Construction of unstable concentrated solutions of the Euler and gSQG equations

Abstract

In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of point-vortices. Those solutions are themselves unstable, in the sense that their localization radius grows from order \eps to order \eps^\beta (with β<1\beta < 1) in a time of order |\ln\eps|. This proves
in particular that the logarithmic lower-bound obtained in previous papers (in particular [P. Buttà and C. Marchioro, \emph{Long time evolution of concentrated Euler flows with planar symmetry}, SIAM J. Math. Anal., 50(1):735–760, 2018]) about vorticity localization in Euler and gSQG equations is optimal. In addition we construct unstable solutions of the Euler equations in bounded domains concentrated around a single unstable stationary point. To achieve this we construct a domain whose Robin's function has a saddle point.

Full text

Construction of unstable concentrated solutions of the Euler and
gSQG equations
Martin Donati∗
Abstract
In this paper we construct solutions to the Euler and gSQG equations that are concentrated
near unstable stationary configurations of point-vortices. Those solutions are themselves unsta-
ble, in the sense that their localization radius grows from order εto order εβ(with β < 1) in a
time of order |ln ε|. This proves in particular that the logarithmic lower-bound obtained in previ-
ous papers (in particular [P. Buttà and C. Marchioro, Long time evolution of concentrated Euler
flows with planar symmetry, SIAM J. Math. Anal., 50(1):735–760, 2018]) about vorticity local-
ization in Euler and gSQG equations is optimal. In addition we construct unstable solutions of
the Euler equations in bounded domains concentrated around a single unstable stationary point.
To achieve this we construct a domain whose Robin’s function has a saddle point.
1 Introduction
We are interested in this paper in different active scalar equations from fluid dynamics: the two-
dimensional incompressible Euler equations, used to describe an inviscid and incompressible fluid;
and the Surface Quasi Geostrophic (SQG) equations, used as a model of geophysical flows. We also
consider the generalized Surface Quasi-Geostrophic equations (gSQG) that interpolates between the
Euler equations and the SQG equations. Let ω:R2→Rbe the active scalar, which we will refer
to as the vorticity as in the Euler equations, and u:R2→R2be the velocity of the fluid. Then the
Euler equations, the SQG equations and the gSQG equations in the plane can all be written in the
form 










∂tω(x, t) + u· ∇ω(x, t)=0,∀(x, t)∈R2×(0,+∞)
u(x, t) = ∇⊥∆−s(ω)(x, t),∀(x, t)∈R2×[0,+∞)
u(x, t)→0,as |x| → +∞,∀t∈[0,+∞)
ω(x, 0) = ω0(x),∀x∈R2,
with s∈[1
2,1]. The Euler equations correspond to the case s= 1, the SQG equations to the case
s=1
2, and the gSQG equations are the family s∈(1
2,1). For details on the geophysical model see
for instance [24,28]. Let us recall that the fundamental solution of ∆sin the plane is given for x=y
by
Gs(x, y) = (1
2πln |x−y|if s= 1,
Γ(1−s)
22sπΓ(s)
1
|x−y|2−2sif 1
2≤s < 1,
∗Univ. Grenoble Alpes, Institut Fourier, F-38000 Grenoble, France. Contact : martin.donati@univ-grenoble-
alpes.fr.
2020 Mathematics Subject Classification : 76B47, 34A34
Keywords : Point-vortex dynamics, Euler equations, SQG equations, Vorticity localization, Long time confine-
ment, Hyperbolic critical point
1
arXiv:2303.14657v3 [math.AP] 18 Apr 2024

where Γis the standard Gamma function. Therefore for every s∈(1
2,1] there exists a constant Cs
such that
∇xGs(x, y) = Cs
x−y
|x−y|4−2s.
This motivates us to define α:= 3 −2s, so that |∇xGα(x, y)| ≤ Cα|x−y|−α, for α∈[1,2) and the
appropriate choice of Cα. In conclusion, the equations that we consider are the family of equations











∂tω(x, t) + u(x, t)· ∇ω(x, t)=0,
u(x, t) = ZR2
Cα
(x−y)⊥
|x−y|α+1 ω(y, t)dy,
ω(x, 0) = ω0(x).
(1)
We recall that the Euler case corresponds to s=α= 1. We observe that we necessarily have that
∇ · u= 0 when it makes sense. When α≥2, the kernel cease to be L1
loc so the Biot-Savart law in
(1) does not make sense anymore. In that case the Biot-Savart law should be expressed differently,
see for instance [19]. In this paper we only consider α∈[1,2).
For the Euler equations, we have global existence and uniqueness of both strong solutions, and
weak solutions in L1∩L∞from the Yudovitch theorem [29]. When α∈(1,2), from [13], we know
the existence, but not uniqueness, of weak solutions in L1∩L∞of equations (1). The existence of
strong solutions is only known locally in time, and the blow-up of the SQG equations is an important
open problem.
In this paper we construct families of particular initial data ω0such that any (since it may not
be unique) solution of (1) satisfies various constraints. These functions ω0always lie in L1∩L∞,
but can also be taken C∞. Hence when α= 1, the solution remains C∞for all time, but in general,
we can only consider global in time L1∩L∞solutions of (1).
We now focus on the particular situation where the active scalar is concentrated into small blobs
as follows. We denote by D(z, r)the disk of radius rcentered in z. We consider solutions ωsatisfying
ω=
N
X
i=1
ωiand supp ωi⊂D(zi(t), r(t)),
with r(t)being small in some sense. A classical model to describe concentrated solutions of equa-
tions (1) is the point-vortex model. The principle is to approximate the blob ωiby the Dirac mass
aiδzi(t), where the intensity
ai=ZR2
ωi(x, t)dx,
is constant in time1. The dynamics of those Dirac masses, that we call point-vortices, is then given
by the system
∀i∈J1, N K,d
dtzi(t) = CαX
1≤j≤N
j=i
aj
(zi(t)−zj(t))⊥
|zi(t)−zj(t)|α+1 .(α-PVS)
This system of equations is often called α-point-vortex system, or α-model. We recall that this model
is mathematically justified, see for instance [22,27,26,2,6] for the Euler case and [13,25,15,5] for
1See [13] Corollary 2.7 for the case of weak solutions. For strong solutions this is a direct consequence of the fact
that ∇ · u= 0, see for instance [21].
2

the gSQG case. In particular, on a finite time interval [0, T ], if no collisions of point-vortices occurs,
then if
ω0,ε −→
ε→0
N
X
i=1
aiδzi(0)
weakly in the sense of measures, then
ωε(t)−→
ε→0
N
X
i=1
aiδzi(t),
where the t7→ zi(t)are the solutions of the point-vortex dynamics. This means that point-vortices
are a singular limit of solutions of the associated PDE. Fore a more detailed introduction of the
point-vortex system, we refer the reader to [21].
2 Vorticity confinement and main results
We now introduce the long time vorticity confinement problem, recall some important theorems on
the subject and state our main results.
2.1 Long time confinement problem
We define the long time confinement problem as the following – see [2]. Let N∈N∗and ε > 0. For
each i∈ {1, . . . , N}let z∗
i∈R2chosen pairwise distinct and ai∈R∗=R\ {0}. Assume that ω0is
such that 




















ω0=
N
X
i=1
ω0,i and supp ω0,i ⊂D(z∗
i, ε),
ω0,i has a sign and ZR2
ω0,i(x)dx=ai,
|ω0| ≤ Cε−ν,for some ν≥2,
|z∗
i(t)−z∗
j(t)|>0,∀t∈[0,+∞),
(2)
were we denote by z∗
i(t)the associated solution of the point-vortex dynamics with initial data
z∗
i(0) = z∗
iand intensities ai. The last hypothesis ensures that the point-vortex dynamics has a
global in time solution: no collision occurs. This is not a very restrictive hypothesis since it is
known that the point-vortex system has a global solution for almost any initial data, in the sense
of the Lebesgue measure. This was proved for the Euler point-vortex dynamics (namely equations
(α-PVS) for α= 1) in the torus [10], in bounded domains [7] and in the plane2[20]. For the general
α-model (α-PVS) it was proved in the plane3[4,14].
Let β < 1. We introduce the exit time:
τε,β = sup (t≥0such that ∀s∈[0, t],supp ω(·, s)⊂
N
[
i=1
D(z∗
i(s), εβ)).
The long time confinement problem consists in obtaining a lower-bound on τε,β in order to describe
how long the approximation of a concentrated solution of equations (1) by the point-vortex model
(α-PVS) remains valid. Results have been obtained in [2,8,5]. In the following, we recall some of
them and state our main results, starting with the case α= 1.
2With an additional hypothesis on the intensities: Pi∈Pai= 0, for any P⊂ {1,...,N}, with P=∅. This
hypothesis was then weakened in [14] to P={1,...,N}.
3With the same additional hypothesis.
3

