Vortex collapses for the Euler and Quasi-Geostrophic Models

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Vortex collapses for the Euler and Quasi-Geostrophic

Models

Ludovic Godard-Cadillac

To cite this version:

Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic Models. 2021. �hal-03111989�

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Vortex collapses for the Euler and Quasi-Geostrophic Models

Ludovic Godard-Cadillac1 January 26, 2021

Abstract

This article studies point-vortex models for the Euler and surface quasi-geostrophic equa-tions. In the case of an inviscid fluid with planar motion, the point-vortex model gives ac-count of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains three main results with several links between each other. In the first part, we provide two uniform bounds on the trajectories for Euler and quasi-geostrophic vortices related to the non-neutral cluster hypothesis. In a second part we focus on the Euler point-vortex model and under the non-neutral cluster hypothesis we prove a convergence result. The third part is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

1 Introduction

1.1 The inviscid surface quasi-geostrophic equation

We are interested in a model of point-vortices for the inviscid surface quasi-geostrophic equation (

∂tω + v · ∇ω = 0,

v = ∇⊥(−∆)−sω, (SQG)

where v : R+× R2 → R2 is the fluid velocity and ω : R+× R2 → R is called the active scalar.

The notation ⊥ refers to the counterclockwise rotation of angle π₂. Typically, the surface quasi-geostrophic equation models the dynamic in a rotating frame of the potential temperature for a stratified fluid subject to Brunt-Väisälä oscillations. This is a standard model for geophysical fluids and it is intensively used for weather forecast and climatology. For more details about this physical model, see e.g. [21] or [25]. Mathematically, the quasi-geostrophic equation has many properties in common with the bi-dimensional Euler equation written in terms of vorticity

(

∂tω + v · ∇ω = 0,

v = ∇⊥(−∆)−1ω. (Euler 2D)

The bi-dimensional Euler equation can be seen as a particular case of the quasi-geostrophic equation where s is equal to 1. Elements of standard theory for Euler equations can be found in [5]. Local well-posedness of classical solutions for (SQG) was established in [8], where are also studied the analogies with the bi and tri-dimensional Euler equation. Solutions with arbitrary Sobolev growth were constructed in [15] in a periodic setting. So far and contrarily to the bi-dimensional Euler equations, establishing global well-posedness of classical solutions for (SQG) is an open problem. Note also that the global existence of weak solutions in L2(R2) was established in [22], but below a certain regularity threshold, these weak solutions show dissipative behaviors and non-uniqueness is possible. The arguments leading to these properties are close to ones related to the Onsager conjecture. See [6] for an example of two weak solutions to (SQG) with the same initial datum. Exhibiting global smooth solutions or patch solutions is a challenging issue as there is no equivalent of the Yudovitch theorem [26]. A first example was recently provided in [7] by developing a bifurcation argument from a specific radially symmetric function. The varia-tional construction of an alternative example in the form of a smooth traveling-wave solution was completed in [13, 11]. Corotating patch solutions with two patches [14] and N patches forming an N -fold symmetrical pattern [9] were recently exhibited with bifurcation argument. The C1

analogous of these solutions has also been investigated independently recently in [1]. Another recent independent result [12] also build corotating solutions with N patches using variational argument. In this last article, the desingularization of the associated point-vortex problem is achieved.

The present work aims at developing the understanding of the links between the quasi-geostrophic equation and the bi-dimensional Euler equation through the study of the point-vortex model. In the case of the bi-dimensional Euler equation, the point-point-vortex model is a

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system of differential equations for points on R2that approximates situations where the vorticity ω is highly concentrated around several points. In such a situation, it is more convenient to see the vorticity as being a pondered sum of Dirac masses evolving in time. This model is widely studied in fluid mechanics of the plane. An extensive presentation of the main results on this system can be found at [16, §4], completed by [17]. The desingularization problem which consists in a rigorous derivation of the point-vortex model is a classical issue for the bi-dimensional Euler equation [24], to our knowledge it is still open for (SQG) vortices although recent results exist [10, 23].

This article generalizes several existing results known for the Euler vortices or extends known results for Euler to the quasi-geostrophic case. The first part of this article is the generalization of a uniform bound result, Theorem 2.1 in [16, §4]. We prove that under the non-neutral cluster hypothesis (defined hereafter) the vortices stays bounded in finite time and this bound does not depend on the singularity of the kernel nor on the initial position of the vortices. We also provide a uniform relative bound for a slight relaxation of the non-neutral cluster hypothesis. In the second part of this work we prove that under the non-neutral cluster hypothesis the trajectories of the vortices for the Euler model are convergent in finite time even in the case of collapses. The quasi-geostrophic case is left open. The third part of this work is devoted to the question of the improbability of collapses. This consists in studying the Lebesgue measure of the initial conditions leading to a collapse, which is expected to be equal to 0. This question has been successfully answered by Theorem 2.2 in [16, §4] under the non-neutral cluster hypothesis implying that the vortex trajectories are uniformly bounded. We generalize this result to (SQG) for all s ∈ (0, 1] and weaken the boundedness assumption for a relative bound on the trajectories (the vortices remain bounded with respect to one another).

1.2 Presentation of the point-vortex model

The point-vortex model on the plane R2 _{consists in assuming that at time t = 0 the vorticity}

can write as a pondered sum of Dirac masses, ω(t = 0, x) =

N

X

i=1

aiδxi. (1)

The points xi are the respective position of the vortices aiδxi and the coefficients ai are their

intensity. The first equation in (SQG) or (Euler 2D) is a transport equation on the vorticity ω. It is expected that the Dirac masses initially located at xi are left unchanged but transported

by the flow. Formally, if we solve the evolution equations (SQG) or (Euler 2D) with initial datum (1), we obtain that the initial speed writes

v(t = 0, x) =

N

X

i=1

ai∇⊥xG |xi− x|, (2)

where G is the profile of the Green function of the fractional Laplace operator (−∆)s in the plane R2_{. Here the parameter s is chosen to be in (0, 1]. In this case, The profiles G : R}∗

+ → R

are given by:

G1(r) := 1 2π log 1 r and Gs(r) := Γ(1 − s) 22s_πΓ(s) 1 r2(1−s), (3)

with Γ the classical Gamma function. Nevertheless, a problem arises from the singularity of the speed in (2). Since the vorticity is concentrated in one point then it is usually assumed that it does not interact with itself but only with the vortices that are at a positive distance. We derive that the differential equation describing the evolution of the position of the vortices is given by

d dtxi(t) = N X j=1 j6=i aj∇⊥G |xi(t) − xj(t)|. (4)

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In the work [16, §4], the authors only consider the case s = 1 corresponding to the bi-dimensional Euler equation. We are going to generalize some of their results to more general kernel profiles G that include the quasi-geostrophic case. In the sequel, the function G is assumed to satisfy

G ∈ C1,1_loc R∗₊_{∩ C}1,1 _{[1, +∞[}_. ₍₅₎

By the Cauchy-Lipschitz theorem, the point-vortex system (4) is well-posed as long as there are no vortex collapses. In other words, as long as the system stays far from the singularity of the kernel G in 0. We add some other hypothesis on the function G later on. These hypothesis will always be satisfied by the Green functions of the fractional laplacian. Hence this work can be viewed as an extension of [16, §4] to the quasi-geostrophic equation and related model. In order to ensure that the manipulated objects are well-defined, we introduce the sets of initial datum leading to a collapse in finite time.

Definition 1.1 (Set of collapses). The set of initial datum such that two or more vortices collapse on the interval of time [0, T )

C_T _:=n_{X ∈ R}2N _{: ∃ T}_X _{∈ [0, T ), lim inf} t→TX min i6=j |xi(t) − xj(t)| = 0 o . (6) We then set C_:= +∞_[ T =1 C_T_. ₍₇₎

The set C is called the set of collapses and TX is the time of collapse associated to the initial

datum X ∈ C. Note that these sets depend on the choice of the kernel G.

The point-vortex differential equation (4) is well-defined for all initial datum X ∈ R2N _{\ C.}

If we restrict the analysis to a bounded interval of time [0, T ) then it is well-defined for all initial datum X ∈ R2N \ CT, which eventually allows us to study a possible vortex collapse at

time t = T. The main element to point-out about this dynamic is its Hamiltonian nature. The Hamiltonian of the point-vortex system is given by

H : R2N _−→ R_,

X = (x1· · · xN) 7−→

X

i6=j

aiajG |xi− xj|. (8)

Considering the singularity of r 7→ G(r) when r → 0, this function is well-defined if and only if xi 6= xj for all i 6= j. In other words, it is well-defined as long as there are no vortex collapse.

The system (4) can be rewritten ai

d

dtxi(t) = ∇

⊥

xiH(X). (9)

The first consequence of this Hamiltonian reformulation is the preservation of the Hamiltonian H along the flow St_{of (4).}

Lemma 1.2 (Conservation of the Hamiltonian). Let X ∈ R2N _{\ C}

T, we have

∀ t ∈ [0, T ), d dtH(S

t_{X) = 0.} ₍₁₀₎

Another consequence of the Hamiltonian of the system is the Liouville theorem that ensures the preservation of the Lebesgue measure by the flow St.

Lemma 1.3 (Liouville theorem). Let V0 ⊆ R2N \ CT measurable. Then, we have

∀ t ∈ [0, T ), d dt L

2N_(St_V

0) = 0, (11)

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For the proof of the Liouville Theorem, we refer to the one given by Arnold in [3, §3]. With the Hamiltonian formulation also comes the Noether theorem [18] that provides the quantities left invariant by the flow corresponding to the geometrical invariances of the Hamiltonian H. The translations invariance implies the conservation of the vorticity vector.

Lemma 1.4(Conservation of the vorticity vector). The vorticity vector is defined for all initial datum X ∈ R2N by M (X) := N X i=1 aixi. (12) Let X ∈ R2N \ CT. We have ∀ t ∈ [0, T ), d dtM (S t_{X) = 0.} ₍₁₃₎

When the system is non-neutral, meaning thatP_iai 6= 0, this lemma implies the preservation

of the center of vorticity of the system defined by B(X) := N X i=1 ai −1XN i=1 aixi. (14)

Similarly, the invariance by the rotations, implies the conservation of the moment of inertia. Lemma 1.5 (Conservation of the moment of inertia). The moment of inertia is defined for all X ∈ R2N _by I(X) := N X i=1 ai|xi|2. (15) Let X ∈ R2N _{\ C} T. Then ∀ t ∈ [0, T ), d dtI(S t_{X) = 0.} ₍₁₆₎

Although these two lemmas are a direct consequence of the Noether theorem, they follow more simply from a direct computation using directly the differential point-vortex system (4). The combination of these two lemmas implies the following invariant

Lemma 1.6 (Collapse constraint). The collapse constraint is defined for all X ∈ R2N _by

C(X) := N X i=1 N X j=1 i6=j aiaj|xi− xj|2. (17) Let X ∈ R2N \ CT. Then ∀ t ∈ [0, T ), d dtC(S t_{X) = 0.} ₍₁₈₎

The preservation of this quantity is referred as a collapse constraint because it is widely used in the study of vortex collapses for small number of vortices [19, 20, 2, 4]. Indeed, a collapse means that |xi− xj|2 vanishes for some values of i and j. Combined with the preservation of C,

this gives a necessary condition for a vortex collapse. For instance, in the case of a collapse for a system of 3 vortices, this gives the constraint C = 0.

Proof. We expand the square in the right-hand side of (17) and obtain by a straight-forward calculation C(X) = 2 N X i=1 ai I(X) − 2M(X)2. (19)

The vorticity vector M (X) and the vorticity momentum I(X) are preserved by the flow as stated respectively in Lemmas 1.4 and 1.5. Therefore (19) implies the preservation of (17) by the vortex flow.

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2 Main results

2.1 Uniform bound results

2.1.1 The uniform bound theorem

The specific case of the Euler point-vortex system corresponds to the Green function of the laplacian (3). This particular case is studied in [16, §4]. More precisely, they focused on a specific situation for which the intensities for the vortices satisfy

∀ A ⊆ {1 · · · N } s.t. A 6= ∅, X

i∈A

ai 6= 0. (20)

A vortex system such that the sum of all the intensities ai is equal to 0 is called in [16, §4] a

“neutral system”. No name to Hypothesis (20) is given and we suggest that to call it “non-neutral clusters hypothesis”. Indeed this assumption exactly means that any vortex cluster is not neutral. The main interest of this hypothesis relies on the properties of the center of vorticity. Under the non-neutral cluster hypothesis, it is well defined, not only for the whole system but also for any subset of vortices. Seen the properties of the centers of vorticity in (14), it can be said intuitively that a vortex cluster is expected to “turn around its center of vorticity”. If the center of vorticity is well-defined for all subsets of vortices then it is possible to claim that the vortices do not diverge in finite time. More precisely, we provide a bound on the trajectories that is uniform with respect to the initial datum X ∈ R2N _{but also with respect to the singularity of}

the kernel profile G near 0. Nevertheless, this bound is not uniform with respect to the behavior of the kernel profile G at +∞.

Theorem 2.1 (Uniform bound on the trajectories). Consider the point-vortex dynamic (4) under the non-neutral clusters hypothesis (20) with a kernel profile G ∈ C_loc1,1 R∗₊_∩C1,1 _{[1, +∞[}_. Then, given any positive timeT > 0, there exist a constant C such that for all initial datum X ∈ R2N \ CT, sup t∈[0,T ) X − St GX ≤ C, (21) where St

G is the (4) flow associated to kernel profile G. Moreover, the constant C depends only

on N , the intensities ai, the final time T and on supremum of r 7→ dG_dr(r) for r ≥ 1. This

constant C does not depend on the initial datum X ∈ R2N nor on the singularity of the kernel r 7→ G(r) when r → 0.

Theorem 2.1 provides an uniform bound on the behaviors of vortices under the non-neutral clusters hypothesis. In [16, §4], a weaker version of this theorem is established only for the Euler case. We extend this result to a general case where it holds no matter what the singularity of the kernel G in 0+ _{is. This theorem may seem quite surprising at first sight since the singularity}

of the kernel governs the situations where the speed of the vortices become very high. Dealing with a boundedness result, we could expect hypothesis on the singularity and not only on the behavior of G at infinity. This is why this result must be understood as a quantitative result on the intuition that a vortex cluster is expected to “turn around its center of vorticity” and not leave whatever its speed is. Remark that the non-neutral cluster hypothesis (20) is essential. Indeed, the simple situation of a vortex pair with intensities +1 and −1 gives raise to a translation motion along two parallel lines at a speed that blows up as the initial distance between the two vortices goes to 0+. The trajectories are bounded in finite time but the bound is not uniform, depending on the initial conditions and on the singularity of the kernel. We underline that Theorem 2.1 apply to the quasi-geostrophic case given by the Green function of the fractional laplacian (3). In this sense, this theorem is the extension of Theorem 2.1 in [16, §4] to the quasi-geostrophic case. Note finally that at a technical level, this theorem is useful for dealing with the singularity of a kernel profile G near 0. Indeed, if G is replaced by a smooth

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approximation Gε that coincides with G on [ε, +∞) then the bound is independent of ε. This

result is used in particular to provide existence results for the evolution problem. 2.1.2 The case of intensities ai all positive

As a further consequence of Theorem 2.1 we can show the the impossibility of collapses in the case where the intensities ai are all positive.

Corollary 2.2. Let G be a profile such that

G(r) −→ +∞ as r → 0. (22)

Assume that the intensities ai are all positive and consider an initial datum X ∈ R2N for the

point-vortex system (4) such that xi 6= xj fori 6= j. Then there is no collapse of vortices at any

time.

Contrarily to the existence result given by Theorem 2.2 in [16, §4] and its generalization to (SQG) at Theorem 2.7, this result is true for all initial datum and not only for almost every one. Hypothesis (22) may appear at first sight as a bit restrictive. Indeed, if we consider the kernels

∀ r > 0, dG dr(r) =

1

rα, (23)

with 0 < α < 1 then the associated kernel G does not satisfy (22) although it is singular enough at 0 to create definition problems. The possibility to extend Corollary 2.2 to this case is an open problem. Nevertheless, the physical relevant cases are α = 1 for the Euler model or 3 < α < 1 for the quasi-geostrophic model. For these values of α, Corollary 2.2 apply.

2.1.3 The uniform relative bound theorem

A natural question concerning Theorem 2.1 is to ask what this result becomes when the non-neutral cluster hypothesis (20) ceases to be satisfied. For instance, we consider instead of (20) the following hypothesis:

∀ A ⊆ {1 · · · N } s.t. A 6= ∅ or {1 · · · N }, X

i∈A

ai 6= 0. (24)

In other words, all the strict sub-clusters must have the sum of their intensities different from 0 but we allow the total sum PN_i=1ai to be equal to 0. This situation is achieved for instance by

the vortex pair of intensities +1 and −1 that are translating at a constant speed. As a corollary of Theorem 2.1 we check that in this situation the vortices remain bounded with respect to one another.

Theorem 2.3 (Uniform relative bound on the trajectories). For a given set of points noted X = (x1· · · xN) ∈ R2N, we define the diameter of this set by

diam(X) := max

i6=j |xi− xj|. (25)

Consider the point-vortex dynamic (4) under hypothesis (24) with a kernel profile G ∈ C_loc1,1 R∗

+

∩ C1,1 _{[1, +∞[}_{. Let} _{T > 0 the final time. Then for all kernel profile G ∈ C}1,1 loc R∗+

∩ C1,1 _{[1, +∞[} _{and for all initial datum} _{X ∈ R}2N _{that are not leading to collapse on}_{[0, T ),}

sup

t∈[0,T )

diam S_GtX≤ diam(X) + C, (26)

where S_Gt is the flow associated to (4) with the kernel profile equal to G. Moreover, the constant C depends only on N , the intensities ai, the final time T and on supremum of r 7→

dG dr(r) for r ≥ 1.

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This constant C does not depend on the initial datum X ∈ R2N nor on the singularity of the kernel r 7→ G(r) when r → 0. The main argument for the proof of this Theorem is the following: if the vortices are separated into two sub-clusters far from each other, then (24) implies that we can apply Theorem (2.1) to these two clusters separately and get a contradiction.

2.2 Convergence result for Euler point-vortices

The systems of vortices that are studied with more details in [16, §4] are the systems for which the non-neutral clusters hypothesis (20) holds. As stated in the previous section, their result concerning uniform bound on the trajectories can be improved to consider a much wider class of point-vortex systems, including the quasi-geostrophic case. Nevertheless, in the particular case of the Euler point-vortex system with the non-neutral clusters hypothesis (20), we are also able to write a convergence result. When vortices come to collapse, their speed may become infinite as a consequence of the kernel profile singularity in 0+. But if their speed blows up, then any pathological behavior near the time of collapse TX is a priori possible. We prove here that the

trajectories are actually convergent in the Euler case.

Theorem 2.4 (Convergence for Euler vortices under non-neutral clusters hypothesis). Consider the point-vortex model (4) under hypothesis (20) with a kernel profile G1 corresponding to the

Green function of the laplacian (3). Let X ∈ C be an initial datum leading to a collapse at time TX. Then, for all i = 1 · · · N , there exists an x∗i ∈ R2 such that

xi(t) −→ x∗i as t → TX−. (27)

The first step of the proof consists in the following idea: For a fixed value of t ∈ [0, TX)

define the distribution Pt:=PN_i=1aiδxi(t). The point-vortex equation gives that this distribution

converges in W−2,∞ as t → TX. In a second time, it is possible to prove that this convergence

is actually stronger and obtain (27) exploiting the non-neutral cluster hypothesis. The proof of this result is specific to the Euler case and we do not know yet whether the conclusion extends to the (SQG) case.

2.3 Improbability of collapses for point-vortices

Understanding the sets of collapses CT is an important issue for the study of point-vortices

since these set give the time of existence of the point-vortex system for a given initial datum. Marchioro and Pulvirenti in their study of the point-vortex problem [16] provided the following improbability result

Theorem 2.5 (Improbability of collapses, Marchioro-Pulvirenti, 1984). Consider the point-vortex problem (4) with kernel profile G1, the kernel associated to the Green function of the

laplacian on the plane (3). Assume that the intensities of the vortices satisfy the non neutral cluster hypothesis (20).

Then, the set of initial datum for the dynamic (4) that lead to collapses in finite time is a set of Lebesgue measure equal to 0.

The non neutral cluster hypothesis (20) is an important hypothesis for this theorem as it allows us to use Theorem 2.1 that provides a uniform bound. Concerning this result, the most natural question consists in removing this assumption on the intensities of the vortices, which eventually leads to the following conjecture.

Conjecture 2.6. The set of collapses C has a Lebesgue measure 0.

It is not yet possible to prove such a conjecture. The main difficulty lays in the understanding of the situations where some vortices collide in such a way that they go to infinity in finite time, or show an unbounded pathological behavior. The existence of unbounded trajectories in finite

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time is also an open problem. Although we are not able to answer to Conjecture (2.6), we are able to improve Theorem 2.5 as stated in the following theorem:

Theorem 2.7 (improbability of collapses for Euler and SQG vortices). Consider the point-vortex problem (4) with kernel profile Gs, the kernel associated to the Green function of the

laplacian or the fractional laplacian on the plane (3) for s ∈ (0, 1]. Assume that the intensities of the vortices satisfy (24).

Then, the set of initial datum leading to collapses has a Lebesgue measure equal to 0. This theorem is slightly more general than Theorem 2.5 for two reasons. First, it is true both for Euler and for quasi-geostrophic point-vortex models. This aspect was already partially improved by [10], where they obtained the result for s > 1/2. Indeed, the value s = 1/2 appear to be a critical value for the quasi-geostrophic equations where integrability problems arise. Our arguments manage to pass trough these difficulties and obtain the result for all s ∈ (0, 1]. The second improvement lays in the fact that we managed to replace the non neutral cluster hypothesis (20) by the weaker hypothesis (24). This weaker hypothesis can be seen at first sight as a small improvement. Nevertheless, it make a quite important difference because the non neutral cluster hypothesis (20) implies that the trajectories are bounded (Theorem 2.1) whereas hypothesis (24) only imply a relative bound (Theorem 2.3). In other words, this improbability result allows a simple unbounded behaviors: the cases where the vortices collectively goes to infinity. Unfortunately, it is not clear whether we can decompose a more complicated diverging situation into sub-clusters of vortices inside which the vortices stay bounded with respect to one another. If such a decomposition is true then, combined with Theorem 2.7, this would provide an important step towards the answer of Conjecture 2.6.

3 Outlines of the proofs

To ease the general reading and understanding of the article, we draw here only the outlines of the proofs of the main Theorems. These larger proofs are decomposed into smaller lemmas stated in this section and forming the main intermediate steps. The links and articulations between the different lemmas are developed in this section but the detailed technical proofs of each different Lemma are all postponed to Section 4. Concerning Theorem 2.1 and Theorem 2.3, the proofs are shorter and directly done in section 4.

3.1 Outline of the proof for Theorem 2.4

The first part of the proof consists in considering Dirac masses located on the vortices and to prove that this converges in the distributional sense as t → TX, the time of collapse. More

precisely, we converge in W−2,∞.

Lemma 3.1 (Convergence of the Dirac measures). Let X ∈ R2N. From the vortices xi(t)

evolving according to the differential equations (4) with initial datum X, define the distribution

PX(t) := N

X

i=1

aiδxi(t), (28)

where δx denotes the Dirac mass at point x ∈ R2. Assume now that the evolution problem

associated to the initial datum X is well defined on an interval [0, T [ for some T > 0. Then, there exist X∗ ∈ R2N _and _{b ∈ {0, 1}}N _{such that}

N X i=1 aiδxi(t) −→ N X i=1 aibiδx∗

i in the weak sense of measure ast → T

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and such that

bi = 0 =⇒ sup

t∈[0,T [

|xi(t)| = +∞.

