nu,3)}}.$

###### Proof.

Recalling that the Laplacian of a radially symmetric function in
$\mathbb{R}^{n}$ is

$\Delta_{Y}={d^{2}\over ds^{2}}+{n-1\over s}{d\over ds},\quad
Y=(Y_{1},\ldots,Y_{n}),\quad s=\sqrt{Y_{1}^{2}+\ldots Y_{n}^{2}}$

we interpret the differential operator in (3.13) as the Laplacian in
$\mathbb{R}^{5}$ and recast Problem (3.13) in $\mathbb{R}^{5}$. Define

$\Psi(Y_{1},\ldots,Y_{4},z)=\psi(\sqrt{Y_{1}^{2}+\ldots Y_{4}^{2}},z),\quad
r=\sqrt{Y_{1}^{2}+\ldots Y_{4}^{2}},\quad Y_{5}=z$

and

$H(Y_{1},\ldots,Y_{4},z)=h(\sqrt{Y_{1}^{2}+\ldots Y_{4}^{2}},z),\quad
r=\sqrt{Y_{1}^{2}+\ldots Y_{4}^{2}},\quad\quad Y_{5}=z.$

To solve Problem (3.13) we find bounded (non-singular) solutions $\Psi$ to

$\Delta_{\mathbb{R}^{5}}\Psi+H=0,\quad{\mbox{in
}}\mathbb{R}^{5},\quad\Psi(Y)\to 0,\quad\quad\mbox{as}\quad|Y|\to\infty.$

We define $\Psi$ using the Newtonian potential in $\mathbb{R}^{5}$ as

$\Psi(Y)={1\over 15\omega_{5}}\int_{\mathbb{R}^{5}}{1\over|Z-Y|^{3}}H(Z)\,dZ,$

with $\omega_{5}$ the volume of the unit $5$-ball. Moreover
$\nabla\Psi(Y)=-{3\over
15\omega_{5}}\int_{\mathbb{R}^{5}}{(Z-Y)\over|Z-Y|^{5}}H(Z)\,dZ.$ From (3.14)
we get that $(1+|Y|^{2+\nu})|H(Y)|\lesssim 1$. Estimating the above integrals
splitting them in the region $|Z|<{|Y|\over 2}$ and its complement we get

$(1+|Y|)|\nabla\Psi(Y)|+|\Psi(Y)|\lesssim{1\over 1+|Y|^{\nu}}.$

Going back to the original variables $x=(r,z)\in\Sigma$ we get the required
estimates. ∎

We denote by
$H^{0}(x;P)={\mathcal{T}}\left(\Delta_{5}\bar{\Psi}^{0}+W^{0}\right)$ the
solution to

$\Delta_{5}H^{0}+\Delta_{5}\bar{\Psi}^{0}+W^{0}=0,\quad{\mbox{in
}}\Sigma,\quad{\partial H^{0}\over\partial r}=0\quad{\mbox{on
}}{\partial}\Sigma$ (3.15)

with $H^{0}(x,P)\to 0$ as $|x|\to\infty$, for all $t\in[0,T]$.

A direct computation gives

$\displaystyle\Delta_{5}\bar{\Psi}^{0}+W^{0}$
$\displaystyle=(1-\eta_{0}(x))W^{0}+(\Delta_{5}\eta_{0}(x))\sum_{j=1}^{k}\Psi_{j}^{0}(x;P_{j})+2\sum_{j=1}^{k}\nabla_{r,z}\eta_{0}(x)\cdot\nabla_{r,z}\Psi_{j}^{0}(x;P_{j}).$
(3.16)

From (2.11) we get, for $y={x-P_{j}\over\varepsilon_{j}}$,

$\displaystyle w_{j}^{0}(y)$
$\displaystyle=U(y)\left(1-{\varepsilon_{j}y_{1}\over
2r_{j}}(\Gamma_{0}+\bar{A}+\Gamma)+\varepsilon^{2}\theta_{1}[P_{j}](y)+\varepsilon^{3}\theta_{2}[P_{j}](y)\right)$
(3.17)

where $\theta_{1}$ and $\theta_{2}$ are functions also described by (2.13) and
(2.14) respectively. Hence the function $\Delta_{5}\bar{\Psi}^{0}+W^{0}$ is
smooth, it is the sum of one term with compact support and size $O(1)$, and a
term which is bounded by $O({\varepsilon^{2}\over 1+|x|^{4}})$,
$(x,t)\in\Sigma\times[0,T]$ uniformly for $\varepsilon\to 0$.

From Lemma 3.1 and (3.16) we get that

$(1+|x|)|\nabla H^{0}(x,P)|+|H^{0}(x;P)|\lesssim{1\over 1+|x|^{2}},\quad$

uniformly for $t\in[0,T]$ as $\varepsilon\to 0$.

We define the first approximate solution to be

$\displaystyle\Psi^{0}(x;P(t))$
$\displaystyle=\eta_{0}(x)\sum_{j=1}^{k}\Psi_{j}^{0}(x;P_{j}(t))+H^{0}(x;P(t)),\quad$
(3.18) $\displaystyle W^{0}(x;P(t))$
$\displaystyle=\sum_{j=1}^{k}W_{j}^{0}(x;P_{j}(t)).$

###  3.3. The very first error

Define the operators

$\displaystyle S_{1}(W,\Psi)$
$\displaystyle=|\log\varepsilon|rW_{t}+{\nabla}^{\perp}(r^{2}(\Psi-\alpha_{0}|\log\varepsilon|))\cdot{\nabla}W,\quad$
(3.19) $\displaystyle S_{2}(W,\Psi)$ $\displaystyle=\Delta_{5}\Psi+W.$

We recast Problem (3.2) as the problem of finding $(W,\Psi)$ with

$S_{1}(W,\Psi)=S_{2}(W,\Psi)=0\quad{\mbox{in}}\quad\Sigma\times[0,T),\quad{\partial\Psi\over\partial
r}=0\quad{\mbox{on }}{\partial}\Sigma\times[0,T)$

with $W$ and $\Psi$ close respectively to $W^{0}$ and $\Psi^{0}$, and
$|\Psi(x,t)|\to 0$ as $|x|\to\infty$. By construction we have

$\displaystyle S_{2}(W^{0},\Psi^{0})$
$\displaystyle=0\quad\Sigma\times[0,T),\quad{\partial\over\partial
r}\Psi^{0}(r,z,t)=0\quad\partial\Sigma\times[0,T),$
$\displaystyle|\Psi^{0}(x,t)|\to 0,\quad\quad\mbox{as}\quad|x|\to\infty.$

We now compute $S_{1}(W^{0},\Psi^{0})$ writing

$\displaystyle S_{1}(W^{0},\Psi^{0})$
$\displaystyle=\sum_{j=1}^{k}E_{j}^{0},\quad{\mbox{where}}$ (3.20)
$\displaystyle E_{j}^{0}$
$\displaystyle=|\log\varepsilon|\,r\,{\partial}_{t}W^{0}_{j}+{\nabla}^{\perp}(r^{2}(\Psi^{0}-\alpha_{0}|\log\varepsilon|))\cdot{\nabla}W_{j}^{0}.$

We want to describe $S_{1}(W^{0},\Psi^{0})$. We start with the following
general remark.

######  Remark 3.1.

For a function $W(x,t)$ given in the form

$W(x,t)=\frac{1}{r_{j}\varepsilon_{j}^{2}}w\Bigl{(}\frac{x-P_{j}}{\varepsilon_{j}},t\Bigr{)},$

for some function $w(y,t)$, $y={x-P_{j}\over\varepsilon_{j}}$ we have

$\varepsilon_{j}^{4}|\log\varepsilon|\,r\,{\partial}_{t}W=|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1})\left[\varepsilon_{j}^{2}w_{t}-\varepsilon_{j}{\partial}_{t}\varepsilon_{j}\nabla
w\cdot y-\varepsilon_{j}\nabla w\cdot{\partial}_{t}P_{j}\right]$ (3.21)

Indeed, for $x=\varepsilon_{j}y+P_{j}$,

$\displaystyle\varepsilon_{j}^{4}|\log\varepsilon|\,r\,{\partial}_{t}W$
$\displaystyle=|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1})\Biggl{[}\varepsilon_{j}^{2}w_{t}-({{\partial}_{t}r_{j}\over
r_{j}}+{2{\partial}_{t}\varepsilon_{j}\over\varepsilon_{j}})\varepsilon_{j}^{2}w-\varepsilon_{j}{\partial}_{t}\varepsilon_{j}\nabla
w\cdot y-\varepsilon_{j}\nabla w\cdot{\partial}_{t}P_{j}\Biggl{]}.$

From our first assumption (3.8) on the parameters $r_{j}$ and
$\varepsilon_{j}$ we get

$\frac{{\partial}_{t}r_{j}}{r_{j}}=-2\frac{{\partial}_{t}\varepsilon_{j}}{\varepsilon_{j}},\quad\quad\mbox{for
all}\quad t$ (3.22)

which gives (3.21).

A further inspection of the expression (3.21) gives a formula which will be
useful later on

$\displaystyle\varepsilon_{j}^{4}|\log\varepsilon|r{\partial}_{t}W=|\log\varepsilon|$
$\displaystyle\Bigg{[}-\varepsilon_{j}\nabla
w\cdot{\partial}_{t}P_{j}+\varepsilon_{j}^{2}{\partial}_{t}w+\varepsilon_{j}{\partial}_{t}\varepsilon_{j}(y_{1}{\partial}_{1}w-y_{2}{\partial}_{2}w)$
(3.23) $\displaystyle\quad-{\varepsilon_{j}^{2}\over
r_{j}}{\partial}_{t}z_{j}y_{1}{\partial}_{2}w+{\varepsilon_{j}^{3}\over
r_{j}}y_{1}{\partial}_{t}w-{\varepsilon_{j}^{2}\over
r_{j}}{\partial}_{t}\varepsilon_{j}y_{1}\nabla w\cdot y\Bigg{]}.$

This relation follows from observing that

$\displaystyle\varepsilon_{j}^{4}|\log\varepsilon|r{\partial}_{t}W=|\log\varepsilon|$
$\displaystyle\Bigg{[}-\varepsilon_{j}\nabla
w\cdot{\partial}_{t}P_{j}+\varepsilon_{j}^{2}{\partial}_{t}w-\varepsilon_{j}{\partial}_{t}\varepsilon_{j}\nabla
w\cdot y$ $\displaystyle-{\varepsilon_{j}^{2}\over
r_{j}}y_{1}{\partial}_{t}P_{j}\cdot\nabla w+{\varepsilon_{j}^{3}\over
r_{j}}y_{1}{\partial}_{t}w-{\varepsilon_{j}^{2}\over
r_{j}}{\partial}_{t}\varepsilon_{j}y_{1}\nabla w\cdot y\Bigg{]}$

and (again (3.22) for $P_{j}(t)=(r_{j}(t),z_{j}(t))$)

$\displaystyle-\varepsilon_{j}{\partial}_{t}\varepsilon_{j}\nabla w\cdot
y-{\varepsilon_{j}^{2}\over r_{j}}y_{1}{\partial}_{t}P_{j}\cdot\nabla
w=\varepsilon_{j}{\partial}_{t}\varepsilon_{j}(y_{1}{\partial}_{1}w-y_{2}{\partial}_{2}w)-{\varepsilon_{j}^{2}\over
r_{j}}{\partial}_{t}z_{j}y_{1}{\partial}_{2}w.$

Writing

$\Psi_{j}^{0}(x;P_{j})={1\over r_{j}}\psi^{0}_{j}(y;P_{j}),\quad
y={x-P_{j}\over\varepsilon_{j}}$

from (3.11) we have

$W_{j}^{0}(x;P_{j})={1\over\varepsilon_{j}^{2}r_{j}}w_{j}^{0}(y;P_{j}),\quad{\mbox{with}}\quad-\Delta_{5,j}\psi_{j}^{0}=w_{j}^{0}$
(3.24)

where

$\Delta_{5,j}={\partial}_{y_{1}}^{2}+{\partial}_{y_{2}}^{2}+{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}\,{\partial}_{y_{1}}.$ (3.25)

Using (3.21) and (3.24) we easily get that the first term in $E_{j}^{0}$ (see
(3.20)) is given by

$\displaystyle\varepsilon_{j}^{4}|\log\varepsilon|\,r\,{\partial}_{t}W^{0}_{j}$
$\displaystyle=-\varepsilon_{j}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1})\,{\partial}_{t}P_{j}\cdot\nabla w^{0}_{j}$
$\displaystyle+\varepsilon_{j}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1})\left[\varepsilon_{j}{\partial}_{t}w^{0}_{j}-{\partial}_{t}\varepsilon_{j}\nabla
w^{0}_{j}\cdot y\right].$

We compute now the second term in $E_{j}^{0}$ given in (3.20) and we analyze
it in the region $\\{x\,:\,|x-P_{j}|<|\log\varepsilon|^{-1}\\}$, and recall
(3.7). In this region we have, for $y={x-P_{j}\over\varepsilon_{j}}$,
$|y|<\varepsilon_{j}^{-1}|\log\varepsilon|^{-1}$,

$\displaystyle\varepsilon^{4}_{j}{\nabla}^{\perp}(r^{2}(\Psi^{0}-\alpha_{0}|\log\varepsilon|))\cdot{\nabla}W_{j}^{0}$
$\displaystyle=\nabla^{\perp}\left((1+{\varepsilon_{j}\over
r_{j}}y_{1})^{2}(\psi_{j}^{0}-r_{j}\alpha_{j}|\log\varepsilon_{j}|)\right)\cdot\nabla
w^{0}_{j}$
$\displaystyle+\nabla^{\perp}(\tilde{\varphi}_{j}(\varepsilon_{j}y+P_{j};P)\,\,)\,\cdot\nabla
w^{0}_{j},$

where

$\displaystyle\tilde{\varphi}_{j}(x;P)$
$\displaystyle=\tilde{\varphi}_{j}(\varepsilon_{j}y+P_{j};P)=r_{j}(1+{\varepsilon_{j}\over
r_{j}}y_{1})^{2}\times$ (3.26)
$\displaystyle\Biggl{(}\alpha_{j}|\log\varepsilon_{j}|-\alpha_{0}|\log\varepsilon|\,\,+H^{0}(\varepsilon_{j}y+P_{j};P)+\sum_{\ell\not=j}{1\over
r_{\ell}}\,\,\psi_{\ell}^{0}({\varepsilon_{j}\over\varepsilon_{\ell}}y+{P_{j}-P_{\ell}\over\varepsilon_{\ell}})\Biggl{)},$

where $H^{0}$ is the correction introduced in (3.15). Define

${\mathcal{R}}_{j}(y,t)=\varepsilon_{j}|\log\varepsilon|\,{\partial}_{t}P_{j}^{\perp}\cdot
y+\tilde{\varphi}_{j}(\varepsilon_{j}y+P_{j};P)-\tilde{\varphi}_{j}(P_{j};P),$

and the operator

$\displaystyle
B_{0}(\phi)=-\varepsilon_{j}{\partial}_{t}\varepsilon_{j}(1+{\varepsilon_{j}\over
r_{j}}y_{1})\nabla\phi\cdot y-{\varepsilon_{j}^{2}\over
r_{j}}y_{1}{\partial}_{t}P_{j}\cdot\nabla\phi$ (3.27)

