Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto-Daido model with inertia

We present the global existence and long-time behavior of measure-valued solutions to the kinetic Kuramoto--Daido model with inertia. 
For the global existence of measure-valued solutions, we employ a Neunzert's mean-field approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto--Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.


1.
Introduction. Synchronization of weakly coupled limit-cycle oscillators appears in many biological systems such as metabolic synchrony in yeast cell suspension, synchronous firing of a cardiac pacemaker, and the flashing of fireflies (see [4,5,16,38] for details). Owing to emerging interest in complex networks in computer and social sciences, the synchronous dynamics of information flow through complex networks has attracted considerable interest among researchers from the nonlinear sciences (e.g., applied mathematics, control theory, and statistical physics). Thus far, several mathematical models have been proposed and used for simulating collective synchronization phenomena in complex networks [1,26]. Among them, we focus on the Kuramoto-type model, which is a minimal prototype model for analytical treatment. The Kuramoto model (KM) [21,22] is a system of first-order ordinary differential equations (ODEs) for the phase of weakly coupled subject to suitable initial data: where m, K, and N are the strength of inertia, coupling, and the system size, respectively, and Ω i is the natural frequency of the ith oscillator, which is randomly extracted from the frequency distribution function g = g(Ω). The phase coupling function Γ = Γ(θ) satisfies the following properties: Γ(θ + 2π) = Γ(θ), |Γ(θ) − Γ(θ * )| ≤ Γ Lip |θ − θ * |, Γ(0) = Γ(π) = 0, Γ(θ) > 0, θ ∈ (0, π), Γ(−θ) = −Γ(θ).
Note that the system (1) and (3) is translational invariant and that Kuramoto's coupling function sin θ satisfies the structural condition (3). In the following, we briefly present previous mathematical results for the KDM with inertia and Γ(θ) = sin θ.
The system (1) with sinusoidal coupling was first introduced by Ermentrout [16] for modeling slow relaxation in the synchronization process in certain biological systems (e.g., fireflies of the Pteroptyx malaccae). In the absence of inertia (m = 0), the system (1) corresponds to the Kuramoto-Daido model [12,13] for a general coupling. The first-order harmonics of Γ correspond to the Kuramoto model with inertia and the system (1) is applied for modeling superconducting Josephson junction arrays [14,33,34,35,36,37] and power networks [15]. Compared to the vast literature on the KM, there are few research papers on the KDM. The system (1) has several distinct dynamic features compared to the KM: For example, it is well known [1] that the KM exhibits a continuous second-order phase transition at the critical coupling strength K cr for unimodal, symmetric, and long-range distribution functions g such as Lorentz and Gaussian distributions. In contrast, the KDM with large inertia shows a discontinuous first-order phase transition for the aforementioned distributions, and exhibits hysteresis [31,32]. It is unknown whether the KDM with small inertia will show similar dynamic behaviors.
However, when the number of oscillators is sufficiently large, the kinetic version of the KDM is often used in the physics literature [2,3] to study phase-transition phenomena. Let f = f (θ, ω, Ω, t) be the one-oscillator distribution function in [0, 2π) := R/(2πZ) with frequency ω, and natural frequency Ω at time t. Then, the formal thermodynamic limit (N → ∞) using the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy yields a Vlasov-type kinetic equation for f : As far as we know, the well-posedness issue and qualitative asymptotic behavior of the kinetic model (4) have not been addressed in the literature. In this paper, we study fundamental mathematical questions such as the existence of solutions and their asymptotic behavior via the finite-dimensional result for the KDM with inertia and a rigorous thermodynamic limit. Next, we briefly discuss our main results. First, we present a measure-theoretic formulation of the kinetic model (4) in Section 2.3. Because we are interested in the concentration of phases or frequencies, i.e., formation of a Dirac delta in phase and frequency, it is natural to include such singular measures in our concept of solutions. Therefore, measurevalued solutions to (4) are a natural class of solutions as far as asymptotics are concerned. Using the measure-theoretic formulation, we provide the existence of measure-valued solutions for an initial Radon measure with finite moments up to second order in Section 4. The approximate measure-valued solutions are empirical measures constructed from the corresponding finite-dimensional KDM with inertia and then the local-in-time stability result in Proposition 3 yields the convergence of the approximate measure-valued solutions. Second, for the asymptotic behavior of measure-valued solutions, we first establish the finite-dimensional result for the corresponding KDM with inertia, and then using the rigorous thermodynamic result in Section 4, we lift the finite-dimensional restriction to provide infinite-dimensional results. The rest of the paper is outlined as follows: In Section 2, we briefly review the basic mathematical structure of the KDM and kinetic KDM with inertia, and present a measure-theoretic formulation of the kinetic KDM with inertia. In Section 3, we present a local-stability estimate of the KDM in terms of a bounded Lipschitz distance. In Section 4, we provide a global well-posedness of measure-valued solutions. In Section 5, we study the asymptotic synchronization property and contraction estimates for the measure-valued solutions. Finally, Section 6 is devoted to a summary of the main results and direction for future work. In Appendix A, we present the detailed proof of Theorem 5.2.

2.
Preliminaries. In this section, we briefly review the basic properties of the kinetic KDM with inertia and present a measure-theoretic formulation of the kinetic KDM.
2.1. The KDM with inertia. Consider an ensemble of many weakly coupled limit-cycle oscillators under the effect of uniform inertia. In this case, the dynamics of θ i is governed by the following system of first-order ODEs: , Note that the system (5) is equivalent to the system (1), and here we added Ω i as a part of the dynamic quantities, although it is invariant under the dynamics (5). We first observe that the system (5) is dissipative. This can be easily seen from the fact that the vector field generated by the first-order system (5), has a negative divergence: We also note that the equilibria to the system (5) correspond to equilibria to the KDM without inertia and vice versa, i.e., We now introduce average quantities and fluctuations around the averaged ones: Note that the system can be completely decoupled as two independent systems for averages and fluctuations: , Then it is easy to see that As far as long-time dynamics is concerned, for some well-prepared initial data with ω c (0) = Ω c , the dynamics of the averages is indistinguishable from that of the KM. In this case, the inertia plays only the role of convergence rate toward the phase-locked states.
2.2. The kinetic KDM with inertia. In this part, we present several a priori estimates for the kinetic KDM with inertia. Consider the initial boundary value problem for the KDM: subject to initial and boundary conditions: ) be a smooth solution to (6) and (7). Then we have Proof. We integrate (6) with respect to (θ, ω) ∈ [0, 2π] × R to get By the normalization condition given in (7), we have For notational simplicity, we also introduce · to denote the integral over the phase space [0, 2π) × R 2 with respect to a measure g(Ω)dθdωdΩ: Note that since θ ∈ [0, 2π) is a 2π-periodic variable, h(θ, ω, Ω) is a 2π-periodic function with respect to θ on [0, 2π) × R × R.
Proposition 2. (Evolution of the moments) Let f be a smooth solution to (6) and (7). Then for t > 0, we have Proof. (i) We integrate (6) using the periodicity to find (ii) We multiply (6) by ωg(Ω) to get This yields (iii) We multiply (6) by θg(Ω) to get We integrate this equation to find the desired result.
(iv) We first multiply (6) by Ωg(Ω) to find This yields the desired result.

2.3.
A measure theoretic formulation. In this part, we present a measuretheoretic formulation of the kinetic KDM (6) with inertia. When the natural frequencies are distributed, i.e., nonidentical oscillators, the oscillators phases cannot collapse to a single phase asymptotically. In contrast, the frequencies of oscillators can be collapsed to a single frequency asymptotically. Hence, when the asymptotic complete-frequency synchronization occurs, the asymptotic limit of measure-valued solutions will have a Dirac measure concentrated on the average natural frequency as its component even for smooth initial data. Therefore, the suitable space for the asymptotic behavior of the solutions will be a measure space instead of the usual Sobolev space. In this manner, a measure-valued solution emerges as a natural concept for the solution to the kinetic KDM with inertia. For this, we adopt a standard Neunzert's framework from [25,29]. Let M([0, 2π) × R 2 ) be the set of nonnegative Radon measures on [0, 2π) × R 2 , and we use a standard duality relation for a Radon measure ν ∈ M and its test function h: ) a space of all weakly continuous time-dependent measures. Next, we present the definition of a measure-valued solution to (6) as follows.
if and only if µ satisfies the following conditions: Let m 0 (t) be the total mass of µ t , i.e., Then it follows from (8) that we have conservation of mass:

