Global existence and collisions for symmetric configurations of nearly parallel vortex filaments

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Ann. I. H. Poincaré – AN 29 (2012) 813–832

www.elsevier.com/locate/anihpc

Global existence and collisions for symmetric configurations of nearly parallel vortex filaments

Valeria Banica

^a,1

, Evelyne Miot

^b,^∗

aLaboratoire Analyse et Probabilités (EA 2172), Département de Mathématiques, Université d’Evry, 23 Bd. de France, 91037 Evry, France bLaboratoire de Mathématiques, Université Paris-Sud 11, Bât. 425, 91405 Orsay, France

Received 19 August 2011; accepted 10 April 2012 Available online 26 May 2012

Abstract

We consider the Schrödinger system with Newton-type interactions that was derived by R. Klein, A. Majda and K. Damodaran (1995)[17]to modelize the dynamics ofNnearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid.

The known large time existence results are due to C. Kenig, G. Ponce and L. Vega (2003)[16] and concern the interaction of two filaments and particular configurations of three filaments. In this article we prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations ofN nearly parallel filaments for anyN2. We also show the existence of travelling wave type dynamics. Finally we describe configurations leading to collision.

1. Introduction

In this paper we study the dynamics ofN interacting vortex filaments in a 3-dimensional homogeneous incompressible fluid. We focus on filaments that are all nearly parallel to the z-axis. They are described by means of complex-valued functions Ψ_j(t, σ )∈C, 1j N, wheret ∈Ris the time,σ∈Rparameterizes thez-axis, and Ψ_j(t, σ )is the position of thej-th filament. A simplified model for the dynamics of such nearly parallel filaments has been derived by R. Klein, A. Majda and K. Damodaran[17]in the form of the following 1-dimensional Schrödinger system of equations

⎧⎪

⎨

⎪⎩

i∂tΨj+Γj∂_σ²Ψj+

k=j

Γk

Ψ_j−Ψ_k

|Ψ_j−Ψ_k|² =0, 1jN, Ψ_j(0, σ )=Ψ_j,0(σ ).

(1.1)

* Corresponding author.

E-mail addresses:[email protected](V. Banica),[email protected](E. Miot).

1 Partially supported by the French ANR project R.A.S. ANR-08-JCJC-0124-01.

http://dx.doi.org/10.1016/j.anihpc.2012.04.005

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HereΓ_jis a real number representing the circulation of thej-th filament.²In the case whereΨ_j(t, σ )=Ψ_j(t)=X_j(t) are exactly parallel filaments, system (1.1) reduces to the well-known point vortex system arising in 2-dimensional homogeneous incompressible fluids

⎧⎪

⎨

⎪⎩ idX_j

dt +

k=j

Γk

X_j−X_k

|X_j−X_k|²=0, 1j N, X_j(0)=X_j,0.

(1.2) The system (1.1) combines on the one hand the linearized self-induction approximation for each vortex filament, given by the linear Schrödinger equation, and on the other hand the interaction of the filaments, for anyσ, by the point vortex system. Solutions of the simplified model (1.1) have remarkable mathematical and physical properties, as described in[20]. The main issue in this context is the possibility of collision of at least two of the filaments in finite time at some pointσ.

Before presenting the known results on nearly parallel vortex filaments let us briefly review some classical facts on the point vortex system (1.2). Its dynamics preserves the center of inertia

jΓjXj(t), the angular momentum

jΓ_j|X_j(t)|²and the quantities

j=k

Γ_jΓ_klnX_j(t)−X_k(t)²,

j=k

Γ_jΓ_kX_j(t)−X_k(t)².

In case of circulations having all the same signs this implies that no collision among the vortices can occur in finite time. Therefore there exists a unique globalC¹ solution (X_j(t))_j to (1.2). ForN =2 global existence still holds independently of the circulation signs since|X₁(t)−X₂(t)| remains constant. When dealing with more than two vortices the single-sign assumption of the circulations really matters – explicit examples of configurations leading to collapse in finite time have been given by self-similar shrinking triangles[1]. For any circulations the equilateral triangle is a rotating or translating configuration, and for identical circulations the ends and the middle of a segment form also a relative equilibrium configuration. ForN 4 and identical circulationsΓ_j=Γ ∀j, vertices of regular polygons also form relative equilibrium configurations. They rotate around the center of inertia with constant angular velocityω=Γ (N−1)/(2R²), whereR is the size of the polygon. These polygon configurations are stable if and only ifN7. The proof of this result, conjectured by Kelvin in 1878, was recently completed by L.G. Kurakin and V.I. Yudovitch in 2002[18](see also[22]). Finally, the configuration formed by adding to anN-polygon configuration one point of arbitrary circulationΓ₀ at the center of inertia, is a relative equilibria rotating with constant angular velocityω= [Γ (N−1)+2Γ0]/(2R²). A natural observation to be done is that asN increases the dynamics gets more and more sophisticated.

The first result on nearly parallel vortex filaments has been given in[17]. The authors proved that forN=2 the linearized system around the exactly parallel filaments solution of (1.2) is stable if the circulations have the same sign and unstable otherwise. Moreover, they made numerical simulations suggesting global existence for (1.1) in the first case and collision in finite time in the second case. Their first conjecture on global existence was proved then by C. Kenig, G. Ponce and L. Vega [16]for filamentsΨ_j obtained as small H¹ perturbations of exactly parallel filamentsX_j,

Ψj(t, σ )=Xj(t)+uj(t, σ ), 1jN. (1.3)

More precisely, it has been proved in[16](see also the survey[23]) that foru_j(0)sufficiently small inH¹(R)– and therefore inL^∞(R)– global existence and uniqueness of the solution to system (1.1) hold for all vortex solutions (X_j)_j of equal circulations and such that|X_j(t)−X_k(t)| =d for 1j =kN. The only such possible configurations areN=2 with any pair(X₁, X₂), andN =3 with (X₁, X₂, X₃)an equilateral triangle. Moreover, local existence and uniqueness hold for any numberN of filaments and any circulationsΓ_j and the solution exists at least up to times of order|ln

juj(0)H¹|.

Finally, let us mention that P.-L. Lions and A. Majda[19]developed an equilibrium statistical theory for nearly parallel filaments using the approximation given by system (1.1).

2 The free Schrödinger operator derived in[17]is actuallyi∂t+α_jΓ_j∂_σ², whereα_j is another vortex core parameter related to thej-th filament.

For simplicity we assume throughout the paper thatα_j=1.

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The purpose of this article is to study other specific configurations of vortex filaments. In order to obtain large time existence results we will strongly use the symmetry properties of the configuration of the straight filaments(X_j)_j in itself, and those of the perturbation(u_j)_j on the other hand.

In the first part of this paper we focus on the case whereN3 and(X_j)_j is a regular rotating polygon of radius 1 withN vertices, with or without its center. The indexj=0 refers to the center of the polygon and 1j Nto the vertices of the polygon. Since (1.1) is invariant under translations, we can suppose that the center of inertia of the polygon is set at the origin, i.e.X₀(t)=0 for allt. We shall impose that the circulations in the vertices have the same valueΓ and thatωhas the same sign asΓ. For simplicity we consider

Γ_j=1, 1jN.

In the cases where the center of the polygon is not considered, the angular speedωis(N−1)/2, hence positive. In the cases when the center of the polygon is considered, the circulationΓ0must be larger than−(N−1)/2.

We will consider very specific perturbations of the configuration(Xj)j, assuming that all the perturbations are the same for each of the straight filaments, a dilation combined with a rotation. More precisely we shall focus on solutions having the form

Ψ_j(t, σ )=X_j(t)Φ(t, σ ), (1.4)

withΨ (t, σ )close toX_j(t)in some sense as|σ| → ∞. Let us notice that this dilation–rotation type of perturbations keeps the symmetry of the polygon for all(t, σ ). A natural example of such perturbations are the ones withΦ−1 small in H¹(R). Our result below allows to handle a larger class of perturbations of the regular rotating polygon, including also for example all small constant rotations of the polygon.

Theorem 1.1.LetN3and(X_j)_j be the equilibrium solution given by a regular rotating polygon of radius1, with or without its center, withΓ_j=1for1j N and positive angular velocityω. Assume that

Ψ_j,0(σ )=X_j,0Φ₀(σ ), withΦ₀such that

E(Φ₀)=1

2 |∂_σΦ₀|²+ω 2

|Φ₀|²−1−ln|Φ₀|²

satisfiesE(Φ₀)η₁, whereη₁is an absolute constant.³ Then there exists a unique global solution(Ψ_j)_j of (1.1), with this initial datum, such that

Ψ_j(t, σ )=X_j(t)Φ(t, σ ), t∈R withΦ−Φ0∈C(R, H¹(R)). Moreover

3

4|Ψj(t, σ )−Ψk(t, σ )|

|X_j(t)−X_k(t)| 5

4, t, σ∈R.