2.2 Result for Euler equations in the plane
A first general result was obtained in [2].
Theorem 2.1 (Marchioro-Buttà, [2]).Let β < 1/2. Then there exists ξ0>0such that for every
ε > 0small enough, for any ω0satisfying (2)for some ν≥2, the solution ωof the Euler equations
satisfies
τε,β > ξ0|ln ε|.
In special cases, this can be improved. For instance, with the same hypotheses than those of
Theorem 2.1, but assuming furthermore that N= 1, one can easily obtain that τε,β ≥ε−ξ0, for some
ξ0>0. An extension of this result is the following.
Theorem 2.2 (Marchioro-Buttà, [2]).Let β < 1/2. Then there exists ξ0>0and a configuration
of point-vortices, namely a choice of aiand z∗
i, such that for any ε > 0small enough and for any
ω0satisfying (2), the solution ωof the Euler equations satisfies
τε,β > ε−ξ0.
The configuration is a self-similar expanding configuration of three point-vortices. The key point
is that as point-vortices move far from each other, their mutual influence decreases with time.
In this paper, we want to prove that the logarithmic bound obtained in Theorems 2.1 is optimal,
namely that there are solutions of (1) satisfying (2) such that τε,β ≤ξ1|ln ε|. We prove the following
result.
Theorem 2.3. There exists β0<1/2,ν≥2and a configuration (z∗
i)i,(ai)iof point-vortices with
N= 3 such that for every β∈(β0,1), for any ξ1>4π
3(1 −β)and for every ε > 0small enough
there exists ω0satisfying (2)such that the solution ωof the Euler equations in the plane satisfies
τε,β ≤ξ1|ln ε|.
This confirms that the logarithmic bound obtained in Theorems 2.1 is optimal.
2.3 Result for the gSQG equations in the plane
A result similar to Theorem 2.1 has been obtained for the gSQG equations.
Theorem 2.4 (Cavallaro-Garra-Marchioro, [5]).Let α∈(1,2) and βsuch that 0< β < 4−2α
5−α<1
2.
Then there exists ξ0>0such that for every ε > 0small enough, for any ω0satisfying (2)with ν= 2
and for any ωa weak solution of (1)we have that
τε,β > ξ0|ln ε|.
Remark 2.5. Please note that in the hypotheses of Theorem 2.4 is assumed that ν= 2, which is
not in the hypotheses of Theorem 2.1. Actually, we claim that in their proof of Theorem 2.4, the
authors of [5] only need to assume the existence of ν≥2, and not ν= 2. This is due to Lemma 2.4
of [5].
We then prove the following.
Theorem 2.6. Let α∈[1,2). Then there exists ν≥2, an initial configuration (z∗
i)i,(ai)iof point-
vortices with N= 3, and β0<4−2α
5−αsuch that for every β∈(β0,1), for every ξ1>1−β
Cα(2−2−α)√αand
for every ε > 0small enough there exists ω0satisfying (2)such that any solution ωof (1)satisfies
τε,β ≤ξ1|ln ε|.
4

This confirms that the logarithmic bound obtained in Theorems 2.4 is optimal.
Remark 2.7. Both in Theorems 2.3 and 2.6, the lower-bound for ξ1is not optimal. Moreover, ω0
is localized initially in a disk of size εbut the size of its support is of order εν/2. In Appendix C,
we give details how concentrated the initial data needs to be depending on the construction, and give
examples constructions involving more point-vortices, which improve the bounds for ξ1and ν.
2.4 Results for the Euler equations in bounded domains
In a second part of this paper, we turn to a new situation. We focus on the Euler equations, namely
the case α= 1, but in a bounded domain Ω. Let us recall the Euler equations in a bounded and
simply connected domain Ω⊂R2:













∂tω+u· ∇ω= 0,
u=∇⊥∆−1ω,
u·n= 0,on ∂Ω,
ω(x, 0) = ω0(x),on Ω.
(Eu)
When being far from the boundary, one can express the effect of the boundary as a Lipschitz exterior
field. This trick makes it very easy to extend Theorem 2.1 to the case of bounded domains – as it
is suggested in [2].
In that same article, the authors proved that when the initial vorticity is concentrated near the
center of a disk, namely that Ω = D(0,1),N= 1 and z1= 0, then we obtain the same power-
law lower-bound τε,β ≥ε−ξ0than with expanding self-similar configurations. This result has been
generalized to other bounded domains in [8]. This is due to a strong stability property induced
by the shape of the boundary. Here we are interested in the opposite situation: we construct a
domain whose boundary creates an instability. We then obtain a third and final result, different
from Theorems 2.3 and 2.6 because it only involves a single blob.
Theorem 2.8. There exists a smooth bounded domain Ωand β0<1/2such that for every β∈
(β0,1), for every ξ1>(1 −β)2π
√3and every ε > 0small enough, there exists ω0satisfying (2)with
N= 1 and ν= 4 such that the solution ωof (Eu)satisfies
τε,β ≤ξ1|ln ε|.
We prove Theorems 2.3,2.6 and 2.8 using the same plan: we construct a solution of the point-
vortex dynamics that move away from its initial position exponentially fast, then construct a solution
concentrated around it in the sense of hypothesis (2).
The paper is organized as follows. In Section 3we give several definitions and expose in details
the plan of the proofs and the main tools. In Section 4we do the explicit construction to prove
Theorems 2.3 and 2.6. Finally in Section 5we prove Theorem 2.8.
3 Outline of the proofs
In this section we expose the main tools required for the proofs of our results. Before going any
further, let us introduce some notation.
In the rest of the paper,
•|z|, for z∈R2designates the usual 2-norm, or modulus,
5

•z⊥= (−z2, z1),
•|Z|∞for Z= (z1, . . . , zN)∈(R2)Ndesignates max1≤i≤N|zi|,
•Cis a name reserved for constants whose value is not relevant, and may change from line to
line,
•D(z, r)is the disk (in R2) or radius rcentered in z.
Please notice that our construction – in particular ω0– depends on ε, though we do not write the
dependence of each quantity in εfor the sake of legibility.
3.1 Plan
The proofs of Theorems 2.3,2.6 and 2.8 rely on two main steps. We first look for an unstable
stationary configuration of point-vortices. Then we control the behaviour of a well prepared solution
initially concentrated around this configuration.
Let us give some details on each step.
Step 1: constructing an unstable vortex configuration.
We consider the dynamical system d
dtZ(t) = f(Z(t)). Then we say that Z∗is a stationary point
of the dynamics if f(Z∗)=0, and that it is unstable if Df(Z∗)has an eigenvalue with positive real
part.
At this step, we first aim to choose N,Ω(when necessary), the family of aiand z∗
i. The trick
is following: we choose intensities ai∈R∗and a point Z∗= (z∗
1, . . . , z∗
N)∈(R2)Nwhich is a
stationary and unstable initial datum of the point-vortex dynamics (α-PVS). Let us notice that
choosing a stationary configuration Z∗ensures that the hypothesis |z∗
i(t)−z∗
j(t)|>0for all t≥0
is always satisfied. The consequence of the instability is that for every ε > 0, there exists an initial
configuration of point-vortices Z0such that
|Z∗−Z0|∞=ε
2,
and the solution t7→ Z(t)such that Z(0) = Z0move away exponentially fast from Z∗, therefore
implying that for any β < 1,
τZ:= sup nt≥0such that ∀s∈[0, t],|Z(s)−Z∗|∞≤2εβo≤ξ0|ln ε|.
This problem is much simpler than the original one since we are investigating the behaviour of
solutions of a system of ordinary differential equations, the point-vortex dynamics, instead of a
solution of a partial derivative equation. More precisely, we have the following proposition, obtained
as a corollary of Theorem 6.1, Chapter 9 of [17], and proved in Section 3.3.
Proposition 3.1. Let f: (R2)N→(R2)N. We consider the differential equation
d
dtZ(t) = f(Z(t)).(3)
Assume that there exists Z∗∈(R2)Nsuch that f(Z∗)=0. Assume furthermore that Df(Z∗)has an
eigenvalue with positive real part λ0>0.
Then for any λ<λ0, for every ε > 0small enough and for any β∈(0,1) there exists and a
choice of Z0such that |Z0−Z∗|∞=ε/2,
τZ≤1−β
λ|ln ε|.
6

