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

Abstract

This paper is concerned with the Keller--Segel system with flux limitation, \begin{align} \tag{$\ast$} \begin{cases} u_t=Δu - \nabla \cdot (uf(|\nabla v|^{2})\nabla v), \\ v_t=Δv - v + u \end{cases} \end{align} in bounded $n$-dimensional domains with homogeneous Neumann boundary conditions, where $f$ generalizes the prototype obtained on letting \[ f(ξ) = k_f(1 + ξ)^{-α}, \quad ξ\ge 0, \] with $k_f>0$ and $\alpha >0$. In this framework, it is shown that if either $n=1$ and $\alpha >0$ is arbitrary, or $n\ge 2$ and $\alpha >\frac{n-2}{2(n-1)}$, then for any nonnegative initial data belonging to the space of Radon measures for the population density and to $W^{1,q}$ with $q\in (\max \{1,(1-2\alpha )n\},\frac{n}{n-1})$ for the signal density, there exists a global classical solution of the Neumann problem for $(\ast )$, which is continuous at $t=0$ in an appropriate sense.