In particular, ifΦ₀(σ )^|^σ−→^|→∞1thenΨ_j(t, σ )^|^σ−→^|→∞X_j(t)∀t, and ifΦ₀∈1+H¹(R)thenΨ_j−X_j∈C(R, H¹(R)).

Remark 1.Theorem1.1does not assert that if initially Φ₀−1H¹ is small thenΦ(t )−1H¹ remains small for allt.

Our analysis is based on the observation that the solution(Ψ_j)_j to system (1.1) satisfies (1.4) if and only ifΦ is solution to the equation

i∂tΦ+∂_σ²Φ+ω Φ

|Φ|²

1− |Φ|²

=0. (1.5)

3 Introduced in Lemma2.1below.

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Eq. (1.5) is a hamiltonian equation, which preserves the energy E(Φ)=1

2 |∂σΦ|²+ω 2

|Φ|²−1−ln|Φ|²

. (1.6)

Note that in the setting of Theorem1.1 the solutions satisfy |Φ| 1, so that Eq. (1.5) is formally similar to the well-known Gross–Pitaevskii equation

i∂tΦ+∂_σ²Φ+ωΦ

1− |Φ|²

=0, (1.7)

with energy given by EGP(Φ)=1

2 |∂_σΦ|²+ω 4

|Φ|²−12

.

In fact we shall see that both functionalsE(Φ)andEGP(Φ)are comparable whenever|Φ| 1. A key point in the proof is, as in[16], the fact that ifE(Φ₀)is small then the solutionΦenjoys the property

sup

t∈R

Φ(t )²−1

L^∞1

4. (1.8)

This allows us to establish Theorem1.1by using the techniques introduced in[24]by P.E. Zhidkov (see also P. Gérard [11,12]) for solving the Gross–Pitaevskii equation in the energy space.

In the case whereΦ₀∈1+H¹(R)we mention that the proof in[16]can be adapted here, by showing that some quantities are still conserved even though|X_j(t)−X_k(t)|are not all the same.

As far as we have seen, global existence and uniqueness of the filaments hold forN=2 with any(X_j)_j and any small perturbations, forN=3 with(X_j)_j the equilateral triangle stable equilibrium and any small perturbations, for anyN2 with(X_j)_j the regular polygon equilibrium and any small perturbations with strong symmetry conditions.

We expect then that global existence might hold for smallN and less restrictive conditions on the perturbations.

In the second part of this paper we study the case N=4, Γ_j=1,

and we assume that(X_j)_j=(X₁, X₂, X₃, X₄)is a square of radius 1 rotating with constant angular speed. Again, since (1.1) is invariant under translations, we can suppose that the square is centered at the origin. Our main result in this case may be formulated as follows.

Theorem 1.2.LetN=4and(X_j)_j be the equilibrium solution given by a rotating square of radius1withΓ_j=1.

Let(u_j,0)_j∈H¹(R)⁴and setΨ_j,0=X_j,0+u_j,0. We introduce the energy⁴

E0=1 2

j

∂_σΨ_j,0(σ )²dσ +1

2

j=k

−ln

|Ψ_j,0(σ )−Ψ_k,0(σ )|²

|Xj,0−Xk,0|²

+

|Ψ_j,0(σ )−Ψ_k,0(σ )|²

|Xj,0−Xk,0|² −1

dσ.

We also introduce the quantity E0=max

E0; u_1,0+u_3,0²_L2

2 +u_2,0+u_4,0²_L2

2

and we assume that E0η₂

4 Note thatE00.

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for an absolute small constantη₂>0. Then there exists an absolute constantC >0, and there exists a timeT, with T Cmin

1 E0

1/4maxj,ku_j,0−u_k,0^1/2_L2

, 1 E0

1/3

,

such that there exists a unique corresponding solution(Ψ_j)_j to system(1.1)on[0, T], satisfyingΨ_j=X_j+u_j, with u_j∈C([0, T], H¹(R)), and such that

3

4|Ψ_j(t, σ )−Ψ_k(t, σ )|

|Xj(t)−Xk(t)| 5

4, t∈ [0, T], σ∈R.

Finally, if the initial perturbation is parallelogram-shaped, namely u_1,0+u_3,0_L²= u_2,0+u_4,0_L²=0,

then the solution(Ψ_j)_j is globally defined.

Remark 2.In the proof of Theorem1.2we shall actually establish a local existence result for anyN, any parallel configuration(X_j)_j, any set of positive circulations(Γ_j)_jand any perturbations with small energy, but not necessarily small inH¹. This is a slight improvement of the result in[16], see also the next two remarks.

Remark 3.As we shall see, we can infer from the smallness of the energyE0and from Sobolev embeddings that the nearly parallel filamentsΨ_j,0are not too far from the straight filamentsX_j,0and thatE0C

ju_j,0²_H1. Conversely, if we assume that

juj,0H¹ is sufficiently small then one can show thatE0C

juj,0²_H1 and the assumptions of Theorem1.2are satisfied. Therefore the hypothesis on the energy is less restrictive than the one on theH¹norm, see also the next remark.

Remark 4.From 0E0C

juj,0²_H1 it follows thatE0C

juj,0²_H1 so the time of existence is a priori larger than in[16]. Moreover, for all >0 Theorem1.2allows for initial perturbations of the form

Ψ_j,0 (σ )=e^iϕ^{(σ )}Xj,0+T(σ ),

withϕ, Tsuch that(ϕ, T)H¹=O(1). This amounts to rotating and translating the square(X_j)_jat each levelσ. By taking oscillating phases of the formϕ(σ )=√

εϕ₀(εσ )with a fixed ϕ₀∈H¹, which impliesϕL² O(1),

∇ϕL²=O()and by choosingTsuch thatTH¹=O()we compute E0=O

²

,

j

uj,0²_H1O(1).

Therefore Theorem1.2 provides a unique solution at least up to time of order 1/√

ε, while theH¹ norm of the perturbations is of order one. This suggests that the energy space is more appropriate for the analysis of (1.1) than classical Sobolev spaces.

The proof of Theorem1.2follows the one of Theorem1.1combined with the one in[16]. In particular, we consider, as in[16], the energy

E(t)=1 2

j

∂σΨj(t, σ )²dσ

+1 2

j=k

−ln

|Ψ_j(t, σ )−Ψ_k(t, σ )|²

|Xj(t)−Xk(t)|²

+

|Ψ_j(t, σ )−Ψ_k(t, σ )|²

|Xj(t)−Xk(t)|² −1

dσ, (1.9)

and show that the solution can be extended as long asE(t)remains small. For this purpose we show thatu_j can be extended locally from a timet₀by a fixed point argument for smallH¹perturbationsw_j of the linear evolutions of the

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initial data, i.e.u_j(t)=e^i(t⁻^t⁰^)∂^σ²u_j(t₀)+w_j(t). In here we use crucially the fact that the deviatione^i(t⁻^t⁰^)∂^σ²u_j(t₀)− u_j(t₀)can be upper-bounded inL^∞ in terms of the energy at the initial time E(t₀). As observed in[16], for any two parallel filaments and for the equilateral triangle configuration the energy is conserved, i.e.E(t)=E(0)=E0, so that global existence follows for small energy perturbations. Unfortunately, under the assumptions of Theorem1.2 the energy is no longer conserved (unless the perturbation(u_j)_j is parallelogram-shaped). Instead, we estimate its evolution in time, showing that it does not increase too fast, and this control enables us to obtain a large time of existence.

We finally mention another collection of dynamics that is governed by the linear Schrödinger equation. For shifted perturbationsΨ_j=X_j +u, for any X_j withΓ_j the same, we obtain thatu is a solution of the linear Schrödinger equation. So ifuis regular enough, it has constantH¹norm, so the filaments remain separated for all time. Moreover, due to the dispersive inequality for the linear Schrödinger equation, the perturbations spread in time along the parallel configuration Xj. Finally, we get examples ofC^∞ perturbations decaying at infinity that generate a singularity in finite time by considering less regular perturbations than H¹that lead to anL^∞ dispersive blow-up for the linear Schrödinger. The self-similar linear Schrödinger solution constructed from homogeneous data|x|⁻^pwith 0< p <1 in [6] leads to solutions blowing up in L^∞ in finite time at one point. Also, the linear Schrödinger evolution of eⁱ^|^x^|²/(1+ |x|²)^mwith 1/2< m1 has been proved in[5]to be anL²solution whose modulus blows up in finite time at one point.

The third part of this work is devoted to travelling waves for system (1.1). Let us recall that in the case of one single filament, a travelling wave dynamics was exhibited by H. Hasimoto [14]and experimentally observed by E.J. Hopfinger and F.K. Browand [15]. Here we construct travelling waves for several filaments via finite energy travelling wave solutions to Eq. (1.5), i.e. solutions of the formΦ(t, σ )=v(σ +ct), withvsolution of the equation

icv+v+ω v

|v|²

1− |v|²

=0 (1.10)

and having finite energy, E(v)=1

2 |∂_σv|²+ω 2

|v|²−1−ln|v|²

<∞. (1.11)

As in Theorem1.1we assume thatω >0. In order to avoid havingvapproaching zero we shall impose that the energy is small.

Existence, stability issues and qualitative behaviour near the speed of sound of travelling waves for Gross–

Pitaevskii-type equations and related problems were extensively studied in the past years (see for instance [9,25,4, 10,13,3,21,8]and the references therein). For the 1-dimensional Gross–Pitaevskii equation (1.7), finite energy travelling waves (referred to as “grey solitons” in the context of non-linear optics) are known to exist for all 0< c <√

2ω, and they have the explicit form (see e.g.[13])

v(σ )=vc(σ )= 1−

1

2ω(2ω−c²) cosh²(

√2ω−c²

2 σ )

eⁱ^arctan

ωe

√2ω−c2σ+c2−ω c√

2ω−c2 −iarctan√ ^c

2ω−c2.

The modulus|v_c|of such maps is close to 1 whencis close to√

2ω, in which caseE(v_c)CEGP(v_c)C(2ω−c²)^3/2 (see[13]), so the energy is finite and as small as needed. Note that therefore the mapsv_c, withcclose to√

2ωenter the class of perturbations presented in Theorem1.1. Our next result in this context is the following.

Theorem 1.3.Letcbe such that0<2ω−c²< η₃for an absolute small constantη₃>0. There exists a travelling wave solution to system(1.1)

Ψ_j(t, σ )=e^{it ω}⁺ⁱ^2πj^N v(σ+ct ),

wherev∈C^∞(R)is a solution to Eq.(1.10), with finite energyE(v)C(2ω−c²)^3/2, such thatvnever vanishes.

The modulus|v|is an even function, increasing on[0,∞)and satisfying onR 0<1−v(σ )²<min

3 2ω

2ω−c² , C

2ω−c²e⁻

√2ω−c²⁻|σ| .

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Finally, we have a limit at infinity

v(σ )→exp(iθ_±), σ→ ±∞, with|θ₊−θ₋|C

2ω−c². HereCdenotes an absolute numerical constant.

It has been noticed in[16]that the Galilean invariance of system (1.1) leads to helix-shaped vortex filaments. In here, on one hand Eq. (1.5) is invariant under Galilean transform, i.e.Φ_ν(t, σ )=e⁻^{it ν}²⁺^iνσΦ(t, σ −2t ν)is also a solution∀ν∈R. On the other handX_j(t)=e^{it ω}⁺ⁱ^2πj^N forj=0, so

Ψ_j,ν(t, σ )=e^{it (ω}⁻^ν²⁾⁺^iνσ⁺ⁱ^2πj^N Φ(t, σ−2t ν)

=e^{it (ω}⁻^ν²⁾⁺^iνσ⁺ⁱ^2πj^N v

σ+t (c−2ν) . Therefore, choosingν=√

ω, we obtain a stationary(θ₊−θ₋)-twistedN-helix filament configuration with some localized perturbation travelling in time on each of its filaments.

Last but not least, in the last part of this paper we describe configurations of nearly parallel filaments that lead to a collision in finite time. They are obtained by the same kind of dilation–rotation perturbations as in Theorem1.1.

Theorem 1.4.LetN2 and(Xj)j be the stationnary configuration given by a regularN-polygon with its center and circulationsΓj=1for1jN,Γ0= −(N−1)/2. Then the initial condition

Ψ_j,0(σ )=X_j(0)

1− e⁻ ^σ

2 1−4i

√1−4i

yields a solution(Ψ_j)_j for system(1.1), withΨ_j−X_j∈C(R, H¹(R)), that collide at timet=1atσ=0.

