Displaying items by tag: hydrodynamic limit

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

McKenzie Black, and Changhui Tan

Kinetic and Related Models, Volume 18, No. 4, pp. 609-632 (2025)

Abstract

We investigate a class of Vlasov-type kinetic flocking models featuring nonlinear velocity alignment. Our primary objective is to rigorously derive the hydrodynamic limit leading to the compressible Euler system with nonlinear alignment. This study builds upon the work by Figalli and Kang [Anal. PDE, 12(3), 843-866, 2018], which addressed the scenario of linear velocity alignment using the relative entropy method. The introduction of nonlinearity gives rise to an additional discrepancy in the alignment term during the limiting process. To effectively handle this discrepancy, we employ the monokinetic ansatz in conjunction with the relative entropy approach. Furthermore, our analysis reveals distinct nonlinear alignment behaviors between the kinetic and hydrodynamic systems, particularly evident in the isothermal regime.

	doi:10.3934/krm.2024028
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219

Razvan C. Fetecau, Weiran Sun, and Changhui Tan

Physica D: Nonlinear Phenomena, Volume 325, pp. 146-163 (2016).

Abstract

We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.

	doi:10.1016/j.physd.2016.03.011
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