Zero-Noise Selection for Point Vortex Dynamics after Collapse

Francesco Grotto∗ and Marco Romito†

Dipartimento di Matematica, Universit`a di Pisa

Largo Pontecorvo 5, 56127 Pisa, Italia

Milo Viviani‡

Scuola Normale Superiore, Classe di Scienze

Piazza dei Cavalieri, 7, 56126 Pisa, Italia

(Dated: July 12, 2023)

The continuation of point vortex dynamics after a vortex collapse is investigated by means of a

regularization procedure consisting in introducing a small stochastic diffusive term, that corresponds

to a vanishing viscosity. In contrast with deterministic regularization, in which a cutoff interaction

selects in the limit a single trajectory of the system after collapse, the zero-noise method produces a

probability distribution supported by trajectories satisfying relevant conservation laws of the point

vortex system.

I. INTRODUCTION

The point vortex (PV) system,

x˙ i =

1

2π

XN

j̸=i

γj

(xi − xj)⊥

|xi − xj |2 , (PV)

describing the evolution of vortices with positions

x1, . . . , xN ∈ R2 and intensities γ1, . . . , γN ∈ R\ {0}, is a

classical model dating back to the works of Helmholtz [1],

describing the dynamics of an idealized 2-dimensional incompressible

fluid for which vorticity is concentrated in a

finite number of points. It is a Hamiltonian ODE system

with singular interactions between particles, and it is a

well-studied model for both its mathematical properties

and physical interest as a discretization method in fluid

dynamics [2].

The vector field driving (PV) is singular when the positions

of two or more vortices coincide. Singular configurations

of point vortices leading to collapse in finite

time, or (equivalently, by time inversion) the evolution

of many vortices starting from the same position

(vortex burst), have been known since the XIX century

[3]. A classical result by D¨urr-Marchioro-Pulvirenti [4, 5]

states that these singular configurations are a negligible

set of phase space with respect to absolutely continuous

invariant measures of the system, that is there exists

a measure-preserving flow on phase space consisting of

smooth solutions [6]. Nevertheless, there exists a large

variety of singular configurations, which are in fact dense

in phase space.

It was proved in [7] that given any configuration of N

point vortices at time t = 0, there exists a solution of

(PV) with N + 2 vortices, with three of them bursting

out of a single one at time t > 0. In the same way, it is

∗ francesco.grotto at unipi.it

† marco.romito at unipi.it

‡ milo.viviani at sns.it

possible to construct configurations of arbitrarily many

vortices leading to collapse in finite time, and this kind of

singular configurations contains codimension-1 submanifolds

of phase space. This reveals the existence of a rich

variety of vortex collapses and bursts, opening the possibility

of PV models for vorticity creation and energy

dissipation in incompressible fluids [8]. Moreover, advection

by the velocity field induced by PV systems close

to collapse can be regarded as a model for the chaotic

Lagrangian movement of particles in turbulent 2D flows

[9, 10], providing a further motivation for the study of

singular vortex configurations.

Considering PV dynamics as a generalized solution to

2D incompressible Euler equations (cf. Section IIIA below),

these facts imply a wild non-uniqueness of the time

evolution: for instance the construction in [7] can be used

to exhibit PV dynamics in which vortices split at completely

arbitrary times. Therefore, the question of what

should be the correct (physical) continuation of the evolution

after a vortex collapse, or before a vortex burst,

naturally arises.

In this paper, we investigate the dynamics of N = 3

vortices after their collapse through regularization by

noise: we will introduce stochastic forcing terms in (PV)

that prevent collision by means of small-scale fluctuations,

and then consider a zero-noise limit, that is the

(nontrivial) vortex dynamics obtained in the limit where

the diffusion coefficient vanishes. In particular, we will

exhibit examples in which the zero-noise limit, after a collapse,

is supported by many trajectories preserving first

integrals of motion. In other terms, we will produce a PV

evolution exhibiting a collapse at finite time, after which

a spontaneously stochastic (in the sense of [11, 12]) continuation

of the dynamics consists in a random choice of

a bursting solution.

Our result will be compared to the (deterministic) regularization

procedure in which the interaction kernel of

vortices is modified so to prevent collapses [13–16], in

which case a single possible trajectory is identified for

the vortices to follow after collapse. More generally, such

a procedure is a relevant tool in the investigation of ODE

arXiv:2307.05133v1 [physics.flu-dyn] 11 Jul 2023

2

models after formation of singularities [17].