In conclusion, proving that τZ≤ξ1|ln ε|simply relies on finding an eigenvalue with positive real
part of the Jacobian matrix of the dynamic’s functional.
Step 2: constructing the approximation
The idea is then to prove that a solution ωwith well prepared initial data ω0satisfying (2)
satisfies that for every t≤τε,β,
|B(t)−Z(t)|∞=o(εβ),
where
Bi(t) = 1
aiZR2
xωi(x, t)dx,
and B(t) = (B1(t), . . . , BN(t)). The conclusion then comes from the fact that by construction
(Step 1), there exists t1≤1−β
λsuch that |Z(t1)−Z∗|∞= 2εβand thus for εsmall enough,
τε,β ≤t1≤ξ1|ln ε|.
In order to obtain this control on |B(t)−Z(t)|∞, we need to estimate the moment of inertia
Ii(t) = 1
aiZR2|x−Bi(t)|2ωi(x, t)dx
of each blob. The constant νin (2) intervenes when estimating I(0). The critical part in this step
is the competition between the growth of Ii, which loosen the control on |B(t)−Z(t)|∞, with the
growth of |Z(t)−Z∗|∞. When we do not have constraints on ν, then one can always choose ν
large enough so that Iiremains small long enough. However the difficulty arises when wanting to
construct ω0with ν= 4. This requires to be able to estimate precisely the growth of each Ii, of
|Z(t)−Z∗|∞and of |B(t)−Z(t)|∞.
All of this is captured in Theorem 3.2 presented in Section 3.2.
3.2 Confinement around an unstable configuration
Once the unstable configuration of point-vortices is obtained in Step 1, most of the work needed in
the second step does not depend on that configuration nor on the specific framework. Therefore, we
establish a general theorem that we will be able to apply for any suitable configuration of vortices,
also including when appropriate the presence of a boundary.
To understand better the dynamics of each blob, we describe the influence of the other blobs or
of the boundary by an an exterior field. We assume that each blob is a solution of a problem











∂tωi(x, t) + ui(x, t) + Fi(x, t)· ∇ωi(x, t)=0,
ui(x, t) = ZR2
Cα
(x−y)⊥
|x−y|α+1 ωi(y, t)dy,
ωi(x, 0) = ωi,0(x),
(4)
where Fiis an exterior field that satisfies ∇ · Fi= 0. Let Z∗∈(R2)Nand f∈C1(R2)N,(R2)N
such that f(Z∗) = 0. We write f= (f1, . . . , fN). For any Z0, let t7→ Z(t)be the solution of the
problem 


d
dtZ(t) = f(Z(t))
Z(0) = Z0.
(5)
7

In this particular setting, for any β∈(0,1), we have
τε,β = sup (t≥0such that ∀s∈[0, t],supp ω(·, s)⊂
N
[
i=1
D(z∗
i, εβ)).
Assuming that |Z0−Z∗|∞=ε/2, we recall that
τZ:= sup nt≥0such that ∀s∈[0, t],|Z(s)−Z∗|∞<2εβo.
We then have the following theorem.
Theorem 3.2. Let N∈N∗,ai∈R∗for every i∈ {1, . . . , N},Z∗∈(R2)N. Let f∈C1(R2)N,(R2)N
and Fisuch that ∇ · Fi= 0. We assume the following.
(i)f(Z∗)=0and Df(Z∗)has an eigenvalue with positive real part λ0,
(ii)There exists Csuch that for all i∈ {1, . . . , N },∀t≤τε,β,
|Fi(Bi(t), t)−fi(B(t))| ≤ C
N
X
j=1 pIj,
(iii)There exists constants κ0,κ1and κ2such that ∀i∈ {1, . . . , N },∀x, x′∈D(z∗
i, εβ),∀t≤τε,β,
|Fi(x, t)−Fi(x′, t)| ≤ κ0|x−x′|,(6)
and (x−x′)·Fi(x, t)−Fi(x′, t)≤κ1|x−x′|2,(7)
and ∀X, X ′∈(R2)Nsuch that |X−Z∗|∞≤2εβand |X′−Z∗|∞≤2εβ,
|f(X)−f(X′)|∞≤κ2|X−X′|∞.(8)
Then there exists ν≥2and β0<4−2α
5−αsuch that for all β∈(β0,1), for every ξ > 1−β
λ0and for every
ε > 0small enough, there exists ω0satisfying (2)such that any ω=PN
i=1 ωisolution of the problem
(4)for every isatisfies
τε,β ≤ξ|ln ε|.
Proof. First, we use Hypothesis (i)to apply Proposition 3.1 and get that for every εsmall enough,
there exists Z0such that |Z0−Z∗|∞=ε/2and for every β∈(0,1), the solution Zof the problem (5)
satisfies
τZ≤1−β
λ|ln ε|.(9)
Now let ω0satisfying (2) for some ν≥2and such that
B(0) = Z0,(10)
and
∀i∈ {1, . . . , N }, Ii(0) ≤εν.(11)
This is always possible as stated in Remark B.1 given in Appendix B. Let ω=PN
i=1 ωisuch that
each ωiis a solution of the problem (4).
8

We observe that if τε,β ≤τZ, then we have the desired result. So for the sake of contradiction,
we can assume that τε,β > τZ.
Recalling that ωisolves (4), we have that
d
dtBi=1
aiZFi(x, t)ωi(x, t)dx
and d
dtIi=2
aiZ(x−Bi(t)) ·(Fi(x, t)−Fi(Bi(t), t))ωi(x, t)dx.
Indeed, if ωiis smooth (when α= 1 or before a possible regularity blow-up if α > 1), these are
classical computations. In general, ωi∈L1∩L∞and these relations hold in the weak sense, see for
instance [13, Corollary 2.8] or [5].
Now using Hypothesis (iii), and observing from (2) that ωi
ai≥0, we get that

d
dtIi≤2κ1Z|x−Bi(t)|2ωi(x, t)
ai
dx= 2κ1Ii(t).
Therefore, we get that
Ii(t)≤Ii(0)e2κ1t.(12)
We now want to estimate |B(t)−Z(t)|∞. For every i∈ {1, .. . , N }we have that

d
dtBi(t)−fi(B(t))=
1
aiZFi(x, t)−fi(B(t))ωi(x, t)dx
=
1
aiZFi(x, t)−Fi(Bi(t), t) + Fi(Bi(t), t)−fi(B(t))ωi(x, t)dx
≤κ0Z|x−Bi(t)|ωi(x, t)
ai
dx+C
N
X
j=1 qIj(t),
where we used hypotheses (ii)and (iii). By the Cauchy Schwartz inequality we have that
Z|x−B(t)|ωi(x, t)
ai
dx=Z|x−B(t)|2ωi(x, t)
ai
dx1/2Zωi(x, t)
ai
dx1/2
≤pIi(t),
and therefore, since the result is now uniform in i,

d
dtB(t)−f(B(t))∞≤C
N
X
j=1 qIj(t).
The value of Cis irrelevant and changes from line to line. The value of κ0is also irrelevant in the
end and is absorbed in C.
We recall that by relation (10), B(0) = Z0=Z(0) and that Zis a solution of the problem (5)
with fbeing a Lipschitz map by Hypothesis (iii). We now use a variant of the Gronwall’s inequality
– Lemma B.2 given in appendix – to obtain that
B(t)−Z(t)∞≤Ceκ2tZt
0
N
X
j=1 qIj(s)ds.
9

