Hydrodynamic limit of a coupled Cucker-Smale system with strong and weak internal variable relaxation
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FlockingHydrodynamic limitKinetic modelMultiscale modelThermomechanical Cucker-Smale modelInternal variableSingular weights
Published version: Kim, J., Poyato, D., & Soler, J. (2021). Hydrodynamic limit of a coupled Cucker–Smale system with strong and weak internal variable relaxation. Mathematical Models and Methods in Applied Sciences, 1-73. [10.1142/S0218202521400042]
SponsorshipEuropean Research Council (ERC) 639638; MECD (Spain) FPU14/06304; MINECO-Feder (Spain) RTI2018-098850-B-I00; Junta de Andalucia European Commission PY18-RT-2422 A-FQM-311-UGR18; Basic Science Research Program through the National Research Foundation of Korea (NRF) - Ministry of Science and ICT NRF-2020R1A4A3079066
In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker–Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes’ law. Whilst the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions.