Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation

We consider the drift-diffusion equation u t − e∆u + ∇ · (u ∇K * u) = 0 in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case K(x) = −|x|. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity e studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.