Using relation (12), we have that
N
X
j=1 Zt
0qIj(s)ds≤
N
X
j=1 Zt
0qIj(0)eκ1sds≤
N
X
j=1 pIj(0)
κ1
eκ1t.
Recalling that by relation (11), Ij(0) ≤εν, we obtain that
B(t)−Z(t)∞≤Cεν /2−(κ1+κ2)t|ln ε|.
We are interested in proving that ∀t≤τZ,
B(t)−Z(t)∞=o(εβ),(13)
as ε→0. For ε < 1and for all t≤τZ, by relation (9), for any λ<λ0,
−t|ln ε| ≥ 1−β
λ.
Therefore, one sufficient condition to obtain (13) is
ν > 2κ1+κ2
λ0
(1 −β)+2β, (14)
since in that case, one can choose λclose enough to λ0such that
ν/2−κ1+κ2
λ(1 −β)> β.
We observe that necessarily, κ2≥λ0so κ1+κ2
λ0>1and therefore the map ϕ:β7→ 2κ1+κ2
λ0(1−β) +2β
is decreasing on [0,1]. Let
ν > ϕ 4−2α
5−α=2
5−α(1 + α)κ1+κ2
λ0
+ 4 −2α.(15)
Since ϕis also continuous, then there exists β0<4−2α
5−αsuch that ∀β∈(β0,1),
ν > ϕ(β).
In particular, for every β∈(β0,1), relation (14) holds true and thus relation (13) too.
We conclude by observing that for εsmall enough,
|B(τZ)−Z∗|∞≥ |Z(τZ)−Z∗|∞− |B(τZ)−Z(τZ)|∞≥2εβ−o(εβ)≥εβ.
Since ∀t < τε,β, we have that |B(t)−Z∗|∞< εβby definition of τε,β , the previous relation is in
contradiction with τε,β > τZ. Therefore,
τε,β ≤τZ≤1−β
λ|ln ε|,
for any λ < λ0. In particular, for every β∈(β0,1), for every ξ1>1−β
λ0, we proved that for εsmall
enough,
τε,β ≤ξ1|ln ε|.
10

Let us mention that Hypothesis (i)is a consequence of the choice of Z∗and the aimade at Step
1, and that Hypothesis (ii)is a consequence of the nature of the relation between the exterior fields
Fiand the map f. Of course, the result could not be true if the Fiand fwere not related to each
other.
The existence of κ0,κ1and κ2in the Hypothesis (iii)is a direct consequence of the fact that
every Fiand fare Lipschitz maps, which will always be the case in our framework. Even though
relation (7) is a consequence of (6), only κ1and κ2intervene in the condition (15) on ν. This is
important when one wants to compute precisely the bounds on ν, as we do in Section 5, and for
several other examples in Appendix C.
Finally, we notice that we actually proved that any solution ωstarting from any initial data ω0
satisfying (2) and relations (10) and (11) satisfies τε,β ≤ξ1|ln ε|. Therefore, there is a large family
of initial data, in terms for instance of the shape of the blob, that solves our problem.
3.3 Proof of Proposition 3.1.
Let us recall Theorem 6.1, Chapter 9 of [17].
Theorem 3.3. Let f: (R2)N→(R2)N. We consider the differential equation
d
dtZ(t) = f(Z(t)).
Assume that there exists Z∗∈(R2)Nis such that f(Z∗)=0. Assume furthermore that Df(Z∗)has
an eigenvalue with positive real part λ0>0. Then there exists a solution of (3)such that Z(t)exists
some fixed neighbourhood of Z∗, that
Z(t)−→
t→−∞ Z∗,
and that 1
tln |Z(t)−Z∗|∞−→
t→−∞ λ0.
We now prove Proposition 3.1. Let β∈(0,1) and let e
Za solution of (3) given by Theorem 3.3.
Since e
Z−→
t→−∞ Z∗and since e
Zexits some fixed neighbourhood of Z∗, for εsmall enough, there exists
t0and t1such that 










−∞ < t0< t1
t1→ −∞ as ε→0
|e
Z(t1)−Z∗|∞= 2εβ
|e
Z(t0)−Z∗|∞=ε/2.
Let Z(t) = e
Z(t+t0), we have that |Z(0) −Z∗|∞=ε/2and that τZ≤t1−t0since Z(t1−t0)=2εβ.
Moreover, since
ln |Z(t)−Z∗|∞=λ0t+ot→−∞(t),
then for any η∈(0,1), for −tbig enough we have that
1−η < ln |Z(t)−Z∗|∞
λ0t<1 + η
Therefore, for εsmall enough, applying in t0and t1(we recall that t1→ −∞ as ε→0) we have
that
t1<ln |Z(t1)−Z∗|∞
λ0(1 + η)=βln ε+ ln 2
λ0(1 + η)=−β|ln ε|+ ln 2
λ0(1 + η)
11

and
−t0<−ln |Z(t0)−Z∗|∞
λ0(1 −η)=|ln ε|+ ln 2
λ0(1 −η)
and thus
t1−t0<|ln ε|1
λ0(1 −η)−β
λ0(1 + η)+ln 2
λ0|ln ε|1
1 + η+1
1−η.
Therefore, by letting η→0, for any λ<λ0, for εsmall enough,
t1−t0≤1−β
λ|ln ε|.
By definition, τZ≤t1−t0≤1−β
λ|ln ε|. This concludes the proof.
4 Unstable configurations of multiple vortices in the plane
In this section we show that we can choose a configuration Z∗and intensities aisuch that we can
apply Theorem 3.2 to estimate the exit time τε,β of a solution of the equations (1). Please notice
that Theorem 2.3 is a direct consequence of Theorem 2.6 by taking α= 1. Therefore, we work with
some general α∈[1,2).
We start by introducing a family of point-vortex configurations Z∗for any N≥3. We then con-
struct the exterior fields Fisuch that the blobs ωiis a solution of the problem (4), and start proving
each of the conditions to apply Theorem 3.2. We then prove Theorem 2.3, and Proposition C.1 by
computing explicitly the properties of our construction with N= 3. In Appendix C, we use the
construction with N= 7 and N= 9.
4.1 Vortex crystals
Let us introduce a family of stationary solutions of the α-point-vortex model (α-PVS) that are part
of the so called vortex crystals family. For a more general study of vortex crystals and their stability,
we refer the reader to [1,23,3]. Fore the sake of legibility, we identify C=R2for the position of
point-vortices. We use the notation x=(x)p,(x)q= (x)p+i(x)q.
Let N≥3, and
Kα(x, y) = Cα
(x−y)⊥
|x−y|α+1 ,
so that the point-vortex equation (α-PVS) becomes
∀i∈J1, N K,d
dtzi(t) = X
1≤j≤N
j=i
ajKαzi(t), zj(t).
In particular, by setting for all Z= (z1, . . . , zN):
f(Z) = 
X
j=i
ajKα(zi, zj)
1≤i≤N
,
then we have that d
dtZ(t) = f(Z(t)).
12

We consider Npoint-vortices in the following configuration. The first N−1points form a regular
(N−1)-polygon, and the N-th vortex is placed at the center. For instance, by letting ζ=ei2π
N−1,
where here idenotes the complex unit, we set
∀j∈ {1, . . . , N −1}, z∗
j=ζj, aj= 1
z∗
N= 0, aN∈R.
Then, (see for instance [1]) the solution of (α-PVS) with initial configuration Z∗= (z∗
1, . . . , z∗
N)
satisfies zj(t) = eiµtz∗
jfor some angular velocity νthat does not depend on j. The motion of the
whole configuration is a rigid rotation around 0, which makes it a so called vortex crystal.
This stands for any choice of aN∈R. Now we make a particular choice. For every N≥3, there
exists aN= 0 in the previous configuration such that the solution is stationary (µ= 0). Indeed, let
us compute the velocity of any point vortex (except the one at the center that is always stationary),
for instance zN−1=ζN−1= 1.
d
dtzN−1(0) =
N−2
X
j=1
Kα(1, ζj) + aNKα(1,0)
=Cα

