Open access peer-reviewed chapter

Relaxation Theory for Point Vortices

Written By

Ken Sawada and Takashi Suzuki

Submitted: 22 March 2016 Reviewed: 28 November 2016 Published: 01 March 2017

DOI: 10.5772/67075

From the Edited Volume

Vortex Structures in Fluid Dynamic Problems

Edited by Hector Perez-de-Tejada

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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. Ω R 2 . Motion of this fluid is described by the Euler-Poisson equation

ω t + u ω = 0 , Δ ψ = ω ,   u = ψ ,     ψ | Ω = 0 E1

where

= ( x 2 x 1 ) ,     x = ( x 1 , x 2 ) ,

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

In the point vortex model

ω ( x , t ) = i = 1 N α i δ x i ( t ) ( d x ) E2

system of Eq. (1) is reduced to

α i d x i d t = x i H N ,       i = 1 , 2 , , N E3

associated with the Hamiltonian

H N ( x 1 , x N ) = 1 2 i α i 2 R ( x i ) + i < j α i α j G ( x i , x j ) , E4

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

R ( x ) = [ G ( x , x ) + 1 2 π log | x x | ] 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 ) = I α ˜ ρ α ˜ ( x ) P ( d α ˜ ) , x Ω       E5

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

Ω ρ α ˜ ( x ) d x = 1 ,       α ˜ I ,

and P ( d α ˜ ) is the numerical density of the vortices with the relative intensity α ˜ . Under H N = E = constant , α 2 N β N = β = constant and N → ∞, mean field equation is derived by several arguments [27], that is,

Δ ψ ¯ = I α ˜ e β α ˜ ψ ¯ Ω e β α ˜ ψ ¯ P ( d α ˜ ) ,         ψ ¯ | Ω = 0 E6

with

ω ¯ = Δ ψ ¯ , ρ α ˜ = e β α ˜ ψ ¯ Ω e β α ˜ ψ ¯

where

ρ α ˜ ( x ) = lim N Ω N 1 μ N β N ( d x , d x 2 , d x N )
μ N β N ( d x 1 , d x N ) = 1 Z ( N , β N ) e β N H N d x 1 d x N
Z ( N , β N ) = Ω N e β N H N d x 1 d x N .

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 = 1 N p σ i 1 Ω i ( t ) ( x ) E7

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 + p u ¯ = D ( p + β p ( σ ω ¯ ) p ψ ¯ ) ,       β p = Ω D ω ¯ ψ ¯ Ω D ( I σ 2 p d σ ω ¯ 2 ) | ψ ¯ | 2 ω ¯ = I σ p d σ = Δ ψ ¯ ,       ψ ¯ | Ω = 0 ,       u ¯ = ψ ¯ E8

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 ρ α ˜ = ρ α ˜ ( x , t ) , α ˜ I , in the form of

ρ α ˜ t + ρ α ˜ u ¯ = D ( ρ α ˜ + β α ˜ ρ α ˜ ψ ¯ ) ,     ω ¯ = I α ˜ ρ α ˜ P ( d α ˜ ) = Δ ψ ¯ ,       ψ ¯ | Ω = 0 ,       u ¯ = ψ ¯ β = Ω D ω ¯ ψ ¯ Ω D I α ˜ 2 ρ α ˜ P ( d α ˜ ) | ψ ¯ | 2 . E9

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

ρ α ˜ ν + β α ˜ ρ α ˜ ψ ¯ ν | Ω = 0 E10

satisfies the requirements of isolated system in thermodynamics.

In fact, averaging Eq. (9) implies

ω ¯ t + ω ¯ u ¯ = D ( ω ¯ + β ω ¯ 2 ψ ¯ ) ,     ω ¯ ν + β ω ¯ 2 ψ ¯ ν | Ω = 0   ω ¯ = Δ ψ ¯ ,       ψ ¯ | Ω = 0 ,       u ¯ = ψ ¯ ,       β = Ω D ω ¯ ψ ¯ Ω D ω ¯ 2 | ψ ¯ | 2 E11