The remainder of this paper is organized as follows. In Section 2 we derive Eq. (1.5). We then present some preliminary lemmas about its energy, which lead to the proof of Theorem 1.1. Section 3 is devoted to the proof of Theorem1.2. Section4contains the construction of travelling waves for Theorem1.3. Finally, in Section5 we construct the collision dynamics in Theorem1.4. In all the following the notation C denotes an absolute constant which can possibly change from a line to another.

2. Proof of Theorem 1.1

We first derive Eq. (1.5). Plugging the ansatzΨ_j(t, σ )=X_j(t)Φ(t, σ )into system (1.1) withΓ_j=1 for 1jN we obtain

iX_j∂_tΦ+i∂_tX_jΦ+X_j∂_σ²Φ+ Φ

|Φ|²

k=j

Xj−Xk

|X_j−X_k|² =0.

Next we use (1.2) to get X_j

i∂_tΦ+∂_σ²Φ

−i∂_tX_j Φ

|Φ|²

1− |Φ|²

=0.

Now if we consider a configuration rotating with speedωaround its steady center of inertiaX₀=0, for 1j N we haveX_j(t)=e^{it ω}⁺^iθ^j, so that−i∂_tX_j=ωX_j and hence we obtain Eq. (1.5),

i∂tΦ+∂_σ²Φj+ω Φ

|Φ|²

1− |Φ|²

=0.

Conversely, assume thatΦis a solution to Eq. (1.5) and setΨ_j=X_jΦ. Reversing the previous arguments, we obtain i∂_tΨ_j+∂_σ²Ψ_j+

k=j

Ψj−Ψk

|Ψ_j−Ψ_k|² =0, 1jN,

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while, sinceΨ₀(t, σ )=0 for all(t, σ ), i∂_tΨ₀=∂_σ²Ψ₀=0 and

N k=1

Ψ0−Ψk

|Ψ0−Ψ_k|² = − Φ

|Φ|² N k=1

Xk

|X_k|²=0 and therefore(Ψ_j)_j is a solution to system (1.1).

2.1. Some preliminary lemmas

Lemma 2.1.There exists an absolute constantη1and a timet1depending only onη1such that:

(i) IfE(f )η1then |f|²−1

L^∞1 4.

(ii) If∂_σfL²η₁then for all0tt₁

√1 2

e^{it ∂}^σ²f−f

L^∞e^{it ∂}^σ²f −f

H¹1 4.

Proof. (i) The functiona(x)=x−1−lnxis positive and convex, and vanishes only atx=1, therefore we can adapt standard arguments already used in the context of Ginzburg–Landau-type functionals (see e.g.[2]). More precisely, we assume by contradiction that||f (σ₀)|²−1|>1/4 for someσ₀∈R. For example,|f (σ₀)|>√

5/4. Next, since ∂_σf²_L22E(f )we have by Cauchy–Schwarz inequality

f (σ )f (σ₀)−

σ

σ₀

∂_xf (x) dx

5 4 −

2E(f )|σ−σ₀|.

It follows that|f|>√

9/8 onI= [σ₀−1/(500E(f )), σ₀+1/(500E(f ))]. Therefore E(f )1

2a 9

8

|I| = 1 500E(f )a

9 8

, a contradiction ifE(f )η₁is sufficiently small.

(ii) The property (ii) is a known one used in the Gross–Pitaevskii study (see Lemma 3 in[11]) to which we recall the short proof: the Fourier transform ofe^{it ∂}^σ²f −f can be written as ^e⁻^itξ

2−1

ξ ξf (ξ ), so theˆ L²norm is bounded by C√

t∂_σfL² and theH˙¹norm is bounded byC∂_σfL², i.e.

e^{it ∂}^σ²f −f

H¹C(1+√

t )∂_σfL²C(1+√ t )η1.

We chooseη1small enough andt1small with respect toη1such that for 0tt1, e^{it ∂}^σ²f −f

H¹1 4,

and the conclusion of the lemma follows. 2

Since(x−1)²/4x−1−lnx10(x−1)²on[3/4,5/4]we immediately obtain the second lemma.

Lemma 2.2.If|f|²−1L^∞1/4then we can compare the energies:

EGP(f )≡1

2∂_σf²_L2+ω

8|f|²−1²

L²E(f )5EGP(f ).

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So, if we consider an initial perturbation such thatΦ₀−1 is sufficiently small inH¹, we infer from Sobolev embedding that EGP(Φ₀) <∞ and that |Φ₀|²−1L^∞ <1/4. Hence Lemma 2.2ensures thatΦ₀ belongs to the energy space associated to Eq. (1.5).

