Coupled Self-organized Hydrodynamics and Navier–Stokes Models: Local Well-posedness and the Limit from the Self-organized Kinetic-Fluid Models

A coupled system of self-organized hydrodynamics and Navier–Stokes equations (SOH-NS), which models self-propelled particles in a viscous fluid, was recently derived by Degond et al. (J Math Fluid Mech 21(1), Art. 6, 36, 2019), starting from a micro-macro particle system of Vicsek–Navier–Stokes model, through an intermediate step of a self-organized kinetic-fluid model by multiple coarse-graining processes. In spherical coordinates, the SOH-NS system is singular. To avoid this coordinate singularity, we first transfer SOH-NS into a non-singular system by stereographic projection, then prove the local in time well-posedness of classical solutions by energy method. Furthermore, by employing the Generalized Collision Invariants (GCI)-based Hilbert expansion approach, we justify the hydrodynamic limit from the self-organized kinetic-fluid model to macroscopic dynamics with optimal convergence rate. This provides the first analytically rigorous justification of the modeling and asymptotic analysis in Degond et al. (2019).