for

ω ¯ = I α ˜ ρ α ˜ P ( d α ˜ ) , ω ¯ 2 = I α ˜ 2 ρ α ˜ P ( d α ˜ ) . E12

Then, we obtain mass and energy conservations

d d t Ω ω ¯ = 0 , ( ω ¯ t , ψ ¯ ) = 1 2 d d t ( ω ¯ , ( Δ ) 1 ω ¯ ) = 0     E13

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

ρ α ˜ t + ρ α ˜ u ¯ = D ρ α ˜ ( log ρ α ˜ + β α ˜ ψ ¯ ) . E14

Then, it follows that

d d t Ω Φ ( ρ α ˜ ) d x + β α ˜ ( ρ t α ˜ , ψ ¯ )   = Ω D ρ α ˜ | ( log ρ α ˜ + β α ˜ ψ ¯ ) | 2 E15

from Eq. (10), where

Φ ( s ) = s ( log s 1 ) + 1 0 ,     s > 0.

Hence, it follows that

d d t Ω ( I Φ ( ρ α ˜ ) P ( d α ˜ ) ) = Ω ( I D ρ α ˜ | ( log ρ α ˜ + β α ˜ ψ ¯ ) | 2 P ( d α ˜ ) ) 0 E16

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 Ω i ( 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 | Ω i | Δ | Ω | . The probability that the average vorticity at x is σ is denoted by p ( x , σ , t ) d x ， which satisfies

p ( x , σ , t ) d σ = 1. E17

Let

Ω p ( x , σ , t ) d x = M ( σ )   E18

be independent of t. Since

| Ω | = p ( x , σ , t ) d x d σ = M ( σ ) d σ E19

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

ω ¯ ( x , t ) = σ p ( x , σ , t ) d σ , E20

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

ω ¯ = Δ ψ ¯ ,       ψ ¯ | Ω = 0 ,       u ¯ = ψ ¯ . E21

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

S ( p ) = p ( x , σ ) log p ( x , σ ) d x d σ E22

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

E = 1 2 Ω ω ¯ ψ ¯ . E23

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

δ S β p δ E c ( σ ) δ M ( σ ) d σ Ω ζ ( x ) ( δ p d σ ) d x = 0 , E24

which is reduced to

p ( x , σ ) = e c ( σ ) ( ζ ( x ) + 1 ) β p σ ψ ¯ . E25

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 ) β p σ ψ ¯ E26

and hence, Eq. (17) implies

p ( x , σ ) = e c ( σ ) β p σ ψ ¯ e c ( σ ' ) β p σ ' ψ ¯ d σ . E27

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

c ( σ ) = log ( Ω p ( x , 0 ) e β p σ ψ ¯ d x Ω p ( x , σ ) d x ) . E28

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

Δ ψ ¯ = σ M ( σ ) p ( x , 0 ) e β p σ ψ ¯ Ω p ( x , 0 ) e β p σ ψ ¯ d σ ,       ψ ¯ | Ω = 0 ω ¯ = I σ p d σ = Δ ψ ¯ , Ω p ( x , σ , t ) d x = M ( σ ) . E29

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

p t + ( p u ¯ ) = J , J ν | Ω = 0 E30

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 Ω p ( x , σ , t ) = 0 E31

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

ω ¯ t + ( ω ¯ u ¯ + J ω ) = 0 , E32

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

Here, we ignore the diffusion energy E d = 1 2 J 2 p d σ d x to take

E = 1 2 Ω ω ¯ ψ ¯ E33

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 , J d σ = 0 , J 2 2 p d σ C ( x , t ) E34

where

S ( p ) = p ( x , σ , t ) log p ( x , σ , t ) d σ d x .