We will also need the following transposition of a standard property of the Gross–Pitaevskii energy (see[7,11,12]).

Lemma 2.3.Letf be such thatE(f )η₁, withη₁defined in Lemma2.1. Leth∈H¹(R)withh_H¹1/2. Then the energyE(f+h)is finite. More precisely we have, for absolute numerical constantsC,C,

E(f+h)CEGP(f+h)C

1+E(f )

1+ h²_H1

. Moreover,

|f +h| −1

L^∞2+√ 2 4 <1.

Proof. We first infer from Lemma2.1(i) that|f|−1L^∞1/4, and from Lemma2.2thatEGP(f ) <∞. Next, applying Gagliardo–Nirenberg inequality we gethL^∞√

2h_H¹√

2/2, so that|f+h| −1L^∞(2+√

2)/4<1.

By Lemma2.2it follows thatE(f+h)CEGP(f+h). Using thatEGP(f ) <∞andh∈H¹as well as Sobolev in- equalities we conclude thatEGP(f+h)is finite, with the corresponding estimate (see also, e.g., Lemma 2 in[11]). 2 2.2. Proof of Theorem 1.1

First we will establish local well-posedness for Eq. (1.5) by performing a fixed point argument for the operator A(w)(t )=i

t

0

e^i(t⁻^{τ )∂}^τ² e^{iτ ∂}^σ²Φ₀+w(τ )

|e^{iτ ∂}^σ²Φ0+w(τ )|²

1−e^{iτ ∂}^σ²Φ0+w(τ )² dτ on the ball

B_T =

w∈C

[0, T], H¹ , sup

0tT

w(t)

H¹1 4

,

withT small to be chosen later. ThenΦ(t )=e^{it ∂}^σ²Φ₀+w(t)will be a solution for (1.5) on[0, T]with initial dataΦ₀. Observe that the proof of Lemma2.1(ii) yields thatt→(e^{it ∂}^σ²Φ0−Φ0)∈C([0, T], H¹(R)). So the mapΦwill belong to the energy space ifΦ₀belongs to the energy space (by Lemma2.3applied tof =Φ₀andh=e^{it ∂}^σ²Φ₀−Φ₀+w(t) forT t₁witht₁from Lemma2.1), and it will belong to 1+H¹(R)ifΦ₀is in 1+H¹(R).

The hypothesis of Theorem1.1is that we start withΦ₀verifying E=E(Φ₀)=1

2∂_σΦ₀²_L2+ω 2

−ln|Φ₀| + |Φ₀|²−1 η₁.

We first imposeT t₁, witht₁defined in Lemma2.1. Letw∈B_T, and set for 0tT Φ(t ) =e^{it ∂}^σ²Φ₀+w(t)=Φ₀+

e^{it ∂}^σ²Φ₀−Φ₀+w(t) .

By Lemma2.1(ii) and by choice ofB_T, we haveΦ(t ) −Φ₀H¹ 1/2 on[0, T]. Therefore, applying Lemma2.3 tof =Φ₀andh=Φ(t ) −Φ₀we obtain that|Φ(t ) | −1L^∞(2+√

2)/4 on[0, T]. In particular, sinceC⁻¹

|Φ|CforC >0 we can estimate the action of the operator as follows A(w)(t )

H¹t sup

0τt

Φ(τ )

|Φ(τ ) |²

1−Φ(τ )²

H¹

C t sup

0τt

1−Φ(τ )²

L²+∂σΦ(τ )

L²

C t sup

0τt

EGP Φ(τ ) .

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We use again Lemma2.3and the boundΦ(τ ) −Φ₀_H¹1/2 to obtain sup

0tT

A(w)(t)

H¹C T (1+E).

Arguing similarly, we readily check that forw₁, w₂∈B_T sup

0tT

A(w1)(t)−A(w2)(t)

H¹C T (1+E) sup

0tT

w1(t)−w2(t)

H¹.

Hence imposing a second smallness condition onT with respect toEwe obtain a fixed pointwforAinBT. Therefore local well-posedness holds for Eq. (1.5) on[0, T]withT depending only onE.

Next, since the energy of Eq. (1.5) is conserved E

Φ(T )

=E Φ(0)

=E,

we re-iterate the local in time argument to get the global existence. Finally, Lemma2.1insures us that sup

t∈R

Φ(t )²−1

L^∞1 4, so the solution satisfies indeed

1

4 Φ(t, σ )5

4, t, σ∈R. 3. Proof of Theorem 1.2

3.1. Some useful quantities

From now on we will writeΨj k=Ψj−Ψk,Xj k=Xj−Xkanduj k=uj−uk.

