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Relaxation Theory for Point Vortices

Written By

Ken Sawada and Takashi Suzuki

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

DOI: 10.5772/67075

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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.


  • 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


=(x2x1),  x=(x1,x2),

and u, ωand ψstand for the velocity, vorticity and stream function, respectively.

In the point vortex model


system of Eq. (1) is reduced to

αidxidt=xiHN,   i=1,2,,NE3

associated with the Hamiltonian


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


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 xwith 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





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


is used. It describes detailed vortex distribution, where Np, σiand Ωi(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


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



Then, we obtain mass and energy conservations

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

where (,) stands for the L2 inner product. Assuming ρα˜>0, we write the first equation of (9) as


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


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


2. Vorticity patch model

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



Ωp(x,σ,t)dx=M(σ) E18

be independent of t. Since


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


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


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


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


which is reduced to


Here, βpand 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


and hence, Eq. (17) implies


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


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


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


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


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, Jωis associated with the detailed fluctuation of (ω, u) from (ω¯,u¯)by Eq. (1).

Here, we ignore the diffusion energy Ed=12J2pdσdxto take


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




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




Eq. (35) is reduced to


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



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




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 σand 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 Ωbe 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 xof which vorticity was originally σand 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,


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





We obtain


recalling Eq. (7). Since


it holds that


From Eq. (48), the related probability




and hence,


We also have


which implies


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


It holds also that




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 σand 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


to put




Finally, we use the identity on local mean vorticity


to assign




These relations are summarized in the following Table 1 :

Vorticity patch modelPoint vortex model

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


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


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 Jfor σ= 0 is


and hence


Flux is thus given by


We reach




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


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




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


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



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


which implies


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


with σ= –μ. If Ωis 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


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


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


with c↑ 1 as μ↑ +∞. It follows that




and hence


If βis 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:


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,



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= Band ω0 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 λbe the total mass defined by Eq. (76). Then, it follows that

limsuptTβ(t)<8πλ  T<+E96


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


and hence


It holds also that


which implies




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


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


Second, it follows that


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


by Eq. (100). Second, we have


and hence


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= Tas 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 εregularity 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.



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