Displaying items by tag: flocking

Roman Shvydkoy, Changhui Tan

Abstract

We study the hydrodynamic limit of kinetic flocking models with nonlinear velocity alignment and strong local Fokker--Planck forcing. In the limit of small Knudsen number the kinetic densities converge to a local Maxwellian centered around macroscopic quantities satisfying the compressible Euler system with isothermal pressure \(p = \sigma\rho\), while macroscopic alignment is obtained by convolution of the microscopic velocity law with the standard Gaussian, resulting in a different nonlinearity. Such discrepancy has been observed already in the work of Black and Tan [KRM, 18(4):609–632, 2025].

In this note we rigorously justify the limit for admissible weak kinetic solutions and non-vacuous limiting macroscopic solution. Our analysis provides a quantitative rate of convergence in terms of relative entropy: \(\mathcal H(f_\varepsilon | \mu) \lesssim \varepsilon \bigl(1+|\log\varepsilon|^{p-2}\bigr)\) provided initially \(\mathcal H(f_\varepsilon(0) | \mu(0)) \leq \varepsilon\), where \(p\) is the order of non-linearity in the alignment force. In the linear case \(p=2\) we recover the known result of Karper, Mellet, and Trivisa [M3AS, 25(01):131–163, 2015].

This work is supported by NSF grants DMS #2238219

Changhui Tan, Liutang Xue

Abstract

We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order \(0<\alpha<1\) and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density \(\rho\) and the potential gradient \(\Gamma=\nabla\Lambda^{-\alpha}u\), both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.

This work is supported by NSF grants DMS #2238219

Vinh Nguyen, Roman Shvydkoy, Changhui Tan

Advances in Mathematics, Volume 494, 110929, 45pp. (2026)

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol \(\phi\) and a non-linear coupling of velocities given by the power law \(A(v) = |v|^{p-2}v\), \(p>2\). The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the \(k\)-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

	doi:10.1016/j.aim.2026.110929
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	This work is supported by NSF grants DMS #2238219

Trevor M. Leslie, and Changhui Tan

Communications in Partial Differential Equations, Volume 48, No. 5, pp. 753-791 (2023)

Abstract

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker-Smale dynamics. Our approach is largely inspired by the work of Brenier and Grenier [SIAM J. Numer. Anal, 35(6):2317-2328, 1998] on the pressureless Euler equations.

	doi:10.1080/03605302.2023.2202720
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan

Archive for Rational Mechanics and Analysis, Volume 228, No 1, pp. 1-37 (2018).

Abstract

We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian \((-\partial_{xx})^{\alpha/2}, \alpha\in(0,1)\). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all \(\alpha\in(0,1)\). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.

	doi:10.1007/s00205-017-1184-2
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