From Kinetic Flocking Model of Cucker-Smale Type to Self-Organized Hydrodynamic model

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker-Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez, Bostan, Carrillo and Degond (2019). In the traditional kinetic equation with collisions, for example, Boltzmann type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by $\mathcal{L}$, is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker($\mathcal{L}$) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator $\mathcal{L}$ is not symmetric, i.e. $\mathcal{L}\neq \mathcal{L}^*$, where $\mathcal{L}^*$ is the dual of $\mathcal{L}$. Moreover, the collision invariants lies in Ker($\mathcal{L}^*$), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear.