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Physics » "Vortex Structures in Fluid Dynamic Problems", book edited by Hector Perez-de-Tejada, ISBN 978-953-51-2944-8, Print ISBN 978-953-51-2943-1, Published: March 1, 2017 under CC BY 3.0 license. © The Author(s).

# Relaxation Theory for Point Vortices

By Ken Sawada and Takashi Suzuki
DOI: 10.5772/67075

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## Overview

Figure 1. Vorticity distribution: vorticity patch model (left). point vortex model (right).

Figure 2. Sketch of localization procedure.

# Relaxation Theory for Point Vortices

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## Abstract

We study relaxation dynamics of the mean field of many point vortices from quasi-equilibrium to equilibrium. Maximum entropy production principle implies four consistent equations concerning relaxation-equilibrium states and patch-point vortex models. Point vortex relaxation equation coincides with Brownian point vortex equation in micro-canonical setting. Mathematical analysis to point vortex relaxation equation is done in accordance with the Smoluchowski-Poisson equation.

Keywords: point vortex, quasi-equilibrium, relaxation dynamics, maximum entropy production, global-in-time solution

## 1. Introduction

The physical object studied in this chapter is non-viscous, noncompressible fluid with high Reynolds number occupied in bounded, simply-connected domain. ΩR2 . Motion of this fluid is described by the Euler-Poisson equation

 ωt+∇⋅uω=0,Δψ=−ω, u=∇⊥ψ,  ψ|∂Ω=0 (1)

where

=(x2x1),  x=(x1,x2),

and u, ω and ψ stand for the velocity, vorticity and stream function, respectively.

In the point vortex model

 ω(x,t)=∑i=1Nαiδxi(t)(dx) (2)

system of Eq. (1) is reduced to

 αidxidt=∇xi⊥HN,   i=1,2,⋯,N (3)

associated with the Hamiltonian

 HN(x1,⋯xN)=12∑iαi2R(xi)+∑i

where G=G(x,x) is the Green’s function of –Δ provided with the Dirichlet boundary condition and

R(x)=[G(x,x)+12πlog|xx|]x=x.

Onsager [1] proposed to use statistical mechanics of Gibbs to Eq. (3). In the limit N → ∞ with αN = 1, local mean of vortex distribution is given by

 ω¯(x)=∫Iα˜ρα˜(x)P(dα˜),x∈Ω (5)

where αi=α˜iα,α˜iI=[1,1] is the intensity of the i-th vortex, ρα˜(x) is the existence probability of the vortex at x with relative intensity α˜ , which satisfies

Ωρα˜(x)dx=1,   α˜I,

and P(dα˜) is the numerical density of the vortices with the relative intensity α˜ . Under HN=E=constant, α2NβN=β=constant and N → ∞, mean field equation is derived by several arguments [27], that is,

 −Δψ¯=∫Iα˜e−βα˜ψ¯∫Ωe−βα˜ψ¯P(dα˜),    ψ¯|∂Ω=0 (6)

with

ω¯=Δψ¯,ρα˜=eβα˜ψ¯Ωeβα˜ψ¯

where

ρα˜(x)=limNΩN1μNβN(dx,dx2,dxN)
μNβN(dx1,dxN)=1Z(N,βN)eβNHNdx1dxN
Z(N,βN)=ΩNeβNHNdx1dxN.

Since Ref. [8], structure of the set of solutions to Eq. (6) has been clarified in accordance with the Hamiltonian given by Eq. (4) (see [9] and the references therein).

Quasi-equilibria, on the other hand, are observed for several isolated systems with many components [10]. Thus, we have a relatively stationary state, different from the equilibrium, which eventually approaches the latter. Relaxation indicates this time interval, from quasi-equilibrium to equilibrium. To approach relaxation dynamics of many point vortices, patch model

 ω(x,t)=∑i=1Npσi1Ωi(t)(x) (7)

is used. It describes detailed vortex distribution, where N p , σ i and Ω i (t) denote the number of patches, the vorticity of the i-th patch and the domain of the i-th patch, respectively. Mean field equations for equilibrium and for relaxation time are derived by the principles of maximum entropy [11, 12] and maximum entropy production [13, 14], respectively. For the latter case, one obtains a system on p=p(x,σ,t) ,

 ∂p∂t+∇⋅pu¯=∇⋅D(∇p+βp(σ−ω¯)p∇ψ¯),   βp=−∫ΩD∇ω¯⋅∇ψ¯∫ΩD(∫Iσ2pdσ−ω¯2)|∇ψ¯|2ω¯=∫Iσpdσ=−Δψ¯,   ψ¯|∂Ω=0,   u¯=∇⊥ψ¯ (8)

with the diffusion coefficient D=D(x,t)>0.