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

δ S ˙ β p δ E ˙ Ω D 1 ( δ J 2 2 p d σ ) d x Ω ζ ( δ J d σ ) d x = 0. E35

Since

E ˙ = d d t E = Ω ψ ¯ ω ¯ t = Ω J ω ψ ¯ = σ J ψ ¯ d σ d x S ˙ = d d t S = p t ( log p + 1 ) d σ d x = J p p d σ d x , E36

Eq. (35) is reduced to

J = D ( p + β p σ p ψ ¯ + p ζ ) . E37

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

0 = J d σ = D ( p + β p σ p ψ ¯ + p ζ ) d σ = D ( β p ω ¯ ψ ¯ + ζ ) E38

and

0 = σ J ψ ¯   d σ d x = σ D ( p + β p σ p ψ ¯ + p ζ ) ψ ¯ d σ d x = σ D ( p + β p ( σ p p ω ¯ ) ψ ¯ ) ψ ¯ d σ d x = Ω D ω ¯ ψ ¯ d x β p Ω D ( σ 2 p d σ ω ¯ 2 ) | ψ ¯ | 2 d x E39

which implies

ζ = β p ω ¯ ψ ¯ E40

and

β p = Ω D ω ¯ ψ ¯ Ω D ( σ 2 p d σ ω ¯ 2 ) | ψ ¯ | 2 E41

Thus, we end up with

p t + ( p u ¯ ) = D ( p + β p ( σ ω ¯ ) p ψ ¯ ) ,       β p = Ω D ω ¯ ψ ¯ Ω D ( σ 2 p d σ ω ¯ 2 ) | ψ ¯ | 2 D ( p + β p ( σ ω ¯ ) p ψ ¯ ) ν | Ω = 0 ,       ω ¯ = I σ p d σ = Δ ψ ¯ ,       ψ ¯ | Ω = 0 ,       u ¯ = ψ ¯ E42

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 α α ˜ , probability ρ α ˜ ( x , t ) and number density P ( d α ˜ ) . 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 ).

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

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 ) d x d σ 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,

N c ( k ) ( Δ ) = N ( k ) ( x , σ ) d σ , E43

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 ) d σ = Ω N ( k ) ( x , σ ) , M ( k ) ( σ ) d σ = | Ω | N ( k ) ( σ ) d σ N ( k ) ( σ ) d σ , E44

and

γ ( k ) ( σ ) d σ = σ ( k ) M ( k ) ( σ ) d σ , E45

respectively.

We obtain

N p = N ( 0 ) ( x , σ ) d σ d x , E46

recalling Eq. (7). Since

σ ( k ) = 2 k σ , E47

it holds that

N ( k ) ( x , σ ) d x d σ = ( 4 k 2 k ) N c ( 0 ) ( Δ ) δ 0 ( d σ ) + 2 k N ( 0 ) ( x , σ ) d x d σ . E48

From Eq. (48), the related probability

p ( k ) ( x , σ ) d x d σ = N ( k ) ( x , σ ) d x d σ N c ( k ) ( Δ ) E49

satisfies

p ( k ) ( x , σ ) d x d σ = ( 4 k 2 k ) N c ( 0 ) ( Δ ) δ 0 ( d σ ) + 2 k N ( 0 ) ( x , σ ) d x d σ ( 4 k 2 k ) N c ( 0 ) ( Δ ) + 2 k N ( 0 ) ( x , σ ) d σ = ( 4 k 2 k ) N c ( 0 ) ( Δ ) δ 0 ( d σ ) + 2 k N ( 0 ) ( x , σ ) d x d σ 4 k N c ( 0 ) ( Δ ) E50

and hence,

lim k p ( k ) ( x , σ ) d x d σ = δ 0 ( d σ ) . E51

We also have

M ( k ) ( σ ) d σ = Ω p ( k ) ( x , σ ) d x = lim Δ 0 i = 1 | Ω | / Δ N ( k ) ( x i , σ ) d x d σ N c ( k ) ( Δ ) Δ E52

which implies

M ( k ) ( σ ) d σ = | Ω | 4 k N p lim Δ 0 i = 1 | Ω / Δ | N ( k ) ( x i , σ ) d σ = | Ω | 4 k N p N ( k ) ( σ ) d σ = | Ω | ( ( 1 2 k ) δ 0 ( d σ ) + 2 k N ( 0 ) ( σ ) d σ N p ) E53

by Δ N c ( k ) ( Δ ) = | Ω | 4 k N p and Eq. (48). We have, therefore,

lim k M ( k ) ( σ ) d σ = | Ω | δ 0 ( d σ ) . E54

It holds also that

γ ( k ) ( σ ) = Ω σ ( k ) p ( k ) ( x , σ ) d x = Ω σ p ( 0 ) ( x , σ ) d x = σ ( k ) M ( k ) ( σ ) d σ = σ | Ω | N p N ( 0 ) ( σ ) d σ E55

and

ω ¯ ( k ) ( x ) = σ ( k ) p ( k ) ( x , σ ) d σ = σ p ( 0 ) ( x , σ ) d σ . E56

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 | 4 k N p = α ˜ α ,   k 1 , E57

and Eq. (47), we reach the ansatz σ | Ω | = α ˜ , 1 2 k N p = α , 2 k N p = N . Similarly, we use

σ | Ω | N p N ( 0 ) ( σ ) d σ = α ˜ P ( d α ˜ ) E58

to put

N ( 0 ) ( σ ) d σ N p = M ( 0 ) ( σ ) d σ | Ω | = P ( d α ˜ ) E59

by

σ | Ω | N p N ( 0 ) ( σ ) d σ = σ | Ω | 1 2 k N p 2 k N p N ( 0 ) ( σ ) d σ N p = α ˜ α N P ( d α ˜ ) = α ˜ P ( d α ˜ ) . E60

Finally, we use the identity on local mean vorticity

σ p ( 0 ) ( x , σ ) d σ = α ˜ ρ α ˜ ( x ) P ( d α ˜ ) E61

to assign

1 | Ω | p ( 0 ) ( x , σ ) d σ = ρ α ˜ ( x ) P ( d α ˜ ) , E62

regarding

σ p ( 0 ) ( x , σ ) d σ = σ | Ω | p ( 0 ) ( x , σ ) | Ω | d σ = α ˜ ρ α ˜ ( x ) P ( d α ˜ ) . E63

These relations are summarized in the following Table 1 :

Vorticity patch model Point vortex model
σ | Ω | α ˜
1 2 k N p α
2 k N p N
N ( 0 ) ( σ ) d σ N p P ( d α ˜ )
1 | Ω | p ( 0 ) ( x , σ ) d σ ρ α ˜ ( x ) P ( d α ˜ )

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

Δ ψ ¯ = σ ( k ) M ( k ) ( σ ) p ( k ) ( x , 0 ) e β p σ ( k ) ψ ¯ Ω p ( k ) ( x , 0 ) e β p σ ( k ) ψ ¯ d σ = σ | Ω | N p N ( 0 ) ( σ ) p ( k ) ( x , 0 ) e β p 2 k σ ψ ¯ Ω p ( k ) ( x , 0 ) e β p 2 k σ ψ ¯ d σ = σ | Ω | p ( k ) ( x , 0 ) e β p 2 k | Ω | σ | Ω | ψ ¯ Ω p ( k ) ( x , 0 ) e β p 2 k | Ω | σ | Ω | ψ ¯ N ( 0 ) ( σ ) N p d σ . E64

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

α ˜ p ( k ) ( x , 0 ) e β N N α ˜ ψ ¯ Ω p ( k ) ( x , 0 ) e β N N α ˜ ψ ¯ P ( d α ˜ ) E65

for β N = 4 k N p | Ω | β p = N · 2 k β p | Ω | . Sending k → ∞, we obtain the first equation of (6) with β = β N N 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 ) ) E66

and hence

ζ ( x , t ) = D ( x , t ) 1 J ( x , 0 , t ) + p ( x , 0 , t ) p ( x , 0 , t ) . E67

Flux is thus given by

J ( x , σ , t ) = D ( x , t ) ( p ( x , σ , t ) + β p ( t ) σ p ( x , σ , t ) ψ ¯ ( x , t ) p ( x , σ , t ) D ( x , t ) 1 J ( x , 0 , t ) + p ( x , 0 , t ) p ( x , 0 , t ) ) . E68

We reach

p t + ( p u ¯ ) = D ( p + β p σ p ψ ¯ p [ D 1 J + p p ] σ = 0 ) E69

with

β p = β p ( t ) = Ω D ω ¯ ψ ¯ Ω D ω ¯ [ D 1 J + p p ] σ = 0 ψ ¯ D σ 2 p | ψ ¯ | 2 d σ d x E70

