  This paper provides a unified mathematical analysis of a family of non-local
diffuse interface models for tumor growth describing evolutions driven by
long-range interactions. These integro-partial differential equations model
cell-to-cell adhesion by a non-local term and may be seen as non-local variants
of the corresponding local model proposed by H. Garcke et al. (2016). The model
in consideration couples a non-local Cahn-Hilliard equation for the tumor phase
variable with a reaction-diffusion equation for the nutrient concentration, and
takes into account also significant mechanisms such as chemotaxis and active
transport. The system depends on two relaxation parameters: a viscosity
coefficient and parabolic-regularization coefficient on the chemical potential.
The first part of the paper is devoted to the analysis of the system with both
regularizations. Here, a rich spectrum of results is presented. Weak
well-posedness is first addressed, also including singular potentials. Then,
under suitable conditions, existence of strong solutions enjoying the
separation property is proved. This allows also to obtain a refined stability
estimate with respect to the data, including both chemotaxis and active
transport. The second part of the paper is devoted to the study of the
asymptotic behavior of the system as the relaxation parameters vanish. The
asymptotics are analyzed when the parameters approach zero both separately and
jointly, and exact error estimates are obtained. As a by-product,
well-posedness of the corresponding limit systems is established.
