On interaction energy and steady states of aggregation-diffusion equations

2022.09.22（星期四）上午10:00-11:00

点：Zoom: 623 7604 7575 Code: 060485

要：In this talk, I will talk about two results related to the interaction energy E[f]= \int f(x)f(y)W(x-y) dxdy for a nonnegative density f and radially decreasing interaction potential W. The celebrated Riesz rearrangement shows that E[f] \le E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to make quantitative stability estimate of E[f^*]-E[f] in terms of some distance between f and f^*. Previously such stability estimates have been obtained for characteristic functions. In this talk, I will discuss a joint work with Yao Yao about the stability for general densities.

I will also discuss another joint work with Matias Delgadino and Yao Yao about the uniqueness and non-uniqueness of steady states of aggregation equations with degenerate diffusion, where the convexity of the interaction energy plays an important role.  For the diffusion power m\ge 2, we constructed a novel interpolation curve between any two radially decreasing densities with the same mass, and showed that the interaction energy is convex along this interpolation. This lead to the uniqueness of the steady state, and the threshold is sharp.