One dimensional singular Cucker–Smale model: Uniform-in-time mean-field limit and contractivity

Choi, Young-Pil; Zhang, Xiongtao

doi:10.1016/j.jde.2021.04.002

Cited by 15 publications

(14 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Roughly speaking, the 1D variant of (1.9) can be recovered from the 1D Cucker-Smale equation by a simple change of variables. This observation was used in many previous works [20,30,67] providing new insight into large-time behavior and well-posedness. It is noteworthy that in the above literature the authors usually treat the weak formulations for (1.9) as a new relaxed formulation for the 1D Cucker-Smale equation.…”

Section: Introductionmentioning

confidence: 89%

“…On the other hand, in [59] the second author derived a Dobrushin-type stability estimate for the Kuramoto-Sakaguchi equation with singular, one-sided Lipschitz kernels. Finally the contribution in [20] deals exactly with a global-in-time contractivity estimate for the 1D version of (1.9) by exploiting the uniform-in-time stability of the associated particle system and an useful reformulation of the system in 1D. We expand this result to the multidimensional case, where the 1D reformulation of the fibered distance W 2,ν is not available by keeping the argumentation purely kinetic.…”

Section: Introductionmentioning

confidence: 99%

“…In the literature of optimal transport, N. Gigli used one in [26] as an auxiliary tool in the construction of the geometric tangent space to the quadratic Wasserstein space (P 2 (R d ), W 2 ), see also [27] and [3]. In the literature of collective dynamics, it was used in [12,20,38] to prove contractivity respectively for the Kuramoto-Sakaguchi equation (with restricted initial data and large coupling strength), the 1D kinetic Cucker-Smale model with weakly singular couplings, and a variant of the multidimensional Cucker-Smale model with a Hessian communication weight. Recently, it has also been proposed in [39,40] with applications to the existence and uniqueness of stationary measures of iterated function system.…”

Section: Introductionmentioning

confidence: 99%

“…It is worthwhile to compare our contribution to previous works in this direction, cf. [12,50,59,20].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Heterogeneous gradient flows in the topology of fibered optimal transport

Peszek¹,

Poyato²

2022

Preprint

View full text Add to dashboard Cite

We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case R d × R d , where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures (P2,ν(R 2d ), W2,ν) with fixed marginal ν ∈ P(R d ) in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.

show abstract

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It is worthwhile to compare our contribution to previous works in this direction, cf. [12,50,59,20].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Heterogeneous gradient flows in the topology of fibered optimal transport

Peszek¹,

Poyato²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…From practical needs, a singular communication weight 𝜓 is also studied, for example, complete clustering predictability and the mean-field limit have been investigated in Refs. 45,46.…”

Section: From Cucker-smale To Consensusmentioning

confidence: 99%

On the stochastic robustness of complete clustering predictability for a first‐order consensus model

Shim

2021

Stud Appl Math

View full text Add to dashboard Cite

We study stochastic noise response on the complete cluster predictability of a first-order stochastic consensus model on the real line with an odd, bounded, Lipschitz, and monotone coupling function. Emergence of multiclusters is one of the novel dynamical features of the deterministic consensus model, and it has been extensively studied in collective dynamics community. Recently, the second author and his collaborators proposed a complete cluster predictability of the first-order nonlinear consensus model that can be derived from the Cucker-Smale flocking model on the real line, and they verified that the proposed clustering algorithm explicitly determines the number of asymptotic clusters and their cluster velocities only in terms of the initial data, coupling strength and supremum of coupling function. In this work, we show that the same cluster algorithm does work for the complete clustering predictability problem when the deterministic consensus model is stochastically perturbed in a multiplicative way.

show abstract

Inevitable monokineticity of strongly singular alignment

Fabisiak,

Peszek

2023

Math. Ann.

View full text Add to dashboard Cite

We prove that certain types of measure-valued mappings are monokinetic i.e. the distribution of velocity is concentrated in a Dirac mass. These include weak measure-valued solutions to the strongly singular Cucker–Smale model with singularity of order greater or equal to the dimension of the ambient space. Consequently, we are able to answer a couple of open questions related to the singular Cucker–Smale model. First, we prove that weak measure-valued solutions to the strongly singular Cucker–Smale kinetic equation are monokinetic, under very mild assumptions that they are uniformly compactly supported and weakly continuous in time. This can be interpreted as a rigorous derivation of the macroscopic fractional Euler-alignment system from the kinetic Cucker–Smale equation without the need to perform any hydrodynamical limit. This suggests the superior suitability of the macroscopic framework to describe large-crowd limits of strongly singular Cucker–Smale dynamics. Second, we perform a direct micro- to macroscopic mean-field limit from the Cucker–Smale particle system to the fractional Euler-alignment model. This leads to the final result—the existence of weak solutions to the fractional Euler-alignment system with almost arbitrary initial data in 1D, including the possibility of a vacuum. Existence can be extended to 2D under the a priori assumption that the density of the mean-field limit has no atoms.

show abstract

One dimensional singular Cucker–Smale model: Uniform-in-time mean-field limit and contractivity

Cited by 15 publications

References 30 publications

Heterogeneous gradient flows in the topology of fibered optimal transport

Heterogeneous gradient flows in the topology of fibered optimal transport

On the stochastic robustness of complete clustering predictability for a first‐order consensus model

Inevitable monokineticity of strongly singular alignment

Contact Info

Product

Resources

About