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} u_t=Δ u - ∇ · (uf(|∇ v|²)∇ v), \\ v_t=Δ v - v + u {cases} {align} in bounded n-dimensional domains with homogeneous Neumann boundary conditions, where f generalizes the prototype obtained on letting \[ f(ξ) = k_f(1 + ξ)-α, ξ ≥ 0, \] with kf > 0 and α > 0. In this framework, it is shown that if either $n = 1$ and α > 0 is arbitrary, or n ≥ 2 and α > n-2/2(n-1), then for any nonnegative initial data belonging to the space of Radon measures for the population density and to W1,q with q ∈ (max\1, (1-2α)n\, n/n-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.