In this chapter, we regard Eq. (2) as a limit of Eq. (7). First, point vortex model valid to the relaxation time is derived from Eq. (8), that is, a system on ρα˜=ρα˜(x,t),α˜I , in the form of

 ∂ρα˜∂t+∇⋅ρα˜u¯=∇⋅D(∇ρα˜+βα˜ρα˜∇ψ¯),  ω¯=∫Iα˜ρα˜P(dα˜)=−Δψ¯,   ψ¯|∂Ω=0,   u¯=∇⊥ψ¯β=−∫ΩD∇ω¯⋅∇ψ¯∫ΩD∫Iα˜2ρα˜P(dα˜)|∇ψ¯|2. (9)

Second, the stationary state of Eq. (9) is given by Eq. (6). Third, Eq. (9) coincides with the Brownian point vortex model of Chavanis [15]. Finally, system of Eq. (9) provided with the boundary condition

 ∂ρα˜∂ν+βα˜ρα˜∂ψ¯∂ν|∂Ω=0 (10)

satisfies the requirements of isolated system in thermodynamics.

In fact, averaging Eq. (9) implies

 ∂ω¯∂t+∇⋅ω¯u¯=∇⋅D(∇ω¯+βω¯2∇ψ¯),  ∂ω¯∂ν+βω¯2∂ψ¯∂ν|∂Ω=0 ω¯=−Δψ¯,   ψ¯|∂Ω=0,   u¯=∇⊥ψ¯,   β=−∫ΩD∇ω¯⋅∇ψ¯∫ΩDω¯2|∇ψ¯|2 (11)

for

 ω¯=∫Iα˜ρα˜P(dα˜),ω¯2=∫Iα˜2ρα˜P(dα˜). (12)

Then, we obtain mass and energy conservations

 ddt∫Ωω¯=0,(ω¯t,ψ¯)=12ddt(ω¯,(−Δ)−1ω¯)=0 (13)

where (,) stands for the L 2 inner product. Assuming ρα˜>0 , we write the first equation of (9) as

 ∂ρα˜∂t+∇⋅ρα˜u¯=∇⋅Dρα˜∇(logρα˜+βα˜ψ¯). (14)

Then, it follows that

 ddt∫ΩΦ(ρα˜)dx+βα˜(ρtα˜,ψ¯) =−∫ΩDρα˜|∇(logρα˜+βα˜ψ¯)|2 (15)

from Eq. (10), where

Φ(s)=s(logs1)+10,  s>0.

Hence, it follows that

 ddt∫Ω(∫IΦ(ρα˜)P(dα˜))=−∫Ω(∫IDρα˜|∇(logρα˜+βα˜ψ¯)|2P(dα˜))≤0 (16)

from Eq. (13), that is, entropy increasing.

## 2. Vorticity patch model

In Eq. (7), the vorticity σ i is uniform in a region with constant area Ωi(t) , called vorticity patch. A patch takes a variety of forms as the time t varies. We collect all the vorticity patches in a small region, called cell. Cell area Δ thus takes the relation |Ωi|Δ|Ω| . The probability that the average vorticity at x is σ is denoted by p(x,σ,t)dx ， which satisfies

 ∫​p(x,σ,t)dσ=1. (17)

Let

 ∫Ωp(x,σ,t)dx=M(σ) (18)

be independent of t. Since

 |Ω|=∬​p(x,σ,t)dxdσ=∫​M(σ)dσ (19)

equality (18) means conservation of total area of patches of the vorticity σ. Then, the macroscopic vorticity is defined by

