Relation between vorticity patch model and point vortex model for .
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
- relaxation dynamics
- maximum entropy production
- global-in-time solution
The physical object studied in this chapter is non-viscous, noncompressible fluid with high Reynolds number occupied in bounded, simply-connected domain. . Motion of this fluid is described by the Euler-Poisson equation
and u, ω and ψ stand for the velocity, vorticity and stream function, respectively.
In the point vortex model
system of Eq. (1) is reduced to
associated with the Hamiltonian
where is the Green’s function of –Δ provided with the Dirichlet boundary condition and
where is the intensity of the i-th vortex, is the existence probability of the vortex at x with relative intensity , which satisfies
Quasi-equilibria, on the other hand, are observed for several isolated systems with many components . 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 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 ,
with the diffusion coefficient
Second, the stationary state of Eq. (9) is given by Eq. (6). Third, Eq. (9) coincides with the Brownian point vortex model of Chavanis . Finally, system of Eq. (9) provided with the boundary condition
satisfies the requirements of isolated system in thermodynamics.
In fact, averaging Eq. (9) implies
Then, we obtain mass and energy conservations
where (,) stands for the L 2 inner product. Assuming , we write the first equation of (9) as
Then, it follows that
from Eq. (10), where
Hence, it follows that
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 , 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 . The probability that the average vorticity at x is σ is denoted by ， which satisfies
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 and the velocity through
With the Lagrange multipliers , it follows that
which is reduced to
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
and hence, Eq. (17) implies
From Eqs. (18) and (26), similarly, it follows that
with the diffusion flux of , where ν denotes the outer unit normal vector. We obtain the total patch area conservation for each σ,
Here, we ignore the diffusion energy to take
as the total energy of this system. Using maximum entropy production principle, we chose the flux J to maximize entropy production rate under the constraint
Using Lagrange multipliers , we obtain
Eq. (35) is reduced to
From the constraint of Eq. (34), it follows that
Thus, we end up with
3. Point vortex model
Point vortex model is regarded as a special case of vorticity patch model, where the patch size shrinks to zero . 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 , probability and number density . 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 are the fundamental quantities ( Figure 1 ).
Here, we use the following localization in order to transform vorticity patch to point vortex ( Figure 2 ):
Divide each patch into two patches with half area and the same vorticity.
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 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,
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
recalling Eq. (7). Since
it holds that
From Eq. (48), the related probability
We also have
by and 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 σ and local mean vorticity. Based on
and Eq. (47), we reach the ansatz . Similarly, we use
Finally, we use the identity on local mean vorticity
These relations are summarized in the following Table 1 :
|Vorticity patch model||Point vortex model|
After k-times localization, the first equation in Eq. (29) takes the form
for . Sending k → ∞, we obtain the first equation of (6) with 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
Flux is thus given by
Therefore, after k-times localization procedure, it holds that
Putting , similarly, we obtain
4. Relaxation dynamics
If , it holds that in Eq. (11). Then, we obtain
assuming D = 1. Conservations of total mass and energy
In the stationary state, we obtain by Eq. (77). Hence, it follows that
from Eq. (76). Here, the third equation implies the fourth equation as
therefore, Eq. (78) is reduced to
In fact, to see the third equality of (81), we note
If μ < 0, system of Eq. (81) except for the third equation is equivalent to the Gel’fand problem
If , it actually holds that Eq. (87). In this case, we have and the result follows from an elementary calculation. More precisely, putting , , we obtain
where . Using , , we have
with c ↑ 1 as μ ↑ +∞. It follows that
If β is constant in Eq. (9), it is the mean field limit of Brownian vortices . 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 . Here, we show the following theorem, where denotes the Green’s function for the Poisson part,
where . It holds that . The proof is similar as in Lemma 5.2 of  for the case of Neumann boundary condition.
Theorem 1: Let Ω = B and ω 0 be a smooth function in the form of with , . 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
In particular, we have
Proof: From the assumption, it follows that and
Then, we obtain
It holds also that
The comparison theorem now guarantees and hence
Second, it follows that
from A(1) = M. Under the hypothesis of Eq. (96), we have δ > 0 such that
Then, T = +∞ gives a contradiction.
Now, we assume T < +∞. First, equality in (106) implies
by Eq. (100). Second, we have
Inequality (110) takes place of the monotonicity formula used for the Smoluchowski-Poisson equation, which guarantees the continuation of up to t = T as a measure on [9, 17]. Thus, there is such that for 0 ≤ t < T. By Eq. (100), therefore, it holds that
with c ≥ 0 and . From the elliptic regularity, we obtain
Then, implies for k = 1,2,. Hence, it holds that c = 0 in Eq. (111).
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.