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Abstract
We consider the Kuramoto-Sakaguchi-Fokker-Planck equation (namely, parabolic Kuramoto-Sakaguchi, or Kuramoto-Sakaguchi equation, which is a nonlinear parabolic integro-differential equation) with inertia and white noise effects. We study the hydrodynamic limit of this Kuramoto-Sakaguchi equation. During showing this main result, as a support, we also prove a Hardy-type inequality over the whole real line.
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We study the rigorous derivation of hydrodynamics of the Kuramoto model for synchronization phenomena, introduced by Choi and Lee (Math Models Methods Appl Sci 30: 2175–2227, 2020), which is pressureless Euler equations with nonlocal interaction forces. We present two different ways of deriving that hydrodynamic model. We first discuss the asymptotic analysis for the inertial kinetic Kuramoto equation with a strong local frequency alignment force. We show that a weak solution to the kinetic equation converges to the classical solution of that hydrodynamic synchronization model under certain assumptions on the initial data. We also provide the derivation from the particle Kuramoto model with inertia as the number of oscillators goes to infinity in the mono-kinetic case. Our proofs are based on a modulated energy-type estimate combined with the bounded Lipschitz distance between local densities.
We prove a one-dimensional Hardy inequality on the halfline with sharp constant, which improves the classical form of this inequality. As a consequence of this new inequality we can rederive known doubly weighted Hardy inequalities. Our motivation comes from the theory of Schrödinger operators and we explain the use of Hardy inequalities in that context.
We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.
We study the dynamics of a general class of coupled oscillators driven by
quenched external torques, which includes thermal noise and inertia. In the
limit of zero noise and inertia, the dynamics reduces to the one of the
Kuramoto model. For unimodal torque distributions, introducing a reduced
parameter space involving dimensionless moment of inertia, temperature, and
width of the distribution, the dynamics is shown to exhibit a nonequilibrium
first-order transition from a synchronized phase at low parameter values to an
incoherent phase at high values. In proper limits, we recover the known
continuous phase transitions in the Kuramoto model and in its noisy extension,
and an equilibrium continuous transition in a related model of long-range
interactions.
We discuss the application to systems of coupled nonlinear oscillators of methods originally developed for the Vlasov-Poisson equations of plasma kinetic theory. These methods lead to a simple but rigorous derivation of the infinite-oscillator limit for the well-known Kuramoto model, and also to a compact proof of global existence and uniqueness of solutions to the resulting Kuramoto-Sakaguchi equations. Vlasov-limit techniques can also be adapted to prove a central limit theorem for statistical ensembles of systems of oscillators. However, we emphasize that the noiseless Kuramoto model itself is strictly deterministic, and that the Kuramoto-Sakaguchi equations can be formulated without any reference to a stochastic distribution of oscillator frequencies.
The paper is devoted to the analysis of a hydrodynamic limit for the Vlasov-Navier-Stokes equations.This system is intended to model the evolution of particles interacting with a fluid. The coupling arises from the force terms. The limit problem is the Navier-Stokes sys- tem with non constant density. The density which is involved in this system is the sum of the (constant) density of the fluid and of the macroscopic density of the particles. The proof relies on a relative entropy method.
This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. The starting point is the kinetic model considered in [10] with the addi-tion of an attraction-repulsion interaction potential. Introducing different scalings than in [10], the non-local effects of the alignment and attraction-repulsion interac-tions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model. The proof relies on the energy method.
The hydrodynamic limit of a kinetic Cucker-Smale model is investigated. In
addition to the free-transport of individuals and the Cucker-Smale alignment
operator, the model under consideration includes a strong local alignment term.
This term was recently derived as the singular limit of an alignment operator
due to Motsch and Tadmor. The model is enhanced with the addition of noise and
a confinement potential. The objective of this work is the rigorous
investigation of the singular limit corresponding to strong noise and strong
local alignment. The proof relies on the relative entropy method and entropy
inequalities which yield the appropriate convergence results. The resulting
limiting system is an Euler-type flocking system.
We study the dynamics of assemblies of interacting neurons. For large fully connected networks,the dynamics of the system can be described by a partial differential equation reminiscent of age-structure models used in mathematical ecology, where the "age" of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network.
We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behavior of solutions; both for small and large J, we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing.
The Chapman-Enskog method of kinetic theory is applied to two problems of synchronization of globally coupled phase oscillators. First, a modified Kuramoto model is obtained in the limit of small inertia from a more general model which includes ``inertial'' effects. Second, a modified Chapman-Enskog method is used to derive the amplitude equation for an O(2) Takens-Bogdanov bifurcation corresponding to the tricritical point of the Kuramoto model with a bimodal distribution of oscillator natural frequencies. This latter calculation shows that the Chapman-Enskog method is a convenient alternative to normal form calculations. Comment: 7 pages, 2-column Revtex, no figures, minor changes
A model for synchronization of globally coupled phase oscillators including ``inertial'' effects is analyzed. In such a model, both oscillator frequencies and phases evolve in time. Stationary solutions include incoherent (unsynchronized) and synchronized states of the oscillator population. Assuming a Lorentzian distribution of oscillator natural frequencies, , both larger inertia or larger frequency spread stabilize the incoherent solution, thereby making harder to synchronize the population. In the limiting case , the critical coupling becomes independent of inertia. A richer phenomenology is found for bimodal distributions. For instance, inertial effects may destabilize incoherence, giving rise to bifurcating synchronized standing wave states. Inertia tends to harden the bifurcation from incoherence to synchronized states: at zero inertia, this bifurcation is supercritical (soft), but it tends to become subcritical (hard) as inertia increases. Nonlinear stability is investigated in the limit of high natural frequencies. Comment: Revtex, 36 pages, submit to Phys. Rev. E
We study the robustness in the nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck (KS-FP for short) equation in the presence of frustrations. For this, we construct a new unstable mode for the corresponding linear part of the perturbation around the incoherent state, and we show that the nonlinear perturbation stays close to the unstable mode in some small time interval which depends on the initial size of the perturbations. Our instability results improve the previous results on the KS-FP with zero frustration [J. Stat. Phys. 160 (2015), pp. 477–496] by providing a new linear unstable mode and detailed energy estimates.
Measure Theory and Fine Properties of Functions, Revised Edition provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Rn, Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions as well as functions of bounded variation. The text provides complete proofs of many key results omitted from other books, including Besicovitch's covering theorem, Rademacher's theorem (on the differentiability a.e. of Lipschitz functions), area and coarea formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Aleksandrov's theorem (on the twice differentiability a.e. of convex functions). This revised edition includes countless improvements in notation, format, and clarity of exposition. Also new are several sections describing the p-? theorem, weak compactness criteria in L1, and Young measure methods for weak convergence. In addition, the bibliography has been updated. Topics are carefully selected and the proofs are succinct, but complete. This book provides ideal reading for mathematicians and graduate students in pure and applied mathematics.
Synchronization of weakly coupled oscillators is ubiquitous in biological and chemical complex systems. Recently, research on collective dynamics of many-body systems has been received much attention due to their possible applications in engineering. In this survey paper, we mainly focus on the large-time dynamics of several synchronization models and review state-of-art results on the collective behaviors for synchronization models. Following a chronological order, we begin our discussion with two classical phase models (Winfree and Kuramoto models), and two quantum synchronization models (Lohe and Schrödinger–Lohe models). For these models, we present several sufficient conditions for the emergence of synchronization using mathematical tools from dynamical systems theory, kinetic theory and partial differential equations in a unified framework.
We treat hydrodynamic limits of the Vlasov-Maxwell-Boltzmann system for one and two species of particles in a viscous incompressible regime.
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
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.
Motivated by recent interest for multiagent systems and smart grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with nontrivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform Kuramoto model. Here, nonuniform Kuramoto oscillators are characterized by multiple time constants, nonhomogeneous coupling, and nonuniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of nonuniform Kuramoto oscillators. These conditions reduce to necessary and sufficient tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization in a power network to the underlying network parameters.
Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis. The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables. The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces. The book contains over 200 exercises.
The Boltzmann equation and its formal hydrodynamic limits.- Mathematical tools for the derivation of hydrodynamic limits.- The incompressible Navier-Stokes limit.- The incompressible Euler limit.- The compressible Euler limit.
The exponential synchronization rate is addressed for Kuramoto oscillators in the presence of a pacemaker. When natural frequencies are identical, we prove that synchronization can be ensured even when the phases are not constrained in an open half-circle, which improves the existing results in the literature. We derive a lower bound on the exponential synchronization rate, which is proven to be an increasing function of pacemaker strength, but may be an increasing or decreasing function of local coupling strength. A similar conclusion is obtained for phase locking when the natural frequencies are non-identical. An approach to trapping phase differences in an arbitrary interval is also given, which ensures synchronization in the sense that synchronization error can be reduced to an arbitrary level.
This chapter focuses on the classical models of fluid mechanics. It introduces the Boltzmann equation and discusses its structure and main formal properties. The dimensionless form of the Boltzmann equation and its main scaling parameters are discussed in the chapter. It discusses in detail the formal derivation of the most classical partial differential equations (PDEs) of fluid mechanics from the Boltzmann equation by several different methods. The known mathematical results on the Cauchy problem for the PDEs of fluid mechanics and the-state-of-the-art on the existence theory for the Boltzmann equation are reviewed in the chapter. It focuses on compactness methods, leading to global results, and especially the derivation of global weak solutions of the incompressible Navier–Stokes equations from renormalized solutions of the Boltzmann equation. In some models that can be found in the literature, the “viscous heating” term is absent from the temperature equation. Whether the “viscous heating” term should be taken into account or not depends in fact on the relative size of the fluctuations of velocity field about its average value and of the fluctuations of temperature field about its average values.
Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.
We consider an Individual-Based Model for self-rotating particles interacting
through local alignment and investigate its macroscopic limit. This model
describes self-propelled particles moving in the plane and trying to
synchronize their rotation motion with their neighbors. It combines the
Kuramoto model of synchronization and the Vicsek model of swarm formation. We
study the mean-field kinetic and hydrodynamic limits of this system within two
different scalings. In the small angular velocity regime, the resulting model
is a slight modification of the 'Self-Organized Hydrodynamic' model which has
been previously introduced by the first author. In the large angular velocity
case, a new type of hydrodynamic model is obtained. A preliminary study of the
linearized stability is proposed.
Using relative entropy estimates about an absolute Maxwellian, it is shown that any properly scaled sequence of DiPerna-Lions renormalized solutions of some classical Boltzmann equations has fluctuations that converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. Moreover, if the initial fluctuations entropically converge to an infinitesimal Maxwellian then the limiting fluid variables satisfy a version of the Leray energy inequality. If the sequence satisfies a local momentum conservation assumption, the momentum densities globaly converge to a solution of the Stokes equation. A similar discrete time version of this result holds for the Navier-Stokes limit with an additional mild weak compactness assumption. The continuous time Navier-Stokes limit is also discussed. © 1993 John Wiley & Sons, Inc.
We present an approach based on Gronwall’s inequalities for the asymptotic complete phase–frequency synchronization of Kuramoto oscillators with finite inertia. For given finite inertia and coupling strength, we present admissible classes of initial configurations and natural frequency distributions, which lead to the complete phase–frequency synchronization asymptotically. For this, we explicitly identify invariant regions for the Kuramoto flow, and derive second-order Gronwall’s inequalities for the evolution of phase and frequency diameters. Our detailed time-decay estimates for phase and frequency diameters are independent of the number of oscillators. We also compare our analytical results with numerical simulations.
The Kuramoto model captures various synchronization phenomena in biological
and man-made systems of coupled oscillators. It is well-known that there exists
a critical coupling strength among the oscillators at which a phase transition
from incoherency to synchronization occurs. This paper features four
contributions. First, we characterize and distinguish the different notions of
synchronization used throughout the literature and formally introduce the
concept of phase cohesiveness as an analysis tool and performance index for
synchronization. Second, we review the vast literature providing necessary,
sufficient, implicit, and explicit estimates of the critical coupling strength
for finite and infinite-dimensional, and for first and second-order Kuramoto
models. Third, we present the first explicit necessary and sufficient condition
on the critical coupling to achieve synchronization in the finite-dimensional
Kuramoto model for an arbitrary distribution of the natural frequencies. The
multiplicative gap in the synchronization condition yields a practical
stability result determining the admissible initial and the guaranteed ultimate
phase cohesiveness as well as the guaranteed asymptotic magnitude of the order
parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto
models consisting of second-order Kuramoto oscillators with inertia and viscous
damping together with first-order Kuramoto oscillators with multiple time
constants. We prove that the multi-rate Kuramoto model is locally topologically
conjugate to a first-order Kuramoto model with scaled natural frequencies, and
we present necessary and sufficient conditions for almost global phase
synchronization and local frequency synchronization. Interestingly, these
conditions do not depend on the inertiae which contradicts prior observations
on the role of inertiae in synchronization of second-order Kuramoto models.
We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry, and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.
We consider the discrete Couzin-Vicsek algorithm (CVA), which describes the interactions of individuals among animal societies such as fish schools. In this article, we propose a kinetic (mean-field) version of the CVA model and provide its formal macroscopic limit. The final macroscopic model involves a conservation equation for the density of the individuals and a non conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.
Fluid dynamic limit of Boltzmann equation for granular hard-spheres in a nearly elastic regime
- Ricardo J Alonso
- Bertrand Lods
- Isabelle Tristani
Ricardo J. Alonso, Bertrand Lods, and Isabelle Tristani. Fluid dynamic limit
of Boltzmann equation for granular hard-spheres in a nearly elastic regime, 2021.
arXiv:2008.05173.
Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models
- Franck Boyer
- Pierre Fabrie
Franck Boyer and Pierre Fabrie. Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Applied Mathematical Sciences.
Springer New York, NY, 2013.
Muckenhoupt's proof of the Hardy inequality in dimension 1
- A José
- Cañizo
José A. Cañizo. Muckenhoupt's proof of the Hardy inequality in dimension 1. Available online: https://canizo.org/page/26 (accessed in 2023, 2024).
Macroscopic models of collective motion and self-organization. Séminaire Laurent Schwartz -EDP et applications
- Pierre Degond
- Amic Frouvelle
- Jian-Guo Liu
- Sebastien Motsch
- Laurent Navoret
Pierre Degond, Amic Frouvelle, Jian-Guo Liu, Sebastien Motsch, and Laurent Navoret. Macroscopic models of collective motion and self-organization. Séminaire
Laurent Schwartz -EDP et applications, pages 1-27, 2012-2013. talk:1.
- Lawrence C Evans
Lawrence C. Evans. Partial Differential Equations: Second Edition, volume 19 of
Graduate Studies in Mathematics. American Mathematical Society, 2010.
Entropic convergence and the linearized limit for the Boltzmann equation with external force
- Tina Mai
Tina Mai. Entropic convergence and the linearized limit for the Boltzmann equation
with external force, 2016. arXiv:1612.05096.