The stochastic PV model we will consider is related (in

the large N Mean Field limit) to 2D Navier Stokes equations,

for which our procedure corresponds to considering

the inviscid limit. Therefore, the coexistence of the zeronoise

selection principle we propose with other regularization

procedures leading to different continuations for

(PV) after collapse indicates an intrinsic non-uniqueness

of PVs as a 2D fluid dynamics model, whose relation with

2D turbulence remains open for future investigation.

II. VORTEX COLLAPSE AND BURST

The Hamiltonian system of N = 3 vortices on the

whole plane R2 is integrable (cf. [18] for a general discussion

on integrability of PV systems), and it admits

singular solutions in which the vortices collapse in –or

burst out of– a single point, with self-similar trajectories,

that is the angles of the triangle having as vertices

the vortices’ position are constants of motion.

Theorem II.1. [19, Section 3] Consider the system

(PV) of N = 3 vortices with intensities γ1, γ2, γ3 ∈

R \ {0} starting from distinct positions x1, x2, x3 ∈ R2

at t = 0. The existence time interval of a maximal solution

of (PV) ends with a collapse of the three vortices,

or (alternatively) it begins with a burst, if and only if

X

j̸=k

γjγk = 0, I =

X

j̸=k

γjγk|xj − xk|2 = 0, (II.1)

and x1, x2, x3 are not the vertices of an equilateral triangle.

Conditions (II.1) also imply that the motion of

vortices is self-similar, and if starting positions form an

equilateral triangle, the configuration is of relative equilibrium.

Let us remark that, while self-similarly collapsing configurations

exist for more than three vortices [20], for PV

systems with N ≥ 4 there also exist configuration leading

to collapse with trajectories that are not self-similar [21].

The moment of inertia I(t) = I is a first integral of

motion. The first condition in (II.1) excludes that collapsing

vortices, and thus bursts, have intensities of the

same sign. Indeed, in the latter case the Hamiltonian of

the system,

H = −

1

2π

X

j̸=k

γjγk log |xj − xk|,

(with conjugate coordinates (xi,1, γixi,2)i=1,...,N ) can be

used to control the minimum distance of vortices.

Invariance of H implies that there are no solutions in

which two vortices collide at finite time and a third one

remains distinct. The above Proposition thus characterises

all singular configurations for N = 3, and it furthermore

reveals that in those cases the motion is selfsimilar.

In particular, any continuation of vortex dynamics after

collapse should be a self-similar burst. We are thus

interested in characterizing the possible collapsing and

bursting vortex configurations so to study how regularization

procedures select among them a continuation of

the dynamics.

A. Classifying Singular Configurations

Let us consider intensities γ1, γ2, γ3 and initial positions

x1(0), x2(0), x3(0) ∈ R2 satisfying (II.1). The center

of vorticity C = γ1x1 + γ2x2 + γ3x3 is a first integral

of PV motion, and we can assume that it is the origin

(0, 0). The self-similar solution then takes the form

xi(t) =

p

a(t∗ − t)Θa,b(t∗ − t)xi(0),

Θa,b(t) =



cos( b

2a log(t)) −sin( b

2a log(t))

sin( b

2a log(t)) cos( b

2a log(t))



, (II.2)

where a > 0, b ̸= 0 are dynamical parameters that are

determined completely by vortex intensities and initial

positions (cf. [7, Section 2]). While we expect that the

number and intensities of vortices be preserved in the

continuation of dynamics after collapse, as we understand

the latter as a limit of a regularized system for which that

is the case, in principle parameters ruling the self-similar

evolution might be influenced by the collapse, namely the

triangle of vortex positions could change its symmetry

class.

Exploiting the integrability of the system, we reduce

the study of those parameters to that of a single point of

R2. Rescaling intensities corresponds to scaling time in

(PV), so we can assume γ2 = 1 without loss of generality,

hence γ3 = − γ1

1+γ1

by the first condition of (II.1). The

collapse condition I = 0 and the assumption C = (0, 0)

of above give:

I = γ1|x1 − x2|2 − γ2

1 |x1−x3|2

1+γ1

− γ1|x2−x3|2

1+γ1

= 0

C = γ1x1 + x2 − γ1

1+γ1

x3 = (0, 0).

