Quantitative Stability Of Aggregation-Diffusion Under Local Attraction

In this note, we establish quantitative stability of aggregation-diffusion energy for compactly supported data. Our consideration focuses on the degenerate diffusion with local attraction that are asymptotically flat at infinity, such as Riesz kernels. We characterize the spectrum lower-bound of the Wasserstein Hessian at steady states. Assumptions are kept as minimal as that needed for uniqueness.