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An Euler-type hyperbolic-parabolic system of chemotactic aggregationdescribing the vascular network formation is investigated in the criticalregularity setting. For small initial data around a constant equilibrium state,the well-posedness of the global classical solution to the Cauchy problem withgeneral pressure laws is established in homogeneous hybrid Besov spaces. Then,the optimal time-decay rates of the global solution are analyzed under anadditional regularity assumption on the initial data. Furthermore, therelaxation limit (large friction limit) of the hyperbolic-parabolic system isjustified rigorously. It is shown that as the friction coefficient tends tozero, the global solution of the hyperbolic-parabolic chemotaxis systemconverges to the global solution of the Keller-Segel equations with an explicitconvergence rate. To capture the dissipative properties of the nonlinearsystem, our approach relies on the introduction of new effective unknowns inlow frequencies and the construction of a Lyapunov functional in the spirit ofBeauchard and Zuazua's in [5] to treat the high frequencies.

The hyperbolic-parabolic chemotaxis system for vasculogenesis: global dynamics and relaxation limit toward a Keller-Segel model