Referring to (3.20), in the region close to the point $P_{j}$ given by
$|x-P_{j}|<|\log\varepsilon|^{-1}$, we have, for
$y={x-P_{j}\over\varepsilon_{j}}$,

$\displaystyle\varepsilon_{j}^{4}E^{0}_{j}(P_{j}+\varepsilon_{j}y)$
$\displaystyle=\nabla^{\perp}{\mathcal{R}}_{j}(y,t)\cdot\nabla
w^{0}_{j}+\varepsilon_{j}^{2}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}w^{0}_{j}+|\log\varepsilon|B_{0}(w_{j}^{0})$
$\displaystyle+\nabla^{\perp}\left((1+{\varepsilon_{j}\over
r_{j}}y_{1})^{2}(\psi_{j}^{0}-r_{j}\alpha_{j}|\log\varepsilon_{j}|)\right)\cdot\nabla
w^{0}_{j}.$

In the same region, for $i\not=j$,

$\displaystyle\varepsilon_{j}^{4}E^{0}_{i}(P_{j}+\varepsilon_{j}y)$
$\displaystyle={\varepsilon_{j}^{4}\over\varepsilon_{i}^{4}}\varepsilon_{i}^{4}E^{0}_{i}(({\varepsilon_{j}\over\varepsilon_{i}}{P_{j}-P_{i}\over\varepsilon_{j}}+{\varepsilon_{j}\over\varepsilon_{i}}y)=O({\varepsilon^{5}|\log\varepsilon|^{2}\over
1+|y|}).$

Hence for
$\rho=|y|=|{x-P_{j}\over\varepsilon_{j}}|<\varepsilon_{j}^{-1}|\log\varepsilon|^{-1}$
we have

$\displaystyle\varepsilon_{j}^{4}$ $\displaystyle
S_{1}(W^{0},\Psi^{0})(P_{j}+\varepsilon_{j}y)=\varepsilon_{j}^{4}\sum_{i=1}^{k}E^{0}_{i}(P_{j}+\varepsilon_{j}y)$
(3.28) $\displaystyle=\nabla^{\perp}{\mathcal{R}}_{j}(y,t)\cdot\nabla
w^{0}_{j}+\varepsilon_{j}^{2}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}w^{0}_{j}+|\log\varepsilon|B_{0}(w^{0}_{j})$
$\displaystyle+{\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}(\rho,\theta,t,\varepsilon)+{\varepsilon^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}[E_{1}(\rho,\theta,t,\varepsilon)+E_{3}(\rho,\theta,t,\varepsilon)]+O\left({\varepsilon^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}\right)\,$

as $\varepsilon\to 0$. To get this estimate we have used (2.7). The functions
$E_{j}(\rho,\theta,t,\varepsilon)$ have the following form

$\displaystyle
E_{j}(\rho,\theta,t,\varepsilon)=E_{j,1}(\rho,\varepsilon,P(t))\cos(j\theta)+E_{j,2}(\rho,\varepsilon,P(t))\sin(j\theta)$

where

$\displaystyle\sum_{i=0}^{2}\Bigl{|}(1+\rho)^{i}\frac{\partial^{i}E_{j,\ell}}{\partial\rho^{i}}\Bigr{|}+\Bigl{|}\nabla_{P}E_{j,\ell}\Bigr{|}\lesssim
1\quad\ell=1,2,\quad\quad\mbox{as}\quad\varepsilon\to 0,$

uniformly for $t\in[0,T)$

###  3.4. Dynamics for $P^{0}_{j}$ and reduction of the very first error.

It is possible to reduce the size of the very first error (3.28) in each of
the regions
$|{x-P_{j}\over\varepsilon_{j}}|<\varepsilon_{j}^{-1}|\log\varepsilon|^{-1}$,
$j=1,\ldots,k$, if we choose properly the points $P_{j}^{0}$ in (3.4). Write
the function $\tilde{\varphi}_{j}$ in (3.26) as

$\tilde{\varphi}_{j}(x;P)=\varphi_{j}(x;P)+\varepsilon^{2}\theta_{j\varepsilon_{j}}(x;P)$

with

$\varphi_{j}(x;P)={r^{2}\over
r_{j}}\Biggl{(}\alpha_{j}|\log\varepsilon_{j}|-\alpha_{0}|\log\varepsilon|\,\,+H^{0}(x;P)+\sum_{\ell\not=j}{1\over
r_{\ell}}\,\,\bar{\psi}_{\ell}^{0}(x;P_{\ell})\Biggl{)}$ (3.29)
$\displaystyle\bar{\psi}_{\ell}^{0}(x;P_{\ell})$
$\displaystyle=\log\frac{1}{|x-P_{\ell}|^{4}}\big{(}1-\frac{3}{2r_{\ell}}(r-r_{\ell})+H(x;P_{\ell})\big{)}+K(x;P_{\ell})+\varepsilon_{\ell}\Gamma(x;P_{\ell})$
$\displaystyle{\mbox{where}}\quad\varepsilon_{\ell}\Gamma(x;P_{\ell})={r-r_{\ell}\over
2r_{\ell}}\Gamma({x-P_{\ell}\over\varepsilon_{\ell}})\quad{\mbox{and}}$
$\displaystyle\theta_{j\varepsilon_{j}}(x;P)$
$\displaystyle={1\over\varepsilon^{2}}\sum_{\ell\not=j}{r^{2}\over
r_{j}r_{\ell}}\log\frac{|x-P_{\ell}|^{4}}{(\varepsilon_{\ell}^{2}+|x-P_{\ell}|^{2})^{2}}\big{(}1-\frac{3}{2r_{\ell}}(r-r_{\ell})+H(x;P_{\ell})\big{)}.$

We refer to (2.3), (2.4) and (2.5) for the definition of $H$, $K$ and
$\Gamma$. We Taylor expand

$\displaystyle\tilde{\varphi}_{j}$ $\displaystyle(P_{j}+\varepsilon_{j}y;P)=\
\tilde{\varphi}_{j}(P_{j};P)+\varepsilon_{j}{\nabla}_{x}\varphi_{j}(P_{j};P)\cdot
y+\sum_{k=2}^{4}\frac{\varepsilon_{j}^{k}}{k!}D_{x}^{k}\varphi_{j}(P_{j};P)[y]^{k}$
(3.30)
$\displaystyle\,+\,\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(P_{j};P)[y]+{\varepsilon_{j}^{4}\over
2}D_{x}^{2}\theta_{j\varepsilon_{j}}(P_{j};P)[y]^{2}+Q(\varepsilon_{j}y,P)$

where $Q(z,\zeta)$ is a function smooth in its arguments that satisfies

$|{\nabla}_{z}Q(z,\zeta)|\ \leq\ C\varepsilon|z|^{4}.$

We choose the points $P^{0}_{j}$ in (3.3)-(3.4) to satisfy the ODEs system

$-|\log\varepsilon|\,{\partial}_{t}P_{j}+\nabla_{x}^{\perp}\varphi_{j}(P_{j};P)=0,\quad
t\in[0,T],\quad j=1,\ldots k.$ (3.31)

We claim that there exist $T$ and a solution to (3.31) of the form

$\bar{P}^{0}(t)=(\bar{P}^{0}_{1},\ldots\bar{P}_{k}^{0})\quad\bar{P}^{0}_{j}(t)=P_{0}+P_{j}^{0}(t),\quad
P_{0}=(r_{0},0)$ (3.32)

satisfying the initial condition

$\bar{P}^{0}(0)=(\bar{P}^{0}_{1}(0),\ldots,\bar{P}_{k}^{0}(0)),\quad\bar{P}_{j}^{0}=(r_{0},0)+p_{j}^{0},$

for some $(p_{1}^{0},\ldots,p_{k}^{0})\in\mathbb{R}^{2k}$ with

${a\over\sqrt{|\log\varepsilon|}}\leq|p_{i}^{0}|\leq{a^{-1}\over\sqrt{|\log\varepsilon}},\forall
i\quad\,{a\over\sqrt{|\log\varepsilon|}}\leq|p_{i}^{0}-p_{j}^{0}|\leq{a^{-1}\over\sqrt{|\log\varepsilon}}\quad\forall
i\not=j,$

with $a>0$ fixed, for all $\varepsilon>0$ small. Besides, the point $P^{0}(t)$
satisfies the bounds in (3.4).

Using (3.10), we compute for all $j=1,\ldots,k$

$\nabla_{x}\varphi_{j}^{\perp}(P_{j};P)=\Theta_{j}(P)=\Theta_{j}^{0}(P)+\Theta_{j}^{1}(P)$

where for $P_{j}=(r_{j},z_{j})$,

$\displaystyle\Theta_{j}^{0}(P)$
$\displaystyle=-4\sum_{\ell\not=j}\,\,\frac{(P_{j}-P_{\ell})^{\perp}}{|P_{j}-P_{\ell}|^{2}}+2\,{r_{0}-r_{j}\over
r_{0}r_{j}}\,|\log\varepsilon|\,{\bf e_{2}}$

and $\Theta_{j}^{1}(P)=O(1)$, as $\varepsilon\to 0$, for $P$ in the range
under consideration. Scaling

$\quad P^{0}(t)=|\log\varepsilon|^{-{1\over 2}}\,Q^{0}(t)$

system (3.31) takes the form

$-{\partial}_{t}Q_{j}+\tilde{\Theta}_{j}(Q)=0,\quad t\in[0,T],\quad j=1,\ldots
k,$

where

$\tilde{\Theta}_{j}(Q)=|\log\varepsilon|^{-{1\over
2}}\Theta_{j}(|\log\varepsilon|^{-{1\over 2}}Q).$

The functions $\tilde{\Theta}_{j}$ are locally Lipschtz continuous for $Q$ in
the set

$\Lambda=\\{Q=(Q_{1},\ldots,Q_{k})\,:\,\|Q_{j}\|_{L^{\infty}[0,T)}\lesssim
1,|Q_{j}(t)-Q_{i}(t)|\geq C\,\forall i\not=j\\},$

with Lipschitz constant bounded above and away from $0$ uniformly as
$\varepsilon\to 0$. In fact, for $Q\in\Lambda$, one has

$\tilde{\Theta}^{0}_{j}=O(1),\quad\tilde{\Theta}^{0}_{j}=O({|\log|\log\varepsilon||\over|\log\varepsilon|^{1\over
2}})$

as $\varepsilon\to 0$. Here
$\tilde{\Theta}_{j}^{a}(Q)=|\log\varepsilon|^{-{1\over
2}}\Theta_{j}^{a}(|\log\varepsilon|^{-{1\over 2}}Q)$ for $a=0,1$. Standard
ODEs theory gives that there exists $T>0$ such that, for all $\varepsilon>0$
small enough there exists a unique solution $Q^{0}$ to the Cauchy problem

$-{\partial}_{t}Q_{j}+\tilde{\Theta}_{j}(Q)=0,\quad t\in[0,T],\quad j=1,\ldots
k,\quad Q_{i}(0)=\sqrt{|\log\varepsilon|}p^{0}_{i}.$

This solution is defined in an interval of time $[0,T]$, with $T=O(1)$ as
$\varepsilon\to 0$.

By construction the points $P_{0}+P_{j}^{0}$ satisfy

$\sup_{t\in[0,T]}\|P_{j}^{0}(t)-{q_{j}(t)\over\sqrt{|\log\varepsilon|}}\|\lesssim{|\log|\log\varepsilon||\over|\log\varepsilon|},$

where $q_{j}$ evolve via the leapfrogging dynamics (1.14).

This choice for $P^{0}$ reduces the size of the term
$\nabla^{\perp}{\mathcal{R}}_{j}(y,t)\cdot\nabla w^{0}_{j}$ in the first line
of the error of approximation described by (3.28) when evaluated in the region
$|x-P_{j}|<|\log\varepsilon|^{-1}$. Let us now analyze the terms in (3.28)
given by

$\varepsilon_{j}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1})\left[\varepsilon_{j}{\partial}_{t}w^{0}_{j}-{\partial}_{t}\varepsilon_{j}\nabla
w^{0}_{j}\cdot
y\right]-\varepsilon_{j}^{2}|\log\varepsilon|y_{1}{\partial}_{t}P_{j}\cdot\nabla
w^{0}_{j}.$

These terms have the same form as the terms in the following line in (3.28),
with the only difference that in this case the functions
$E_{j}(\rho,\theta,t,\varepsilon)$ do also depend on ${\partial}_{t}P$ and not
only on $P$. This fact is consequence of (3.23) and our assumptions on the
points $P_{j}$.

We are thus in a position to conclude that if we choose the points $P^{0}$ to
satisfy (3.31), the error of approximation (3.28) in the region
$|x-P_{j}|<|\log\varepsilon|^{-1}$ can be described as, for
$y={x-P_{j}\over\varepsilon_{j}}$, $\rho=|y|$

$\displaystyle\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})$
$\displaystyle(P_{j}+\varepsilon_{j}y)=\nabla^{\perp}{\mathcal{R}}_{j}(y,t;P)\cdot\nabla
w_{j}^{0}+{\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}(\rho,\theta,t,\varepsilon)$ (3.33)
$\displaystyle+{\varepsilon^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}[E_{1}(\rho,\theta,t,\varepsilon)+E_{3}(\rho,\theta,t,\varepsilon)]+O\left({\varepsilon^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}\right)\,$

as $\varepsilon\to 0$, where the functions $E_{j}(\rho,\theta,t,\varepsilon)$
now depends also on ${\partial}_{t}P$, and have the form

$\displaystyle E_{j}[P,{\partial}_{t}P](\rho,\theta,t,\varepsilon)$
$\displaystyle=E_{j,1}(\rho,\varepsilon,P(t),{\partial}_{t}P(t))\cos(j\theta)$
(3.34)
$\displaystyle+E_{j,2}(\rho,\varepsilon,P(t),{\partial}_{t}P(t))\sin(j\theta),$

where

$\displaystyle\sum_{i=0}^{2}\Bigl{|}(1+\rho)^{i}\frac{\partial^{i}E_{j,\ell}}{\partial\rho^{i}}\Bigr{|}+\Bigl{|}\nabla_{P}E_{j,\ell}\Bigr{|}+\Bigl{|}\nabla_{{\partial}_{t}P}E_{j,\ell}\Bigr{|}\lesssim
1\quad\ell=1,2,\quad\quad\mbox{as}\quad\varepsilon\to 0.$

Besides,

$\displaystyle{\mathcal{R}}_{j}(y,t;P)$
$\displaystyle=\varepsilon_{j}|\log\varepsilon|{\partial}_{t}(P_{j}-P_{j}^{0})^{\perp}\cdot
y+\tilde{\varphi}_{j}(\varepsilon_{j}y+P_{j};P)-\tilde{\varphi}_{j}(P_{j};P)$
(3.35)
$\displaystyle-\nabla_{x}\varphi_{j}(\bar{P}_{j}^{0};\bar{P}^{0})\,\varepsilon_{j}y.$

where $\bar{P}^{0}_{j}(t)=P_{0}+P_{j}^{0}(t)$, see (3.32).