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

σ ( k ) p ( k ) t + ( σ ( k ) p ( k ) u ¯ ) = D ( σ ( k ) p ( k ) + β p ( σ ( k ) ) 2 p ( k ) ψ ¯ σ ( k ) p ( k ) [ D 1 J ( k ) + p ( k ) p ( k ) ] σ = 0 ) . E71

Putting β N = 4 k N p | Ω | β p , similarly, we obtain

t ( α ˜ ρ α ˜ P ( d α ˜ ) ) + ( α ˜ ρ α ˜ P ( d α ˜ ) u ¯ ) = ( D ( ( α ˜ ρ α ˜ P ( d α ˜ ) ) + β α ˜ 2 ρ α ˜ P ( d α ˜ ) ψ ¯ ) ) , E72

from

lim k p ( k ) ( x , σ , t ) = δ 0 ( d σ ) , lim k J ( k ) ( x , 0 , t ) = 0 σ ( k ) p ( k ) ( x , σ , t ) = σ p ( 0 ) ( x , σ , t ) = σ | Ω | p ( 0 ) ( x , σ , t ) | Ω | α ˜ ρ α ˜ ( x , t ) P ( d α ˜ ) ( σ ( k ) ) 2 p ( k ) ( x , σ , t ) = 2 k σ σ p ( 0 ) ( x , σ , t ) = 2 k | Ω | ( σ | Ω | ) 2 p ( 0 ) ( x , σ , t ) | Ω | 2 k | Ω | α ˜ 2 ρ α ˜ ( x , t ) P ( d α ˜ ) E73

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

4. Relaxation dynamics

If P ( d α ˜ ) = δ 1 ( d α ˜ ) , it holds that ω ¯ = ω ¯ 2 in Eq. (11). Then, we obtain

ω t + ω ψ = ( ω + β ω ψ ) ,       ω ν + β ω ψ ν | Ω = 0 , ω | t = 0 = ω 0 ( x ) 0 E74
Δ ψ = ω ,       ψ | Ω = 0 , β = Ω ω ψ Ω ω | ψ | 2 E75

assuming D = 1. Conservations of total mass and energy

ω ( , t ) 1 = λ ,   ( ψ ( , t ) , ω ( , t ) ) = e , E76

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

d d t Ω Φ ( ω ) = Ω ω | ( log ω β ψ ) | 2 0 , E77

where Φ ( s ) = s ( log s 1 ) + 1 .

In the stationary state, we obtain log ω + β ψ = constant by Eq. (77). Hence, it follows that

Δ ψ = ω ,       ψ | Ω = 0 ,     ω = λ e β ψ Ω e β ψ ,     β = Ω ω ψ Ω ω | ψ | 2 ,       e = Ω ω ψ E78

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

( ω , ψ ) = β Ω ω | ψ | 2 . E79

Using

v = β ψ , μ = β λ Ω e β ψ ,   E80

therefore, Eq. (78) is reduced to

Δ v = μ e v ,   v | Ω = 0 , e λ 2 = Ω | v | 2 ( Ω v ν ) 2   . E81

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

e = ( ω , ψ ) = β 1 λ Ω e v v Ω e v E82

which implies

μ = λ Ω e v λ e Ω e v v Ω e v = λ 2 e Ω e v v ( Ω e v ) 2 E83

and hence

e λ 2 = 1 μ Ω e v v ( Ω e v ) 2 = v 2 2 ( Ω v ν ) 2   . E84

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

Δ w = σ e w , w | Ω = 0 E85

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

lim μ 0 e λ 2 = + E86

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

lim μ + v μ 2 2 ( Ω v μ ν ) 2 = 0 E87

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 { x R 2 |   | 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 = v log μ , s = log r , we obtain

u s s + e u + 2 s = 0 ,     s < 0 ,     u ( 0 ) = log μ ,     lim s u s e s = 0 ,     v 2 2 ( Ω v ν ) 2 = I 2 π , E88

where I = 0 u s 2 d s u s ( 0 ) 2 . Using w = u 2 s , p = 1 2 ( e w + 2 ) 1 / 2 , we have

p = 1 + 2 ( 1 c e 2 s ) 1 E89

with c ↑ 1 as μ ↑ +∞. It follows that

I = ( 1 c ) 2 0 e 4 s ( 1 c e 2 s ) 2 d s E90

with

0 e 4 s ( 1 c e 2 s ) 2 d s = 1 2 c ( 1 c ) + 1 2 c 2 log ( 1 c ) E91

and hence

lim c 1 I = 0. E92

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:

d F d t = Ω ω | ( log ω + β ψ ) | 2 , F ( ω ) = Ω Φ ( ω ) 1 2 ( ( Δ ) 1 ω , ω ) . E93