N−2
X
j=1
(1 −ζj)
|1−ζj|α+1 +aN

⊥
.
Since ζj=ζN−1−j, the quantity PN−2
j=1
(1−ζj)
|1−ζj|α+1 is a non vanishing real number and thus letting
aN=−
N−2
X
j=1
(1 −ζj)
|1−ζj|α+1
enforces that d
dtzN−1(0) = 0. By symmetry, d
dtzj(0) = 0 for every j∈1, . . . , N −1. As for zN, it is
always stationary, again by a symmetry argument, or by a simple computation.
In conclusion, for any N≥3, we constructed a N-vortex configuration that is stationary. In
order to study the stability of the equilibrium Z∗, we compute Df(Z∗). To this end, let us compute
1
CαKα(zi+x, zj+y)−Kα(zi, zj)
=(zi+x)−(zj+y)⊥
(zi+x)−(zj+y)α+1 −(zi−zj)⊥
|zi−zj|α+1
=(zi−zj)⊥
(zi+x)−(zj+y)α+1 −(zi−zj)⊥
|zi−zj|α+1 +(x−y)⊥
(zi+x)−(zj+y)α+1
=(zi−zj)⊥
|zi−zj|α+1 1−(α+ 1)(x−y)·zi−zj
|zi−zj|2+o|x|+|y|−1+(x−y)⊥
|zi−zj|α+1 +o|x|+|y|,
and finally, we obtain that
1
CαKα(zi+x, zj+y)−Kα(zi, zj)
=−(zi−zj)⊥
|zi−zj|α+3 (α+ 1)(x−y)·(zi−zj) + (x−y)⊥
|zi−zj|α+1 +o|x|+|y|.(16)
13

Since each coordinate ziof fis of dimension two, we need some clarification on the notations. We
now think of fas the map
e
f:(R2N→R2N
p1, q1, . . . , pN, qN7→ e
fp1,e
fq1,..., e
fpN,e
fqN=fp1+iq1, . . . , pN+iqN.
Relation (16) yields for i=jthat

























1
ajCα
∂pje
fpi=−(zi−zj)q
|zi−zj|α+3 (α+ 1)(zi−zj)p
1
ajCα
∂pje
fqi=(zi−zj)p
|zi−zj|α+3 (α+ 1)(zi−zj)p−1
|zi−zj|α+1
1
ajCα
∂qje
fpi=−(zi−zj)q
|zi−zj|α+3 (α+ 1)(zi−zj)q+1
|zi−zj|α+1
1
ajCα
∂qje
fqi=(zi−zj)p
|zi−zj|α+3 (α+ 1)(zi−zj)q,
and for i=j,


























∂pie
fpi=−X
j=i
∂pje
fpi
∂pie
fqi=−X
j=i
∂pje
fqi
∂qie
fpi=−X
j=i
∂qje
fpi
∂qie
fqi=−X
j=i
∂qje
fqi.
In order to keep the notations as light as possible, we will write Dfin place of De
f.
4.2 Defining the exterior fields
We recall that in order to apply Theorem 3.2, we need to show that every blob ωiis a solution of a
problem (4) with some exterior field Fi.
Let ω0satisfying the general hypotheses (2) for N≥3and Z∗the stationary vortex crystal
presented in Section 4.1. Let ωbe a solution of (1). We observe that each blob ωiis solution of (1)
by letting
F(x, t) = Fi(x, t) = ZX
j=i
Kα(x, y)ωj(y, t)dy.
Since ∇x·Kα(x, y) = 0, then ∇ · F= 0. Moreover, we have the following lemma, that proves that
Hypothesis (ii)of Theorem 3.2 is satisfied.
Lemma 4.1. Let i∈ {1, . . . , N }. We have for all t≤τε,β that
Fi(Bi(t), t)−fi(B(t))≤C
N
X
j=1 pIj,
where Cdepends only on α, the aiand Z∗.
14

Proof.
|Fi(Bi(t), t)−fi(B(t))|=X
j=iZKα(Bi(t), y)ωj(y, t)dy−X
j=i
ajKα(Bi(t), Bj(t))
=X
j=iZKα(Bi(t), y)−Kα(Bi(t), Bj(t))ωj(y, t)dy
≤X
j=iZKα(Bi(t), y)−Kα(Bi(t), Bj(t))|aj|ωj(y, t)
aj
dy
≤CX
j=i|aj|Zy−Bj(t)ωj(y, t)
aj
dy
≤CX
j=iqIj(t),
where we used that Kαand its derivatives are smooth on R2×R2\ {x=y}, and go to 0 when
|x−y| → +∞, so in particular Kαis a Lipschitz map on Si=jD(z∗
i, εβ)×D(z∗
j, εβ).
The existence of κ0,κ1and κ2is trivial since Fand fare Lipschitz maps while the blobs and
point-vortices remain far from each other, which is always the case when t≤τε,β and t≤τZ, so
Hypothesis (iii)is always satisfied.
Therefore, in order to apply Theorem 3.2, we come down to prove that there is indeed an
eigenvalue of Df(Z∗)with positive real part. Then the only remaining difficult task is to estimate
the constants κ1,κ2and λ0to obtain an information on the lowest possible choice of ν.
4.3 Proof of Theorems 2.3 and 2.6
We now prove Proposition C.1, which in turns proves Theorem 2.3. We construct explicitly the
vortex crystal as described in Section 4.1 with N= 3, namely the configuration:





z∗
1= (−1,0), a1= 1
z∗
2= (1,0), a2= 1
z∗
3= (0,0), a3=−1
2α.
We now compute Df(Z∗)using the method previously described at the end of Section 4.1 to
obtain the 6×6matrix:
Df(Z∗) = Cα








0 2−(α+1) 0 2−(α+1) 0−2−α
2−(α+1)α0 2−(α+1)α0−2−αα0
0 2−(α+1) 0 2−(α+1) 0−2−α
2−(α+1)α0 2−(α+1)α0−2−αα0
0 1 0 1 0 −2
α0α0−2α0








The eigenvalues of this matrix are 0, with multiplicity 4, and ±Cα(2 −2−α)√α. So by letting
λ0=Cα(2 −2−α)√αwe have here a positive eigenvalue, associated with the eigenvector
vλ0=−1,√α, −1,√α, −2α+1,√α2α+1 .
15

This conclude Step 1.
We now apply Theorem (3.2) and obtain the existence for any α∈[1,2) of ν≥2and β0<4−2α
5−α
such that for every β∈(β0,1), every
ξ1>1−β
λ0
=1−β
Cα(2 −2−α)√α
and every ε > 0small enough, there exists ω0satisfying (2) such that any solution ωof (1) satisfies
τε,β ≤ξ1|ln ε|.
We proved Theorem 2.6, and by taking α= 1 we proved Theorem 2.3 as well.
5 Unstable point-vortex in a bounded domain
In this section we construct a bounded domain Ωand an initial data ω0satisfying (2) with N= 1,
such that the solution of (Eu) satisfies that
τε,β ≤ξ1|ln ε|,
for εsmall enough and for some ξ1. Since N= 1, we denote by z∗=z∗
1=Z∗and a=a1.
We start by recalling some facts about the point-vortex dynamics in bounded domain, then we
construct the domain. Finally, we prove Theorem 2.8 using again Theorem 3.2.
5.1 Euler equations and point-vortices in bounded domains
For the rest of the paper, we consider the Euler equations (Eu), which differs from before in the
sense that now α=s= 1, and ω: Ω →R, where Ωis a bounded simply connected subset of R2.
We now recall that the problem (∆Ψ = ωon Ω
Ψ=0 on ∂Ω
has a unique solution
Ψ(x) = ZΩ
GΩ(x, y)ω(y)dy,
where GΩ: Ω ×Ω\ {x=y} → Ris the Green’s function of −∆with Dirichlet condition in the
domain Ω. Therefore, we have the Biot-Savart law:
u(x, t) = ZΩ∇⊥
xGΩ(x, y)ω(y, t)dy.
An important property of the Green’s function GΩis that it decomposes as
GΩ=GR2+γΩ,
where
GR2(x, y) = 1
2πln |x−y|,
16

