Existence of Weak Solutions to Kinetic Flocking Models

Karper, Trygve K.; Mellet, Antoine; Trivisa, Konstantina

doi:10.1137/120866828

Cited by 92 publications

(97 citation statements)

References 12 publications

(27 reference statements)

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“…The global-in-time existence of weak solutions for the Vlasov equation with local alignment forces was studied in [20]. In the presence of diffusion, the global-in-time existence classical solutions around the global Maxwellian was obtained in [12].…”

Section: 2mentioning

confidence: 99%

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Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

Carrillo

Choi

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329-361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.

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Section: 2mentioning

confidence: 99%

“…In the presence of diffusion, the global-in-time existence classical solutions around the global Maxwellian was obtained in [12]. We basically take a similar strategy as in [20] and develop it to handle the additional terms, confinement and interaction potentials, in order to provide the details of proof of Theorem 4.1.…”

Section: 2mentioning

confidence: 99%

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

Carrillo

Choi

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…) be a given initial probability measure on R 2d , and let h 0 be the initial approximation given by (34) with a uniform grid size h. Then we have…”

Section: Existence and Stability Of Measure Valued Solutionsmentioning

confidence: 99%

“…Also, there are lots of literature about the kinetic equations for CS type models and their hydrodynamic limits. 11,[30][31][32][33][34][35][36] In this paper, we first analyze the unconditional flocking behavior for a weighted MT model (see (6)). We also consider flocking behavior for a model with a "tail" (see (21) and Lemmas 2.1 and 2.2).…”

Section: Introductionmentioning

confidence: 99%

Global existence and uniqueness of measure valued solutions to a Vlasov‐type equation with local alignment

Gao

Xue

2017

Math Methods in App Sciences

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We use a particle method to study a Vlasov-type equation with local alignment, which was proposed by Sebastien Motsch and Eitan Tadmor [J. Statist. Phys., 141(2011), pp. 923-947]. For N-particle system, we study the unconditional flocking behavior for a weighted Motsch-Tadmor model and a model with a "tail".When N goes to infinity, global existence and stability (hence uniqueness) of measure valued solutions to the kinetic equation of this model are obtained. We also prove that measure valued solutions converge to a flock. The main tool we use in this paper is Monge-Kantorovich-Rubinstein distance. KEYWORDSflocking, Monge-Kantorovich-Rubinstein distance, particle method Equation (1) is the continuous model of the following particle system (for 1 ⩽ i ⩽ N) 7640

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“…An interesting feature of the C-S model is that it exhibits a phase-like transition from disordered states to ordered states, depending on the spatial decay rate β of the communication weight ψ. Indeed, Cucker and Smale [10] derived sufficient conditions for global flocking in terms of the initial configuration and communication weight, and these results were further improved in [15,16] (see [18,19,20] for generalized particle and kinetic C-S models). However, when the number of particles is sufficiently large, system (2.1) can be described effectively by the one-particle density function f = f (x, ξ, t) at the spatial velocity position (x, ξ) ∈ R d × R d at time t > 0, which has been obtained in the mean-field limit.…”

Section: Hydrodynamic C-s Modelmentioning

confidence: 99%

Emergent Dynamics for the Hydrodynamic Cucker--Smale System in a Moving Domain

Ha¹,

Kang²,

Kwon³

2015

SIAM J. Math. Anal.

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Abstract. We study the emergent dynamics for the hydrodynamic Cucker-Smale system arising in the modeling of flocking dynamics in interacting many-body systems. Specifically, the initial value problem with a moving domain is considered to investigate the global existence and time-asymptotic behavior of classical solutions, provided that the initial mass density has bounded support and the initial data are in an appropriate Sobolev space. In order to show the emergent behavior of flocking, we make use of an appropriate Lyapunov functional that measures the total fluctuation in the velocity relative to the mean velocity. In our analysis, we present the local well-posedness of the smooth solutions via Lagrangian coordinates, and we extend to the global-in-time solutions by establishing the uniform flocking estimates. 1. Introduction. The purpose of this paper is to continue our study [14] on the large-time dynamics of the hydrodynamic Cucker-Smale (C-S) model that arises from the macroscopic description of flocking modeling. Consider an ensemble of many C-S flocking particles that occupy a finite region in R d . When the number of particles is sufficiently large and the system is in a close-to-flocking state, the dynamics of the ensemble can be described effectively by a fluid-type model, i.e., the hydrodynamic C-S model with a moving vacuum boundary. Let ρ and u be the local mass density and bulk velocity, respectively, of the ensemble, the support of which is denoted by a time-varying set Ω(t) := {x ∈ R d | ρ(x, t) = 0} for a given initially bounded open set Ω := Ω(0). In this case, the macroscopic dynamics of the ensemble is governed by the initial value problem of the hydrodynamic C-S system:

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Existence of Weak Solutions to Kinetic Flocking Models

Cited by 92 publications

References 12 publications

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

Global existence and uniqueness of measure valued solutions to a Vlasov‐type equation with local alignment

Emergent Dynamics for the Hydrodynamic Cucker--Smale System in a Moving Domain

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