From BGK-alignment model to the pressured Euler-alignment system with singular communication weights

This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights ${\phi }_{\alpha }(x)=\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ for some $\alpha >0$. We consider a kinetic BGK-alignment model consisting of a kinetic BGK-type equation with a singular Cucker-Smale alignment force. By taking into account a small relaxation parameter, which corresponds to the asymptotic regime of a strong effect from BGK operator, we quantitatively derive the isentropic Euler-alignment system with pressure $p(\rho )={\rho }^{\gamma }$, $\gamma =1+\frac{2}{d}$ from that kinetic equation.