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

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