Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation

We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$ 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 $$\varepsilon $$ studied in our previous paper (Biler et al. in J Differ Equ 271:1092–1108, 2021), obtaining sharp upper and lower bounds for Sobolev norms.