 ω¯(x,t)=∫​σp(x,σ,t)dσ, (20)

which is associated with the stream function ψ¯=ψ¯(x,t) and the velocity u¯=u¯(x,t) through

 ω¯=−Δψ¯,   ψ¯|∂Ω=0,   u¯=∇⊥ψ¯. (21)

To formulate equilibrium, we apply the principle of maximum entropy [11, 12], seeking the maximal state of

 S(p)=−∬​p(x,σ)logp(x,σ)dxdσ (22)

under the constraint Eqs. (17), (18) and

 E=12∫Ωω¯ψ¯. (23)

With the Lagrange multipliers (βp,c(σ),ζ(x)) , it follows that

 δS−βpδE−∫​c(σ)δM(σ)dσ−∫Ωζ(x)(δ∫​pdσ)dx=0, (24)

which is reduced to

 p(x,σ)=e−c(σ)−(ζ(x)+1)−βpσψ¯. (25)

Here, β p and c(σ) may be called inverse temperature and chemical potential, respectively. We put c(0) = 0 because of the degree of freedom of c(σ) admitted by Eq. (19). Then, it follows that

 p(x,σ)=p(x,0)e−c(σ)−βpσψ¯ (26)

and hence, Eq. (17) implies

 p(x,σ)=e−c(σ)−βpσψ¯∫​e−c(σ')−βpσ'ψ¯dσ′. (27)

From Eqs. (18) and (26), similarly, it follows that

 c(σ)=log(∫Ωp(x,0)e−βpσψ¯dx∫Ωp(x,σ)dx). (28)

The equilibrium mean field equation of vorticity patch model is thus given by Eqs. (20), (21), (27) and (28), which is reduced to

 −Δψ¯=∫​σM(σ)p(x,0)e−βpσψ¯∫Ωp(x,0)e−βpσψ¯dσ,   ψ¯|∂Ω=0ω¯=∫Iσpdσ=−Δψ¯,∫Ωp(x,σ,t)dx=M(σ). (29)

One may use the principle of maximum entropy production to describe near from equilibrium dynamics [13, 14]. We apply the transport equation

 ∂p∂t+∇⋅(pu¯)=−∇⋅J,J⋅ν|∂Ω=0 (30)

with the diffusion flux J=J(x,σ,t) of p=p(x,σ,t) , where ν denotes the outer unit normal vector. We obtain the total patch area conservation for each σ,

 ∂M∂t=∂∂t∫Ωp(x,σ,t)=0 (31)

because u¯ν|Ω=0 follows from Eq. (21). Eq. (30) implies

 ∂ω¯∂t+∇⋅(ω¯u¯+Jω)=0, (32)

where Jω=σJ(x,σ,t)dσ stands for the local mean vorticity flux. Since Jων=0 on Ω , Eq. (32) implies conservation of circulation Γ=Ωω¯ . Furthermore, J ω is associated with the detailed fluctuation of (ω, u) from (ω¯,u¯) by Eq. (1).

Here, we ignore the diffusion energy Ed=12J2pdσdx to take

 E=12∫Ωω¯ψ¯ (33)

as the total energy of this system. Using maximum entropy production principle, we chose the flux J to maximize entropy production rate S.  under the constraint

 E˙=0,∫​Jdσ=0,∫​J22pdσ≤C(x,t) (34)

where

S(p)=p(x,σ,t)logp(x,σ,t)dσdx.

Using Lagrange multipliers (βp,D,ζ)=(βp(t),D(x,t),ζ(x,t)) , we obtain

 δS˙−βpδE˙−∫ΩD−1(δ∫​J22pdσ)dx−∫Ωζ(δ∫​Jdσ)dx=0. (35)

Since

 E˙=ddtE=∫Ωψ¯∂ω¯∂t=∫ΩJω⋅∇ψ¯=∬​σJ⋅∇ψ¯dσdxS˙=ddtS=−∬​∂p∂t(logp+1)dσdx=−∬​J⋅∇ppdσdx, (36)

Eq. (35) is reduced to

 J=−D(∇p+βpσp∇ψ¯+pζ). (37)

