Open access peer-reviewed chapter

Relaxation Theory for Point Vortices

By Ken Sawada and Takashi Suzuki

Submitted: March 22nd 2016Reviewed: November 28th 2016Published: March 1st 2017

DOI: 10.5772/67075

Downloaded: 524

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=0E1

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)E2

system of Eq. (1) is reduced to

αidxidt=xiHN,   i=1,2,,NE3

associated with the Hamiltonian

HN(x1,xN)=12iαi2R(xi)+i<jαiαjG(xi,xj),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)+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˜),xΩ   E5

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=constantand N → ∞, mean field equation is derived by several arguments [27], that is,

Δψ¯=Iα˜eβα˜ψ¯Ωeβα˜ψ¯P(dα˜),    ψ¯|Ω=0E6

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)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),

pt+pu¯=D(p+βp(σω¯)pψ¯),   βp=ΩDω¯ψ¯ΩD(Iσ2pdσω¯2)|ψ¯|2ω¯=Iσpdσ=Δψ¯,   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 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=ΩDω¯ψ¯ΩDIα˜2ρα˜P(dα˜)|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

ρα˜ν+βα˜ρα˜ψ¯ν|Ω=0E10

satisfies the requirements of isolated system in thermodynamics.

In fact, averaging Eq. (9) implies

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

for

ω¯=Iα˜ρα˜P(dα˜),ω¯2=Iα˜2ρα˜P(dα˜).E12

Then, we obtain mass and energy conservations

ddtΩω¯=0,(ω¯t,ψ¯)=12ddt(ω¯,(Δ)1ω¯)=0  E13

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¯).E14

Then, it follows that

ddtΩΦ(ρα˜)dx+βα˜(ρtα˜,ψ¯) =ΩDρα˜|(logρα˜+βα˜ψ¯)|2E15

from Eq. (10), where

Φ(s)=s(logs1)+10,  s>0.

Hence, it follows that

ddtΩ(IΦ(ρα˜)P(dα˜))=Ω(IDρα˜|(logρα˜+βα˜ψ¯)|2P(dα˜))0E16

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.E17

Let

Ωp(x,σ,t)dx=M(σ) E18

be independent of t. Since

|Ω|=p(x,σ,t)dxdσ=M(σ)dσE19

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,E20

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¯.E21

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)dxdσE22

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

E=12Ωω¯ψ¯.E23

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

δSβpδEc(σ)δM(σ)dσΩζ(x)(δpdσ)dx=0,E24

which is reduced to

p(x,σ)=ec(σ)(ζ(x)+1)βpσψ¯.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)ec(σ)βpσψ¯E26

and hence, Eq. (17) implies

p(x,σ)=ec(σ)βpσψ¯ec(σ')βpσ'ψ¯dσ.E27

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

c(σ)=log(Ωp(x,0)eβpσψ¯dxΩp(x,σ)dx).E28

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)eβpσψ¯Ωp(x,0)eβpσψ¯dσ,   ψ¯|Ω=0ω¯=Iσpdσ=Δψ¯,Ωp(x,σ,t)dx=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

pt+(pu¯)=J,Jν|Ω=0E30

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,

Mt=tΩp(x,σ,t)=0E31

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

ω¯t+(ω¯u¯+Jω)=0,E32

where Jω=σJ(x,σ,t)dσstands for the local mean vorticity flux. Since Jων=0on 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=12J2pdσdxto take

E=12Ωω¯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,Jdσ=0,J22pdσC(x,t)E34

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˙ΩD1(δJ22pdσ)dxΩζ(δJdσ)dx=0.E35

Since

E˙=ddtE=Ωψ¯ω¯t=ΩJωψ¯=σJψ¯dσdxS˙=ddtS=pt(logp+1)dσdx=Jppdσdx,E36

Eq. (35) is reduced to

J=D(p+βpσpψ¯+pζ).E37

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

0=Jdσ=D(p+βpσpψ¯+pζ)dσ=D(βpω¯ψ¯+ζ)E38

and

0=σJψ¯ dσdx=σD(p+βpσpψ¯+pζ)ψ¯dσdx=σD(p+βp(σppω¯)ψ¯)ψ¯dσdx=ΩDω¯ψ¯dxβpΩD(σ2pdσω¯2)|ψ¯|2dxE39

which implies

ζ=βpω¯ψ¯E40

and

βp=ΩDω¯ψ¯ΩD(σ2pdσω¯2)|ψ¯|2E41

Thus, we end up with

pt+(pu¯)=D(p+βp(σω¯)pψ¯),   βp=ΩDω¯ψ¯ΩD(σ2pdσω¯2)|ψ¯|2D(p+βp(σω¯)pψ¯)ν|Ω=0,   ω¯=Iσpdσ=Δψ¯,   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 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)dxdσbe 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,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)dσ=ΩN(k)(x,σ),M(k)(σ)dσ=|Ω|N(k)(σ)dσN(k)(σ)dσ,E44

and

γ(k)(σ)dσ=σ(k)M(k)(σ)dσ,E45

respectively.