##  4\. Improvement of the approximation

In the previous section we introduced the functions $\Psi^{0}(x;P)$ and
$W^{0}(x;P)$ defined in (3.18), where $P$ is a collection of $k$ points of the
form

$P=(P_{1},\ldots,P_{k}),\quad P_{j}(t)=P_{0}+P^{0}_{j}(t)+P^{1}_{j}(t)+{\bf
a}_{j}(t),\quad P_{0}=(r_{0},0).$

So far we have defined $P_{j}^{0}$ explicitly to solve (3.31), while
$P^{1}_{j}$ and ${\bf a}_{j}$ are still parameters that are assumed to satisfy
(3.5) and (3.6) respectively.

The next step in our argument is to modify $(W^{0},\Psi^{0})$ in order to
produce a better approximate solution $(W^{*},\Psi^{*})$. We do it taking
$(W^{*},\Psi^{*})$ of the form

$\displaystyle\Psi^{*}(x,t;P)$
$\displaystyle=\Psi^{0}+\sum_{j=1}^{k}{\eta_{j2}\over
r_{j}}\psi_{j}^{*}({x-P_{j}\over\varepsilon_{j}},t)+\psi^{*,out}(x,t)$ (4.1)
$\displaystyle W^{*}(x,t;P)$
$\displaystyle=W^{0}+\sum_{j=1}^{k}{\eta_{j1}\over
r_{j}\varepsilon_{j}^{2}}\phi_{j}^{*}({x-P_{j}\over\varepsilon_{j}},t)+\phi^{*,out}(x,t)$
(4.2)

where

$\eta_{jN}(x,t)=\eta({|\log\varepsilon|^{\zeta}\,|x-P_{j}|\over N}),$ (4.3)

and $\eta$ is given by (3.12). Here $\zeta>1$ is a positive number,
independent of $\varepsilon$. Since the relative distance between $P_{j}$ and
$P_{i}$, $j\not=i$, is of the order $|\log\varepsilon|^{-{1\over 2}}$, we have

$P_{i}\in{\mbox{support}}\,(\eta_{i2}),\quad{\mbox{support}}\,(\eta_{i2})\cap{\mbox{support}}\,(\eta_{j2})=\emptyset.$

It will be convenient to choose $\zeta=3$. This will guarantee that $\Psi^{0}$
and $W^{0}$ are the main terms in the decomposition of $\Psi^{*}$ and $W^{*}$
in the region where the corresponding cut-off functions $\eta_{j2}$,
$\eta_{j1}$ are non-zero.

If we insert the expressions of $\Psi^{*}$ and $W^{*}$ given by (4.1) and
(4.2) in the Euler operator $S_{1}$ (see (3.19)) we get

$\displaystyle
S_{1}(W^{*},\Psi^{*})=\sum_{j=1}^{k}{\eta_{1j}\over\varepsilon_{j}^{4}}E_{j}^{in}[\phi_{j}^{*},\psi_{j}^{*}\,\psi^{*,out},P]+E^{out}[\phi^{*,out},\psi^{*,out},\phi^{*,in},\psi^{*,in},P],$
(4.4)

where

$\phi^{*,in}=(\phi_{1}^{*},\ldots,\phi_{k}^{*}),\quad\psi^{*,in}=(\psi_{1}^{*},\ldots,\psi_{k}^{*}).$

The inner-operators $E_{j}^{in}$, $j=1,\ldots,k$, are defined as

$\displaystyle E_{j}^{in}$
$\displaystyle[\phi_{j},\psi_{j}\,\psi^{out},P](y,t):=|\log\varepsilon|\varepsilon_{j}^{2}(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}\phi_{j}+|\log\varepsilon|B_{0}(\phi_{j})$
$\displaystyle+\nabla^{\perp}\left((1+{\varepsilon_{j}\over
r_{j}}y_{1})^{2}(\psi_{j}^{0}-r_{j}\alpha_{j}|\log\varepsilon_{j}|+\psi_{j}+r_{j}\psi^{out})+{\mathcal{R}}_{j}(y,t;P)\right)\cdot\nabla\phi_{j}$
$\displaystyle+\nabla^{\perp}\left((1+{\varepsilon_{j}y_{1}\over
r_{1}})^{2}(\psi_{j}+r_{j}\psi^{out})\right)\nabla(w^{0}_{j}+\sum_{i\not=j}{r_{j}\varepsilon_{j}^{2}\over
r_{i}\varepsilon_{i}^{2}}w^{0}_{i})$
$\displaystyle+\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})(\varepsilon_{j}y+P_{j}),\quad|y|<3R_{j},\quad
R_{j}:={1\over\varepsilon_{j}|\log\varepsilon|^{\zeta}}$ (4.5)

with $y={x-P_{j}\over\varepsilon_{j}}$, $t\in[0,T)$, and ${\mathcal{R}}_{j}$
defined in (3.35). Moreover $B_{0}$ is the operator defined in (3.27), which
can be equivalently written as

$\displaystyle
B_{0}(\phi)=\varepsilon_{j}{\partial}_{t}\varepsilon_{j}(y_{1}{\partial}_{1}\phi-
y_{2}{\partial}_{2}\phi)-{\varepsilon_{j}^{2}\over
r_{j}}{\partial}_{t}\varepsilon_{j}y_{1}y\cdot\nabla\phi-{\varepsilon_{j}^{2}\over
r_{j}}{\partial}_{t}z_{j}y_{1}{\partial}_{2}\phi$

see (3.21) and (3.23) for the derivation of $B_{0}$, and its equivalent form.
The outer-operator $E^{out}$

$\displaystyle E^{out}$
$\displaystyle[\phi^{out},\psi^{out},\phi^{in},\psi^{in},P](x,t):=|\log\varepsilon|\,r\,\phi^{out}_{t}$
(4.6)
$\displaystyle+\nabla_{x}^{\perp}(r^{2}(\Psi^{0}+\sum_{j=1}^{k}{\eta_{j2}\over
r_{j}}\psi_{j}({x-P_{j}\over\varepsilon_{j}})+\psi^{out}-\alpha_{0}|\log\varepsilon|))\cdot\nabla_{x}\phi^{out}$
$\displaystyle+\sum_{j=1}^{k}\left[r\,|\log\varepsilon|\,{\partial}_{t}\bar{\eta}_{j1}+\nabla_{x}^{\perp}(r^{2}(\Psi^{0}+\sum_{j=1}^{k}{\eta_{j2}\over
r_{j}}\psi_{j}({x-P_{j}\over\varepsilon_{j}})+\psi^{out}-\alpha_{0}|\log\varepsilon|))\nabla\bar{\eta}_{1j}\right]{\phi_{j}\over\varepsilon_{j}^{2}r_{j}}$
$\displaystyle+\left[\sum_{j=1}^{k}(\eta_{2j}-\eta_{1j})\nabla_{x}^{\perp}(r^{2}({\psi_{j}\over
r_{j}}+\psi^{out}))+{r^{2}\psi_{j}\over
r_{j}}\nabla_{x}^{\perp}\eta_{2j}\right]\cdot\nabla_{x}W^{0}$
$\displaystyle+(1-\sum_{j=1}^{k}\eta_{2j})\nabla^{\perp}(r^{2}\psi^{out})\cdot\nabla
W^{0}+(1-\sum_{j=1}^{k}\eta_{j1})S_{1}(W^{0},\Psi^{0})=0\quad(x,t)\in\Sigma\times[0,T).$

If we now insert the expressions of $\Psi^{*}$ and $W^{*}$ given by (4.1) and
(4.2) in the Euler operator $S_{2}$ (see (3.19)) we get

$\displaystyle S_{2}$
$\displaystyle[\Psi^{*},W^{*}]=S_{2}[\phi^{*,in},\psi^{*,in},\phi^{*,out},\psi^{*,out},P],\quad{\mbox{with}}$
(4.7) $\displaystyle S_{2}$
$\displaystyle[\phi^{in},\psi^{in},\phi^{out},\psi^{out},P]=\sum_{j=1}^{k}{\eta_{1j}\over
r_{j}\varepsilon_{j}^{2}}[\Delta_{5,j}\psi_{j}+\phi_{j}]+\Delta_{5}\psi^{out}+\phi^{out}$
$\displaystyle+\sum_{j=1}^{k}(\eta_{j1}-\eta_{j2}){\Delta_{5,j}\psi_{j}\over
r_{j}\varepsilon_{j}^{2}}+\sum_{j=1}^{k}({\psi_{j}\over
r_{j}}\Delta_{5}\eta_{j2}+2\nabla_{x}\eta_{j2}\nabla_{x}{\psi_{j}\over
r_{j}}).$

Here $\Delta_{5,j}$ is the differential operator in the expanded $y$-variable
defined in (3.25), while $\Delta_{5}$ is the differential operator defined in
the original $x$-variable as given in (1.6). We require conditions on the
boundary and at infinity on $\psi^{*,out}$: for all $t\in[0,T]$

${\partial\psi^{*,out}\over\partial
r}(x,t)=0,\quad{\mbox{on}}\quad\partial\Sigma\times[0,T],\quad|\psi^{*,out}(x,t)|\to
0,\quad\quad\mbox{as}\quad|x|\to\infty.$

We have the following result:

######  Proposition 4.1.

There exist constants $\sigma>0$ in (3.5)-(3.6), $\sigma_{*}>0$, $b>0$, points
$P^{1}=(P_{1}^{1},\ldots,P_{k}^{1})$ satisfying (3.5), and functions

$\phi^{*,in}:=(\phi_{1}^{*}(y,t),\ldots,\phi_{k}^{*}(y,t)),\quad\psi^{*,in}:=(\psi_{1}^{*}(y,t),\ldots,\psi_{k}^{*}(y,t)),\quad\phi^{*,out},\quad\psi^{*,out}$

in (4.1), (4.2) such that, for any points ${\bf a}$ satisfying (3.6), the
following facts hold. For

$P=\tilde{P}+{\bf
a},\quad\tilde{P}=(\tilde{P}_{1},\ldots,\tilde{P}_{k}),\quad\tilde{P}_{j}=P_{0}+P^{0}_{j}+P^{1}_{j},\quad
P_{0}=(r_{0},0),$

we have

$\displaystyle E_{j}^{in}[\phi_{j}^{*},\psi_{j}^{*}\,\psi^{*,out},{\bf
a}](y,t)$
$\displaystyle=\varepsilon_{j}\nabla^{\perp}\left[\left(|\log\varepsilon|{\partial}_{t}{\bf
a}^{\perp}_{j}+D_{x}{\nabla}_{x}\varphi_{j}(\tilde{P}_{j};\tilde{P})[{\bf
a}]\right)\cdot y\right]\nabla U$
$\displaystyle+{\mathcal{E}}_{j}[\phi_{j}^{*},\psi_{j}^{*}\,\psi^{*,out},{\bf
a}](y,t)$

where $\varphi_{j}$ is given in (3.29) and

$\Bigl{|}{\mathcal{E}}_{j}[\phi_{j}^{*},\psi_{j}^{*}\,\psi^{*,out},{\bf
a}](y,t)\Bigl{|}=O\left({\varepsilon^{5-\sigma_{*}}\over
1+|y|^{3}}\right),\quad|y|<3R_{j},\quad
R_{j}:={1\over\varepsilon_{j}|\log\varepsilon|^{3}}$

for all $j=1,\ldots,k$, and

$\left|E^{out}[\phi^{*,out},\psi^{*,out},\phi^{*,in},\psi^{*,in},{\bf
a}](x,t)\right|\lesssim{\varepsilon^{4-{\sigma_{*}}}\over
1+|x|^{4}},\quad(x,t)\in\Sigma\times[0,T].$

We refer to (4) and (4.6) for the explicit definitions of $E_{j}^{in}$ and
$E^{out}$. Besides,

$\displaystyle(1+|y|)|\nabla\psi_{j}^{*}(y,t)|+|\psi_{j}^{*}(y,t)|$
$\displaystyle\lesssim\varepsilon^{2}|\log\varepsilon|^{b},\quad|y|<3R_{j},\,t\in[0,T]$
$\displaystyle(1+|y|^{2})|\phi_{j}^{*}(y,t)|$
$\displaystyle\lesssim\varepsilon^{2}|\log\varepsilon|^{b},\quad|y|<3R_{j},\,t\in[0,T]$
$\displaystyle(1+|x|^{4})|\phi^{*,out}(x,t)|+(1+|x|^{2})|\psi^{*,out}(x,t)|$
$\displaystyle\lesssim\varepsilon^{2}|\log\varepsilon|^{b},\quad(x,t)\in\Sigma\times[0,T].$

We observe that for $(W^{*},\Psi^{*})$ in $(\ref{f110})$-$(\ref{f220})$ with
the parameter functions considered above, we have the total errors in
$(\ref{fullS10})$ and (4.7) estimated as

$\displaystyle|S_{1}(W^{*},\Psi^{*})(x,t)|\ \leq$ $\displaystyle\
C\varepsilon^{1-\sigma_{*}}\sum_{j=1}^{k}\frac{1}{1+|y_{j}|^{3}}+C{\varepsilon^{4-\sigma_{*}}\over
1+|x|^{4}},\quad y_{j}=\frac{x-P_{j}(t)}{\varepsilon},$
$\displaystyle|S_{2}(W^{*},\Psi^{*})(x,t)|\ \leq$ $\displaystyle\
C\varepsilon^{4-\sigma}\eta_{1}\left({4|x-(r_{0},0)|\over r_{0}}\right).$

Besides, for all $j=1,\ldots,k$

$W^{*}(x,t;P)={1\over
r_{j}\varepsilon_{j}^{2}}\,U({x-P_{j}\over\varepsilon_{j}})\,(1+o(1))$

where $U$ is given by (1.17), in the slightly smaller region
$|x-P_{j}|<|\log\varepsilon|^{-3}$. Next section will be devoted to building
this approximate solution and to proving Proposition 4.1.