Excluding the trivial solution x1 = x2 = x3 = 0, we

can exploit invariance under rotations and homotheties

of those conditions by applying R ∈ O+(2) (an orthogonal

matrix with positive determinant) such that

Rx3 = (1, 0). The second equation now gives

Rx2 =



γ1

1 + γ1

, 0



− γ1Rx1,

so we find the possible configurations in terms of Z =

Rx1 by intersecting the two conditions I = I(Z) = 0 and

H0 = H(Z) = γ1 log |Z − Rx2|

−

γ2

1

1 + γ1

log |Z − (1, 0)| −

γ1

1 + γ1

log |Rx2 − (1, 0)|

3

FIG. 1: Intersections between the level sets H = H0 (dashed black) and I = 0 (solid green) for PV configurations

with different intensities. On the left a case with only two intersections: the point C indicates an intersection

corresponding to a collapsing configurations, and point B to a bursting configuration. On the left a case with four

intersections, red and blue points marking respectively collapsing and bursting configurations.

for any fixed choice of H0 and γ1. Such intersection of

curves only consists of isolated points Z encoding 1) the

similarity class of the triangle of vertices x1, x2, x3, 2)

the angle of rotation of such triangle around (0, 0), 3)

the order with which x1, x2, x3 appear as vertices of the

triangle.

Notice in particular that, given γ1,H0, if Z = (z1, z2)

is an intersection so is ˜ Z = (z1,−z2), the two points

corresponding to similar triangles with vertices x1, x2, x3

in a different order, one leading to a vortex collapse and

the other to a burst.

The self-similar evolution of vortices away from the

burst/collapse time leaves Z = Z(t) invariant. In other

words, this procedure classifies the types of self-similar

evolution leading to a burst or collapse of vortices. Figure

1 reports two different cases, in which H, γ1 are chosen

so that there are 2 or 4 intersections.

Remark II.2. As suggested by the comparison between

the plots in fig. 1, it is possible that curves H = H0 and

I = 0 in the Z-plane intersect in three points, as a limit

case in which B2 and C2 coincide. In that case, the third

point corresponds to a relative equilibrium, in which the

PV configuration rotates periodically around its center

of vorticity. Such dynamics, when perturbed by additive

noise, persists in the zero-noise limit.

We refer to [22, 23] concerning procedures for determining

singular PV configurations.

B. Deterministic Regularization

A natural way of investigating continuation after a vortex

collapse is to regularize the vortex interaction, so that

the modified ODE system is well-posed for all times, and

then study the limit behavior in the regularization parameter.

Consider a convolution kernel h : R → R and,

for ϵ > 0, the modified PV dynamics

˙ xϵi

=

XN

j̸=i

γjKϵ(xϵi

− xϵj

), (II.3)

Kϵ = ∇⊥ 

hϵ ∗

􀀀

− 1

2π log



, hϵ(x) =

1

ϵ2 h

x

ϵ



,

with K(x) = x⊥

2π|x|2 being the original interaction kernel.

The most relevant examples of kernels h are: bump

functions (that is, smooth compactly supported functions);

the Gaussian probability distribution, 1

2π e−|x|2/2;

the Cauchy probability distribution, (1 + |x|2)−2/π (in

this case eq. (II.3) is the blob-vortex system); 1

2πK0(|x|)

with K0 the modified Bessel function of second kind (in

this case (II.3) is the α-Euler vortex system, cf. [15] ).

More generally, one can consider the a kernel h satisfying

the assumptions in [16, Theorems 1,6].

For ϵ > 0, any choice of intensities γ1, . . . , γN ∈ R \

{0} and initial positions x1(0), . . . , xN(0) ∈ R2, the ODE

system (II.3) is globally well-posed, the solution flow is

a group of homeomorphisms that preserve the product

Lebesgue measure on (R2)N, and for all times t ∈ R and

i ̸= j, xi(t) ̸= xj(t), that is there are no collapses of

vortices, [16, Proposition 3].

Let now N = 3, and consider a configuration

x1(0), x2(0), x3(0) ∈ R2 that leads to a collapse of the system

(PV) at time t∗ > 0.