Abstract. The self-organized hydrodynamic models can be derived from the kinetic version of the Vicsek model. The formal derivations and local well-posedness of the macroscopic equations are done by Degond and his collaborators. In this paper, we rigorously justify this hydrodynamic limit.

In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for C ∞ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto's recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index γ satisfying γ + 2s ∈ (−1, 1), s ∈ (0, 1/2) and in this case we consider the kinetic factor in the form of Φ(v) = |v| γ instead of v γ as Morimoto did before.

In this paper we first employ the energetic variational method to derive a micromacro model for compressible polymeric fluids. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We then prove the global in time existence of the smooth solution near the global equilibrium.

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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. [14], starting from a micro-macro particle system of Vicsek-Navier-Stokes model, through an intermediate step of a self-organized kinetic-kinetic model by multiple coarse-graining processes. 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, employing the Hilbert expansion approach, we justify the hydrodynamic limit from the self-organized kinetic-fluid model to macroscopic dynamics. This provides the first analytically rigorous justification of the modeling and asymptotic analysis in [14].