Note that the proof makes no use of the non-neutral clusters hypothesis (20). In the particular case where this hypothesis is satisfied, theorem 2.1 implies that the vortices stay bounded on bounded intervals of time. Therefore, in this particular case all the coefficients bi given by this

lemma are equal to 1.

The next step of the proof exploits more precisely Hypothesis (20) and the continuity of the trajectories to obtain that a given vortex xi(t) can only converge as t → T− to one element

of the set {x∗₁· · · x∗

N} given by Lemma 3.1. More precisely, if a given point xi(t) have at least

two adherence points, it is possible to extract a third adherence point that does not belongs to {x∗₁· · · x∗_N} and this provides a contradiction with Lemma 3.1. See Section 4 for a detailed proof.

3.2 Outline of the proof for Theorem 2.7

3.2.1 The modified system

Let i be fixed in {1 · · · N }. The modified system consists in studying the evolution of yij := xi−xj

for j ∈ {1 · · · N } \ {i}. The point of this modification, since i is fixed, is that it decreases the dimension of the studied system because the vector Yi= (yij)j6=i belongs to R2(N −1) and reduce

the problem to study relative movement. This modification can also be interpreted in a way as quotienting the space by the translations of the plane. The initial problem (4) implies that

d dt(xi− xj)(t) = X k6=i ak∇⊥Gs |xi− xk|− X k6=j ak∇⊥Gs |xj− xk|. (30)

Therefore, the evolution of yij is given by

d dtyij = (ai+ aj)∇ ⊥_G s |yij|+ X k6=i,j a_k∇⊥Gs |yik| + ∇⊥Gs |yij− yik| . (31)

The main interest of this new system is that Theorem 2.7 can be reformulated using only the differences yij.

Lemma 3.2 (Reformulation of Theorem 2.7). Denote by Yi(t) := yij(t)_j6=ithe solution to(31)

at time t with initial datum Yi ∈ R2(N −1). Assume that for all i ∈ {1 · · · N } and for all T > 0

and ρ > 0 the set n

Yi:= (yij)j6=i∈ B(0, ρ)2(N −1): ∃ TX ∈ [0, T ], lim inf t→T− X min j6=i yij(t) = 0o. (32) has its Lebesgue measure L2(N −1) _{equal to}_{0. Then in this case the conclusion of Theorem 2.7}

holds.

The notation B(x0, ρ) refers to the euclidian ball of R2 of center x0 and radius ρ. The rest of

the work consists then in studying this system (31) and to establish that (32) does have measure 0. This modified dynamics has many properties in common with the original one. In particular, this new dynamics still satisfies the Liouville property. Define the function Hij : R2(N −1) → R

by Hij h (yik)k6=i i := (ai+ aj)Gs |yij| + yij · X k6=i,j ak∇Gs |yik| + X k6=i,j akG |yij− yik| . (33)

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Combining this equation with (31) gives d dtyij = ∇ ⊥ yijHij h (yik)k6=i i . (34)

This equation says that, in a certain sense, the dynamic of the vector Yi := (yij)j6=i shows an

Hamiltonian structure. It is not an Hamiltonian system because the function Hij does depend

on j but there is still a structure with an operator ∇⊥. Therefore, the Schwartz theorem gives

divyij d dtyij = divyij ∇⊥yijHij h (yik)k6=i i = 0. (35)

Since the velocity is divergent-free, a Liouville theorem holds for this dynamics.

Lemma 3.3 (Liouville theorem). The Lebesgue measure on the space R2(N −1) given by Y

j6=i

dyij (36)

is preserved by the flow St_i associated to (34).

A detailed proof of the Liouville theorem in a more general setting can be found at [3, §3] and we refer to it for the proof of Lemma 3.3. The main argument for the proof of the Liouville theorem that is proved at [3, §3] is that for a dynamical system ˙x = f (x), the following formula holds d dtL d _St_Ω₌Z St_Ω div f (x) dx. (37)

where St is the flow of the equation ˙x = f (x) and Ω any measurable set. Since in our case we have a divergent-free dynamic, we deduce Lemma 3.3.

3.2.2 Estimate the collapses

From the kernel Gs given at (3), define now the regularized profiles Gs,ε for ε ∈ (0, 1]. The

objective here is to drop the singularity of the kernel Gs near 0+. We ask Gs,ε to be C1 on R+

(until the boundary) and to verify

• Gs,ε(q) = Gs(q) when ε ≤ q, (38) • |Gs,ε(q)| ≤ |Gs(q)| for all q ∈ R+, (39) • d dqGs,ε(q) ≤ _dqdGs(ε) for all q ≤ ε, (40) • |Gs,ε(q)| ≤ 2|Gs(ε)| for all q ∈ R+. (41)

Since Gs,ε is of class C1,1 on R+, the dynamic defined by (4) for the kernel Gs,ε is always

well-defined and is Hamiltonian. In the sequel we denote St

i,ε the flow at time t associated

to the evolution equation (31) with the kernel profile Gs replaced by the regularization Gs,ε.

The motion induced by the kernel profile Gs,ε coincides with the original motion provided that

the distances between the vortices remain higher than ε. Theorem 2.7 can be reformulated as follows.

Lemma 3.4 (Reformulation of Theorem 2.7 with ε-Regularized dynamic). Assume that for all T > 0 and for all ρ > 0 we have the following convergence:

L2(N −1)nYi = (yij)j6=i∈ B(0, ρ)2(N −1) : min j6=i t∈[0,T ]inf yε ij(t) ≤ εo−−−−→ ε→0+ 0. (42)

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This lemma is nothing more than Lemma 3.2 where we added this parameter ε > 0 allowing us to regularize the kernel. Therefore, combined with Lemma 3.2, this lemma gives Theorem 2.7 provided that the convergence (42) holds.

Lemma 3.5. Let i ∈ {1 · · · N }, ε > 0 and ρ > 0. Then

L2(N −1)nYi= (yij)j6=i∈ B(0, ρ)2(N −1) : min j6=i t∈[0,T ]inf yε ij(t) ≤ εo≤ C    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε2s _if _{s < 0.5.} (43) where the constant C only depends on N , |ai|, T and ρ.

The proof of this last lemma is reminiscent from the proof of Theorem 2.5 with some new arguments. The main idea is to rely on a Bienaym´e-Tchebycheff inequality applied to a well-chosen function. This last estimate (43) with Lemma 3.4 concludes the proof of Theorem 2.7.

4 Technical proofs

4.1 Proofs of the Lemmas for the Hamiltonian formulation

4.1.1 Proof of Lemma 1.2 Let X ∈ R2N \ CT, we have d dtH(S t_{X) =} N X i=1 N X j=1 j6=i aiaj∇G |xi(t) − xj(t)|) · d dt xi(t) − xj(t) . (44)

Using the equations of motion (4) gives d dtH(S t_{X) =} N X i=1 N X j=1 j6=i N X k=1 k6=i,j aiajak∇G |xi(t) − xj(t)|) · ∇⊥G|xi(t) − xk(t)|) − N X i=1 N X j=1 j6=i N X k=1 k6=i,j aiajak∇G |xi(t) − xj(t)|) · ∇⊥G |xj(t) − xk(t)|), (45)

where we used the identity ∇G(|x|) · ∇⊥G(|x|) = 0. If we relabel the indices of the second sum above by swapping the roles as follow: i → k → j → i, we get

d dtH(S t_{X) =} N X i=1 N X j=1 j6=i N X k=1 k6=i,j aiajak∇G |xi(t) − xj(t)|) · ∇⊥G|xi(t) − xk(t)|) − N X i=1 N X j=1 j6=i N X k=1 k6=i,j aiajak∇G |xk(t) − xi(t)|) · ∇⊥G |xi(t) − xj(t)|). (46)

Using now the identity ∇G(| − x|) · ∇⊥G(|y|) = ∇G(|y|) · ∇⊥G(|x|), we obtain that the two sum appearing in the expression above are equal and therefore

d dtH(S

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4.1.2 Proof of Lemma 1.4 Let X ∈ R2N _{\ C} T, we have d dtM (S t_{X) =} N X i=1 ai d dtxi(t). (47)

Using the equations of motion (4) gives d dtM (S t_{X) =} N X i=1 X j=1 j6=i aiaj∇⊥G |xi(t) − xj(t)|. (48)

If we relabel the sum above by swapping the roles of the indices: i ↔ j, then d dtM (S t_{X) = −} N X i=1 X j=1 j6=i aiaj∇⊥G |xi(t) − xj(t)|, (49)

where we used the identity ∇⊥G(| − x|) = −∇⊥G(|x|). Thus, combining these two last equalities gives d dtM (S t_{X) = 0.} ₍₅₀₎ 4.1.3 Proof of Lemma 1.5 Let X ∈ R2N _{\ C} T, we have d dtI(S t_{X) = 2} N X i=1 aixi(t) · d dtxi(t). (51)

Using the equations of motion (4) gives d dtI(S t_{X) = 2} N X i=1 N X j=1 j6=i aiajxi(t) · ∇⊥G |xi(t) − xj(t)|. (52)

If we relabel the sum above by swapping the roles of the indices: i ↔ j, then d dtI(S t_{X) = −2} N X i=1 N X j=1 j6=i aiajxj(t) · ∇⊥G |xi(t) − xj(t)|, (53)

where we used the identity ∇⊥G(| − x|) = −∇⊥G(|x|). Combining these two equalities leads to

d dtI(S t_{X) =} N X i=1 N X j=1 j6=i aiaj xi(t) − xj(t)· ∇⊥G |xi(t) − xj(t)|. (54)

Observing now that the vector ∇⊥G(|x|) is orthogonal to the vector x, we deduce that all the scalar products appearing in the expression above are 0. Thus,

d dtI(S

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4.2 Proofs for the uniform bound results

4.2.1 Proof of Theorem 2.1

The proof is done with a recursive argument on the number of vortices. Denote xG,N_i (t) for i = 1 · · · N the solution to the N point-vortices problem (4) that interact through the rotating potential which profile is given by G. We recall here that G ∈ C_loc1,1(R∗₊) ∩ C1,1([1, +∞)). We add to this system the action of a smooth exterior field F .

d dtx N i = N X j=1 j6=i aj∇⊥G |xNi (t) − xNj (t)| + Fi(xNi (t), t). (56)

This field is here to model the interaction with distant vortices. When vortices are far enough, it is possible to remove them from the system and replace them by this smooth external field. This idea permits to do the recursive argument. Assume that

|Fi(x, t)| ≤ λ where λ > 0 is fixed. (57)

The recursive hypothesis consists in assuming that for all k ≤ N there exists a constant Ck that

satisfies

sup

t∈[0,T ]

|SG,kt X − X| ≤ Ck, (58)

where S_G,kt designates the flow of the equation (56) with k vortices and associated to the profile G. This constant Ck depends on the number of vortices N , the intensities ai, the final time T

and on sup_r≥1|dG_dr(r)|. It also depends on the choice of the parameter λ. This constant Ck is

independent on the initial datum X ∈ R2N\ CT and independent on the singularity of kernel G.

We are going to prove that there exist a constant CN +1 such that

sup

t∈[0,T ]

|SG,N +1t X − X| ≤ CN +1. (59)

We also prove that this constant CN +1only depends on the number of vortices N , the intensities

ai, the final time T , on supr≥1|dGdr(r)| and on λ. We now fix the value of the bound for

the external field Fi. This value is denoted λ0. Assume for the sake of contradiction that

“CN +1= +∞”. More precisely, assume that for S chosen arbitrarily large, there exist a profile

G, a time t∗ _{∈ [0, T ] and an index j such that}

|xG,N +1_j (t∗) − xj| = S. (60) Define a = max i=1···N +1|ai|, (61) A = min P⊆{1···N +1} P6=∅ X i∈P ai , (62) M_{N +1}G (t) := N +1_X i=1 aixG,N +1_i (t). (63)

Using Lemma 1.4, we get that the vorticity vector varies only under the action of the external field F . This external field is bounded by λ0 as stated in (57) and then

MG N +1(t) − MN +1G (0) = N +1_X i=1 ai Z t 0 FxG,N +1_i (t′), t′dt′ ≤ (N + 1) λ0a T =: b. (64)

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On the other hand, MG N +1(t∗) − MN +1G (0) = N +1_X i=1 ai xG,N +1j (t∗) − xj + N +1_X i=1 ai zG,N +1i (t∗) − z G,N +1 i (65) with zi := xG,N +1i − x G,N +1

j and where t∗and j are given by (60). Combining (60), (64) and (65)

with a triangular inequality S N +1_X i=1 ai ≤ b + N +1_X i=1 ai ziG,N +1(t∗) − ziG,N +1 . (66)

As a consequence and since |PN +1_i=1 ai| ≥ A,

SA − b ≤ a(N + 1) max i=1···N +1 zG,N +1 i (t∗) − z G,N +1 i . (67)

We note k the index for which is realized the maximum above and then we obtain zG,N +1 k (t∗) − z G,N +1 k ≥ SA − b a(N + 1). (68)

The first consequence is the existence of a time t† for which

|xj(t†) − xk(t†)| ≥

SA − b 2a(N + 1).

At such a time, the particles are divided into two clusters separated by a distance d that is bounded from below:

d ≥ SA − b

2a(N + 1)2.

Indeed, the least favorable case consists in the situation where all the remaining vortices forms a rectilinear chain made of N + 1 − 2 points spaced with intervals of same length and that link the vortex of index j to the vortex of index k.

The non-neutral clusters hypothesis given by (20) implies that A > 0. Therefore, if S is chosen very large then so is d. We can for instance choose S large enough so that d ≥ 1. This implies that the interaction between these two clusters is bounded by

(N + 1) a sup

r≥1

dG_dr(r)_. (69)

Now consider one of the two clusters. Since the interaction term is bounded by (69), then the dynamic restricted to this cluster is the point-vortex dynamic plus an external field, as in (56). This external field is bounded by

λ := λ0+ (N + 1) a sup r≥1

dG_dr(r)_.

This means that we can apply the recursive hypothesis (58) to each of these clusters which are made of a number of vortices equal or lower than N + 1.

At time t†, the vortices of one cluster evolve under the action of the other vortices of the cluster and under the action of an exterior field which strength is bounded by λ. The recursive hypothesis provides a constant CN such that on an interval of time [0, T ] the vortices do not

move on a distance larger than CN. We now choose the parameter S much bigger in such a way

that the distance d between the clusters cannot be filled during the interval of time [0, T ). This gives a contradiction because in this configuration the displacement of the vortices is uniformly bounded by CN whereas it is assumed that at least one moves on a distance S and S can be

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4.2.2 Proof of Corollary 2.2

Assume for the sake of contradiction that there exist T > 0 such that lim inf

t→T mini6=j |xi(t) − xj(t)| = 0. (70)

Then it is possible to extract an increasing sequence of time (tn) and two indices i0 6= j0 such

that

lim

n→+∞ |xi0(tn) − xj0(tn)| = 0. (71)

Consider two indices k06= l0. Two cases occurs:

either lim inf

n→+∞ |xk0(tn) − xl0(tn)| = 0 or ∀ n ∈ N, |xk0(tn) − xl0(tn)| ≥ c > 0.

(72) In the first case, it is possible to obtain that

lim

n→+∞ |xk0(tn) − xl0(tn)| = 0, (73)

provided that we proceed to another omitted extraction of the sequence (tn). We considering

now successively all the possible pairs of indices i 6= j and we proceed iteratively to such an extraction if the first case of (72) happens. We eventually end up with a set of couples P ⊆ {(i, j) : i, j = 1 · · · N with i 6= j} and an increasing sequence of time (tn) such that

∀ (i, j) ∈ P, xi(tn) − xj(tn) −→ 0+ as n → +∞ (74)

and

∀ (i, j) /∈ P, i 6= j, xi(tn) − xj(tn) ≥ c > 0, (75)

with c > 0 a constant independent of n. Since the ai are all positive, the non-neutral cluster

hypothesis is satisfied and then Theorem 2.1 in apply. The trajectories remain bounded by a constant C. Combining this with (75) gives

∀ (i, j) /∈ P, i 6= j, G |xi(tn) − xj(tn)| ≥ min

r∈[c,C]G(r). (76)

Since the ai are all positive, we deduce that

X i6=j aiajG |xi(tn) − xj(tn)| ≥ X (i,j)∈P i6=j aiajG |xi(tn) − xj(tn)|+ min r∈[c,C]G(r) X (i,j) /∈P i6=j aiaj. (77)

Using again the positivity of the coefficients ai, we obtain that hypothesis (22) combined

with (74) imply that X

(i,j)∈P i6=j

aiajG |xi(tn) − xj(tn)|−→ +∞ for n → +∞. (78)

Thus, in the view of (77), X

i6=j

aiajG |xi(tn) − xj(tn)|−→ +∞ for n → +∞. (79)

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Let S > 0 parameter very large fixed later. Suppose by the absurd that there exists an initial datum X ∈ R2N \ CT, two indices i0 and j0 and a time t0 such that

|xi0(t0) − xj0(t0)| ≥ max

i,j |xi− xj| + S. (80)

By continuity of the trajectories, there exists an interval of time [t1, t2] such that

|xi0(t1) − xj0(t1)| = max

i,j |xi− xj| + S, |xi0(t2) − xj0(t2)| = maxi,j |xi− xj| +

S

2, (81)

and such that

∀ t ∈ [t1, t2], |xi0(t2) − xj0(t2)| ≥ max

i,j |xi− xj| +

S

2. (82)

As a consequence of this last property and similarly as in the proof of Theorem 2.1, we can split the system of vortices into two non-empty subsets P and Q with P ∪ Q = {1 · · · N } such that for all t ∈ [t1, t2],

min

i∈P minj∈Q|xi(t) − xj(t)| ≥

S

2N. (83)

It can be assumed for instance that i0 ∈ P and j0 ∈ Q. Therefore, the vortices in the set P

evolves according to equation d dtxi(t) = X j∈P j6=i aj∇⊥G |xNi (t) − xNj (t)| + Fi(xNi (t), t). (84)

where the external fields Fi is the interaction with the vortices that belongs to Q. As a

conse-quence of (83), and if S is chosen large enough, this field satisfies |Fi(x, t)| ≤ sup

r≥1

_drd G(r)_. (85)

The analogous equation holds for the vortices of the set Q. We now want to apply Theorem 2.1 to the dynamic of the cluster P on [t1, t2] given by (84). The non-neutral clusters hypothesis is

satisfied for the cluster P as a consequence of (24) because P is a strict subset of {1 · · · N }. We obtain from Theorem 2.1 a constant C such that the dynamic of the cluster P without external field Fi is bounded by C. If we add the smooth external field Fi(x, t) and since the bound given

by Theorem 2.1 is uniform, we end up with sup t∈[t1,t2] max i∈P |xi(t) − xi(t1)| ≤ C + Z t2 t1 sup x∈R2N |Fi(x, t′)| dt′. (86)

Combining this with (85) gives sup t∈[t1,t2] max i∈P |xi(t) − xi(t1)| ≤ C + T sup_r≥1 _drdG(r)_. (87)

Doing an analogous argument on the cluster Q gives sup t∈[t1,t2] max i∈Q |xi(t) − xi(t1)| ≤ C + T sup_r≥1 _drdG(r)_. (88)

On the other hand, the conditions (81) imply that |xi0(t2) − xi0(t1)| ≥

S

4 or |xj0(t2) − xj0(t1)| ≥

S

4. (89)

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4.3 Proofs of the lemmas for Theorem 2.4

4.3.1 Proof of Lemma 3.1

Let PX(t) be the distribution defined by (28), assumed well-defined for all t ∈ [0, T [. Let

ϕ ∈ D(R2). Then, d dt PX(t), ϕ_D′_D = d dt N X i=1 aiϕ xi(t)= N X i=1 ai∇ϕ xi(t)· d dtxi(t). (90)

Then, t 7→ PX(t), ϕ_D′_D is Lipschitz and converges as t → T

−_{. Since this holds for any ϕ ∈}

D(R2_{), this implies that P}

X(t) converges in the sense of distributions towards some PX ∈ D′(R2)

as t → T−. There remains to prove that PX is actually a measure that takes the form given

by (29). Consider now an increasing sequence (tn) converging towards T−. Remark first that it

is always possible, up to an omitted extraction of the sequence, to reduce the problem to either xi(tn) −→ x∗i or

xi(tn)

−→ +∞ (93)

for some X∗ ∈ R2N_{. Indeed, if} _x

i(tn) −→ +∞ is not satisfied then there exists an extraction

such that xi(tn) stays bounded. But if it stays bounded then another extraction makes this

sequence converge towards some x∗_i. Repeting this process for i from 1 to N gives (93). Now that (93) holds, define

bi = 0 if xi(tn) −→ +∞, 1 either. (94) Therefore it holds N X i=1 aiδxi(tn)−→ N X i=1 aibiδx∗ i as n → +∞. (95)

By uniqueness of the limit, it is possible to identify PX = N X i=1 aibiδx∗ i. (96)

The fact that the convergence of PX(t) towards PX in D′ is actually a convergence in the weak

sense of measure comes from the fact that the measure PX(t) is bounded byP_i|ai| for all t.

• Step 1: Consider the X∗ ∈ R2N _{given by Lemma 3.1. Let z ∈ R}2_{such that for all i = 1 · · · N ,}

z 6= x∗_i. We are going to prove that for all i = 1 · · · N , lim inf

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Suppose by the absurd that there exists A ⊆ {1 · · · N } with A 6= ∅ such that for all i ∈ A, lim inf

t→T− |xi(t) − z| = 0. (98)

This set A can be chosen such that for all i /∈ A, lim inf

t→T− |xi(t) − z| > 0. (99)

Define

d1_z:= min|x∗_i − z| : i = 1 · · · N > 0, d2_z:= minlim inf

t→T− |xi(t) − z| : i /∈ A

> 0, d∗_z:= mind1_z, d2_z > 0,

(100)

where by convention, for the definition of d2_z, the minimum of the empty set is +∞. Let ϕ be a C∞ function supported on the ball B(z, d∗_z/2) and equal to 1 on the ball B(z, d∗_z/4). As a consequence of Lemma 3.1 (and Theorem 2.1 to obtain that bi = 1 for all i) and by definition

of d∗_z we have, D_XN i=1 aiδxi(t), ϕ E D′_,D −→ D_XN i=1 aiδx∗ i, ϕ E D′_,D = 0 as t → T −_. ₍₁₀₁₎

Using now (98) and (99), we infer the existence of an increasing sequence (tn)n∈N converging

towards T− such that for all n ∈ N,

∀ i ∈ A, xi(tn) ∈ B z, d ∗ z 4 and ∀ i /∈ A, xi(tn) /∈ B z, d ∗ z 2 . (102)

With the definition of ϕ and the definition of d∗_z given at (100), holds for all n ∈ N, D_XN i=1 aiδxi(tn), ϕ E D′_,D = X i∈A ai. (103)

As a consequence of the non-neutral clusters hypothesis (20) we have P_i∈Aai 6= 0. Therefore,

Equations (101) and (103) are in contradiction and then (97) holds.

• Step 2: Assume now that a given vortex xi(t) has two adherence points. By (97), these

two points must be some x∗_i. For instance, x∗_j and x∗_k with j and k such that x∗_j 6= x∗_k. Define the smallest distance between x∗_j and any other possible adherence point

r∗_j := minnr > 0 : ∃ l = 1 · · · N, |x∗_j − x∗_l| = ro. (104)

Consider then the circle

S :=nx ∈ R2 : |x − x∗_j(t)| = 1 2r ∗ j o . (105)

Since x∗_j is inside the ball of radius r_j∗/2 and x∗_k is outside, since these two points are adherence points for the dynamics of xi(t) as t → T−and since the trajectories are continuous, there exist

an increasing sequence of time (tn)n∈N converging towards T− such that

∀ n ∈ N, xi(tn) ∈ S. (106)

By compactness of S, it can be assumed that, up to an extraction, xi(tn) → x∗ ∈ S as n → ∞.

By definition of r∗_j, for all l = 1 · · · N , x∗_l ∈ S. These two facts together are in contradiction/ with (97) and this concludes the proof of Theorem 2.4.

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4.4 Proofs of the lemmas for Theorem 2.7

First, the conclusion of Theorem 2.7 can be formulated as follows

L2NnX ∈ R2N : ∃ TX ∈ R+, lim inf t→T− X min i6=j xi(t) − xj(t) = 0o = 0, (107)

where Ld _{refers to the Lebesgue measure of dimension d. It is possible to reduce the problem}

to bounded intervals of time and bounded regions of space by rewriting (107) as follows.

L2N +∞_[ T =1 +∞_[ ρ=1 n X ∈ R2N : ∃ TX ∈ [0, T ], lim inf t→T− X min i6=j xi(t) − xj(t) = 0 and max i6=j xi(t = 0) − xj(t = 0) ≤ ρ o = 0. (108)

Indeed, we can directly check that the two sets appearing respectively in (107) and (108) are equal. Since the reunion in (108) is a countable reunion, then to conclude that (107) holds it is enough to prove that for all T > 0 and ρ > 0,

L2NnX ∈ R2N : ∃ TX ∈ [0, T ], lim inf t→T− X min i6=j xi(t) − xj(t) = 0 and max i6=j xi(t = 0) − xj(t = 0) ≤ ρo= 0. (109)

Now, let T > 0 and ρ > 0. For i fixed in {1 · · · N } denote by Ti the isomorphism that gives

the position of the point vortices (xk)Nk=1knowing the position of xi and knowing the differences

(yij)j6=i. In other words define,

Ti : R2× R2(N −1) → R2N

(x, Y ) 7→ (x + yi1· · · x + yi(i−1), x, x + yi(i+1)· · · x + yiN). (110)

Thus, the following inclusion holds X ∈ R2N : ∃ TX ∈ [0, T ], lim inf t→T− X min j=1···N j6=i xi(t) − xj(t) = 0 and max i6=j xi(t = 0) − xj(t = 0) ≤ ρ. ⊆ Ti

R2_×n_Y_i _{:= (y}_ij₎_j6=i _{∈ R}2(N −1)_{: ∃ T}_X _{∈ [0, T ],} _{lim inf}

t→T− X min j6=i yij(t) = 0 and max i6=j yij(t = 0) ≤ ρ.o. = Ti

R2_×n_Y_i _{:= (y}_ij₎_j6=i _{∈ B(0, ρ)}2(N −1)_{: ∃ T}_X _{∈ [0, T ],} _{lim inf}

t→T− X min j6=i yij(t) = 0o. (111)

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We conclude that the studied set satisfy n X ∈ R2N : ∃ TX ∈ [0, T ], lim inf t→T− X min i=1···N j=1···Nmin j6=i xi(t) − xj(t) = 0 and max i6=j xi(t = 0) − xj(t = 0) ≤ ρ. o = N \ i=1 X ∈ R2N : ∃ TX ∈ [0, T ], lim inf t→T− X min j=1···N j6=i xi(t) − xj(t) = 0 and max i6=j xi(t = 0) − xj(t = 0) ≤ ρ. ⊆ N \ i=1 Ti

R2_×n_Y_i_{:= (y}_ij₎_j6=i_{∈ B(0, ρ)}2(N −1) _{: ∃ T}_X _{∈ [0, T ],} _{lim inf}

t→T− X min j6=i yij(t) = 0 o . (112) Using now hypothesis (32) and the Fubini theorem,

L2N

R2_× n

Yi := (yij)j6=i ∈ B(0, ρ)2(N −1): ∃ TX ∈ [0, T ], lim inf t→T− X min j6=i yij(t) = 0 o = 0. (113) Since Ti is a linear map, it is absolutely continuous and therefore maps any sets of Lebesgue

measure 0 into sets of Lebesgue measure 0. Therefore, combining this fact with (112) and (113) gives (109).

Let T > 0, ρ > 0 and ε > 0. For i 6= j, we define the set of initial datum such that occurs an ε-collapse between the two vortices xi and xj

Γε,ρ_ij :=nYi= (yil)l6=i ∈ B(0, ρ)2(N −1): ∃ t ∈ [0, T ],

yij(t) ≤ ε

o

. (114)

We also define the time at which occurs the ε-collapse. Let Yi ∈S_j6=iΓε,ρ_ij ,

T_Yε_i := infnt ∈ [0, T ] : min j6=i yij(t) ≤ ε o . (115)

We are also interested in the situations where other collapses occur, far from xi. This corresponds

to the ε-collapses of vector yjk:= xj− xk= yij− yik. Let k 6= i, j, define

Γε,ρ_ijk:=nYi ∈ Γε,ρij : ∃ t < TYεi,

yij(t) − yik(t)

≤ εo. (116)

The fact that Γε,ρ_ijk gives information on whether another ε-collapse occurs far from xi with xj

before the expected ε-collapse between xi and xj implies the following inclusion.

Γε,ρ_ijk ⊆ Γε,2ρ_kj \ Γε,2ρ_kji . (117)

This inclusion must be understood as follows. If occurs an ε-collapse between xj and xk before

the first ε-collapse between xi and xj (left-hand side of the inclusion above), then in particular

we have an ε-collapse between xj and xk(the set Γε,2ρkj in the right-hand side above). Yet, since

we do not have an ε-collapse between xj and xi before the varepsilon-collapse between xj and

xk, we can remove the set Γε,2ρkji in the right-hand side of the inclusion above. Another important

inclusion is

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These two definitions (114) and (116) study the ε-collapse on the exact system with kernel Gs.

We need the same definitions with the regularized kernels Gs,ε.

b Γε,ρ_ij :=nYi= (yil)l6=i ∈ B(0, ρ)2(N −1): ∃ t ∈ [0, T ], yε ij(t) ≤ εo, b T_Yε_i := infnt ∈ [0, T ] : min j6=i yε ij(t) ≤ εo, b Γε,ρ_ijk :=nYi∈ bΓε,ρij : ∃ t < bTYεi, yε ij(t) − yikε(t) ≤ εo. (119)

Remark now that as long as the quantities yij andyij− yik remain higher than ε for all j 6= i

and k 6= i, j, then the dynamics of yij and yijε coincide as a consequence of (38). This property

implies in particular, using the sets defined at (114), (116) and (119), Γε,ρ_ij \ [ k6=i,j Γε,ρ_ijk = bΓε,ρ_ij \ [ k6=i,j b Γε,ρ_ijk . (120)

The hypothesis of Lemma 3.4 can be rephrased as follows: for all ρ > 0, and for all i 6= j,

L2NbΓε,ρ_ij −→ 0 as ε → 0+. (121)

Concerning the conclusion, it is enough to prove that for all ρ > 0, and for all i 6= j,

L2NΓε,ρ_ij −→ 0 as ε → 0+, (122)

because for all i = 1 · · · N the following equality holds: n

Yi:= (yij)j6=i∈ B(0, ρ)2(N −1) : ∃ TX ∈ [0, T ], lim inf t→T− X min j6=i yij(t) = 0o= +∞_\ n=1 [ j6=i Γ 1 n,ρ ij . (123)

The fact that the convergences (121) imply the convergences (122) is given by the following computations. First, using (117) we get

Γε,ρ_ij = " Γε,ρ_ij \ [ k6=i,j Γε,ρ_ijk # ∪ [ k6=i,j Γε,ρ_ijk ⊆ " Γε,ρ_ij \ [ k6=i,j Γε,ρ_ijk # ∪ [ k6=i,j Γε,2ρ_kj \ Γε,2ρ_kji . (124) Remark that it is possible to do the same computation with the remaining term on the very right using again (117) and this gives

Γε,2ρ_kj \ Γε,2ρ_kji = " Γε,2ρ_kj \ [ l6=k,j Γε,2ρ_kjl # ∪ [ l6=i,j,k Γε,2ρ_kjl ⊆ " Γε,2ρ_kj \ [ l6=k,j Γε,2ρ_kjl # ∪ [ l6=i,j,k Γε,4ρ_lj \ Γε,4ρ_ljk . (125)

Doing the same computation as above recursively until the residual term is empty and using (118) transforms (124) into Γε,ρ_ij ⊆ [ k6=j " Γε, 2_kj Nρ\ [ l6=j,k Γε, 2_kjlNρ # . (126)

Using now (120), we finally get Γε,ρ_ij ⊆ [ k6=j " b Γε, 2_kj Nρ\ [ l6=j,k b Γε, 2_kjlNρ # ⊆ [ k6=j b Γε, 2_kj Nρ. (127)

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4.4.3 Proof of Lemma 3.5 Let i ∈ {1 · · · N }, ε > 0 and ρ > 0.

Step 1. Let a > 0. We define a kernel La by

La(q) := q−2−a. (128)

and we associate to the kernel La its ε-regularization La,εas defined by (38)-(41). From this we

define the function

Φ(Yi) :=

X

j6=i

La,ε |yij|. (129)

This function is all the most high valued as the system is close to collapse with vortex xi. Denote

by St_i,ε the flow of the modified system (31) with the regularized kernel Gs,ε. This gives

functions. Thus, d dtΦ St_i,ε_Y_i ≤ Ψ St_i,ε_Y_i_, ₍₁₃₁₎ where Ψ Yi:= X j6=i X k6=i,j ak∇La,ε |yij| ∇Gs,ε |yik| + ∇Gs,ε |yij − yik| . (132)

We now observe, recalling the definition of Gs given at (3), that the ε- regularization (38)-(41)

implies, by a direct computation using polar coordinates, Z B(0,ρ) ∇Gs,ε |y||dy ≤ C    1 if s > 0.5, log(1/ε) if s = 0.5, ε2s−1 if s < 0.5, (133) where B(0, ρ) is the euclidean ball on R2_{. T he constant C depends on ρ and s. Similarly with}

the definition of the kernel La at (128) and since a > 0,

Z

B(0,ρ)

La,ε |y|dy ≤ Cε−a and

Z

B(0,ρ)

∇La,ε |y||dy ≤ Cε−1−a. (134)

Therefore using (134) with the definition of Φ gives Z

B(0,ρ)N −1

Φ(Yi) dYi ≤ Cε−a. (135)

Similarly, the definition of Ψ given at (132) gives Z B(0,ρ)N −1 Ψ(Yi) dYi = Z B(0,ρ)N −1 " X j6=i X k6=i,j ak ∇La,ε |yij| ∇Gs,ε |yik| + ∇Gs,ε |yij− yik| #_YN l=1 l6=i dyil = 2 X j6=i X k6=i,j ak Z B(0,ρ) dy N −3 Z B(0,ρ) ∇La,ε |y|dy Z B(0,ρ) ∇Gs,ε |y|dy , (136)

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and then, using (133) and (134), Z B(0,ρ)N −1 Ψ(Yi) dYi ≤ Cε−2−a    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε2s if s < 0.5. (137)

Step 2. It is now possible to integrate Φ along the flow. We obtain Z B(0,ρ)N −1 sup t∈[0,T ] Φ St_i,εYidYi≤ Z B(0,ρ)N −1 Φ(Yi) dYi+ Z B(0,ρ)N −1 Z T 0 d dtΦ St_i,ε_Y_i dt dYi (138) Using (131) in the estimate above gives

Z B(0,ρ)N −1 sup t∈[0,T ] Φ St_i,εYidYi ≤ Z B(0,ρ)N −1 Φ(Yi) dYi+ Z B(0,ρ)N −1 Z T 0 ΨSt_i,ε_Y_i dt dYi. (139)

Using the Fubini theorem in (139) and the Liouville theorem 3.3 leads to Z B(0,ρ)N −1 sup t∈[0,T ] Φ St_i,εYidYi ≤ Z B(0,ρ)N −1 Φ(Yi) dYi+ Z T 0 Z St i,εB(0,ρ)N −1 Ψ YidS−t_i,εYidt = Z B(0,ρ)N −1 Φ(Yi) dYi+ Z T 0 Z St i,εB(0,ρ)N −1 Ψ YidYidt. (140) We now make use of hypothesis (24) on the intensities of the vorticies. Indeed, this hypothesis allows us to use Theorem 2.3 which states the existence of a constant C′ independent on ε (but dependent on ρ, T , s and the ai) such that

St_i,ε_{B(0, ρ)}N −1_{⊆ B(0, C}′₎N −1_. ₍₁₄₁₎

Thus, the estimate (140) above becomes Z B(0,ρ)N −1 sup t∈[0,T ] Φ St_i,εYidYi ≤ Z B(0,ρ)N −1 Φ(Yi) dYi+ Z T 0 Z B(0,C′₎N −1 Ψ YidYidt. ≤ Cε−a+ C T ε−2−a    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε2s if s < 0.5. , (142)

where for the last estimate we used (135) and (137).

Step 3. By definition of the function Φ, there exists a constant c > 0 such that, n Yi= (yij)j6=i ∈ B(0, ρ)N −1: min j6=i t∈[0,T ]inf yε ij(t) ≤ εo ⊆nYi = (yij)j6=i ∈ B(0, ρ)N −1: sup t∈[0,T ] Φ St_i,εYi≥ c ε−2−a o . (143)

Combining this inclusion with the Bienaym´e-Tchebycheff inequality gives L2(N −1)nYi= (yij)j6=i∈ B(0, ρ)N −1: min j6=i t∈[0,T ]inf yε ij(t) ≤ εo≤ ε 2+a c Z B(0,ρ)N −1 sup t∈[0,T ] Φ St_i,εYidYi. (144)

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Using now (142) in (144), L2(N −1)nYi= (yij)j6=i∈ B(0, ρ)N −1: min j6=i t∈[0,T ]inf yε ij(t) ≤ εo≤ C    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε2s if s < 0.5. , (145) where C is a constant that depends on T , ρ, N , s and on the ai. The lemma is proved.

Acknowledgments

I would like to acknowledge my PhD advisors Philippe GRAVEJAT and Didier SMETS for their confidence and their scientific support, constructive criticism and suggestions during all my work on the point-vortex model. I would like also to acknowledge them for their meticulous rereadings and their advises during the redaction of this article.

The author acknowledges grants from the Agence Nationale de la Recherche (ANR), for the project “Ondes Dispersives Al´eatoires” (ANR-18-CE40-0020-01). The problem considered in this paper was inspired from the Workshop of this ANR ODA on 27-28 June 2019 in Laboratoire Paul Painlev´e (Lille, France).

Contact by e-mail: ludovic.godardcadillac@unito.it

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Vortex collapses for the Euler and Quasi-Geostrophic Models

HAL Id: hal-03111989

https://hal.archives-ouvertes.fr/hal-03111989

Vortex collapses for the Euler and Quasi-Geostrophic

Models

Ludovic Godard-Cadillac

To cite this version:

Vortex collapses for the Euler and Quasi-Geostrophic Models

Contents

1

Introduction

2

Main results

3

Outlines of the proofs

4

Technical proofs

Acknowledgments

References