and γΩ: Ω ×Ω→R+harmonic in both variables. We denote by eγΩ:x7→ γΩ(x, x)the Robin’s
function of the domain Ω. This map plays a crucial role in the study of the point-vortex dynamics
in bounded domain. Indeed, the point-vortices in move according to the system
∀i∈J1, N K,d
dtzi(t) = X
1≤j≤N
j=i
ajzi(t)−zj(t)⊥
zi(t)−zj(t)2+
N
X
j=1
aj∇⊥
xγΩzi(t), zj(t),
which in the case N= 1 reduces to
d
dtz(t) = a
2∇⊥eγΩ(z(t)).
Our plan to prove Theorem 2.8 is the same as the proof of Theorem 2.3. The aim is to apply
Proposition3.1 and Theorem 3.2. We then straight away notice that when ω0satisfy (2) with N= 1,
and thus ω(which we can extend by 0 to R2) is constituted of a unique blob that solves (4) by setting
F(x, t) = Z∇⊥
xγΩ(x, y)ω(y, t)dy,
and zis a solution of (5) by setting
f(z) = a
2∇⊥eγΩ(z(t)).
Therefore, we are looking for z∗an unstable critical point of the Robin’s function eγΩ, namely such
that ∇eγΩ(z∗) = 0 and D∇⊥eγΩ(z∗)has a positive eigenvalue. However the Robin’s function is not
known explicitly in general, and the existence of such a point depends on the domain Ω. Fortunately,
we recall that for any simply connected domain Ωand for any z∗∈Ω, there exists a biholomorphic
map T: Ω →D:= D(0,1) such that T(z0) = 0. Such a map also satisfy that
∀x=y∈Ω, GΩ(x, y) = GDT(x), T (y),
and thus
eγΩ(x) = eγD(x) + 1
2πln |T′(x)|.
5.2 Known results on critical points of the Robin’s function
In this section we refer to [8] and recall the following results. For more details on the Robin’s
function we refer the reader to [16,12].
Proposition 5.1 ([8], Proposition 2.4).If T: Ω →Dis a biholomorphic map such that T(z∗)=0,
then
∇eγΩ(z∗) = 0 ⇐⇒ T′′(z∗)=0.
In particular, if the domain Ωhas two axes of symmetry, then the intersection z∗necessarily
satisfy ∇eγΩ(z∗) = 0 and thus T′′(z∗) = 0. For the time being, we assume the existence of such z∗
and state some of its properties.
Lemma 5.2 ([8], Lemma 4.1).Let T: Ω →D(0,1) and z∗∈Ωsuch that T(z∗) = 0 and T′′(z∗)=0,
then for every ε > 0small enough,
∀x, y, z ∈D(z∗, εβ),∇⊥
xγΩ(x, y)− ∇⊥
xγΩ(z, y)=|x−z||T′′′(z∗)|
6π|T′(z∗)|+Oεβ.
17

The direct corollary of this lemma is that Fsatisfies (7) with
κ1=|a||T′′′(z∗)|
6π|T′(z∗)|+o(1).
Proposition 5.3 ([8], Sections 3.2 and 3.3).Let Ω⊂R2be a bounded simply connected domain.
Then for any biholomorphism T: Ω →Dmapping z∗to 0such that T′′(z∗)=0, the hessian matrix
D2eγΩ(z∗)has non degenerate eigenvalues of opposite signs if and only if |T′′′(z∗)|>2|T′(z∗)|3. In
that case, these eigenvalues are
λ±=2|T′(z∗)|2± |T′′′(z∗)|/|T′(z∗)|
2π,
and the eigenvalues of D∇⊥eγΩ(z∗)are ±p−λ+λ−, so that
λ0=|a|
4π|T′(z∗)|p|T′′′(z∗)|2−4|T′(z∗)|6
is a positive eigenvalue of Df(z∗).
From this we deduce two things. First, the condition |T′′′(z∗)|>2|T′(z∗)|3is a criteria to estab-
lish that Df(z∗)has an eigenvalue with positive real part. Second, since D2eγΩis a real symmetric
matrix, and since D∇⊥(eγΩ) = Rπ/2D2eγΩwith Rπ/2=0−1
1 0 , then
|||Df(z∗)||| =|a|
2D2eγΩ=|a|
2λ+,
so that fsatisfies (8) with
κ2=|a|2|T′(z∗)|2+|T′′′(z∗)|/|T′(z∗)|
4π.
Therefore, in view of relation (15), we will be able in our construction to choose any νsuch that
ν >
5
3|T′′′(z∗)|+ 2|T′(z∗)|3
p|T′′′(z∗)|2−4|T′(z∗)|6+ 1.
which satisfies in particular that
|T′′′(z∗)|
|T′(z∗)|3>15 + 9√65
28 =⇒
5
3|T′′′(z∗)|+ 2|T′(z∗)|3
p|T′′′(z∗)|2−4|T′(z∗)|6+ 1 <4.
In conclusion of this section, in order to prove Theorem 2.8, we need in particular to construct a
domain Ωsatisfying the existence of a point z∗and a biholomorphic map T: Ω →Dsuch that
T(z∗) = T′′(z∗)=0 and |T′′′(z∗)|
|T′(z∗)|3>2
for the construction to be possible with some ν≥2, and that
|T′′′(z∗)|
|T′(z∗)|3>15 + 9√65
28 := c0≈3.12,
for the construction to be possible with ν= 4.
18

Figure 1: The domain Ω3
4.
5.3 Construction of the domain
Let δ∈[1
2,1). Let ζ=eiπ
3. Using the Schwartz Christoffel formula (see for instance [9]), we define
the conformal map Sδ:D→Ωδ:= Sδ(D)mapping 0to 0such that
S′
δ(z) := 1
(z−1)(1−2δ)(z−ζ)δ(z−ζ2)δ(z+ 1)(1−2δ)(z−ζ4)δ(z−ζ5)δ
=(z2−1)3δ−1
(z6−1)δ.
Let Tδ=S−1
δ. We compute that |T′
δ(0) = |S′
δ(0)|= 1,T′′
δ(0) = S′′
δ(0) = 0 and
|T′′′
δ(0)|=|S′′′
δ(0)|= 6δ−2.
Therefore, we have first of all that
|T′′′
δ(0)|>2|T′
δ(0)|3⇐⇒ δ > 2
3.
So the domain Ωδsatisfies that Df(0) has an eigenvalue with positive real part and its Robin’s
function has a saddle point. In Figure 1is plotted the domain Ω3
4. We can then do our construction
of ω0in Ω3
4, but it’s not enough to it with ν= 4.
However, we have that
|T′′′
δ(0)|
|T′
δ(0)|3= 6δ−2.
In particular we observe that for δ∈c0+2
6,1,ν= 4 satisfies (15). Since 0.9>c0+2
6, the domain
Ω0.9is a suitable domain to do the construction of ω0with ν= 4. More details and illustrations
about the family of domains (Ωδ)δare given in Appendix A.
To obtain a smooth domain e
Ωdwith the same properties, one takes an increasing sequence Ωn
of smooth domains which are symmetric with respect to both axes and converge towards Ω9/10.
By symmetry, 0 is necessarily a critical point of the Robin’s function of every domain. Then, we
introduce Tnthe sequence of conformal maps mapping Ωnto D(0,1) satisfying Tn(0) = 0 and
T′
n(0) ∈R+. The construction can be done so that Tn→T9/10 locally in every Ck,k≥0, so that
|T′′′
n(0)|
|T′
n(0)|3−→
n→+∞
|T′′′
9/10(0)|
|T′
n(0)|3> c0,
19

so there exists n0∈N, such that the smooth domain Ωn0is such that
4>
5
3|T′′′
n0(0)|+ 2|T′
n0(0)|3
p|T′′′
n0(0)|2−4|T′
n0(0)|6+ 1.
5.4 Proof of Theorem 2.8
Let δ∈c0+2
6,1, let Ω := Ωδ(or Ωn0as described in the previous section to work with a smooth
domain), z∗= 0 and a= 1.
From Sections 5.2 and 5.3,Df(0) has an eigenvalue λ0=1
4πp12δ(3δ−2) >0,so Hypothesis (i)
of Theorem 3.2 is satisfied. We recall that Hypothesis (iii)is satisfied since fand Fare Lipschitz
maps far from the boundary.
Therefore, there only remains to prove that Hypothesis (ii)is satisfied to apply Theorem 3.2
and conclude the proof of Theorem 2.8.
Let us compute.
F(B(t), t)−f(B(t))=Z∇⊥
xγΩ(B(t), y)ω(y, t)dy−a
2eγΩ(B(t))
=Z∇⊥
xγΩ(B(t), y)−(∇⊥
xγΩ(B(t), B(t))ω(y, t)dy
≤C|a|Z|y−B(t)|ω(y, t)
ady
≤CpI(t),
where on the last line the constant Cdepends only on Ω. Letting δ→1, we observe that λ0→√3
2π.
Now applying Theorem (3.2), we have that for any β0<1/2, one can construct a suitable ω0
satisfying (2) with N= 1 and ν= 4. Theorem 2.8 is now completely proved.
A The family of biconvex hexagonal domains and their Robin’s
function
Let us mention that such domains were drawn and studied already in [11].
We used Wolfram Mathematica to plot several domains in Figure 2. In those cases, the Robin’s
function is not a very nice function to plot. Instead we introduce the conformal radius:
rΩ(x) = e−2πeγΩ(x).
It satisfies the transfer formula (see [11]):
rΩ(S(x)) = |S′(x)|rD(x) = |S′(x)|(1 − |x|2).
This map is a lot easier to draw, see Figure 3. We see that the Robin’s function of the domains
obtained with δ > 2/3have a saddle point in 0.
20

