Tightness of radially-symmetric solutions to 2D aggregation-diffusion equations with weak interaction forces

We prove the tightness of radially-symmetric solutions to 2D aggregation-diffusion equations, where the pairwise attraction force is possibly degenerate at large distance. We first reduce the problem into the finiteness of a time integral in the density on a bounded region, by introducing a new assumption on the interaction potential called the essentially radially contractive (ERC) property. Then we prove this finiteness by using the 2-Wasserstein gradient flow structure, combining with the continuous Steiner symmetrization curves and the local clustering curve. This is the first tightness result on the 2D aggregation-diffusion equations for a general class of weakly confining potentials, i.e., those $W(r)$ with $\lim_{r\rightarrow\infty}W(r)<\infty$, and serves as an important step towards the study of equilibration.