3.
A local-in-time stability estimate. In this section, we provide an a priori local-in-time stability estimate for the measure-valued solution to (6) and (7) using a bounded Lipschitz distance.
3.1. Estimates on particle trajectories. In this part, we first study several a priori estimates of particle trajectories generated from (6).
to be the particle trajectory passing through (θ, ω, Ω) at time t, i.e., subject to initial data We now introduce a priori conditions (P) for measure-valued solutions to (6) and (7): where supp(µ) and B r (z) denote support of measure µ and ball of radius r around z, respectively, i.e., supp(µ) : ) be a measure-valued function with the property (P). Then, we have the following assertions:
3. Let P (t), Q(t), and R(t) be the bounds for supports of the phase, velocity, and natural frequency variables, respectively. Then, we have Proof.
(1) The first assertion directly follows from the boundedness of Γ L ∞ . For the second assertion, we use Lipschitz continuity of Γ to get (2) The proof is obtained from standard ordinary differential equation theory. For more details, we refer the reader to [18].
Direct differentiation of (12) with respect to τ and Lemma 3.1 implỹ Hence, the relation (11) implies

3.2.
Local-in-time stability estimate. In this part, we provide a local-in-time stability of measure-valued solutions to (6) and (7). We first introduce an admissible set Λ of test functions: ≤ max{a, b}d(µ, ν).
It follows from (9) that We note that where C Γ is a constant given by Therefore, we have from (13) |x(s)| ≤ s t |y(τ )|dτ, We add these inequalities to obtain Then Gronwall lemma yields Proof. Let h be a test function in Λ, then we have from Lemmas 3.2 and 3.4 where we employed the following notations for simplicity.

4.
A global existence of measure-valued solutions. In this section, we provide a global existence of a measure-valued solution to (6) following the approach in [18,25,29]. 4.1. Construction of approximate solutions. In this part, we present a construction of approximate solutions using the particle method [27].
Suppose that the initial Radon measure µ 0 has compact support in [0, 2π) × R 2 , and it is included in a square R, i.e., Then for a given positive integer n, we can divide the square R into n 3 subsquares R i , i.e., Let z i = (θ i , ω i , Ω i ) be the center of R i . Then we construct the initial approximation µ n 0 as and we define the approximate solution as ) be a given initial Radon measure on [0, 2π) × R 2 with compact support: and let µ n 0 be the initial approximation given by (14). Then there exists a positive constant C such that where C is a constant proportional to the diameter of the rectangle R.
Lemma 4.2. Let µ n be the approximate measure-valued solution to (6) constructed by the procedure (14)- (15). Then we have

YOUNG-PIL CHOI, SEUNG-YEAL HA AND SEOK-BAE YUN
Proof. It follows from Remark 1 and Lemma 3.1 that we have m n 0 (t) = m n 0 (0), and However, from the construction of the initial approximation µ h 0 , it is easy to prove that m n 0 (0) ≤ m 0 + C n , We then substitute these estimates into Lemma 3.1, (3) to obtain the desired estimates.

4.2.
Convergence of approximate solutions. In this part, we present the convergence of the approximate measure-valued solutions constructed in the previous subsection and establish the well-posedness of the global measure-valued solutions to the kinetic KDM.
) is a Radon measure with compact support satisfying and let µ n t be the approximate solution constructed by the procedure (14)- (15). Then there exists a unique measure-valued solution µ ∈ C w ([0, T ); M([0, 2π) × R 2 )) to (4) with initial data µ 0 such that µ t is the weak-* limit of the approximate solutions, i.e., d(µ t , µ n t ) = 0 as n → ∞. Proof. We divide the estimates into several steps. • Step A. (Select candidates for a measure-valued solution): We apply the localin-stability results in Proposition 3 to µ n1 and µ n2 : This yields that the sequence of approximate solutions {µ n t } is a Cauchy sequence in the complete metric space (M([0, 2π] × R 2 ), d(·, ·)). Therefore there exists a limit measure µ t ∈ M([0, 2π] × R 2 ). However, since d convergence is equivalent to weak-* convergence, we know that µ t is the weak-* limit of µ n t . The estimate (16) also implies d(µ t , µ n t ) ≤ C n .
• Step B. (Use the weak-limit measure µ t as the measure-valued solution to (4) in the sense of Definition 2.1) Step B.1 (Check for weak Lipschitz continuity): We check (1) of Definition 2.1. We first observe from Lemma 3.1 that This yields Step B.2 (Check the defining condition (8) of Definition 2.1): We have Since d convergence is equivalent to weak-* convergence, we have as n → ∞. We will prove the following stronger estimate: Note that and hence it is enough to show that To prove this claim, we first observe that µ n s , We now estimate I 1 and I 2 separately.
• (Estimate I 1 ): We first recall Lemma 4.1 to see that This yields • (Estimate I 2 ): To estimate the term I 2 , we use Lemma 3.1 to get These two estimates lead to Step C. (Verify the uniqueness of the measure-valued solution): Let µ and µ be the two measure-valued solutions in the sense of Definition 2.1 corresponding to the given initial Radon measure µ 0 . Then it follows from Proposition 3 that Thus we have d(µ t , µ t ) = 0, i.e., µ t = µ t , t ∈ (0, T ).
Therefore, we have the uniqueness of measure-valued solution.
Remark 3. 1. For the KM, similar results have been studied in [7,18,23,24]. 2. Note that the measure-valued solution µ has a bounded first moment for each time slice: Moreover, µ has compact support for each time slice: where P (t), Q(t), and R(t) satisfy

