From BGK-alignment model to the pressured Euler-alignment system with singular communication weights. (English) Zbl 1531.35238
Summary: This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights \(\phi_\alpha(x) = | x |^{- \alpha}\) 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 the 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.
MSC:
| 35Q31 | Euler equations |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
isentropic Euler-alignment system; Cucker-Smale model; BGK model; hydrodynamic limit; relative entropy; singular communication weightReferences:
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