We first introduce some useful quantities. In the general case whereN1 andΓj∈R, the dynamics of system (1.1) preserves the following quantities:

The energy 1 2

j

Γ_j² ∂_σΨ_j(t, σ )²dσ−1 2

j=k

Γ_jΓ_k lnΨ_{j k}(t, σ )²dσ, the angular momentum

j

Γ_j Ψ_j(t, σ )²dσ, and

j=k

Γ_jΓ_k Ψ_{j k}(t, σ )²dσ.

However the previous quantities are not well defined in the framework of Theorem1.2, not even formally, since Ψ_j(t, σ )andΨ_{j k}(t, σ )do not tend to zero at infinity. As in[16], we modify them in order to get well-defined quantities, introducing

H=1 2

j

Γ_j² ∂_σΨ_j(t, σ )²dσ−1 2

j=k

Γ_jΓ_k ln

|Ψ_{j k}(t, σ )|²

|Xj k(t)|²

dσ, A=

j

Γj

Ψj(t, σ )²−Xj(t)² dσ, T =

j=k

Γ_jΓ_k Ψ_{j k}(t, σ )²−X_{j k}(t)² dσ.

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Note that, in view of the properties of the point vortex system (1.2) mentioned in the introduction, the renormalized quantitiesH,AandT are still formally preserved in time.

Finally, we also introduce the time-dependent quantity I(t)=1

2

j=k

Γ_jΓ_k

|Ψj k(t)|²

|X_{j k}(t)|²−1

dσ, and we consider the energy

E(t)=H+I(t), (3.1)

which have been already introduced in (1.9) in the introduction.

As noticed in[16], a useful consequence of the convexity estimate(x−1)²/4x−1−lnx10(x−1)²on [3/4,5/4]is the inequality

1 2

j

Γ_j² ∂σΨj(t, σ )²dσ+1 8

j=k

ΓjΓk

|Ψ_{j k}(t, σ )|²

|X_{j k}(t)|² −1 2

dσE(t), (3.2)

which holds as long as the filaments satisfy 3/4|Ψ_{j k}(t)|²/|X_{j k}(t)|²5/4.

3.2. The approach

In this subsection we briefly sketch how to combine elements from[16]and from Section 2 to prove local existence and uniqueness of a solution to system (1.1) in the general case ofN filaments, withN2, and the way to extend this solution as long as the energyE(t)remains sufficiently small. Here we take positive circulations

Γ_j>0, 1jN.

Therefore there exists a unique global solution(X_j)_j to system (1.2). We denote by d >0 the minimal distance between the point vortices for all time. Here we shall make the extra-assumption that

u_j,0=Ψ_j,0−X_j,0∈H¹(R).

We look for a solutionu=(u_j)_j∈C([0, T], H¹(R))^Nto the system

⎧⎪

⎨

⎪⎩

i∂_tu_j+Γ_j∂_σ²u_j+

k=j

Γ_k

Xj k+uj k

|X_{j k}+u_{j k}|²− Xj k

|X_{j k}|²

=0, uj(0)=uj,0, 1jN.

(3.3)

By similar arguments as in Section2, our purpose is to find a fixed point in the Banach space B_T =

w=(w1, . . . , w_N)∈C

[0, T], H¹N

, sup

0tT

w(t)

H¹d 4

for the operatorA(w)=(A_j(w))_j defined by

A_j(w)(t)=i

t

0

k=j

Γ_k

Xj k(τ )+e^{iτ Γ}^j^∂^σ²uj,0+wj(τ )−e^{iτ Γ}^k^∂^σ²uk,0−wk(τ )

|Xj k(τ )+e^{iτ Γ}^j^∂^σ²uj,0+wj(τ )−e^{iτ Γ}^k^∂^σ²uk,0−wk(τ )|²− Xj k(τ )

|X_{j k}(τ )|²

dτ,

and forT sufficiently small with respect toη₂,

ju_j,0H¹,(Γ_j)_j andd. Then as in Section2the solution will be given by

uj(t)=e^{it Γ}^j^∂^σ²uj,0+wj(t).

By transposing the arguments of Section2we obtain the following local well-posedness result.