Figure 2: Plot of the domain Sδ(D), for, left to right: δ=2
5,δ=2
3and δ=3
4.
Figure 3: Plot of the map rSδ(D)◦Sδ:x7→ |S′
δ(x)|(1 − |x|2), for, left to right: a stable case (δ=2
5), the
critical case (δ=2
3), and an unstable case (δ=9
10 ).
B Technical lemmas
B.1 Actual construction of the initial data
We formulate the following remark.
Remark B.1. Let N≥1,Z∗∈(R2)Nand ai∈R∗. For any Z0such that |Z0−Z∗|=ε
2, one can
always chose ω0∈C∞(R2)such that
•ω0satisfies (2),
•B(0) = Z0,
•∀i∈ {1, . . . , N }, Ii(0) ≤εν.
Proof. For N= 1,z0∈R2,a∈R∗, we introduce the vortex patch ω0= 16aε−ν
π1
D(z0,εν/2
4)for ε
small enough. We verify that
|ω0| ≤ Cε−ν
with C=4a
π,Zω0(x)dx=a,
B(0) = 1
aZR2
xω0(x)dx=z0,
21

and
I(0) = 1
aZR2|x−B(0)|2ω0(x)dx=Zεν/2
4
0Z2π
0
16r2ε−ν
πrdθdr= 8ε−νε2ν
256 =1
32εν.
For N > 1we sum patches of this exact form. One can also construct a smooth ω0satisfying those
constraints, by taking a convenient radially mollified version of these vortex patches such that their
support lies within D(zi, ε). Since εν/2
4< ε/2as ν≥2, it is always possible.
B.2 Variant of the Gronwall’s inequality
From the Gronwall’s inequality, we can write the following.
Lemma B.2. Let fbe a C1map and gbe positive non decreasing such that
|f(x)−f(y)| ≤ κ|x−y|.
Let zis a solution of
z′(t) = f(z(t)),
and let ysuch that (|y′(t)−f(y(t))| ≤ g(t)
y(0) = z(0),
where g:R+→R+is smooth. Then
|y(t)−z(t)| ≤ eκt Zt
0
g(s)ds.
Proof. On has that
|y(t)−z(t)|=Zt
0y′(s) + z′(s)ds
≤Zt
0
g(s)ds+Zt
0f(y(s)) −f(z(s))ds
≤Zt
0
g(s)ds+κZt
0|y(s)−z(s)|ds,
so using now the classical Gronwall’s inequality, since t7→ Rt
0g(s)dsis non decreasing, we have that
|y(t)−z(t)| ≤ eκt Zt
0
g(s)ds.
C Computation of the constants
In Theorem 3.2, we proved that the construction is possible as long as
ν > 2
5−α(1 + α)κ1+κ2
λ0
+ 4 −2α,
and
ξ1>1−β
λ0
.
Since κ1,κ2and λ0ultimately depend on the chosen configuration of point-vortices, so do the bounds
on νand ξ1.
We now give details and improvements on those bounds.
22

C.1 Results
We state a few results that will be proved in the following sections.
By computing the constants κ1and κ2for the construction done in Section 4with N= 3, we
obtain the following details on the bound on ν.
Proposition C.1. One can achieve the construction of Theorem 2.6 for any α∈[1,2) with any ν
satisfying
ν > 2
5−α (1 + α)2+2−ααp3(1 + 21+2α)
(2 −2−α)√α+ 4 −2α!,
which is greater than 4.
Proposition C.2. Let ν= 4. There exists α0>1such that for any α∈[1, α0), there exists an
initial configuration (z∗
i)i,(ai)iof point-vortices with N= 7 and β0<4−2α
5−αsuch that for every
β∈(β0,1) there exists ξ1such that for every ε > 0small enough, there exists ω0satisfying (2)such
that any solution ωof (1)satisfies
τε,β ≤ξ1|ln ε|.
In particular, if α= 1, one can take ξ1>4π
9(1 −β).
Unfortunately, we fail to obtain a rigorous estimate of α0. However, in Section C.4, we numeri-
cally check that a construction using 9 blobs can be done with ν= 4 for any α∈[1,2). The constant
ν= 4 is not optimal.
C.2 How to compute the constants
We start by giving a general method on how to obtain κ1and κ2. Let N≥3.
We recall that for all t≤τε,β ,supp ω⊂SN
j=1 D(z∗
j, εβ). Therefore applying (16) to ezi=x,
ex=x′−x,ezj=y,ey= 0, we have ∀x, x′∈D(z∗, εβ), and ∀t≤τε,β,
Fi(x, t)−Fi(x′, t) = ZX
j=iKα(x, y)−Kα(x′, y)ωj(y, t)dy
=X
j=i
CαZ(x−y)⊥
|x−y|α+3 (α+ 1)(x′−x)·(x−y)ω(y, t)dy+o(|x−x′|)
so that
(x−x′)·Fi(x, t)−Fi(x′, t)≤ |x−x′|2
X
j=iZCα
|x−y|α+1 |ω(y, t)|dy+o(1)

≤ |x−x′|2
X
j=i
Cα|aj|
|z∗
i−z∗
j|α+1 +o(1)
.
Therefore we obtain that for every i∈ {1, . . . , N}, for every x, x′∈D(z∗, εβ)and for every t≤τε,β,
(x−x′)·Fi(x, t)−Fi(x′, t)≤κ1|x−x′|2
with
κ1=Cαmax
1≤i≤NX
j=i
|aj|
|z∗
i−z∗
j|α+1 +o(1).
23

For κ2, we need a Lipschitz type estimate for fon SN
j=1 D(z∗
j,2εβ), and thus have that
κ2=

|||Df(Z)|||

∞,SN
j=1 D(z∗
j,2εβ)=|||Df(Z∗)||| +o(εβ) = qρDf(Z∗)[Df(Z∗)]t+o(εβ)
where ρis the spectral radius, that is in our case the greatest eigenvalue in absolute value of the
real symmetric matrix Df(Z∗)[Df(Z∗)]t.
C.3 Proof of Proposition C.1
We now take again N= 3 and compute. First, we directly have that
κ1= 2Cα+o(1).
We now compute the eigenvalues of Df(Z∗)[Df(Z∗)]tand observe that
|||Df(Z∗)||| =Cα2−ααp3(1 + 21+2α).
Therefore, it is possible to choose νsuch that
ν > 2
5−α(1 + α)κ1+κ2
λ0
+ 4 −2α
for εsmall enough as soon as
ν > 2
5−α (1 + α)2+2−ααp3(1 + 21+2α)
(2 −2−α)√α+ 4 −2α!:= g(α).
The plot of α7→ g(α)is given in Figure 4. This concludes the proof of Proposition C.1.
Figure 4: Plot of g(α)in the range α∈[1,2).
C.4 Proof of Proposition C.2
Using the exact same method, we now construct the vortex crystal with N= 7.
Since ∀i=j,|z∗
i−z∗
j| ≥ 1, then we obtain directly that one can take κ1=3
π+o(1). However
in general, we are not able to compute κ2and ν.
24