##  5\. Improvement of the approximation: Proof of Proposition 4.1

The approximate solution $(W^{*},\Psi^{*})$ predicted by Proposition 4.1 is
constructed improving the inner errors $E_{j}^{in}$ defined in (4) ten
successive times, and improving the outer error $E^{out}$ defined in (4.6)
once.

It is useful to have at hand a more explicit expression of the inner-operators
$E_{j}^{in}$. The key observation is that in the region
$|x-P_{j}|<3|\log\varepsilon|^{-\zeta}$, we have, for
$y={x-P_{j}\over\varepsilon_{j}}$,

$\displaystyle(1+{\varepsilon_{j}\over r_{j}}y_{1})^{2}$
$\displaystyle(\psi_{j}^{0}-r_{j}\alpha_{j}|\log\varepsilon_{j}|)+{\mathcal{R}}_{j}(y,t;P)$
(5.1) $\displaystyle=\Gamma_{0}(y)-(4-\alpha r_{0})\log\varepsilon_{j}-\log
8+K(P_{j};P_{j})$
$\displaystyle+\frac{1}{2r_{j}}\Big{(}\Gamma_{0}(y)+\bar{A}+\Gamma(y)\Big{)}\,\varepsilon_{j}y_{1}+{\mathcal{R}}^{0}_{j}(y,t;P),\quad{\mbox{where}}$
$\displaystyle{\mathcal{R}}^{0}_{j}(y,t;P)$
$\displaystyle=\varepsilon_{j}|\log\varepsilon|{\partial}_{t}(P_{j}-P_{j}^{0})^{\perp}\cdot
y+\varepsilon_{j}[{\nabla}_{x}\varphi_{j}(P_{j};P)-{\nabla}_{x}\varphi_{j}(\bar{P}_{j}^{0};\bar{P}^{0})]\cdot
y$
$\displaystyle+\,\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(P_{j};P)[y]$
$\displaystyle+\varepsilon^{2}|\log\varepsilon|{\mathcal{Q}}_{1}[P](y)+\varepsilon^{3}|\log\varepsilon|^{3\over
2}{\mathcal{Q}}_{2}[P](y)$

with $\bar{P}^{0}_{j}=P_{0}+P_{j}^{0}$,
$\bar{P}^{0}=(\bar{P}^{0}_{1},\ldots,\bar{P}_{k}^{0})$, see (3.32),
${\mathcal{Q}}_{1}$, ${\mathcal{Q}}_{2}$ denote functions of the form, for
$y=\rho e^{i\theta}$,

$\displaystyle{\mathcal{Q}}_{1}[P](y)$
$\displaystyle=\Biggl{[}a(P)+b_{1}(P)\cos 2\theta+b_{2}(P)\sin
2\theta\Biggl{]}\,O(|y|^{2})$ (5.2) $\displaystyle{\mathcal{Q}}_{2}[P_{0}](y)$
$\displaystyle=a(P)O(|y|^{3}),$

for $a$, $b_{1}$ and $b_{2}$ smooth functions of the points $P$, which are
uniformly bounded as $\varepsilon\to 0$. Also: $\bar{A}$ is the explicit
constant defined by (2.12) and $\Gamma=\Gamma(y)$ is the function introduced
in (2.5).

###### Proof of (5.1)..

In order to prove (5.1), we combine an expansion of the term
$(1+{\varepsilon_{j}\over
r_{j}}y_{1})^{2}(\psi_{j}^{0}-r_{j}\alpha_{j}|\log\varepsilon_{j}|)$ following
the lines to get (2.16). From (3.30) and (3.35) we get

$\displaystyle{\mathcal{R}}_{j}(y,t;P)$
$\displaystyle=\varepsilon_{j}|\log\varepsilon|{\partial}_{t}(P_{j}-P_{j}^{0})^{\perp}\cdot
y+\varepsilon_{j}[{\nabla}_{x}\varphi_{j}(P_{j};P)-{\nabla}_{x}\varphi_{j}(\bar{P}_{j}^{0};\bar{P}^{0})]\cdot
y$
$\displaystyle+\,\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(P_{j};P)[y]+\sum_{k=2}^{4}\frac{\varepsilon_{j}^{k}}{k!}D_{x}^{k}\varphi_{j}(P_{j};P)[y]^{k}$
$\displaystyle\,+{\varepsilon_{j}^{4}\over
2}D_{x}^{2}\theta_{j\varepsilon_{j}}(P_{j};P)[y]^{2}+Q(\varepsilon_{j}y,P)$

Under our assumptions (3.3), (3.5) and (3.6) on the form and size of the
points $P_{j}$, we have that

$\displaystyle{\mathcal{R}}_{j}(y,t;P)$
$\displaystyle=\varepsilon_{j}|\log\varepsilon|{\partial}_{t}(P_{j}-P_{j}^{0})^{\perp}\cdot
y+\varepsilon_{j}[{\nabla}_{x}\varphi_{j}(P_{j};P)-{\nabla}_{x}\varphi_{j}(\bar{P}_{j}^{0};\bar{P}^{0})]\cdot
y$ (5.3)
$\displaystyle+\,\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(P_{j};P)[y]+\varepsilon^{2}|\log\varepsilon|{\mathcal{Q}}_{1}[P](y)+\varepsilon^{3}|\log\varepsilon|^{3\over
2}{\mathcal{Q}}_{2}[P](y)$

where ${\mathcal{Q}}_{1}$, ${\mathcal{Q}}_{2}$ satisfy (5.2). This fact gives
(5.1).

∎

###  5.1. Strategy for the improvement

Replacing (5.1) in (4) and using (3.8) we re-write the inner operator
$E^{in}_{j}$ as

$\displaystyle E_{j}^{in}[\phi_{j},\psi_{j},\psi^{out},P](y,t)$
$\displaystyle:=L_{1}(\phi_{j})+L_{2}(\phi_{j})+L_{3}(\psi_{j})+Q(\psi_{j},\phi_{j})+L_{3}(r_{j}\psi^{out})+Q(\psi^{out},\phi_{j})$
$\displaystyle+\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})(\varepsilon_{j}y+P_{j}),$
(5.4)

where

$\displaystyle L_{1}(\phi)$
$\displaystyle:=|\log\varepsilon|\,\varepsilon_{j}^{2}(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}\phi+|\log\varepsilon|B_{0}(\phi)$ (5.5)
$\displaystyle L_{2}(\phi)$
$\displaystyle:=\nabla^{\perp}\left(\Gamma_{0}+\frac{\varepsilon_{j}}{2r_{j}}\,y_{1}\,\Big{(}\Gamma_{0}(y)+\bar{A}+\Gamma(y)\Big{)}\,+{\mathcal{R}}^{0}_{j}(y,t;P)\right)\cdot\nabla\phi$
$\displaystyle L_{3}(\psi)$
$\displaystyle:=\nabla^{\perp}((1+{\varepsilon_{j}y_{1}\over
r_{1}})^{2}\psi)\cdot\nabla(w^{0}_{j}+\sum_{i\not=j}w_{i}^{0})$ $\displaystyle
Q(\psi,\phi)$ $\displaystyle:=\nabla^{\perp}\left((1+{\varepsilon_{j}\over
r_{j}}y_{1})^{2}\psi\right)\cdot\nabla\phi.$

Let us now explain the strategy to improve the inner error. This is done in
ten successive steps. Each time we improve with a mechanism dictated by the
form of the part of the error we aim at removing. Suppose the inner error
$E(y,t),$ with $y=\rho e^{i\theta}$, has a Fourier decomposition in the
$\theta$-variable given by

$E(y,t)=E(\rho
e^{i\theta},t)=\sum_{n\in\mathbb{Z}}E_{n}(\rho,t)e^{i\,n\,\theta},\quad
E_{n}(\rho,t)=\int_{0}^{2\pi}E(\rho e^{in\theta})e^{in\theta}\,d\theta.$

We call $E_{n}$ the $n-$th mode in the Fourier decomposition of $E$. If the
part of the error we want to remove has no $0$-th mode in its Fourier
decomposition then we will solve using the elliptic operator

$L[\psi]\,:=\,\nabla^{\perp}\Gamma_{0}\cdot\nabla\phi+\nabla^{\perp}\psi\cdot\nabla
U,\quad\phi=-\Delta_{5,j}\psi.$

You recover $L[\psi]$ from $L_{2}(\phi_{j})$ and $L_{3}(\psi_{j})$ just
formally taking $\varepsilon_{j}=0$, $\omega_{j}^{0}=U$ and $\omega_{i}^{0}=0$
for $i\not=j$. In other words, $L[\psi]$ selects the main terms in
$L_{2}(\phi)$ and $L_{3}(\psi)$. Using the fact that
$-\Delta\Gamma_{0}=f(\Gamma_{0})=U$ where $f(u)=e^{u}$ and
$-\Delta_{5,j}\psi=\phi$ we see that

$\displaystyle L[\psi]\ =$
$\displaystyle-\nabla^{\perp}\Gamma_{0}\cdot\nabla[\Delta_{y}\psi+{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{y_{1}}\psi+f^{\prime}(\Gamma_{0})\psi].$

In polar coordinates $y=\rho e^{i\theta}$ in $\mathbb{R}^{2}$, we check that

$L[\psi]=-\frac{4}{\rho^{2}+1}\frac{{\partial}}{{\partial}\theta}[\Delta_{y}\psi+{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{y_{1}}\psi+f^{\prime}(\Gamma_{0})\psi].$

It is enough to use a simplified version of this operator given by

$L_{0}[\psi]:=-\frac{4}{\rho^{2}+1}\frac{{\partial}}{{\partial}\theta}[\Delta_{y}\psi+f^{\prime}(\Gamma_{0})\psi],\quad\phi=-\Delta_{5,j}\psi.$
(5.6)

For terms in the error of Fourier mode $1$ or higher, we will solve with the
elliptic operator in (5.6). In case the error has mode $1$, it will be
possible to solve only with a proper adjustment of the points $P_{j}^{1}$. We
call this procedure the elliptic improvement.

If the part of inner error $E(y,t)$ we want to remove has $0$-th mode in its
Fourier decomposition, we will solve using the operator

$|\log\varepsilon|\,\varepsilon_{j}^{2}{\partial}_{t}\phi=E.$

This is the main part of $L_{1}(\phi)$. We can solve the above ODEs at the
expenses of losing two power of $\varepsilon_{j}$ (and gaining one power of
$|\log\varepsilon|$) in the estimates for the solution. We will see that the
construction is decided by powers of $\varepsilon$, in the sense that the
exact number of powers of $|\log\varepsilon|$ we gain will be easily absorbed
in a slightly smaller power of $\varepsilon$. We call this procedure the ODEs
improvement. It has the advantage of keeping track of the Fourier modes of the
solutions.

After the fifth inner improvement, we will need to reduce not only the size of
the error, expressed in terms of powers of $\varepsilon$, but also its decay
rate in the $\rho=|y|$-variable. In this case we will solve the transport-type
equation

$L_{1}(\phi)+L_{2}(\phi)=E.$

We call this procedure the transport improvement. After solving this linear
equation, we will not have any more control on the Fourier modes of the
solution $\phi$, but the structure of the problem will automatically give a
new error whose main term has no $0$-th mode in its Fourier decomposition.
This will be crucial to keep improving.

The process of improvement is summarized by the following diagram: for
$\rho={|x-P_{j}|\over\varepsilon_{j}}$

$e_{1}={\varepsilon^{2}|\log\varepsilon|\over
1+\rho^{4}}E_{2}\,\underset{{\mathcal{E}}}{\Rightarrow}\,e_{2}={\varepsilon^{3}|\log\varepsilon|^{2}\over
1+\rho^{3}}E_{1}\,\underset{{\mathcal{E}\&P^{1}}}{\Rightarrow}\,e_{3}={\varepsilon^{4}|\log\varepsilon|^{4}\over
1+\rho^{2}}E_{0}\,\underset{{\mathcal{O}DEs}}{\Rightarrow}\,e_{4}={\varepsilon^{3}|\log\varepsilon|^{6}\over
1+\rho^{3}}E_{12}\sin\theta\,\underset{{\mathcal{E}}\&P}{\Rightarrow}$
$\,e_{5}={\varepsilon^{4}|\log\varepsilon|^{8}\over
1+\rho^{2}}E_{2}\,\underset{{\mathcal{E}}}{\Rightarrow}\,e_{6}={\varepsilon^{5}|\log\varepsilon|^{8}\over
1+\rho}E_{0}\,\underset{{\mathcal{T}}}{\Rightarrow}\,e_{7}={\varepsilon^{3}|\log\varepsilon|^{8}\over
1+\rho^{5}}E_{1}\,\underset{{\mathcal{O}ut}}{\Rightarrow}\,e_{8}={\varepsilon^{3}|\log\varepsilon|^{b}\over
1+\rho^{5}}E_{1}\,\underset{{\mathcal{E}\&P^{1}}}{\Rightarrow}\,$
$e_{9}={\varepsilon^{4}|\log\varepsilon|^{b}\over
1+\rho^{4}}E_{0}\,\underset{{\mathcal{O}DEs}}{\Rightarrow}\,e_{10}={\varepsilon^{3}|\log\varepsilon|^{b}\over
1+\rho^{5}}E_{12}\sin\theta\,\underset{{\mathcal{E}}\&P^{1}}{\Rightarrow}\,e_{11}={\varepsilon^{4}|\log\varepsilon|^{b}\over
1+\rho^{4}}E_{2}\,\underset{{\mathcal{E}}}{\Rightarrow}\,e_{12}={\varepsilon^{5}|\log\varepsilon|^{b}\over
1+\rho^{3}}E_{0}.\,$

Here ${\mathcal{E}}$ stands for inner elliptic improvement,
${\mathcal{E}}\&P^{1}$ stands for inner elliptic improvement with adjustment
of the points $P^{1}$, ${\mathcal{T}}$ stands for inner transport improvement,
${\mathcal{O}DEs}$ stands for inner ODEs improvement, and ${\mathcal{O}ut}$
stands for outer improvement. Besides $b$ is a positive number whose value may
change from line to line and within the same line. Let us explain how to
interpret the diagram: we start with an initial error of which we aim at
eliminating the part of size $\varepsilon^{2}|\log\varepsilon|$, decay in
space ${1\over 1+\rho^{4}}$ and Fourier mode $2$:
$e_{1}={\varepsilon^{2}|\log\varepsilon|\over 1+\rho^{4}}E_{2}$. To do so, we
proceed with the inner elliptic improvement. After this correction is done, we
have a new error. Of the new error we aim now at eliminating the main term,
which has size $\varepsilon^{3}|\log\varepsilon|^{2}$, decay in space ${1\over
1+\rho^{3}}$ and Fourier mode $1$:
$e_{2}={\varepsilon^{3}|\log\varepsilon|^{2}\over 1+\rho^{3}}E_{1}$. And so
on. Notice that the errors $e_{4}$ and $e_{10}$ has mode $1$, but only with
$\sin\theta$ (mode $1$ and odd in $y_{2}$).

