Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces

José A. Carrillo, Young Pil Choi, Jinwook Jung

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8 Citations (Scopus)


In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker-Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulomb interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. The existence of weak and strong solutions to the kinetic and fluid equations is also obtained. We emphasize that the existence of global weak solution with the needed free energy dissipation for the kinetic model is established.

Original languageEnglish
Pages (from-to)327-408
Number of pages82
JournalMathematical Models and Methods in Applied Sciences
Issue number2
Publication statusPublished - 2021 Feb

Bibliographical note

Funding Information:
JAC was partially supported by EPSRC grant number EP/P031587/1 and the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363). YPC was supported by NRF grant (No. 2017R1C1B2012918), POSCO Science Fellowship of POSCO TJ Park Foundation, and Yonsei University Research Fund of 2019-22-021. JJ was supported by NRF grant (No. 2019R1A6A1A10073437).

Publisher Copyright:
© 2021 The Author(s).

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Applied Mathematics


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