On the Existence of Global Smooth Solutions to the Parabolic-Elliptic Keller-Segel System with Irregular Initial Data

We consider the parabolic-elliptic Keller-Segel system {u(t) = Delta u - chi del . (u del nu), 0 = Delta nu - nu + u (*) in a smooth bounded domain Omega subset of R-n, n is an element of N, with Neumann boundary conditions. We look at both chemotactic attraction (chi > 0) and repulsion (chi < 0) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data mu is an element of M+(<(Omega)over bar>) that, if either n = 2, chi < 0 or n = 2, chi > 0 and the initial mass is small or n = 3, chi < 0 and mu = f is an element of L-p(Omega), p > 1 holds, it is still possible to construct global classical solutions to (star), which are continuous in t = 0 in the vague topology on M+((Omega) over bar).