Before starting the process of improvement of the approximation, we observe
that the expression for the initial error
$\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})(P_{j}+\varepsilon_{j}y)$ in (3.33)
gets a simpler form if we use (5.1). Using (5.3) and (2.11), we write

$\displaystyle{\mathcal{R}}_{j}(y,t;P)$
$\displaystyle={\mathcal{R}}_{j}^{00}(y,t;P)+\varepsilon^{2}|\log\varepsilon|{\mathcal{Q}}_{1}[P](y)+\varepsilon^{3}|\log\varepsilon|^{3\over
2}{\mathcal{Q}}_{2}[P](y)$ (5.7) $\displaystyle{\mathcal{R}}_{j}^{00}(y,t;P)$
$\displaystyle:=\varepsilon_{j}|\log\varepsilon|{\partial}_{t}(P_{j}-P_{j}^{0})\cdot
y+\varepsilon_{j}[{\nabla}_{x}\varphi_{j}(P_{j};P)-{\nabla}_{x}\varphi_{j}(P_{j}^{0};P^{0})]\cdot
y$
$\displaystyle+\,\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(P_{j};P)[y],$

and

$\displaystyle w_{j}^{0}$
$\displaystyle=U(y)\left(1-{\varepsilon_{j}y_{1}\over
2r_{j}}(\Gamma_{0}+\bar{A}+\Gamma)+\varepsilon^{2}|\log\varepsilon|{\mathcal{Q}}_{1}[P](y)+\varepsilon^{3}|\log\varepsilon|^{3\over
2}{\mathcal{Q}}_{2}[P](y)\right),$

with ${\mathcal{Q}}_{i}$, $i=1,2$ as in (5.2). Under the constraints
(3.3)-(3.4)-(3.5)-(3.6) on the points $P_{j}$, we get

$\displaystyle{\mathcal{R}}_{j}(y,t;P)\cdot\nabla w_{j}^{0}$
$\displaystyle={\mathcal{R}}^{00}_{j}(y,t;P)\cdot\nabla
U+{\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}(\rho,\theta,t,\varepsilon)$
$\displaystyle+{\varepsilon^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}(E_{1}+E_{3})(\rho,\theta,t,\varepsilon)+O\left({\varepsilon^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}\right)\,;$

hence $\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})(P_{j}+\varepsilon_{j}y)$ in
(3.33) becomes

$\displaystyle\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})$
$\displaystyle(P_{j}+\varepsilon_{j}y)=\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)\cdot\nabla
U+{\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}(\rho,\theta,t,\varepsilon)$ (5.8)
$\displaystyle+{\varepsilon^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}(E_{1}+E_{3})(\rho,\theta,t,\varepsilon)+O\left({\varepsilon^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}\right)\,$

as $\varepsilon\to 0$, where the functions $E_{i}(\rho,\theta,t,\varepsilon)$
now depends also on ${\partial}_{t}P$, and have the form (3.34). These
estimates are valid for $|y|<3\varepsilon_{j}^{-1}|\log\varepsilon|^{-\zeta}$.

We are now ready to start improving. At each step of improvement we analyze
the term of the error we want to remove, we describe the strategy to do it and
we compute the new error produced by the correction.

###  5.2. First inner improvement

The first improvement of the error will remove part of the mode-2 term of size
$\varepsilon^{2}|\log\varepsilon|$ given by

${\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}(\rho,\theta,t,\varepsilon)$

in the error
$\varepsilon_{j}^{4}S_{1}(W^{0},\Psi^{0})(\varepsilon_{j}y+P_{j})$ computed in
(5.8). This term can be decomposed as the sum of two parts, one depending on
${\partial}_{t}P$ and one not:

${\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}={\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{*}[P]+{\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{**}[P,{\partial}_{t}P].$

In fact the origin of ${\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{**}[P,{\partial}_{t}P]$ is
$\varepsilon_{j}^{2}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}w^{0}_{j}$, $|\log\varepsilon|B_{0}(w_{j}^{0})$ and
${\mathcal{R}}_{j}(y,t;P)\cdot\nabla w_{j}^{0}$, and the dependence on
${\partial}_{t}P$ is linear. We check that

${\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{**}[P,{\partial}_{t}P]={\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{**}[r_{0}{\bf
e}_{1}+P^{0}+P^{1},{\partial}_{t}(P^{0}+P^{1})]+{\varepsilon^{5}|\log\varepsilon|\over
1+|y|^{4}}E_{2}$

with $E_{2}$ satisfying (3.34). See (3.3) for the assumptions on $P$, $P^{0}$
and $P^{1}$. We will remove the part of the error
${\varepsilon^{2}|\log\varepsilon|\over 1+|y|^{4}}E_{2}$ given by

$e_{1}:={\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{*}[P]+{\varepsilon^{2}|\log\varepsilon|\over
1+|y|^{4}}E_{2}^{**}[(r_{0},0)+P^{0}+P^{1},{\partial}_{t}(P^{0}+P^{1})],$
(5.9)

thus leaving out what depends on $\partial_{t}{\bf a}$. We will do it solving
the simplified linear elliptic operatot

$L_{0}[\psi]:=-\frac{4}{\rho^{2}+1}\frac{{\partial}}{{\partial}\theta}[\Delta_{y}\psi+f^{\prime}(\Gamma_{0})\psi].$

introduced in (5.6). We freeze the time variable and consider the problem

$\displaystyle{4\over 1+\rho^{2}}\frac{{\partial}}{{\partial}\theta}$
$\displaystyle\left[\Delta\psi+e^{\Gamma_{0}(y)}\psi\right]+g(y)=0{\quad\hbox{in
}}B(0,{8\over\varepsilon|\log\varepsilon|^{\zeta}}),\quad$ (5.10)
$\displaystyle\psi$ $\displaystyle=0{\quad\hbox{in }}\partial
B(0,{8\over\varepsilon|\log\varepsilon|^{\zeta}}),\quad\phi=-\Delta_{5,j}\psi,$

for a bounded function
$g:B(0,{8\over\varepsilon|\log\varepsilon|^{\zeta}})\subset\mathbb{R}^{2}\to\mathbb{R}$.
A necessary condition for the solvability of (5.10) is that

$\int_{0}^{2\pi}g(\rho e^{i\theta})\,d\theta=0\quad\mbox{for
all}\quad\rho\in(0,8R_{\varepsilon}),\quad
R_{\varepsilon}={1\over\varepsilon|\log\varepsilon|^{\zeta}}.$ (5.11)

As

$L_{0}[Z_{\ell}]=0,\quad Z_{\ell}(y)={\partial}_{y_{\ell}}\Gamma_{0}(y).$

we also assume the orthogonality conditions

$\int_{B(0,8R_{\varepsilon})}(1+|y|^{2})\,g(y)\,Z_{\ell}(y)\,dy\,=\,0,\quad\ell=1,2.$
(5.12)

We have the validity of the following result.

######  Lemma 5.1.

Assume that $3\leq m\leq 5$ and

$|g(y)|\ \leq\ (1+|y|)^{-m}$ (5.13)

There exists a constant $C>0$ such that for all $\varepsilon>0$ sufficiently
small and $g\in L^{\infty}(B(0,8R_{\varepsilon}))$ that satisfies conditions
$(\ref{ort0})$, $(\ref{ort2})$ and $(\ref{decay})$, there exists a unique
solution $(\psi,\phi)$ of equation $(\ref{linear1})$ that satisfies

$\int_{0}^{2\pi}\psi(\rho e^{i\theta})\,d\theta=0\quad\mbox{for
all}\quad\rho\in(0,8R_{\varepsilon})$

and the estimate

$\displaystyle|\psi(y)|\,+\,(1+|y|)\,|{\nabla}\psi(y)|\,+\,(1+|y|^{2})\,|\phi(y)|$
$\displaystyle\leq\
C{(1+|y|)^{4-m}}\,\begin{cases}\log\Big{(}\frac{16\varepsilon^{-1}}{|y|+1}\Big{)}&\hbox{
if \quad}m=5\\\ 1&\hbox{ if \quad}3<m<5\\\
\log\Big{(}\frac{16\varepsilon^{-1}}{|y|+1}\Big{)}&\hbox{ if
\quad}m=3\end{cases}$

###### Proof.

Let $y=\rho\,e^{i\theta}$ and decompose $\psi$ and $g$ in Fourier series in
the $\theta$-variable

$g(\rho
e^{i\theta})=\sum_{n\in\mathbb{Z}}g_{n}(\rho)e^{i\,n\,\theta},\quad\psi(\rho
e^{i\theta})=\sum_{n\in\mathbb{Z}}p_{n}(\rho)e^{i\,n\,\theta}.$

Condition (5.11) amounts to $g_{0}\equiv 0$. Imposing $p_{0}\equiv 0$,
equation (5.10) decouples into the infinitely many problems.

$\mathcal{L}_{n}[p_{n}]:=\partial^{2}_{\rho}p_{n}+\frac{1}{\rho}\partial_{\rho}p_{n}-\frac{n^{2}}{\rho^{2}}p_{n}+\frac{8p_{n}}{(1+\rho^{2})^{2}}=\frac{i(1+\rho^{2})}{4n}g_{n}(\rho),\quad
p_{n}(8R_{\varepsilon})=0.$ (5.14)

For each $n\neq 0$, there exists a positive function $\zeta_{n}(\rho)$ such
that $\mathcal{L}_{n}[\zeta_{n}]=0$ and

$\zeta_{n}(\rho)=\rho^{|n|}(1+o(1))\quad\text{as }\rho\to 0$

For $n=\pm 1$ we explicitly have $\zeta_{n}(\rho)=\frac{\rho}{1+\rho^{2}}$,
while for $|n|\geq 2$ we have

$\zeta_{n}(\rho)=\rho^{|n|}(1+o(1))\quad\text{as }\rho\to+\infty.$

Problem (5.14) is uniquely solved by the formula

$p_{n}(\rho)=\mathcal{L}_{n}^{-1}[g_{n}]\,:=\frac{i}{4n}\zeta_{n}(\rho)\int_{\rho}^{8R_{\varepsilon}}\frac{dr}{r\zeta_{n}(r)^{2}}\int_{0}^{r}(1+s^{2})g_{n}(s)\zeta_{n}(s)\,s\,ds.\quad$

Let us consider the case $|n|\geq 2$. We have

$|p_{n}(\rho)|\ \leq\ \mathcal{L}_{n}^{-1}[(1+\rho)^{-\alpha}]\leq
c_{n}(1+\rho)^{|n|}\int_{\rho}^{8R_{\varepsilon}}(1+r)^{-|n|+3-\alpha}dr$

so that

$|p_{n}(\rho)|\ \leq\ c_{n}(1+\rho)^{4-\alpha}$

since $m+|n|>4$. Let us denote
$\bar{P}(\rho)=\mathcal{L}_{2}^{-1}[(1+\rho)^{\alpha}]$ we claim that for some
$\gamma>0$ and all $|n|\geq 2$ we have the validity of the estimate

$|p_{n}(\rho)|\ \leq\ \frac{\gamma}{n^{3}}\bar{P}(\rho).$

That follows from the fact that the right hand side defines a positive
supersolution for the real and imaginary parts of (5.14). Indeed, if $\gamma$
is taken sufficiently large we get

$\mathcal{L}_{n}\big{[}\frac{\gamma}{n^{3}}\bar{P}(\rho)\big{]}+\frac{1+\rho^{2}}{4n}|g_{n}(\rho)|\leq\gamma\frac{4-n^{2}}{10\rho^{2}}(1+\rho)^{4+\alpha}$

Fourier modes $\pm 1$ need to be separately treated because $\zeta_{1}(\rho)$
decays at infinity. At this point we observe that

$Z_{1}(y)=\zeta_{1}(\rho)\cos\theta,\quad Z_{2}(y)=\zeta_{1}(\rho)\sin\theta$

and that the orthogonality conditions (5.12) assumed are equivalent to

$\int_{0}^{8R_{\varepsilon}}(1+\rho^{2})\,g_{\pm
1}(\rho)\zeta_{1}(\rho)\,\rho\,d\rho=0.$

Therefore we can write

$p_{\pm
1}(\rho)=\mp\frac{i}{4}\zeta_{1}(\rho)\int_{\rho}^{8R_{\varepsilon}}\frac{dr}{r\zeta_{1}(r)^{2}}\int_{r}^{8R_{\varepsilon}}(1+s^{2})g_{\pm
1}(s)\zeta_{1}(s)\,s\,ds.\quad$

and we obtain, if we now assume $3<m\leq 5$,

$|p_{\pm 1}(\rho)|\ \lesssim\
\begin{cases}(1+\rho)^{-1}\log{\frac{16R_{\varepsilon}}{\rho+1}}&\hbox{ if
}m=5\\\ (1+\rho)^{4-m}&\hbox{ if }3<m<5\\\
(1+\rho)\log{\frac{16R_{\varepsilon}}{\rho+1}}&\hbox{ if }m=3.\end{cases}$

The desired result then follows from addition of the above estimates since

$|\psi(y)|\ \leq\ \sum_{n\in\mathbb{Z}}|p_{n}(|y|)|.$

Finally, since $\psi$ satisfies the equation

$\Delta\psi+e^{\Gamma}_{0}\psi=-\frac{1}{4}(1+\rho^{2})\int_{0}^{\theta}g(\rho,\theta)\,d\theta,{\quad\hbox{in
}}B(0,8R_{\varepsilon}),\quad\psi=0{\quad\hbox{on
}}{\partial}B(0,8R_{\varepsilon}),$

the bounds for $\phi$ and ${\nabla}\psi$ follow from standard elliptic
estimates.

∎

We extend the function $e_{1}$ in (5.9) to be equal to $0$ outside
$B(0,3R_{j})$. Under the assumptions (3.3) on $P$, $P^{0}$ and $P^{1}$, we
have that

$(1+|y|^{4})\left(|e_{1}(y,t)|+|\log\varepsilon|^{1\over
2}|{\partial}_{t}e_{1}(y,t)|\right)\lesssim\varepsilon^{2}|\log\varepsilon|.$

Moreover the function $e_{1}$ satisfies authomatically the orthogonality
conditions (5.11) and (5.12). Let $\psi^{1}_{j}=\psi_{j}^{1}(y,t)$ and
$\phi_{j}^{1}(y,t)=-\Delta_{5,j}\psi_{j}^{1}$ be the solution to (5.10) when
$g=e_{1}$, as predicted by Lemma 5.1. We have

$|\psi^{1}_{j}(y,t)|\,+\,(1+|y|)\,|{\nabla}\psi_{j}^{1}(y,t)|\,\lesssim\varepsilon^{2}|\log\varepsilon|,\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

and

$\,(1+|y|^{3})\,|\nabla_{y}\phi_{j}^{1}(y,t)|+\,(1+|y|^{2})\,|\phi_{j}^{1}(y,t)|\lesssim\varepsilon^{2}|\log\varepsilon|\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

Having left out what depends on ${\partial}_{t}{\bf a}$ in the part of the
error $e_{1}$ we are considering (see (5.9)), we can harmlessly differentiate
in time equations (5.10), to also get

$\,(1+|y|^{2})\,|{\partial}_{t}\phi_{j}^{1}(y,t)|\lesssim\varepsilon^{2}|\log\varepsilon|^{1\over
2}\quad{\quad\hbox{in }}B(0,3R_{j})\times[0,T].$

Besides, a consequence of the proof of Lemma 5.1 is that the Fourier
decomposition of $\psi_{j}^{1}$ only contains mode-2 terms. Hence the Fourier
decomposition of the function $\phi_{j}^{1}(y,t)$ has a mode-$2$ term of size
$\varepsilon_{j}^{2}|\log\varepsilon|$, mode-$1$ and mode-$3$ terms of size
$\varepsilon_{j}^{3}|\log\varepsilon|$ or smaller.

Using these properties of the functions $\psi_{j}^{1}$, $\phi_{j}^{1}$, and
the notations introduced in (5.5) we get

$\displaystyle
L_{1}(\phi_{j}^{1})={\varepsilon_{j}^{4}|\log\varepsilon|^{3\over 2}\over
1+|y|^{2}}E_{2}+{\varepsilon_{j}^{5}|\log\varepsilon|^{3\over 2}\over
1+|y|}E_{1},$
$\displaystyle\nabla^{\perp}\Gamma_{0}\cdot\nabla[{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{y_{1}}\psi_{j}^{1}]={\varepsilon_{j}^{3}|\log\varepsilon|\over
1+|y|^{3}}(E_{1}+E_{3})+{\varepsilon_{j}^{4}|\log\varepsilon|\over
1+|y|^{2}}(E_{0}+E_{2}),$ and $\displaystyle
L_{2}[\phi_{j}^{1}]+L_{3}[\psi_{j}^{1}]-L[\psi_{j}^{1}]+Q(\psi_{j}^{1},\phi_{j}^{1})={\varepsilon_{j}^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}(E_{1}+E_{3})+{\varepsilon_{j}^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}E_{i\geq 0}.$

In the above expression $E_{j}$ denotes the $j$-th mode in the Fourier
decomposition of $E(y,t)$, in accordance with the notation introduced in
(2.8). Inserting this information in (5.1) we get the description of the new
error, in the region $|x-P_{j}|<3|\log\varepsilon|^{-\zeta}$, for
$y={x-P_{j}\over\varepsilon_{j}}$,

$\displaystyle E_{j}^{in}$
$\displaystyle[\phi_{j}^{1},\psi_{j}^{1},0,P](y,t):=\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)\cdot\nabla
U+{\varepsilon_{j}^{3}|\log\varepsilon|^{2}\over 1+|y|^{3}}(E_{1}+E_{3})$
$\displaystyle+{\varepsilon_{j}^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}(E_{0}+E_{2})+{\varepsilon_{j}^{5}|\log\varepsilon|^{2}\over
1+|y|}\,E_{i\geq 0},$

as $\varepsilon\to 0$, where the functions $E_{j}(\rho,\theta,t,\varepsilon)$
have the form described in (3.34). For our next improvement it is convenient
to write the error as a term of size
$\varepsilon_{j}^{3}|\log\varepsilon|^{2}$-size in Fourier mode $1$ or higher,
with spacial decay bounded by ${1\over 1+|y|^{3}}$, and that does depend on
$P$, ${\partial}_{t}P^{0}$, but not on ${\partial}_{t}P^{1}+{\partial}_{t}{\bf
a}$. We use the fact that the error depends in a linear way on
${\partial}_{t}P$ and assumptions (3.5)-(3.6) to conclude that

$\displaystyle E_{j}^{in}$
$\displaystyle[\phi_{j}^{1},\psi_{j}^{1},0,P](y,t):=\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)\cdot\nabla
U+{\varepsilon^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}(E_{1}+E_{3})[P,{\partial}_{t}P^{0}]++O\left({\varepsilon^{4}|\log\varepsilon|^{2}\over
1+|y|^{2}}\right)$ (5.15)

###  5.3. Second inner improvement

Our next step is to eliminate the terms in the error (5.15) of size
$\varepsilon^{3}|\log\varepsilon|^{2}$. A difference with the first
improvement is that this time these terms posses a mode $1$. We will solve
again an elliptic problem of the form (5.10), but only at the expenses of
asking that the orthogonality conditions (5.12) for the right-hand side are
satisfied. We shall see that this is possible with an adjustment of the points
$P$.

In the process of the construction of the approximation $(\Psi^{*},W^{*})$, we
will need to correct these points several times. All these corrections are
encoded in the point we called $P^{1}$ in (3.3)-(3.5). The final definition of
$P^{1}$ is given by

$P^{1}(t)=\sum_{\ell=1}^{4}P^{1\ell}(t).$ (5.16)

where $P^{1\ell}$ correspond to successive explicit adjustments of $P^{1}$.
Take $\ell=1$, for the first adjustment, and for

$\bar{P}^{1}=(\bar{P}^{1}_{1},\ldots,\bar{P}^{1}_{k}),\quad\bar{P}^{1}_{j}=(r_{0},0)+P^{0}_{j}+P_{j}^{11},$

we take

$e_{2}(y,t):=\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{1})\cdot\nabla
U+{\varepsilon_{j}^{3}|\log\varepsilon|^{2}\over
1+|y|^{3}}(E_{1}+E_{3})[\bar{P}^{1},{\partial}_{t}P^{0}](\rho,\theta,t,\varepsilon).$

We directly check that $\int_{0}^{2\pi}e_{2}(\rho e^{i\theta},t)\,d\theta=0$,
and $e_{2}$ satisfies the decay (5.13) with $m=3$. In fact, under the
assumptions (3.3) on $P$, we have

$(1+|y|^{3})(|e_{2}(y,t)|+|\log\varepsilon|^{1\over
2}|{\partial}_{t}e_{2}(y,t)|)\lesssim\varepsilon^{3}|\log\varepsilon|^{2},\quad|y|<3R_{j}.$

The orthogonality conditions

$\int_{B_{8R_{j}}}(1+|y|^{2})e_{2}(y,y)Z_{\ell}(y)\,dy=0,\quad\ell=1,2$

become a system of ODEs for the point $P^{11}=(P_{1}^{11},\ldots,P_{k}^{11})$
that has the form

$\varepsilon_{j}|\log\varepsilon|\left[{\partial}_{t}P_{j}^{11}+AP^{11}+|\log\varepsilon|^{-{1\over
2}}B(P^{11})\right]=\varepsilon^{3}|\log\varepsilon|^{3}f(\bar{P}^{1})$ (5.17)

where $A$ is the $2\times 2$ matrix defined by

$A=|\log\varepsilon|^{-1}D^{2}_{x}\varphi_{j}(\bar{P}_{j}^{0};P^{0}),\quad
t\in[0,T),$

and $B$

$B(P^{11})=|\log\varepsilon|^{-1}[{\nabla}_{x}\varphi_{j}(\bar{P}_{j}^{1};\bar{P}^{1})-{\nabla}_{x}\varphi_{j}(P_{j}^{0};P^{0})-D^{2}_{x}\varphi_{j}(\bar{P}_{j}^{0};P^{0})P^{11}].$

See (3.29) for the definition of $\varphi_{j}$. Under our assumptions (3.4) on
the points $P^{0}$, we have

$A=O(1),\quad B(P^{11})=(|\log\varepsilon|^{-{1\over 2}}),\quad t\in[0,T),$

uniformly as $\varepsilon\to 0$. Besides, $f(\bar{P}^{1})$ is a smooth
function of $P^{11}$, which is uniformly bounded, together with its
derivative, for $t\in[0,T)$, uniformly as $\varepsilon\to 0$. Standard ODEs
theory gives that, for all $\varepsilon>0$ small enough there exists a unique
solution $P^{11}$ to (5.17) with initial condition $P^{11}(0)=0$, which
satisfy the bounds

$\|P^{11}_{j}\|_{L^{\infty}[0,T)}+\|{\partial}_{t}P^{11}_{j}\|_{L^{\infty}[0,T)}\lesssim\varepsilon^{2}|\log\varepsilon|^{2}.$

We denote by $\psi_{j}^{2}=\psi_{j}^{2}(y,t)$ and
$\phi_{j}^{2}(y,t)=-\Delta_{5,j}\psi_{j}^{2}$ the solution to (5.10), with
$g=e_{2}$, whose existence and estimates are given by Lemma 5.1. We have that
$\int_{0}^{2\pi}\psi_{j}^{2}(\rho e^{i\theta},t)\,d\theta=0$ and

$|\psi^{2}_{j}(y,t)|\,+\,(1+|y|)\,|{\nabla}\psi_{j}^{2}(y,t)|\,\lesssim\varepsilon^{3}|\log\varepsilon|^{3}\,(1+|y|),\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

We also have

$\,(1+|y|^{2})\,|\nabla_{y}\phi_{j}^{2}(y,t)|+\,(1+|y|)\,|\phi_{j}^{2}(y,t)|\lesssim\varepsilon^{3}|\log\varepsilon|^{3}\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

For the same reason we did it before, we can differentiate in time the
equation, to also get

$\,(1+|y|)\,|\log\varepsilon|^{1\over
2}\,|{\partial}_{t}\phi_{j}^{2}(y,t)|\lesssim\varepsilon^{3}|\log\varepsilon|^{3}\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

From Lemma 5.1 we also get that the Fourier decomposition of $\psi_{j}^{2}$
only contains Fourier mode-$1$ and mode-$3$ terms. Hence the Fourier
decomposition of the function $\phi_{j}^{2}(y,t)$ has mode-$1$ and mode-$3$
terms of size $\varepsilon^{3}|\log\varepsilon|^{3}$, terms of mode-$0$,
mode-$2$ or higher of size $\varepsilon^{4}|\log\varepsilon|^{3}$ or smaller.
Using these properties and the notations introduced in (5.5) we get

$\displaystyle L_{1}(\phi_{j}^{2})={\varepsilon^{5}|\log\varepsilon|^{7\over
2}\over 1+|y|}E_{i\geq 0},$
$\displaystyle\nabla^{\perp}\Gamma_{0}\cdot\nabla[{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{y_{1}}\psi^{2}_{j}]={\varepsilon^{4}|\log\varepsilon|^{3}\over
1+|y|^{2}}(E_{0}+E_{i\geq 2})+{\varepsilon^{5}|\log\varepsilon|^{3}\over
1+|y|}E_{i\geq 0},$ $\displaystyle
L_{2}[\phi_{j}^{2}]+L_{3}[\psi_{j}^{2}]-L[\psi_{j}^{2}]+Q(\psi_{j}^{2},\phi_{j}^{2})={\varepsilon^{4}|\log\varepsilon|^{4}\over
1+|y|^{2}}(E_{0}+E_{i\geq 2})+{\varepsilon^{5}|\log\varepsilon|^{4}\over
1+|y|}E_{i\geq 0}$ and $\displaystyle
Q(\psi_{j}^{1},\phi_{j}^{2})+Q(\psi_{j}^{2},\phi_{j}^{1})={\varepsilon^{5}|\log\varepsilon|^{4}\over
1+|y|}E_{i\geq 0}.$

Define

$\tilde{\phi}_{j}^{2}=\sum_{i=1}^{2}\phi_{j}^{i},\quad\tilde{\psi}_{j}^{2}=\sum_{i=1}^{2}\psi_{j}^{i}.$

We get the new error

$\displaystyle E_{j}^{in}$
$\displaystyle[\tilde{\phi}_{j}^{2},\tilde{\psi}_{j}^{2},0,P](y,t):=[\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)-\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{1})]\cdot\nabla
U$ $\displaystyle+{\varepsilon^{4}|\log\varepsilon|^{4}\over
1+|y|^{2}}\,(E_{0}+E_{i\geq
2})[P,{\partial}_{t}P]+{\varepsilon^{5}|\log\varepsilon|^{4}\over
1+|y|}\,E_{i\geq 0}[P,{\partial}_{t}P].$ (5.18)

Using the definition of ${\mathcal{R}}^{00}_{j}$ in (5.7), we check that

$\displaystyle{\mathcal{R}}_{j}^{00}(y,t;P)$
$\displaystyle-{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{1})=\varepsilon_{j}|\log\varepsilon|{\partial}_{t}(P_{j}-\bar{P}_{j}^{1})^{\perp}\cdot
y$
$\displaystyle+\varepsilon_{j}[{\nabla}_{x}\varphi_{j}(P_{j};P)-{\nabla}_{x}\varphi_{j}(\bar{P}_{j}^{1};\bar{P}^{1})]\cdot
y$
$\displaystyle+\,\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(P_{j};P)[y]-\varepsilon_{j}^{3}D_{x}\theta_{j\varepsilon_{j}}(\bar{P}^{1}_{j};\bar{P}^{1})[y].$

###  5.4. Third inner improvement

Our next step is the elimination of the Fourier mode-$0$ term of size
$\varepsilon^{4}|\log\varepsilon|^{4}$ in formula (5.3). We define
$\phi_{j}^{3}$ as follows

$\phi_{j}^{3}(y,t)=-{\varepsilon^{4}\,|\log\varepsilon|^{3}\over
1+|y|^{2}}\,\int_{0}^{t}\varepsilon_{j}^{-2}(s)E_{0}(\rho,\theta,s,\varepsilon)\,ds.$

It solves

$\displaystyle|\log\varepsilon|\varepsilon_{j}^{2}{\partial}_{t}\phi_{j}^{3}$
$\displaystyle+{\varepsilon^{4}|\log\varepsilon|^{4}\over
1+|y|^{2}}\,E_{0}=0\quad(y,t)\in B(0,3R_{j})\times[0,T)$
$\displaystyle\phi_{j}^{3}(y,0)$ $\displaystyle=0,$

it satisfies

$(1+|y|)|\nabla_{y}\phi_{j}^{3}(y,t)|+|\phi_{j}^{3}(y,t)|\lesssim{\varepsilon_{j}^{2}|\log\varepsilon|^{3}\over
1+|y|^{2}}\,,\quad$

and its Fourier decomposition only has mode $0$. Let $\psi_{j}^{3}$ be the
solution to

$-\Delta_{5,j}\psi_{j}^{3}=\phi_{j}^{3}\quad(y,t)\in
B(0,4R_{j})\times[0,T),\quad\psi_{j}^{3}=0\quad(y,t)\in\partial
B(0,4R_{j})\times[0,T).$

Let $p(y,t)$, with $p=0$ on $\partial B(0,4R_{j})$ be the positive radial
solution to

$\Delta_{y}p+{4\over 1+|y|^{2}}=0,\quad y\in B(0,4R_{j}),\quad
p(\rho,t)=4\int_{\rho}^{4R_{j}}{\log(1+s^{2})\over s}\,ds.$

Then

$\Delta_{y}p+{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{1}p+{2\over 1+|y|^{2}}\leq 0,$

thanks to the fact that
$R_{j}={\delta\over\varepsilon_{j}|\log\varepsilon|^{\zeta}}$ with $\zeta>1$.
Take $\bar{\psi}(y,t)=M\,\varepsilon^{2}|\log\varepsilon|^{3}\,p$, for some
$M>0$. This is a super solution for

$\Delta_{5,j}\psi+\phi_{j}^{3}\leq 0,$

and gives that

$|\psi_{j}^{3}(y,t)|+|(1+|y|)\nabla\psi_{j}^{3}(y,t)|\lesssim\varepsilon_{j}^{2}\,|\log\varepsilon|^{5}.$

Besides the Fourier decomposition of the function $\psi_{j}^{3}$ has mode-$0$
terms of size $\varepsilon^{2}|\log\varepsilon|^{5}$, terms of mode-$1$ or
higher of size $\varepsilon^{3}|\log\varepsilon|^{5}$ or smaller.

It is important to notice that $B_{0}(g)$ only contains Fourier mode-$2$ or
mode-$1$ terms if $g$ is a Fourier mode-$0$ function. We refer to (3.27) and
Remark 3.1 for the definition of $B_{0}$ and equivalent formulations. This
observation yields that

$L_{1}(\phi_{j}^{3})-|\log\varepsilon|\varepsilon_{j}^{2}{\partial}_{t}\phi_{j}^{3}={\varepsilon^{4}|\log\varepsilon|^{4}\over
1+|y|^{2}}E_{2}+{\varepsilon^{5}|\log\varepsilon|^{4}\over 1+|y|}E_{1}.$

If $f$ is a Fourier mode-$2$ function and $g$ is a Fourier mode-$0$ function,
then $\nabla^{\perp}f\cdot\nabla g$ is a Fourier mode-$2$ function. We use
this, together with the explicity expression of ${\mathcal{R}}_{j}^{0}$ in
(5.1) and the explicit form of the operator $L_{2}$ in (5.5) to get

$L_{2}(\phi_{j}^{3})={\varepsilon^{3}|\log\varepsilon|^{4}\over
1+|y|^{3}}E_{1,2}\sin\theta+{\varepsilon^{4}|\log\varepsilon|^{4}\over
1+|y|^{2}}E_{2}+{\varepsilon^{5}|\log\varepsilon|^{4}\over 1+|y|}E_{i\geq 0}.$

The key fact is that the main error of size
$\varepsilon^{3}|\log\varepsilon|^{4}$ has a Fourier mode-$1$ term, but only
containing $\sin\theta$, not $\cos\theta$: it is of the form described in
(3.34) for $j=1$ and $E_{j,1}=0$. A similar expression is valid for
$L_{3}(\psi_{j}^{3})$, with two more powers of $|\log\varepsilon|$:

$L_{3}(\psi_{j}^{3})={\varepsilon^{3}|\log\varepsilon|^{6}\over
1+|y|^{3}}E_{1,2}\sin\theta+{\varepsilon^{4}|\log\varepsilon|^{6}\over
1+|y|^{2}}E_{2}+{\varepsilon^{5}|\log\varepsilon|^{6}\over 1+|y|}E_{i\geq 0}.$

We also have

$Q(\psi_{j}^{3},\phi_{j}^{3})={\varepsilon^{5}|\log\varepsilon|^{8}\over
1+|y|}E_{i\geq 0}$
$Q(\psi_{j}^{1},\phi_{j}^{2})+Q(\psi_{j}^{2},\phi_{j}^{1})={\varepsilon^{4}|\log\varepsilon|^{6}\over
1+|y|^{4}}E_{2}+{\varepsilon^{5}|\log\varepsilon|^{6}\over 1+|y|}E_{i\geq 0}$

With this in mind, calling

$\tilde{\phi}_{j}^{3}=\sum_{i=1}^{3}\phi_{j}^{i},\quad\tilde{\psi}_{j}^{3}=\sum_{i=1}^{3}\psi_{j}^{i}$

we get the new error

$\displaystyle E_{j}^{in}$
$\displaystyle[\tilde{\phi}_{j}^{3},\tilde{\psi}_{j}^{3},0,P](y,t):=[\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)-\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{1})]\cdot\nabla
U$ $\displaystyle+{\varepsilon_{j}^{3}|\log\varepsilon|^{6}\over
1+|y|^{3}}\,E_{1,1}\sin\theta+{\varepsilon_{j}^{4}|\log\varepsilon|^{6}\over
1+|y|^{2}}\,E_{2}[P,{\partial}_{t}P]+{\varepsilon_{j}^{5}|\log\varepsilon|^{8}\over
1+|y|}\,E_{i\geq 0}[P,{\partial}_{t}P].$ (5.19)

If we compare this error
$E_{j}^{in}[\tilde{\phi}_{j}^{3},\tilde{\psi}_{j}^{3},0,P]$ with the error
$E_{j}^{in}[\phi_{j}^{1},\psi_{j}^{1},0,P]$ in (5.15), we observe a crucial
difference: even though their main term have size $\varepsilon^{3}$
(multiplied by powers of $|\log\varepsilon|$) and are in Fourier mode-$1$, in
$E_{j}^{in}[\tilde{\phi}_{j}^{3},\tilde{\psi}_{j}^{3},0,P]$ Fourier mode-$1$
enters only with a $\sin\theta$. We shall proceed as in the second improvement
of the approximation, with adjusting the points $P^{1}$ and solving the same
elliptic linear problem, but this time the new error will not have Fourier
mode-$0$ of size $\varepsilon^{4}$. This subtle fact allows us to proceed with
the construction.

###  5.5. Fourth inner improvement

Our next step is to eliminate the term
${\varepsilon_{j}^{3}|\log\varepsilon|^{6}\over 1+|y|^{3}}\,E_{1,1}\sin\theta$
in the error (5.4). Take $\ell=2$ in the decomposition (5.16) of $P^{1}(t)$
and for $\bar{P}^{2}=(\bar{P}^{2}_{1},\ldots,\bar{P}^{2}_{k})$,
$\bar{P}^{2}_{j}=P_{0}+P_{j}^{0}+P_{j}^{11}+P_{j}^{12}$, we take

$e_{4}(y,t):=\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{2})\cdot\nabla
U+{\varepsilon_{j}^{3}|\log\varepsilon|^{6}\over
1+|y|^{3}}\,E_{1,2}[\bar{P}^{2},{\partial}_{t}(P^{0}+P^{11})]\sin\theta.$

We have $\int_{0}^{2\pi}e_{4}(\rho e^{i\theta},t)\,d\theta=0$, and $e_{4}$
satisfies the decay (5.13) with $m=3$. In fact, under the assumptions (3.3) on
$P$, we have

$(1+|y|^{3})(|e_{4}(y,t)|+|\log\varepsilon|^{1\over
2}|{\partial}_{t}e_{4}(y,t)|)\lesssim\varepsilon^{3}|\log\varepsilon|^{5},\quad|y|<3R_{j}.$

The orthogonality conditions

$\int_{B_{8R_{j}}}(1+|y|^{2})e_{4}(y,y)Z_{\ell}(y)\,dy=0,\quad\ell=1,2$

become a system of ODEs for the point $P^{12}=(P_{1}^{12},\ldots,P_{k}^{12})$
of the same form as (5.17). Standard ODEs theory gives that, for all
$\varepsilon>0$ small enough there exists a unique solution $P^{12}$ to (5.17)
with initial condition $P^{12}(0)=0$, which satisfy the bounds

$\|P^{12}_{j}\|_{L^{\infty}[0,T)}+\|{\partial}_{t}P^{12}_{j}\|_{L^{\infty}[0,T)}\lesssim\varepsilon^{2}|\log\varepsilon|^{6}.$

We denote by $\psi_{j}^{4}=\psi_{j}^{4}(y,t)$ and
$\phi_{j}^{4}(y,t)=-\Delta_{5,j}\psi_{j}^{4}$ the solution to (5.10), with
$g=e_{4}$, whose existence and estimates are given by Lemma 5.1. We have that
$\int_{0}^{2\pi}\psi_{j}^{4}(\rho e^{i\theta},t)\,d\theta=0$ and

$|\psi^{4}_{j}(y,t)|\,+\,(1+|y|)\,|{\nabla}\psi_{j}^{4}(y,t)|\,\lesssim\varepsilon^{3}|\log\varepsilon|^{7}\,(1+|y|),\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

We also have

$\,(1+|y|^{2})\,|\nabla_{y}\phi_{j}^{4}(y,t)|+\,(1+|y|)\,|\phi_{j}^{4}(y,t)|\lesssim\varepsilon^{3}|\log\varepsilon|^{7}\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

We can differentiate in time the equation, to also get

$\,(1+|y|)\,|\log\varepsilon|^{1\over
2}\,|{\partial}_{t}\phi_{j}^{4}(y,t)|\lesssim\varepsilon^{3}|\log\varepsilon|^{7}\quad{\quad\hbox{in
}}B(0,3R_{j})\times[0,T].$

From Lemma 5.1 we infer that is that the Fourier decomposition of
$\psi_{j}^{4}$ contains Fourier mode-$1$ terms, but only with $\cos\theta$.
Hence the Fourier decomposition of the function $\phi_{j}^{4}(y,t)$ has
mode-$1$ terms with $\cos\theta$ of size
$\varepsilon^{2}|\log\varepsilon|^{7}$, terms of mode-$0$, mode-$2$ or higher
of size $\varepsilon^{4}|\log\varepsilon|^{7}$ or smaller. Using these
properties and the notations introduced in (5.5) we get

$\displaystyle L_{1}(\phi_{j}^{4})={\varepsilon^{5}|\log\varepsilon|^{7}\over
1+|y|}E_{i\geq 0},$
$\displaystyle\nabla^{\perp}\Gamma_{0}\cdot\nabla[{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{y_{1}}\psi^{4}_{j}]={\varepsilon^{4}|\log\varepsilon|^{7}\over
1+|y|^{2}}E_{i\geq 2}+{\varepsilon^{5}|\log\varepsilon|^{7}\over
1+|y|}E_{i\geq 0},$ $\displaystyle
L_{2}[\phi_{j}^{4}]+L_{3}[\psi_{j}^{4}]-L[\psi_{j}^{4}]+Q(\psi_{j}^{4},\phi_{j}^{4})={\varepsilon^{4}|\log\varepsilon|^{8}\over
1+|y|^{2}}E_{i\geq 2}+{\varepsilon^{5}|\log\varepsilon|^{4}\over
1+|y|}E_{i\geq 0}$ and $\displaystyle
Q(\tilde{\psi}_{j}^{3},\phi_{j}^{4})+Q(\psi_{j}^{4},\tilde{\phi}_{j}^{3})={\varepsilon^{5}|\log\varepsilon|^{8}\over
1+|y|}E_{i\geq 0}.$

Define

$\tilde{\phi}_{j}^{4}=\sum_{i=1}^{4}\phi_{j}^{i},\quad\tilde{\psi}_{j}^{4}=\sum_{i=1}^{4}\psi_{j}^{i}.$

We get the new error, for $y\in B(0,3R_{j})$,

$\displaystyle E_{j}^{in}$
$\displaystyle[\tilde{\phi}_{j}^{4},\tilde{\psi}_{j}^{4},0,P](y,t):=[\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)-\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{2})]\cdot\nabla
U$ $\displaystyle+{\varepsilon^{4}|\log\varepsilon|^{8}\over
1+|y|^{2}}\,E_{i\geq 2}+{\varepsilon^{5}|\log\varepsilon|^{8}\over
1+|y|}\,E_{i\geq 0}[P,{\partial}_{t}P].$ (5.20)

The $0$-th Fourier mode of the new error comes with size $\varepsilon^{5}$
(and powers of $|\log\varepsilon|$).

###  5.6. Fifth inner improvement

We now remove part of the Fourier mode-$2$ or higher terms of size
$\varepsilon^{4}|\log\varepsilon|^{8}$ in (5.5). As in the first inner
improvement, we take just the parts that depend on $P$ and on
${\partial}_{t}P^{0}$, ${\partial}_{t}P^{1}$, but not the ones depending on
${\partial}_{t}{\bf a}$, and we call it $e_{5}$. We extend the function
$e_{5}$ in (5.9) to be equal to $0$ outside $B_{j}=B(0,3R_{j})$. Under the
assumptions (3.3) on $P$, $P^{0}$ and $P^{1}$, we have that

$(1+|y|^{2})(|e_{5}(y,t)|+|\log\varepsilon|^{1\over
2}|{\partial}_{t}e_{5}(y,t)|)\lesssim\varepsilon^{4}|\log\varepsilon|^{8}.$

Moreover the function $e_{5}$ satisfies authomatically the orthogonality
conditions (5.11) and (5.12). Let $\psi^{5}_{j}$ and
$\phi_{j}^{5}(y,t)=-\Delta_{5,j}\psi_{j}^{5}$ be the solution to (5.10) when
$g=e_{5}$, as predicted by Lemma 5.1. We have

$|\psi^{5}_{j}(y,t)|\,+\,(1+|y|)\,|{\nabla}\psi_{j}^{5}(y,t)|\,\lesssim\varepsilon^{4}|\log\varepsilon|^{8}(1+|y|^{2}),\quad{\quad\hbox{in
}}B(0,8R_{j})\times[0,T].$

and

$\,(1+|y|)\,|\nabla_{y}\phi_{j}^{5}(y,t)|+\,\,|\phi_{j}^{5}(y,t)|\lesssim\varepsilon^{4}|\log\varepsilon|^{8}\quad{\quad\hbox{in
}}B(0,8R_{j})\times[0,T].$

We can differentiate in time equations (5.10), to also get

$\,\,|{\partial}_{t}\phi_{j}^{5}(y,t)|\lesssim\varepsilon^{4}|\log\varepsilon|^{8}\quad{\quad\hbox{in
}}B(0,8R_{j})\times[0,T].$

Define

$\tilde{\phi}_{j}^{5}=\sum_{i=1}^{5}\phi_{j}^{i},\quad\tilde{\psi}_{j}^{5}=\sum_{i=1}^{5}\psi_{j}^{i}.$

We get the new error, for $y\in B(0,3R_{j})$

$\displaystyle E_{j}^{in}$
$\displaystyle[\tilde{\phi}_{j}^{5},\tilde{\psi}_{j}^{5},0,P](y,t):=[\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)-\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{2})]\cdot\nabla
U+{\varepsilon^{5}|\log\varepsilon|^{8}\over 1+|y|}\,E_{i\geq 0}.$ (5.21)

