The Hyperbolic-Parabolic Chemotaxis System for Vasculogenesis: Global Dynamics and Relaxation Limit Toward a Keller–Segel Model

An Euler type hyperbolic-parabolic system of chemotactic aggregation describing vascular network formation is investigated in the critical regularity setting. For initial data near a constant equilibrium state, the global well-posedness of the classical solution to the Cauchy problem with general pressure laws is proved in critical hybrid Besov spaces, and qualitative regularity estimates uniform with respect to the relaxation parameter are established. Then, the optimal time-decay rates of the global solution are analyzed under an additional regularity assumption on the initial data. Furthermore, the relaxation limit of the hyperbolic-parabolic system toward a parabolic-elliptic Keller–Segel model is justified rigorously. It is shown that as the relaxation parameter tends to zero, the solutions of the hyperbolic-parabolic chemotaxis system converge to the solutions of the Keller–Segel model at an explicit rate of convergence. Our approach relies on the introduction of new effective unknowns in low frequencies and the construction of a Lyapunov functional in the spirit of that of Beauchard and Zuazua [ Arch. Ration. Mech. Anal., 199 (2011), pp. 177–227] to treat the high frequencies.