Hydrodynamic limit of a coupled Cucker-Smale system with strong and weak internal variable relaxation Kim, Jeongho Poyato Sánchez, Jesús David Soler Vizcaino, Juan Segundo Flocking Hydrodynamic limit Kinetic model Multiscale model Thermomechanical Cucker-Smale model Internal variable Singular weights This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 639638), the MECD (Spain) research grant FPU14/06304, the MINECO-Feder (Spain) research grant number RTI2018- 098850-B-I00, the Junta de Andalucia (Spain) Projects PY18-RT-2422 & A-FQM-311-UGR18 (D.P, J.S). 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. 2021-09-15T11:19:39Z 2021-09-15T11:19:39Z 2021-01-12 info:eu-repo/semantics/article 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] http://hdl.handle.net/10481/70217 10.1142/S0218202521400042 eng info:eu-repo/grantAgreement/EC/H2020/639638 http://creativecommons.org/licenses/by-nc-nd/3.0/es/ info:eu-repo/semantics/openAccess Atribución-NoComercial-SinDerivadas 3.0 España World Scientific