###  5.7. Sixth inner improvement

Our next step is the elimination of the
$\varepsilon_{j}^{5}|\log\varepsilon|^{8}$-term in (5.21),
${\varepsilon_{j}^{5}|\log\varepsilon|^{8}\over 1+|y|}\,E_{i\geq 0}$, to get
faster decay in the $y$ variable. To do so, rather than solving an elliptic
problem or an ODEs, we solve the transport-type equation

$\displaystyle L_{1}(\phi_{j})+L_{2}(\phi_{j})$
$\displaystyle=|\log\varepsilon|\,\varepsilon_{j}^{2}(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}\phi_{j}+|\log\varepsilon|B_{0}(\phi_{j})$
$\displaystyle+\nabla^{\perp}\left(\Gamma_{0}+\frac{\varepsilon_{j}}{2r_{j}}\,y_{1}\,\Big{(}\Gamma_{0}(y)+\bar{A}+\Gamma(y)\Big{)}\,+{\mathcal{R}}^{0}_{j}(y,t;P)\right)\cdot\nabla\phi_{j}=E(y,t),$

in $B_{3R_{j}}\times[0,T]$, with initial condition $\phi(y,0)=0$ in
$B(0,3R_{j})$. We write the operator $B_{0}$ defined in (3.27) as follows

$\displaystyle|\log\varepsilon|B_{0}(\phi)$
$\displaystyle={\mathcal{B}}_{j}^{0}(y,t;P)\cdot\nabla\phi$ (5.22)
$\displaystyle{\mathcal{B}}_{j}^{0}(y,t;P)$
$\displaystyle=-{|\log\varepsilon|\over
2}{\partial}_{t}\varepsilon_{j}^{2}(1+{\varepsilon_{j}\over
r_{j}}y_{1})\,y-|\log\varepsilon|{\varepsilon_{j}^{2}\over
r_{j}}y_{1}{\partial}_{t}P_{j}.$

We will need uniform differentiability in $t$ of the coefficients
${{\mathcal{R}}}_{j}^{0}$ and ${\mathcal{B}}_{j}^{0}$. Thus we consider the
following slightly-modified transport equation

$\displaystyle\left\\{\begin{aligned}
\varepsilon_{j}^{2}|\log\varepsilon|(1+{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}\phi&+\nabla_{y}^{\perp}(\Gamma_{0}(y)+\tilde{{\mathcal{R}}}_{j}(y,t;P))\cdot\nabla_{y}\phi+{\mathcal{B}}_{j}(y,t;P)\cdot\nabla_{y}\phi\\\
&=E(y,t),\quad\text{in }B(0,3R_{j})\times[0,T]\\\ \phi(y,0)&=0,\quad\text{in
}B(0,3R_{j})\end{aligned}\right.$ (5.23)

Here

$\tilde{{\mathcal{R}}}_{j}(y,t)=\frac{\varepsilon_{j}}{2r_{j}}\,y_{1}\,\Big{(}\Gamma_{0}(y)+\bar{A}+\Gamma(y)\Big{)}+{\mathcal{R}}^{1}_{j}(y,t;P)$

where

${\mathcal{R}}^{0}_{j}(y,t;P)={\mathcal{R}}^{1}_{j}(y,t;P)+\varepsilon_{j}|\log\varepsilon|\,{\partial}_{t}{\bf
a}_{j}\cdot y$

see (5.1). In other words, we leave out the term involving ${\partial}_{t}{\bf
a}$. We do the same to ${\mathcal{B}}_{j}^{0}$: using the fact that
${\partial}_{t}\varepsilon_{j}^{2}=-r_{0}\varepsilon^{2}{{\partial}_{t}r_{j}\over
r_{j}^{2}}$, we write

${\mathcal{B}}_{j}^{0}(y,t;P)={\mathcal{B}}_{j}(y,t;P)+{|\log\varepsilon|\varepsilon^{2}r_{0}\over
2}{{\partial}_{t}{\bf a}_{j1}\over r_{j}^{2}}(1+{\varepsilon_{j}\over
r_{j}}y_{1})y-|\log\varepsilon|{\varepsilon_{j}^{2}\over
r_{j}}y_{1}{\partial}_{t}{\bf a}_{j}.$

It is straightforward to check that, under our assumptions (3.3), (3.4) and
(3.5), for $y\in B(0,3R_{j})$ we have

$\displaystyle|{{\mathcal{R}}}^{1}_{j}(y,t)|+|{\partial}_{t}{{\mathcal{R}}}^{1}_{j}(y,t)|$
$\displaystyle\ \leq\ M\varepsilon^{2}(1+|y|^{2}),$ (5.24)
$\displaystyle|\nabla_{y}{{\mathcal{R}}}^{1}_{j}(y,t)|+|{\partial}_{t}\nabla_{y}{{\mathcal{R}}}^{1}_{j}(y,t)|$
$\displaystyle\leq\ M\varepsilon^{2}(1+|y|),$
$\displaystyle|D^{2}_{y}{{\mathcal{R}}}^{1}_{j}(y,t)|$ $\displaystyle\leq\
M\varepsilon^{2},$
$\displaystyle|{\mathcal{B}}_{j}(y,t)|+|{\partial}_{t}{\mathcal{B}}_{j}(y,t)|$
$\displaystyle\leq\ M\varepsilon^{2}|\log\varepsilon|^{1\over 2}(1+|y|),$
$\displaystyle|\nabla_{y}{\mathcal{B}}_{j}(y,t)|$ $\displaystyle\leq\
M\varepsilon^{2}|\log\varepsilon|^{1\over 2},$

for some positive constant $M$ independent of $\varepsilon$.

We consider a smooth cut-off function $\eta_{4}(s)$ as in (3.12), and take

$\eta_{4\varepsilon}(y)=\eta_{4}(|\log\varepsilon|^{\zeta}\,\varepsilon\,|y|).$
(5.25)

We will then have a solution to (5.23) by restricting to $B(0,3R_{j})$ the
solution of the Cauchy problem

$\displaystyle\left\\{\begin{aligned}
\varepsilon_{j}^{2}|\log\varepsilon|(1+\eta_{4\varepsilon}{\varepsilon_{j}\over
r_{j}}y_{1}){\partial}_{t}\phi&+\nabla_{y}^{\perp}(\Gamma_{0}(y)+\eta_{4\varepsilon}\tilde{{\mathcal{R}}}_{j}(y,t;\xi))\cdot\nabla_{y}\phi+\eta_{4\varepsilon}{\mathcal{B}}_{j}(y,t;P)\cdot\nabla_{y}\phi\\\
&=\eta_{4\varepsilon}E(y,t),\quad\text{in }\mathbb{R}^{2}\times[0,T]\\\
\phi(y,0)&=0,\quad\text{in }\mathbb{R}^{2}\end{aligned}\right.$ (5.26)

In § 6 we prove the following result

######  Lemma 5.2.

Let us assume that $\tilde{{\mathcal{R}}}_{j}$ and ${\mathcal{B}}_{j}$ satisfy
$(\ref{papa-new})$. Then there exist numbers $C,\delta>0$ such that for all
sufficiently small $\varepsilon$ and any function $E(y,t)$ that satisfies for
some $A,\alpha\in\mathbb{R}$

$(1+|y|)|{\nabla}_{y}E(y,t)|+|E(y,t)|+|{\partial}_{t}E(y,t)|\ \leq\
A\,(1+|y|)^{\alpha},$

the solution of $(\ref{pico1})$ satisfies

$(1+|y|)|{\nabla}_{y}\phi(y,t)|+|\phi(y,t)|\ \leq\
C\varepsilon^{-2}|\log\varepsilon|^{-1}A(1+|y|)^{\alpha}$

for all $y\in\mathbb{R}^{2}$, with
$|y|<4\varepsilon^{-1}|\log\varepsilon|^{-\zeta}$, $t\in[0,T)$.

We want to apply Lemma 5.2 for the right hand side

$E(y,t)=e_{6},\quad e_{6}={\varepsilon^{5}|\log\varepsilon|^{8}\over
1+|y|}\,E_{i\geq 0}(\rho,\theta,t,\varepsilon)$

where we freeze this term at $P_{j}=P_{0}+P_{j}^{0}+P_{j}^{1}$. We define
$\phi_{j}^{6}$ to be the solution to problem (5.23), with $E$ as above,
predicted by Lemma 5.2. It satisfies

$(1+|y|)|{\nabla}_{y}\phi^{6}_{j}(y,t)|+|\phi^{6}_{j}(y,t)|\ \leq\
C{\varepsilon^{3}|\log\varepsilon|^{7}\over 1+|y|}.$

Let $\psi_{j}^{6}$ be the solution to

$-\Delta_{5,j}\psi_{j}^{6}=\phi_{j}^{6}\quad(y,t)\in
B(0,4R_{j})\times[0,T),\quad\psi_{j}^{6}=0\quad(y,t)\in\partial
B(0,4R_{j})\times[0,T).$

Let $p(y,t)$, with $p=0$ on $\partial B(0,4R_{j})$ be the positive smooth
radial solution to

$\Delta_{y}p+{4\over 1+|y|}=0,\quad y\in B(0,4R_{j}).$

Then $p(y)=16(R_{j}-|y|)-4\int_{|y|}^{4R_{j}}{\log(1+s)\over s}\,ds$, and
$|p(y)|\lesssim(1+|y|)$ and

$\Delta_{y}p+{3\varepsilon_{j}\over
r_{j}+\varepsilon_{j}y_{1}}{\partial}_{1}p+{2\over 1+|y|}\leq 0.$

Take $\bar{\psi}(y,t)=M\,\varepsilon^{3}|\log\varepsilon|^{7}\,p$, for some
$M>0$. It is a super solution for

$\Delta_{5,j}\psi+\phi_{j}^{6}\leq 0,$

and gives that

$(1+|y|^{2})|D^{2}_{y}\psi^{6}_{j}(y,t)|+(1+|y|)|{\nabla}_{y}\psi^{6}_{j}(y,t)|+|\psi^{6}_{j}(y,t)|\
\leq\ C\varepsilon^{3}|\log\varepsilon|^{8}(1+|y|).$

We use this information to compute the following terms

$\displaystyle
L_{3}(\psi_{j}^{6})+Q(\psi_{j}^{6},\phi_{j}^{6})={\varepsilon^{3}|\log\varepsilon|^{8}\over
1+|y|^{5}}E_{i\geq 1},\quad
Q(\psi_{j}^{6},\tilde{\phi}_{j}^{5})+Q(\tilde{\psi}_{j}^{5},\phi_{j}^{6})={\varepsilon_{j}^{5}|\log\varepsilon|^{8}\over
1+|y|^{3}}E_{i\geq 0}$

Define

$\tilde{\phi}_{j}^{6}=\sum_{i=1}^{6}\phi_{j}^{i},\quad\tilde{\psi}_{j}^{6}=\sum_{i=1}^{6}\psi_{j}^{i}.$

We get the new error

$\displaystyle E_{j}^{in}$
$\displaystyle[\tilde{\phi}_{j}^{6},\tilde{\psi}_{j}^{6},0,P](y,t):=[\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;P)-\nabla^{\perp}{\mathcal{R}}_{j}^{00}(y,t;\bar{P}^{2})]\cdot\nabla
U$ $\displaystyle+{\varepsilon^{3}|\log\varepsilon|^{8}\over
1+|y|^{5}}E_{i\geq 1}+{\varepsilon^{5}|\log\varepsilon|^{8}\over
1+|y|^{3}}\,E_{i\geq 0}.$ (5.27)

If we compare this error with the one in (5.15), we see that both their main
terms have size $\varepsilon^{3}$ multiplied by a power of
$|\log\varepsilon|$, and have Fourier mode-$1$. The difference is in their
decay in the $y$-variable: the error in (5.7) has a much faster decay, which
will be crucial to make the final argument of our construction work. We
explain this in Section 8 where the inner-outer scheme is described in detail.

At this point of our construction we choose $\zeta$ in (4.3). Take

$\zeta=3.$

Then

$W^{0}(x)+{\eta_{j1}\over
r_{j}\varepsilon_{j}^{2}}\tilde{\phi}_{j}^{6}({x-P_{j}\over\varepsilon_{j}},t)={1\over
r_{j}\varepsilon_{j}^{2}}U({x-P_{j}\over\varepsilon_{j}})(1+o(1)),\quad\mbox{as}\quad\varepsilon\to
0$

for $|x-P_{j}|<|\log\varepsilon|^{-3}$.

###  5.8. The outer improvement

So far we have modified the approximate solution in the inner regions, namely
at a small distance from the points $P_{j}$. We will now improve the outer
error given by

$\displaystyle E_{0}(x,t;P)$
$\displaystyle=E^{out}[0,0,\tilde{\phi}^{6},\tilde{\psi}^{6},P](x,t)$
$\displaystyle:=\sum_{j=1}^{k}\left[r\,|\log\varepsilon|\,{\partial}_{t}\bar{\eta}_{j1}+\nabla_{x}^{\perp}(r^{2}(\Psi^{0}+\sum_{j=1}^{k}{\eta_{j2}\over
r_{j}}\tilde{\psi}_{j}^{6}({x-P_{j}\over\varepsilon_{j}})-\alpha_{0}|\log\varepsilon|))\nabla\bar{\eta}_{1j}\right]{\tilde{\phi}_{j}^{6}\over\varepsilon_{j}^{2}r_{j}}$
$\displaystyle+\left[\sum_{j=1}^{k}(\eta_{2j}-\eta_{1j})\nabla_{x}^{\perp}(r^{2}{\tilde{\psi}_{j}^{6}\over
r_{j}})+{r^{2}\tilde{\psi}_{j}^{6}\over
r_{j}}\nabla_{x}^{\perp}\eta_{2j}\right]\nabla_{x}W^{0}$
$\displaystyle+(1-\sum_{j=1}^{k}\eta_{j1})S_{1}(W^{0},\Psi^{0})\quad(x,t)\in\Sigma\times[0,T)$

We also want to reduce the size of
$S_{2}[\tilde{\phi}^{6},\tilde{\psi}^{6},0,0,P]$. We refer to (3.19) and also
(4.7). Observe that the function $\Psi^{0}+\sum_{j=1}^{k}{\eta_{j2}\over
r_{j}}\tilde{\psi}_{j}^{6}({x-P_{j}\over\varepsilon_{j}})$ satisfies the
conditions on the boundary and at infinity, see (3.18): for all $t\in[0,T]$

${\partial\psi\over\partial
r}(x,t)=0,\quad{\mbox{on}}\quad{\partial}\Sigma,\quad|\psi(x,t)|\to