We rigorously show the mean-field limit for a large class of swarming individual based models with local sharp sensitivity regions. For instance, these models include nonlocal repulsive-attractive forces locally averaged over sharp vision cones and Cucker–Smale interactions with discontinuous communication weights. We define global-in-time solutions through a differential inclusion system corresponding to the particle descriptions. We estimate the error between the solutions to the differential inclusion system and weak solutions to the expected limiting kinetic equation by employing tools from optimal transport theory. Quantitative bounds on the expansion of the 1-Wasserstein distance along flows based on a weak-strong stability estimate are obtained. We also provide various examples of realistic sensitivity sets satisfying the assumptions of our main results.
Bibliographical noteFunding Information:
Acknowledgments. JAC was partially supported by the project MTM2011-27739-C04-02 DGI (Spain), from the Royal Society by a Wolfson Research Merit Award and from the EPSRC grant number EP/P031587/1. YPC was supported by NRF grant (No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation. YPC also acknowledges the support of the ERC-Starting Grant HDSPCONTR “High-Dimensional Sparse Optimal Control”. JAC and YPC were supported by EPSRC grant EP/K008404/1. SS acknowledges the support of Fondation des Sciences Mathématiques de Paris and Paris Sciences & Lettres Université.
© European Mathematical Society 2019.
All Science Journal Classification (ASJC) codes
- Applied Mathematics