We obtain

Np=N(0)(x,σ)dσdx,E46

recalling Eq. (7). Since

σ(k)=2kσ,E47

it holds that

N(k)(x,σ)dxdσ=(4k2k)Nc(0)(Δ)δ0(dσ)+2kN(0)(x,σ)dxdσ.E48

From Eq. (48), the related probability

p(k)(x,σ)dxdσ=N(k)(x,σ)dxdσNc(k)(Δ)E49

satisfies

p(k)(x,σ)dxdσ=(4k2k)Nc(0)(Δ)δ0(dσ)+2kN(0)(x,σ)dxdσ(4k2k)Nc(0)(Δ)+2kN(0)(x,σ)dσ=(4k2k)Nc(0)(Δ)δ0(dσ)+2kN(0)(x,σ)dxdσ4kNc(0)(Δ)E50

and hence,

limkp(k)(x,σ)dxdσ=δ0(dσ).E51

We also have

M(k)(σ)dσ=Ωp(k)(x,σ)dx=limΔ0i=1|Ω|/ΔN(k)(xi,σ)dxdσNc(k)(Δ)ΔE52

which implies

M(k)(σ)dσ=|Ω|4kNplimΔ0i=1|Ω/Δ|N(k)(xi,σ)dσ=|Ω|4kNpN(k)(σ)dσ=|Ω|((12k)δ0(dσ)+2kN(0)(σ)dσNp)E53

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

limkM(k)(σ)dσ=|Ω|δ0(dσ).E54

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)dσE55

and

ω¯(k)(x)=σ(k)p(k)(x,σ)dσ=σp(0)(x,σ)dσ.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|4kNp=α˜α, k1,E57

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˜)E58

to put

N(0)(σ)dσNp=M(0)(σ)dσ|Ω|=P(dα˜)E59

by

σ|Ω|NpN(0)(σ)dσ=σ|Ω|12kNp2kNpN(0)(σ)dσNp=α˜αNP(dα˜)=α˜P(dα˜).E60

Finally, we use the identity on local mean vorticity

σp(0)(x,σ)dσ=α˜ρα˜(x)P(dα˜)E61

to assign

1|Ω|p(0)(x,σ)dσ=ρα˜(x)P(dα˜),E62

regarding

σp(0)(x,σ)dσ=σ|Ω|p(0)(x,σ)|Ω|dσ=α˜ρα˜(x)P(dα˜).E63

These relations are summarized in the following Table 1 :

Vorticity patch modelPoint vortex model
σ|Ω|α˜
12kNpα
2kNpN
N(0)(σ)dσNpP(dα˜)
1|Ω|p(0)(x,σ)dσρα˜(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)eβpσ(k)ψ¯Ωp(k)(x,0)eβpσ(k)ψ¯dσ=σ|Ω|NpN(0)(σ)p(k)(x,0)eβp2kσψ¯Ωp(k)(x,0)eβp2kσψ¯dσ=σ|Ω|p(k)(x,0)eβp2k|Ω|σ|Ω|ψ¯Ωp(k)(x,0)eβp2k|Ω|σ|Ω|ψ¯N(0)(σ)Npdσ.E64

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

α˜p(k)(x,0)eβNNα˜ψ¯Ωp(k)(x,0)eβNNα˜ψ¯P(dα˜)E65

for βN=4kNp|Ω|βp=N·2kβp|Ω|. Sending k → ∞, we obtain the first equation of (6) with β=βNNby 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)1J(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)+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)).E68

We reach

pt+(pu¯)=D(p+βpσpψ¯p[D1J+pp]σ=0)E69

with

βp=βp(t)=ΩDω¯ψ¯ΩDω¯[D1J+pp]σ=0ψ¯Dσ2p|ψ¯|2dσdxE70

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)[D1J(k)+p(k)p(k)]σ=0).E71

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¯)),E72

from

limkp(k)(x,σ,t)=δ0(dσ),limkJ(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)=2kσσp(0)(x,σ,t)=2k|Ω|(σ|Ω|)2p(0)(x,σ,t)|Ω|2k|Ω|α˜2ρα˜(x,t)P(dα˜)E73

Here, we assume limkJ(k)(x,0,t)=0, because J(k)(x,σ,t)dσ=0and 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¯2in Eq. (11). Then, we obtain

ωt+ωψ=(ω+βωψ),   ων+βωψν|Ω=0,ω|t=0=ω0(x)0E74
Δψ=ω,   ψ|Ω=0,β=ΩωψΩω|ψ|2E75

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

ddtΩΦ(ω)=Ωω|(logωβψ)|20,E77

where Φ(s)=s(logs1)+1.