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

Δ G ( , x ) = δ x , G ( , x ) | Ω = 0 , x Ω E94

and

ρ φ ( x , x ) = φ ( x ) x G ( x , x ) + φ ( x ) x ' G ( x , x ) ,   φ X , E95

where X = { φ C 2 ( Ω ¯ ) | φ ν | Ω = 0 } . It holds that ρ φ L ( 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 ω 0 = ω 0 ( r ) > 0 with ω 0 r < 0 , 0 < r 1 . 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

lim sup t T β ( t ) < 8 π λ     T < + E96

and

T < +     lim inf t T β ( t ) = . E97

In particular, we have

lim inf t T β ( t ) >   T = + ,   lim sup t T β ( t ) 8 π λ   . E98

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

ω r ,   ψ r < 0 , 0 < r 1.

Then, we obtain

M λ 2 π 0 r r ω d r ω ( r , t ) 0 r r d r = r 2 2 ω E99

and hence

ω ( r , t ) 2 M r 2 , 0 < r 1. E100

It holds also that

β = 0 1 ω r ψ r r d r 0 1 ω ψ r 2 r d r > 0 E101

which implies

ω t = Δ ω + β ψ ω + β ω Δ ψ = Δ ω + β ψ ω β ω 2 Δ ω + β ψ ω E102

with

ω ν = β ω ψ ν > 0 on Ω × ( 0 , T ) . E103

The comparison theorem now guarantees ω δ min Ω ¯   ω 0 > 0 and hence

Ω ω | ψ | 2 δ Ω | ψ | 2 = δ e . E104

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

r ψ r = 0 r r ω d r A ( r ) . E105

Second, it follows that

d d t 0 1 ω r 3 d r = 0 1 ( ω r + β ω ψ r ) 2 r r d r = 2 r 2 ω | r = 0 r = 1 + 0 1 4 r ω 2 β ω ψ r r 2 d r = 2 ω | r = 1 + 4 M + 2 β 0 1 A A r d r = 2 ω | r = 1 + 4 M + β M 2 4 M + β M 2 E106

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

4 M + β M 2 δ ,       t T . E107

Then, T = +∞ gives a contradiction.

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

0 T β ( t ) d t C E108

by Eq. (100). Second, we have

d d t Ω ω φ = Ω ω Δ φ + β 2 Ω × Ω ρ φ ω ω E109

and hence

0 T | d d t Ω ω φ | d t C φ , φ X . E110

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

ω ( x , t ) d x c δ 0 ( d x ) + f ( x ) d x in M ( Ω ¯ ) ,       t T , E111

with c ≥ 0 and 0 f = f ( x ) L 1 ( Ω ) . From the elliptic regularity, we obtain

lim inf t T ψ ( x , t ) c 2 π log 1 | x |       loc .  unif .  in     Ω ¯ { 0 } . E112

Then, e = ( ω ( , t ) , ψ ( , t ) ) ( ω ( , t ) , m i n { k , ψ ( , t ) } ) implies e c 2 π min { k , log 1 | 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 ε 0 = ε 0 k > 0 , such that

lim sup t T ω ( , t ) L 1 ( Ω B ( x 0 , R ) ) < ε 0     lim sup t T ω ( , t ) L ( Ω B ( x 0 , R / 2 ) ) < + E113

for 0 < R 1 . The hypothesis in Eq. (113) is valid for x 0 = 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.

Acknowledgments

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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Written By

Ken Sawada and Takashi Suzuki

Submitted: 22 March 2016 Reviewed: 28 November 2016 Published: 01 March 2017