Microscopic, mesoscopic, and macroscopic descriptions of the Euler alignment system with adaptive communication strength (2409.02409v4)

Abstract: This is a continuation of our previous joint work on the $\st$-model in[\textit{Well-posedness and long time behavior of the Euler Alignment System with adaptive communication strength}, accepted at the Abel Symposium Proceedings, also arXiv:2310.00269, 2023]. The $\st$-model, introduced by the first author in [\textit{Environmental averaging}. EMS Surv. Math. Sci., 11 (2024), no. 2, 277413],is an alignment model with the property that the strength of the alignment force, $\st$, is transported along an averaged velocity field. The transport of the strength is designed so that it admits an $e$-quantity, $e = \partial_x u + \st$, which controls regularity in 1D similarly to the classical Cucker-Smale case. The utility of the $\st$-model is that it has the versatility to behave qualitatively like the Motsch-Tadmor model, for which global regularity theory is not known. This paper aims to put the $\st$-model on firmer physical grounds by formulating and justifying the microscopic and mesoscopic descriptions from which it arises. A distinctive feature of the microscopic system is that it is a discrete-continuous system: the position and velocity of the particles are discrete objects, while the strength is an active continuum scalar function. We establish a rigorous passage from the microscopic to the mesoscopic description via the Mean Field Limit and from the mesoscopic to the macroscopic description in the monokinetic and Maxwellian limiting regimes. We also address the long-time behavior of the kinetic Fokker-Planck-Alignment equation by establishing the relaxation to the Maxwellian in 1D when the velocity averaging is given by the Favre filtration. As a supplement to the numerical results already presented in our previous work, we provide additional numerical evidence, via a particle simulation, that the $\st$-model behaves qualitatively like the Motsch-Tadmor model.