Analysis of a geometrical Multiscale Model based on the coupling of ODE's and PDE's for blood flow simulations

In hemodynamics, local phenomena, such as the perturbation of flow pattern in a specific vascular region, are strictly related to the global features of the whole circulation (see, e.g., [L. Formaggia et al., Comput. Vis. Sci., 2 (1999), pp. 75--83]). In [A. Quarteroni, S. Ragni, and A. Veneziani, Comput. Vis. Sci., 4 (2001), pp. 111--124] we have proposed a heterogeneous model, where a local, accurate, three-dimensional description of blood flow by means of the Navier--Stokes equations in a specific artery is coupled with a systemic, zero-dimensional, lumped model of the remainder of circulation. This is a geometrical multiscale strategy, which couples an initial-boundary value problem to be used in a specific vascular region with an initial-value problem in the rest of the circulatory system. It has been successfully adopted to predict the outcome of a surgical operation (see [K. Laganà et al., Biorheology, 39 (2002), pp. 359--364, G. Dubini et al., Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, 2000]). However, its interest goes beyond the context of blood flow simulations. In this paper we provide a well-posedness analysis of this multiscale model by proving a local-in-time existence result based on a fixed-point technique. Moreover, we investigate the role of matching conditions between the two submodels for the numerical simulation