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This paper is devoted to a rigorous derivation of the isentropicEuler-alignment system with singular communication weights $\phi_\alpha(x) =|x|^{-\alpha}$ for some $\alpha > 0$. We consider a kinetic BGK-alignment modelconsisting of a kinetic BGK-type equation with a singular Cucker-Smalealignment force. By taking into account a small relaxation parameter, whichcorresponds to the asymptotic regime of a strong effect from BGK operator, wequantitatively derive the isentropic Euler-alignment system with pressure$p(\rho) = \rho^\gamma$, $\gamma = 1 + \frac2d$ from that kinetic equation.

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