5.
Large-time behavior of the measure-valued solutions. In this section, we present an asymptotic complete-frequency estimate for the measure-valued solutions whose existence is guaranteed by Theorem 4.3 in the previous section. For the desired synchronization estimates to the measure-valued solutions, we first establish the corresponding results at the oscillator level, and then using the rigorous meanfield limit, we obtain a synchronization estimate for the measure-valued solution.
Without loss of generality, we assume that As a preliminary step for the complete synchronization, we consider the initial phase configuration consisting of a finite number of Dirac measures. For definiteness, we set where z i0 is defined as in Section 4.1. Then the unique measure-valued solution to (4) with the initial datum (17) is given by is the unique solution of the Kuramoto-Daido model: In the following, we present asymptotic complete-frequency synchronization estimates and the contraction property of the system (18) with distributed natural frequencies. For the nonidentical Kuramoto oscillators, the phase-space support of µ t does not collapse to a single point. However, we will show that the projected support of µ t in frequency (ω) space will collapse to a single point as in the identical case.
Remark 4. If we consider the initial Radon measure that is absolutely continuous with respect to the Lebesgue measure dθdωdΩ, i.e., µ 0 dθdωdΩ, then we can choose the following approximation for µ 0 the following using the similar argument in [23]. Later in Theorem 5.3 we will use this argument.
For convenience, we recall the following second-order differential inequality: where a > 0, b, c, and d are constants.
Proof. Although the proof is almost the same as in Theorem 5.1 [10], for the reader's convenience, we briefly sketch the proof below. For the detailed proof, see Appendix A.
Case A (Small-inertia regime). Suppose that Framework A holds, and we set We first show that there exists a trapping region for D θ (µ n t ). For this we use the following second-order differential inequality.
Then, from this inequality, we obtain D θ (µ n t ) ≤ D ∞ , t ≥ 0. Next, we differentiate Equation (18) with respect to time t to get By using the lower bound of Γ , we have We now apply Lemma 5.1 to (21) to obtain D ω (µ n t ) ≤ Ce −γt , where γ is a positive constant. Case B (Large-inertia regime). Suppose that Framework B holds. In a manner similar to Case A, we have Then we obtain the trapping region of D θ (µ n t ) such that D θ (µ n t ) ≤ 4mD Ω , t ≥ 0, and from this we have the complete-frequency synchronization: where η is a positive constant. Hence by letting n → ∞, we have the desired results.
2. In contrast to [10], we cannot estimate the limit of the phase and frequency of the system (18), since the momentum of the system (18) is not conserved. However, from Theorem 5.2, we know that the supports of µ t go to one point, as t goes to infinity. This implies that µ t converges to the Dirac measure in the sense of the weak-* limit.
Finally, we present a contraction property of the kinetic Kuramoto-Daido model with finite inertia. In the absence of inertia, it is shown in [6] that the Kuramoto model has a contraction property in Wasserstein distance. The optimal mass transport approach for the contraction relies on the one-dimensional nature of the phase space. However, in our setting, our dynamic phase space is two-dimensional, i.e., [0, 2π] × R in (θ, ω). Hence it seems that we cannot use the optimal mass transport technique directly as in [6].
We now combine Step A and Step B to obtain the following differential inequality: Since 1 − 4mKR 1 > 0, we obtain By assumption, we have This yields where we usedḊ This is a contradiction to (26).
Step A (Maximal frequency fluctuation): Since ω M is Lipschitz continuous, it is almost everywhere differentiable in time t. More precisely, there exist at most countable number of times 0 := t 0 < t 1 < · · · < t ∞ ≤ ∞ such that ω M is differentiable in the time interval (t k−1 , t k ), k = 1, 2, · · · .
We also obtain the following equation from (18): This yields Step B (Minimal frequency fluctuation): In this case, we apply the same argument as Step A to find

We combine Step A and
Step B to obtain the following differential inequality: By condition (P3) and the assumption on Γ Γ(D ∞ ) D ∞ ≥ Γ (D ∞ ). The determinant of (27) satisfies . Hence we obtain the desired result.