From the constraint of Eq. (34), it follows that

 0=∫​Jdσ=−∫​D(∇p+βpσp∇ψ¯+pζ)dσ=−D(βpω¯∇ψ¯+ζ) (38)

and

 0=∬​σJ⋅∇ψ¯ dσdx=∬​−σD(∇p+βpσp∇ψ¯+pζ)⋅∇ψ¯dσdx=∬​−σD(∇p+βp(σp−pω¯)∇ψ¯)⋅∇ψ¯dσdx=−∫ΩD∇ω¯⋅∇ψ¯dx−βp∫ΩD(∫​σ2pdσ−ω¯2)|∇ψ¯|2dx (39)

which implies

 ζ=−βpω¯∇ψ¯ (40)

and

 βp=−∫ΩD∇ω¯⋅∇ψ¯∫ΩD(∫​σ2pdσ−ω¯2)|∇ψ¯|2 (41)

Thus, we end up with

 ∂p∂t+∇⋅(pu¯)=∇⋅D(∇p+βp(σ−ω¯)p∇ψ¯),   βp=−∫ΩD∇ω¯⋅∇ψ¯∫ΩD(∫​σ2pdσ−ω¯2)|∇ψ¯|2D(∇p+βp(σ−ω¯)p∇ψ¯)⋅ν|∂Ω=0,   ω¯=∫Iσpdσ=−Δψ¯,   ψ¯|∂Ω=0,   u¯=∇⊥ψ¯ (42)

by Eqs. (30), (37), (40) and (41), where D=D(x,t)>0 .

## 3. Point vortex model

Point vortex model is regarded as a special case of vorticity patch model, where the patch size shrinks to zero [16]. Here, we give a quantitative description of this limit process, using localization. First, we derive the equilibrium mean field equation of point vortices from that of vorticity patches. Then, we derive relaxation equation for the point vortex model. Fundamental quantities of point vortex model are circulation αα˜ , probability ρα˜(x,t) and number density P(dα˜) . Circulation of each vortex is set to be small to preserve total energy and total circulation in the mean field limit. In the vorticity patch model, on the other hand, vorticity σ and probability p(x,σ,t) are the fundamental quantities ( Figure 1 ).

#### Figure 1.

Vorticity distribution: vorticity patch model (left). point vortex model (right).

Here, we use the following localization in order to transform vorticity patch to point vortex ( Figure 2 ):

#### Figure 2.

Sketch of localization procedure.

1. Divide each patch into two patches with half area and the same vorticity.

2. Again, divide each patch into two patches with half area: one has doubled vorticity and the other has 0 vorticity.

Under this procedure, the number of nonzero patches is doubled and their vorticities are also doubled. At the same time, the area of each patch becomes 1/4 and the number of total patches is quadrupled, while the total circulation is preserved. First, we describe the detailed process for the stationary state of Eq. (7).

Let Ω be divided into many cells with uniform size Δ and let each cell be composed of many patches. Let N(k)(x,σ)dxdσ be the number of patches in the cell after k-times of the above procedure centered at x of which vorticity was originally σ and let σ (k) be the vorticity of these patches after k-times localization. We assume that the number of total vorticity patches in the cell,

 Nc(k)(Δ)=∫​N(k)(x,σ)dσ, (43)

is independent of x. Then, the number of total patches in Ω, the total area of the patches and the total circulation of the patches after k-times localization procedures, with original vorticity σ, are given by

 N(k)(σ)dσ=∫ΩN(k)(x,σ),M(k)(σ)dσ=|Ω|N(k)(σ)dσ∫​N(k)(σ)dσ, (44)

and

 γ(k)(σ)dσ=σ(k)M(k)(σ)dσ, (45)

respectively.

We obtain

 Np=∬​N(0)(x,σ)dσdx, (46)

recalling Eq. (7). Since

 σ(k)=2kσ, (47)

it holds that

 N(k)(x,σ)dxdσ=(4k−2k)Nc(0)(Δ)δ0(dσ)+2kN(0)(x,σ)dxdσ. (48)

From Eq. (48), the related probability

 p(k)(x,σ)dxdσ=N(k)(x,σ)dxdσNc(k)(Δ) (49)

satisfies

 p(k)(x,σ)dxdσ=(4k−2k)Nc(0)(Δ)δ0(dσ)+2kN(0)(x,σ)dxdσ(4k−2k)Nc(0)(Δ)+2k∫​N(0)(x,σ)dσ=(4k−2k)Nc(0)(Δ)δ0(dσ)+2kN(0)(x,σ)dxdσ4kNc(0)(Δ) (50)

and hence,

 limk→∞p(k)(x,σ)dxdσ=δ0(dσ). (51)

We also have

 M(k)(σ)dσ=∫Ωp(k)(x,σ)dx=limΔ→0∑i=1|Ω|/ΔN(k)(xi,σ)dxdσNc(k)(Δ)⋅Δ (52)

which implies

 M(k)(σ)dσ=|Ω|4kNplimΔ→0∑i=1|Ω/Δ|N(k)(xi,σ)dσ=|Ω|4kNpN(k)(σ)dσ=|Ω|((1−2−k)δ0(dσ)+2−kN(0)(σ)dσNp) (53)

by ΔNc(k)(Δ)=|Ω|4kNp and Eq. (48). We have, therefore,

 limk→∞M(k)(σ)dσ=|Ω|δ0(dσ). (54)

It holds also that

 γ(k)(σ)=∫Ωσ(k)p(k)(x,σ)dx=∫Ωσp(0)(x,σ)dx=σ(k)M(k)(σ)dσ=σ|Ω|NpN(0)(σ)dσ (55)

and

 ω¯(k)(x)=∫​σ(k)p(k)(x,σ)dσ=∫​σp(0)(x,σ)dσ. (56)

Fundamental quantities constituting of the mean field limit of point vortex model thus arise as k → ∞.

To explore the relationship between the quantities in two models, we take regards to circulation of one patch, total circulation of patches with original vorticity σ and local mean vorticity. Based on

 σ(k)⋅|Ω|4kNp=α˜⋅α, k≫1, (57)

and Eq. (47), we reach the ansatz σ|Ω|=α˜,12kNp=α,2kNp=N . Similarly, we use

 σ|Ω|NpN(0)(σ)dσ=α˜P(dα˜) (58)

to put

 N(0)(σ)dσNp=M(0)(σ)dσ|Ω|=P(dα˜) (59)

by

 σ|Ω|NpN(0)(σ)dσ=σ|Ω|⋅12kNp⋅2kNp⋅N(0)(σ)dσNp=α˜αNP(dα˜)=α˜P(dα˜). (60)

Finally, we use the identity on local mean vorticity

 ∫​σp(0)(x,σ)dσ=∫​α˜ρα˜(x)P(dα˜) (61)

to assign

 1|Ω|p(0)(x,σ)dσ=ρα˜(x)P(dα˜), (62)

regarding

 ∫​σp(0)(x,σ)dσ=∫​σ|Ω|⋅p(0)(x,σ)|Ω|dσ=∫​α˜ρα˜(x)P(dα˜). (63)

These relations are summarized in the following Table 1 :

Vorticity patch modelPoint vortex model
σ|Ω| α˜
12kNp α
2kNp N
N(0)(σ)dσNp P(dα˜)
1|Ω|p(0)(x,σ)dσ ρα˜(x)P(dα˜)

#### Table 1.

Relation between vorticity patch model and point vortex model for α˜ .

After k-times localization, the first equation in Eq. (29) takes the form

 −Δψ¯=∫​σ(k)M(k)(σ)p(k)(x,0)e−βpσ(k)ψ¯∫Ωp(k)(x,0)e−βpσ(k)ψ¯dσ=∫​σ|Ω|NpN(0)(σ)p(k)(x,0)e−βp2kσψ¯∫Ωp(k)(x,0)e−βp2kσψ¯dσ=∫​σ|Ω|p(k)(x,0)e−βp2k|Ω|⋅σ|Ω|⋅ψ¯∫Ωp(k)(x,0)e−βp2k|Ω|⋅σ|Ω|⋅ψ¯N(0)(σ)Npdσ. (64)

From Table 1 , the right-hand side on Eq. (64) is replaced by

 ∫​α˜p(k)(x,0)e−βNNα˜ψ¯∫Ωp(k)(x,0)e−βNNα˜ψ¯P(dα˜) (65)

for βN=4kNp|Ω|βp=N·2kβp|Ω| . Sending k → ∞, we obtain the first equation of (6) with β=βNN by Eq. (51). This means that the vorticity patch model is transformed to the point vortex model applied to the mean field limit by taking the localization procedure.

