Relaxation Dynamics of Point Vortices

We study a model describing relaxation dynamics of point vortices, from quasi-stationary state to the stationary state. It takes the form of a mean field equation of Brownian point vortices derived from Chavanis, and is formulated by our previous work as a limit equation of the patch model studied by Robert-Someria. This model is subject to the micro-canonical statistic laws; conservation of energy, that of mass, and increasing of the entropy. We study the existence and nonexistence of the global-in-time solution. It is known that this profile is controlled by a bound of the negative inverse temperature. Here we prove a rigorous result for radially symmetric case. Hence E = M 2 large and small imply the global-in-time and blowup in finite time of the solution, respectively. Where E and M denote the total energy and the total mass, respectively.


Introduction
Our purpose is to study the system with where Ω ⊂ R 2 is a bounded domain with smooth boundary ∂Ω, ν is the outer unit normal vector on ∂Ω, and The unknown ω ¼ ω x, t ð Þ∈ R stands for a mean field limit of many point vortices, It was derived, first, for Brownian point vortices by [1,2], with β ¼ β t ð Þ standing for the inverse temperature. Then, [3,4] reached it by the Lynden-Bell theory [5] of relaxation dynamics, that is, as a model describing the movement of the mean field of many point vortices, from quasi-stationary state to the stationary state. This model is consistent to the Onsager theory [6][7][8][9][10][11][12] on stationary states and also the patch model proposed by [13,14], that is, where N p , σ i , and Ω i t ð Þ denote the number of patches, the vorticity of the i-th patch, and the domain of the i-th patch, respectively [15][16][17].
This chapter is concerned on the one-sided case of If this initial value is smooth, there is a unique classical solution to (1)-(4) local in time, denoted by ω ¼ ω x, t ð Þ, with the maximal existence time T ¼ T max ∈ 0, þ∞ ð . More precisely, the strong maximum principle to (1) guaranttes Then, the Hopf lemma to the Poisson equation in (2) ensures ∂ψ ∂ν and hence the well-definedness of We confirm that system (1)-(3) satisfies the requirements of isolated system of thermodynamics. First, the mass conservation is derived from (1) as d dt because ν Á ∇ ⊥ ψ ∂Ω ¼ 0 (11) holds by (2). Second, the energy conservation follows as (1) and (2), because where , ð Þ denotes the L 2 inner product. Third, the entropy increasing is achieved, writing (1) as In fact, it then follows that ð from (11) and We thus end up with the mass conservation the energy conservation and the entropy increasing Henceforth, C > 0 stands for a generic constant. In the previous work [4] we studied radially symmetric solutions and obtained a criterion for the existence of the solution global in time. Here, we refine the result as follows, where B 0, 1 ð Þ denotes the unit ball. Theorem 1 Let Then there arise the mass conservation d dt and the free energy decreasing d dt The system (31) without vortex term, is called the Smoluchowski-Poisson equation. This model is concerned on the thermodynamics of self-gravitating Brownian particles [18] and has been studied in the context of chemotaxis [19][20][21][22][23]. We have a blowup threshold to (34) as a consequence of the quantized blowup mechanism [19,23]. The results on the existence of the bounded global-in-time solution [24][25][26] and blowup of the solution in finite time [27] are valid even to the case that β is a function of t as in β ¼ β t ð Þ. provided with the vortex term ∇ Á ω∇ ⊥ ψ on the right-hand side. We thus obtain the following theorems.

Theorem 4 It holds that
in (31), where δ > 0 is arbitrary. Remark 4 In the context of chemotaxis in biology, the boundary condition of ψ is required to be the form of Neumann zero. The Poisson equation in (34) is thus replaced by by [28] and [29], respectively. In this case there arises the boundary blowup, which reduces the value 8π in Theorems 3-4 to 4π. The value 8π in Theorems 3-4, therefore, is a consequence of the exclusion of the boundary blowup [30]. This property is valid even for (37) or (38) of the Poisson part, if (22) is assumed.
Remark 5 The requirement to ω 0 in Theorem 4 is the concentration at an interior point, which is not necessary in the case of (22). Hence Theorems 3 and 4 are refined as if (22) holds in (35). The main task for the proof of Theorems 1 and 2, therefore, is a control of β ¼ β t ð Þ in (1). This paper is composed of four sections and an appendix. Section 2 is devoted to the study on the stationary solutions, and Theorems 1 and 2 are proven in Sections 3 and 4, respectively. Then Theorem 4 is confirmed in Appendix.

Stationary states
First, we take the canonical system (31) with β independent of t. By (32) and (33), its stationary state is defined by Then it holds that and hence There arises an oredered structure arises in β < 0, as observed by [11], as a consequence of a quantized blowup mechanism [19,20,31]. In the microcanonical system (1) and (2), the value β in (43) has to be determined by E besides M.
Equality (21), however, still ensures (41) and hence (42) in the stationary state even for (1) we obtain by (30) and (43). This system is the stationary state of (1) and (2) introduced by [4]. The first two equalities comprise a nonlinear elliptic eigenvalue problem and the unknown eigenvalue μ is determined by the third equality, 6 Vortex Dynamics -From Physical to Mathematical Aspects The elliptic theory ensures rather deailed features of the set of solutions to (46). Here we note the following facts [31].
We show the following theorem, consistent to Theorem 2.

Proof of Theorem 1
The first observation is the following lemma. Lemma 1 Under the assumption of (22), it holds that Proof: We have (7) and hence by (49), which implies, in particular, at t ¼ 0 by (22).
The inequality β < 0, on the other hand, is sufficient for the following arguments.
Proof: Since (17) we obtain by (8). Then the result follows from the comparison theorem. □ Lemma 3 Under the assumption of the previous lemma, there is Proof: Using (11) and (17), we obtain ð Hence (1) with (2) implies by β ≤ 0 and (88). Since ð follows from (8), furthermore, it holds that Then ineqality (80) induces Here we use the Gagliardo-Nirenberg inequality (see (4.16) of [19]) in the form of to obtain and hence Then, Poincaré-Wirtinger's inequality ensures we obtain and therefore, if and hence by (82). The condition y 0 ð Þ < μ 2 means for C 0 > 0 sufficiently large, and hence we obtain the conclusion. □ Proof of Theorem 1: By the parabolic regularity, it suffices to show that under the assumption. We have readily shown by Lemma 3. Then, the conclusion (95) is obtained similarly to (34). See [26] for more details.

Proof of Theorem 2
We begin with the following lemma. Lemma 4 Under the assumption of (22), it holds that Proof: We have ω ¼ ω r, t ð Þ and ψ ¼ ψ r, t ð Þ for r ¼ |x| under the assumption, which implies ∇ ⊥ ψ ¼ 0. Then we obtain by (17). It holds also that and therefore, there arises that from (31). Then (102) implies d dt Here we use (50) derived from the Poisson part of (31), that is, Putting λ ¼ we obtain d dt Since Àδ þ M 8π < 0, therefore, T ¼ þ∞ is impossible, and we obtain T < þ ∞. □ Lemma 5 Under the assumption (22), there is δ > 0 such that Proof: First, Lemma 1 implies and hence Àβ ¼ Here, we use the Gagliardo-Nirenberg inequality in the form of which implies by the elliptic estimate of the Poisson equation in (2), We have, on the other hand, by (110), and therefore, provided that Then the conclusion follows. □ Proof of Theorem 2: By Lemma 5, there is δ 0 > such that E M 2 < δ ) Àβ ≥ and then, Lemma 4 ensures The assumption in (118) means and hence we obtain the conclusion. □

Appendix Proof of Theorem 4
This theorem is valid to the general case of Ω and ω 0 without (22). We assume δ ¼ 1 without loss of generation, so that β ≤ À 1: (120) We follow the argument [27] concerning (34) with the Poisson part replaced by (42) or (43). Thus we have to take case of the vortex term ∇ Á ω∇ ⊥ ψ, time varying β ¼ β t ð Þ, and the Dirichlet boundary condition in (31). We recall the cut-off function used in [34] (see also Chapter 5 of [19]). Hence and In more details, we take a cut-off function, denoted by ψ, satisfying (121), using a local conformal mapping, and then put φ ¼ ψ 4 . Let be given. First, we have d dt by (11). It holds that Let, furthermore, x 0 ∈ Ω and 0 < R ≪ 1 in the above equality. Then, satisfies the requirement (123). It holds that and hence We obtain, furthermore, and hence in this case. Then it follows that where We have, on the other hand, where G ¼ G x, x 0 ð Þis the Green's function to and Here we use the local property of the Green's function where stands for the fundamental solution to ÀΔ. Let Since (128) implies it holds that Then, we obtain where Here, we have and therefore, x À x 0 x À x 0 Since (128) implies there arises that with similarly.
We have, furthermore, and hence and The residual terms are thus treated similarly, and it follows that which results in We can argue similarly to the vortex term in (124). This time, from Concerning the principal term of (124), we use From and it follows that Let M 1 ¼ M x 0 ,R and M 2 ¼ M x 0 ,2R for Then, using (120), we end up with Inequalilty (166) implies T < þ ∞ if A 0 ð Þ ≪ 1, as is observed by [27] (see also Chapter 5 of [19]). Here we describe the proof for completeness.
The first observation is the monotoniity formula d dt derived from (124) and the symmetry of the Green's function: The proof is the same as in (34) and is omitted.