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

TL;DR

This paper establishes sharp Sobolev estimates for solutions to a nonlinear aggregation-diffusion equation, revealing the precise behavior of mass concentration phenomena in radially symmetric solutions with small diffusion.

Contribution

It provides the first sharp upper and lower bounds for Sobolev norms of solutions, advancing understanding of nonlinear mass concentration in aggregation-diffusion models.

Findings

Optimal bounds for Sobolev norms are derived.

Mass concentration effects are quantitatively characterized.

Results apply to radially symmetric solutions with small diffusivity.

Abstract

We consider the drift-diffusion equation $u_t-\epsilon\Delta u + \nabla \cdot(u\nabla K^*u)=0$ 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 $\epsilon$ studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.