We can derive also relaxation equation of point vortex model from that of vorticity patch model. By Eq. (37), the value of the diffusion flux J for σ = 0 is

 J(x,0,t)=−D(x,t)(∇p(x,0,t)+p(x,0,t)ζ(x,t)) (66)

and hence

 ζ(x,t)=−D(x,t)−1J(x,0,t)+∇p(x,0,t)p(x,0,t). (67)

Flux is thus given by

 J(x,σ,t)=−D(x,t)(∇p(x,σ,t)+βp(t)σp(x,σ,t)∇ψ¯(x,t)−p(x,σ,t)D(x,t)−1J(x,0,t)+∇p(x,0,t)p(x,0,t)). (68)

We reach

 ∂p∂t+∇⋅(pu¯)=∇⋅D(∇p+βpσp∇ψ¯−p[D−1J+∇pp]σ=0) (69)

with

 βp=βp(t)=−∫ΩD∇ω¯⋅∇ψ¯−∫ΩDω¯[D−1J+∇pp]σ=0⋅∇ψ¯∫∫​Dσ2p|∇ψ¯|2dσdx (70)

Therefore, after k-times localization procedure, it holds that

 ∂σ(k)p(k)∂t+∇⋅(σ(k)p(k)u¯)=∇⋅D(∇σ(k)p(k)+βp(σ(k))2p(k)∇ψ¯−σ(k)p(k)[D−1J(k)+∇p(k)p(k)]σ=0). (71)

Putting βN=4kNp|Ω|βp , similarly, we obtain

 ∂∂t(α˜ρα˜P(dα˜))+∇⋅(α˜ρα˜P(dα˜)u¯)=∇⋅(D(∇(α˜ρα˜P(dα˜))+βα˜2ρα˜P(dα˜)∇ψ¯)), (72)

from

 limk→∞p(k)(x,σ,t)=δ0(dσ),limk→∞J(k)(x,0,t)=0σ(k)p(k)(x,σ,t)=σp(0)(x,σ,t)=σ|Ω|⋅p(0)(x,σ,t)|Ω|≈α˜ρα˜(x,t)P(dα˜)(σ(k))2p(k)(x,σ,t)=2kσ⋅σp(0)(x,σ,t)=2k|Ω|⋅(σ|Ω|)2⋅p(0)(x,σ,t)|Ω|≈2k|Ω|α˜2ρα˜(x,t)P(dα˜) (73)

Here, we assume limkJ(k)(x,0,t)=0 , because J(k)(x,σ,t)dσ=0 and the 0-vorticity patch becomes dominant in the system. Then, we obtain Eq. (9) by Eq. (72).

## 4. Relaxation dynamics

If P(dα˜)=δ1(dα˜) , it holds that ω¯=ω¯2 in Eq. (11). Then, we obtain

 ωt+∇⋅ω∇⊥ψ=∇⋅(∇ω+βω∇ψ),   ∂ω∂ν+βω∂ψ∂ν|∂Ω=0,ω|t=0=ω0(x)≥0 (74)
 −Δψ=ω,   ψ|∂Ω=0,β=−∫Ω∇ω⋅∇ψ∫Ωω|∇ψ|2 (75)

assuming D = 1. Conservations of total mass and energy

 ‖ω(⋅,t)‖1=λ, (ψ(⋅,t),ω(⋅,t))=e, (76)

are derived from Eq. (13), while increase in entropy of Eq. (16) is reduced to

 ddt∫ΩΦ(ω)=−∫Ωω|∇(logω−βψ)|2≤0, (77)

where Φ(s)=s(logs1)+1 .