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

Δψ=ω,   ψ|Ω=0,  ω=λeβψΩeβψ,  β=ΩωψΩω|ψ|2,   e=ΩωψE78

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

(ω,ψ)=βΩω|ψ|2.E79

Using

v=βψ,μ=βλΩeβψ, E80

therefore, Eq. (78) is reduced to

Δv=μev, v|Ω=0,eλ2=Ω|v|2(Ωvν)2 .E81

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

e=(ω,ψ)=β1λΩevvΩevE82

which implies

μ=λΩevλeΩevvΩev=λ2eΩevv(Ωev)2E83

and hence

eλ2=1μΩevv(Ωev)2=v22(Ωvν)2 .E84

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

Δw=σew,w|Ω=0E85

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

limμ0eλ2=+E86

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+vμ22(Ωvμν)2=0E87

for any orbit to Eqs. (74), (75) to be global-in-time and compact, for any λ,e>0in 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+eu+2s=0,  s<0,  u(0)=logμ,  limsuses=0,  v22(Ωvν)2=I2π,E88

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

p=1+2(1ce2s)1E89

with c ↑ 1 as μ ↑ +∞. It follows that

I=(1c)20e4s(1ce2s)2dsE90

with

0e4s(1ce2s)2ds=12c(1c)+12c2log(1c)E91

and hence

limc1I=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:

dFdt=Ωω|(logω+βψ)|2,F(ω)=ΩΦ(ω)12((Δ)1ω,ω).E93

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,xΩE94

and

ρφ(x,x)=φ(x)xG(x,x)+φ(x)x'G(x,x), φX,E95

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)>0with 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

limsuptTβ(t)<8πλ  T<+E96

and

T<+  liminftTβ(t)=.E97

In particular, we have

liminftTβ(t)> T=+, limsuptTβ(t)8πλ .E98

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

ωr, ψr<0,0<r1.

Then, we obtain

Mλ2π0rrωdrω(r,t)0rrdr=r22ωE99

and hence

ω(r,t)2Mr2,0<r1.E100

It holds also that

β=01ωrψrrdr01ωψr2rdr>0E101

which implies

ωt=Δω+βψω+βωΔψ=Δω+βψωβω2Δω+βψωE102

with

ων=βωψν>0onΩ×(0,T).E103

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

Ωω|ψ|2δΩ|ψ|2=δe.E104

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

rψr=0rrωdrA(r).E105

Second, it follows that

ddt01ωr3dr=01(ωr+βωψr)2rrdr=2r2ω|r=0r=1+014rω2βωψrr2dr=2ω|r=1+4M+2β01AArdr=2ω|r=1+4M+βM24M+βM2E106

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

4M+βM2δ,   tT.E107

Then, T = +∞ gives a contradiction.

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

0Tβ(t)dtCE108

by Eq. (100). Second, we have

ddtΩωφ=ΩωΔφ+β2Ω×ΩρφωωE109

and hence

0T|ddtΩωφ|dtCφ,φX.E110

Inequality (110) takes place of the monotonicity formula used for the Smoluchowski-Poisson equation, which guarantees the continuation of ω(x,t)dxup 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)dxfor 0 ≤ t < T. By Eq. (100), therefore, it holds that

ω(x,t)dxcδ0(dx)+f(x)dxinM(Ω¯),   tT,E111

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

liminftTψ(x,t)c2πlog1|x|   loc. unif. in  Ω¯{0}.E112

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

limsuptTω(,t)L1(ΩB(x0,R))<ε0  limsuptTω(,t)L(ΩB(x0,R/2))<+E113

for 0<R1. The hypothesis in Eq. (113) is valid for x0=0by 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.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ken Sawada and Takashi Suzuki (March 1st 2017). Relaxation Theory for Point Vortices, Vortex Structures in Fluid Dynamic Problems, Hector Perez-de-Tejada, IntechOpen, DOI: 10.5772/67075. Available from:

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