Displaying items by tag: kinetic equation

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

Alina Chertock, Changhui Tan, and Bokai Yan

Kinetic and Related Models, Volume 11, No 4, pp. 735-756 (2018).

Abstract

We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors, and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.

	doi:10.3934/krm.2018030
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