In the stationary state, we obtain logω+βψ=constant by Eq. (77). Hence, it follows that

 −Δψ=ω,   ψ|∂Ω=0,  ω=λe−βψ∫Ωe−βψ,  −β=∫Ω∇ω⋅∇ψ∫Ωω|∇ψ|2,   e=∫Ωωψ (78)

from Eq. (76). Here, the third equation implies the fourth equation as

 (∇ω,∇ψ)=−β∫Ωω|∇ψ|2. (79)

Using

 v=βψ,μ=βλ∫Ωe−βψ, (80)

therefore, Eq. (78) is reduced to

 −Δv=μe−v, v|∂Ω=0,eλ2=∫Ω|∇v|2(∫∂Ω−∂v∂ν)2 . (81)

In fact, to see the third equality of (81), we note

 e=(ω,ψ)=β−1λ∫Ωe−vv∫Ωe−v (82)

which implies

 μ=λ∫Ωe−v⋅λe∫Ωe−vv∫Ωe−v=λ2e⋅∫Ωe−vv(∫Ωe−v)2 (83)

and hence

 eλ2=1μ⋅∫Ωe−vv(∫Ωe−v)2=‖∇v‖22(∫∂Ω−∂v∂ν)2 . (84)

If μ < 0, system of Eq. (81) except for the third equation is equivalent to the Gel’fand problem

 −Δw=σew,w|∂Ω=0 (85)

with σ = –μ. If Ω is simply connected, there is a non-compact family of solutions as μ ↑ 0, which are uniformly bounded near the boundary [8, 9]. Hence, there arises

 limμ↑0eλ2=+∞ (86)

for this family. For μ ≥ 0, on the contrary, system of Eq. (81) except for the third equation admits a unique solution v=vμ(x) . Regarding Eq. (76), therefore, it is necessary that

 limμ↑+∞‖∇vμ‖22(∫∂Ω−∂vμ∂ν)2=0 (87)

for any orbit to Eqs. (74), (75) to be global-in-time and compact, for any λ,e>0 in Eq. (76).

If Ω=B{xR2| |x|<1} , it actually holds that Eq. (87). In this case, we have v=v(r), r=|x|, and the result follows from an elementary calculation. More precisely, putting u=vlogμ , s=logr , we obtain

 uss+e−u+2s=0,  s<0,  u(0)=−logμ,  lims↓−∞use−s=0,  ‖∇v‖22(∫∂Ω−∂v∂ν)2=I2π, (88)

where I=0us2dsus(0)2 . Using w=u2s , p=12(ew+2)1/2 , we have

 p=−1+2(1−ce2s)−1 (89)

with c ↑ 1 as μ ↑ +∞. It follows that

 I=(1−c)2∫−∞0e4s(1−ce2s)2ds (90)

with

 ∫−∞0e4s(1−ce2s)2ds=12c(1−c)+12c2log(1−c) (91)

and hence

 limc↑1I=0. (92)

If β is constant in Eq. (9), it is the mean field limit of Brownian vortices [15]. It is nothing but the Smoluchowski-Poisson equation [9, 17] and obeys the feature of canonical ensemble, provided with total mass conservation and decrease of free energy:

 dFdt=−∫Ωω|∇(logω+βψ)|2,F(ω)=∫ΩΦ(ω)−12((−Δ)−1ω,ω). (93)

Then, there arises the blowup threshold β=8π/λ [18]. Here, we show the following theorem, where G=G(x,x') denotes the Green’s function for the Poisson part,

 −ΔG(⋅,x′)=δx′,G(⋅,x′)|∂Ω=0,x′∈Ω (94)

and

 ρφ(x,x′)=∇φ(x)⋅∇xG(x,x′)+∇φ(x′)⋅∇x'G(x,x′), φ∈X, (95)

where X={φC2(Ω¯)|φν|Ω=0} . It holds that ρφL(Ω×Ω) . The proof is similar as in Lemma 5.2 of [17] for the case of Neumann boundary condition.

Theorem 1: Let Ω = B and ω 0 be a smooth function in the form of ω0=ω0(r)>0 with ω0r<0 , 0<r1 . Let T ∈ (0, + ∞] be the maximal existence time of the classical solution to Eqs. (74), (75) and λ be the total mass defined by Eq. (76). Then, it follows that

 limsupt↑Tβ(t)<−8πλ  ⇒T<+∞ (96)

and

 T<+∞ ⇒ liminft↑Tβ(t)=−∞. (97)

In particular, we have

 liminft↑Tβ(t)>−∞⇒ T=+∞, limsupt↑Tβ(t)≥−8πλ . (98)

Proof: From the assumption, it follows that (ω,ψ)=(ω(r,t),ψ(r,t)) and

ωr, ψr<0,0<r1.

Then, we obtain

 M≡λ2π≥∫0rrωdr≥ω(r,t)∫0rrdr=r22ω (99)

and hence

 ω(r,t)≤2Mr2,0

It holds also that

 −β=∫01ωrψrrdr∫01ωψr2rdr>0 (101)

which implies

 ωt=Δω+β∇ψ⋅∇ω+βωΔψ=Δω+β∇ψ⋅∇ω−βω2≥Δω+β∇ψ⋅∇ω (102)

with

 −∂ω∂ν=βω∂ψ∂ν>0on∂Ω×(0,T). (103)

The comparison theorem now guarantees ωδminΩ¯ ω0>0 and hence

 ∫Ωω|∇ψ|2≥δ∫Ω|∇ψ|2=δe. (104)

For Eq. (96) to prove, we use the second moment. First, the Poisson part of Eq. (75) is reduced to

 −rψr=∫0rrωdr≡A(r). (105)

Second, it follows that

 ddt∫01ωr3dr=−∫01(ωr+βωψr)2r⋅rdr=−2r2ω|r=0r=1+∫014rω−2βωψrr2dr=−2ω|r=1+4M+2β∫01AArdr=−2ω|r=1+4M+βM2≤4M+βM2 (106)

from A(1) = M. Under the hypothesis of Eq. (96), we have δ > 0 such that

 4M+βM2≤−δ,   t↑T. (107)

Then, T = +∞ gives a contradiction.

Now, we assume T < +∞. First, equality in (106) implies

 ∫0T−β(t)dt≤C (108)

by Eq. (100). Second, we have

 ddt∫Ωωφ=∫ΩωΔφ+β2∬Ω×Ωρφω⊗ω (109)

and hence

 ∫0T|ddt∫Ωωφ|dt≤Cφ,φ∈X. (110)

Inequality (110) takes place of the monotonicity formula used for the Smoluchowski-Poisson equation, which guarantees the continuation of ω(x,t)dx up to t = T as a measure on Ω¯ [9, 17]. Thus, there is μ=μ(dx,t)C*([0,T],M(Ω¯)) such that μ(dx,t)=ω(x,t)dx for 0 ≤ t < T. By Eq. (100), therefore, it holds that

 ω(x,t)dx⇀cδ0(dx)+f(x)dxinM(Ω¯),   t↑T, (111)

with c ≥ 0 and 0f=f(x)L1(Ω) . From the elliptic regularity, we obtain

 liminft↑Tψ(x,t)≥c2πlog1|x|   loc. unif. in  Ω¯∖{0}. (112)

Then, e=(ω(,t),ψ(,t))(ω(,t),min{k,ψ(,t)}) implies ec2πmin{k,log1|x|} for k = 1,2,. Hence, it holds that c = 0 in Eq. (111).

If the conclusion in Eq. (97) is false, we have the ε regularity in Eqs. (74), (75) [9, 17]. Thus, there is ε0=ε0k>0 , such that

 limsupt↑T‖ω(⋅,t)‖L1(Ω∩​B(x0,R))<ε0 ⇒ limsupt↑T‖ω(⋅,t)‖L∞(Ω∩​B(x0,R/2))<+∞ (113)

for 0<R1 . The hypothesis in Eq. (113) is valid for x0=0 by Eq. (111), c = 0, which contradicts to T < + ∞.

## 5. Conclusion

We study the relaxation dynamics of the point vortices in the incompressible Euler fluid, using the vorticity patch which varies with uniform vorticity and constant area. The mean field limit equation is derived, which has the same form as the one derived for the Brownian point vortex model. This equation governs the last stage of self-organization, not only in the point vortices but also in the two-dimensional center guiding plasma and the rotating superfluid helium, from quasi-equilibrium to equilibrium. Mathematical analysis assures this property for radially symmetric case, provided that the inverse temperature is bounded below.

## Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (A) 26247013 and Grant-in-Aid for Challenging Exploratory Research 15K13448.

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