We now assume that α= 1. We then compute that
Df(Z∗) = 1
2π×

























c1−35
24 0 1 −1
2√3
1
6−√3
8−1
80−1
3
√3
2−1
2
5√3
4
5
4
−35
24 −c11 0 1
6
1
2√3−1
8
√3
8−1
30−1
2−√3
2
5
4−5√3
4
0 1 −c1−35
24 −√3
2−1
20−1
3
√3
8−1
8
1
2√3
1
6−5√3
4
5
4
1 0 −35
24 c1−1
2
√3
2−1
30−1
8−√3
8
1
6−1
2√3
5
4
5√3
4
−1
2√3
1
6−√3
2−1
2035
12
√3
2−1
2
1
2√3
1
601
40−5
2
1
6
1
2√3−1
2
√3
2
35
12 0−1
2−√3
2
1
6−1
2√3
1
40−5
20
−√3
8−1
80−1
3
√3
2−1
2c1−35
24 0 1 −1
2√3
1
6
5√3
4
5
4
−1
8
√3
8−1
30−1
2−√3
2−35
24 −c11 0 1
6
1
2√3
5
4−5√3
4
0−1
3
√3
8−1
8
1
2√3
1
60 1 −c1−35
24 −√3
2−1
2−5√3
4
5
4
−1
30−1
8−√3
8
1
6−1
2√31 0 −35
24 c1−1
2
√3
2
5
4
5√3
4
√3
2−1
2
1
2√3
1
601
4−1
2√3
1
6−√3
2−1
2035
12 0−5
2
−1
2−√3
2
1
6−1
2√3
1
401
6
1
2√3−1
2
√3
2
35
12 0−5
20
−√3
2−1
2
√3
2−1
20 1 −√3
2−1
2
√3
2−1
20 1 0 0
−1
2
√3
2−1
2−√3
21 0 −1
2
√3
2−1
2−√3
21 0 0 0

























with c1=1
2√3−13√3
8.
Eigenvalues are 0(with multiplicity 4), ±i√35
4π(each with multiplicity 2), ±2
π(each with multi-
plicity 2) and ±9
4π, so on can let λ0=9
4π. Computations show that
κ2=qρDf(Z∗)[Df(Z∗)]t+o(1) = 5√7
2+o(1).
Finally, it is easy to check that it is possible to choose νsuch that relation (15) holds, for εsmall
enough as soon as
ν > 12 + 5√7
9+ 1.
Observing that 12+5√7
9+ 1 <4, we can conclude that we can construct ω0satisfying (2) with N= 7
and ν= 4 for α= 1 such that
τε,β ≤ξ1|ln ε|,
for any ξ1>4π1−β
9.
In order to conclude the proof of Proposition C.2, we observe that aNand thus κ1
Cα,1
CαDf(Z∗)
and thus λ0
Cαand κ2
Cαare all depending continuously on α. Therefore, it holds true for εsmall enough
that 2
5−α(1 + α)κ1+κ2
λ0
+ 4 −2α<4.
at least on a small interval [1, α0). This ends the proof.
For a general value of α, the coefficients of the matrix Df(Z∗)are too complicated to compute
mathematically the eigenvalues. However, we use Wolfram Mathematica [18] to plot the map
h:α7→ 2
5−α






(1 + α)
max
1≤i≤NX
j=i
|aj|
|z∗
i−z∗
j|α+1 +qmax EigenvaluesDf(Z∗)[Df(Z∗)]t
max Re(Eigenvalues[Df(Z∗)]+ 4 −2α





25

Figure 5: Plot of h(α)in the range α∈[1,2), with N= 7 (left) and N= 9 (right).
to obtain Figure 5, which shows that letting N= 9, we have that 2
5−α(1 + α)κ1+κ2
λ0+ 4 −2α<4
for every α∈[1,2). Therefore, we have very strong numerical evidence of the fact that a construction
is possible for every α∈[1,2) with ν < 4(thus in particular with ν= 4). Please keep in mind that
our method does not yield optimal constants. In particular, ν= 4 is not optimal.
Acknowledgments.
The author whishes to acknowledge useful discussions with Thierry Gallay, Pierre-Damien Thizy
and Mickaël Nahon. This work was partly conducted when the author was working at the Université
Claude Bernard Lyon 1, Institut Camille Jordan.
References
[1] H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. Vainchtein. Vortex Crystals,
volume 39 of Advances in Applied Mechanics. Elsevier, 2003.
[2] P. Buttà and C. Marchioro. Long time evolution of concentrated Euler flows with planar
symmetry. SIAM J. Math. Anal., 50(1):735–760, 2018.
[3] H. E. Cabral and D. S. Schmidt. Stability of Relative Equilibria in the Problem of N+1 Vortices.
SIAM Journal on Mathematical Analysis, 31(2):231–250, 2000.
[4] G. Cavallaro, R. Garra, and C. Marchioro. Localization and stability of active scalar flows. Riv.
Mat. Univ. Parma 4, pages 175–196, 2013.
[5] G. Cavallaro, R. Garra, and C. Marchioro. Long time localization of modified surface quasi-
geostrophic equations. Discrete & Continuous Dynamical Systems - B, 26(9):5135–5148, 2021.
[6] J. Davila, M. Del Pino, M. Musso, and J. Wei. Gluing Methods for Vortex Dynamics in Euler
Flows. Archive for Rational Mechanics and Analysis, 235:1467–1530, 2020.
[7] M. Donati. Two-dimensional point vortex dynamics in bounded domains: Global existence for
almost every initial data. SIAM Journal on Mathematical Analysis, 54(1):79–113, 2022.
[8] M. Donati and D. Iftimie. Long time confinement of vorticity around a stable stationary point
vortex in a bounded planar domain. Annales de l’Institut Henri Poincaré C, Analyse non
linéaire, 38(5):1461–1485, 2020.
26

[9] Tobin A. Driscoll and Lloyd N. Trefethen. Schwarz-Christoffel Mapping. Cambridge Mono-
graphs on Applied and Computational Mathematics. Cambridge University Press, 2002.
[10] D. Dürr and M. Pulvirenti. On the vortex flow in bounded domains. Communications in
Mathematical Physics, pages 265–273, 1982.
[11] M. Flucher. Extremal functions for the trudinger-moser inequality in 2 dimensions. Commen-
tarii mathematici Helvetici, 67(3):471–497, 1992.
[12] M. Flucher. Variational problems with concentration, volume 36 of Progress in Nonlinear Dif-
ferential Equations and their Applications. Birkhäuser Verlag, Basel, 1999.
[13] C. Geldhauser and M. Romito. Point vortices for inviscid generalized surface quasi-geostrophic
models. Am. Ins. Math. Sci., 25(7):2583–2606, 2020.
[14] L. Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete
and Continuous Dynamical Systems, 2022.
[15] L. Godard-Cadillac, P. Gravejat, and D. Smets. Co-rotating vortices with N fold symmetry for
the inviscid surface quasi-geostrophic equation. Preprint, 2020. arXiv:2010.08194.
[16] B. Gustafsson. On the Motion of a Vortex in Two-dimensional Flow of an Ideal Fluid in Simply
and Multiply Connected Domains. Trita-MAT-1979-7. Royal Institute of Technology, 1979.
[17] P. Hartman. Ordinary Differential Equations: Second Edition. Secaucus, New Jersey, U.S.A.:
Birkhauser, 1982.
[18] Wolfram Research, Inc. Mathematica, Version 12.0. Champaign, IL, 2019.
[19] F. Marchand. Existence and regularity of weak solutions to the quasi-geostrophic equations in
the spaces Lpor ˙
H−1/2.Comm. Math. Phys., 277(1):45–67, 2008.
[20] C. Marchioro and M. Pulvirenti. Vortex methods in two-dimensional fluid dynamics. Lecture
notes in physics. Springer-Verlag, 1984.
[21] C. Marchioro and M. Pulvirenti. Mathematical Theory of Incompressible Nonviscous Fluids.
Applied Mathematical Sciences. Springer New York, 1993.
[22] C. Marchioro and M. Pulvirenti. Vortices and localization in Euler flows. Comm. Math. Phys.,
154(1):49–61, 1993.
[23] G. K. Morikawa and E. V. Swenson. Interacting Motion of Rectilinear Geostrophic Vortices.
Physics of Fluids, 14(6):1058–1073, June 1971.
[24] J. Pedlosky. Geophysical Fluid Dynamics. Springer-Verlag, New-York, 1987.
[25] M. Rosenzweig. Justification of the point vortex approximation for modified surface quasi-
geostrophic equations. Preprint, 2020. arXiv:1905.07351.
[26] D. Smets and J. Van Schaftingen. Desingularization of vortices for the Euler equation. Arch.
Ration. Mech. Anal., 198(3):869–925, 2010.
[27] B. Turkington. On the evolution of a concentrated vortex in an ideal fluid. Archive for Rational
Mechanics and Analysis, 97:75–87, 1987.
27

[28] G.K. Vallis. Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 2006.
[29] V.I. Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational
Mathematics and Mathematical Physics, 3(6):1407 – 1456, 1963.
28