Instantaneous regularization of measure-valued population densities in a Keller--Segel system with flux limitation

This paper is concerned with the Keller--Segel system with flux limitation, align cases uₜ= u - (uf (| v|^2) v), \\ vₜ= v - v + u cases align in bounded n-dimensional domains with homogeneous Neumann boundary conditions, where f generalizes the prototype obtained on letting \ f () = kf (1 +) ^-, 0, \ with kf > 0 and > 0. In this framework, it is shown that if either n = 1 and > 0 is arbitrary, or n 2 and > n-22 (n-1), then for any nonnegative initial data belonging to the space of Radon measures for the population density and to W^1, q with q (\1, (1-2) n\, nn-1) for the signal density, there exists a global classical solution of the Neumann problem for (), which is continuous at t = 0 in an appropriate sense.